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the discovery of superconductivity in ferromagnetic ( fm ) uge@xmath4 under pressure gave great impact for condensed - matter community @xcite , since ferromagnetism and superconductivity have been considered to be mutually exclusive phenomena . after the discovery , superconductivity has been found in several u - based fm compounds @xcite . ucoge is one of the compounds among the fm superconductors discovered so far which can be readily explored in experiments , because of its high superconducting ( sc ) transition temperature ( @xmath0 ) and low curie temperature ( @xmath5 ) at ambient pressure . actually , important and intriguing experimental results have been reported for ucoge @xcite , and some of them are summarized as follows . 1 . microscopic measurements of @xmath6sr [ fig.1(a ) ] @xcite and co - nqr [ fig.1(b ) ] @xcite have shown that the superconductivity occurs within the fm region , resulting in homogeneous microscopic coexistence of the ferromagnetism and superconductivity . the superconductivity is also found in the paramagnetic state and @xmath0 shows a broad maximum around a critical pressure of ferromagnetism [ fig.1(c ) ] @xcite , which is in sharp contrast with uge@xmath4 where the sc is seen only in the fm phase @xcite . + ( a)temperature variation of the normalized asymmtries ( @xmath7 ) of the fm and paramagnetic phases measured with zero - field muon - spin rotation on ucoge @xcite . below @xmath8 k , magnetism is observed in the whole sample volume although superconductivity occurs at @xmath9 0.8 k , indicative of coexistence of ferromagnetism and superconductivity on a microscopic scale . ( b ) temperature dependence of @xmath10co @xmath11 in the single - crystal sample @xcite . @xmath11 above 2.3 k , shown by the blue dots , was measured at the paramagnetic ( pm)-frequency ( 8.3 mhz ) peak arising from the @xmath12 transitions , and @xmath11 below 2.3 k , shown by the red dots , was measured at the fm - frequency ( 8.1 mhz ) peak shifted from the pm peak due to the appearance of the internal field at the co site . the inset shows the nqr signal arising from the @xmath12 transitions obtained at 4.2 k ( pm ) and 0.9 k ( fm ) . two @xmath11 components were observed in the sc state , the faster ( slower ) component denoted by red open ( closed ) squares ; the red broken curve below @xmath0 represents the temperature dependence calculated assuming a line - node gap with @xmath13 . the sc anomaly observed from the fm signal is clear evidence that superconductivity occurs in the fm region . ( c ) pressure - temperature phase diagram of ucoge @xcite . ferromagnetism is denoted as blue area , superconductivity is as yellow area . @xmath0(@xmath14 ) extrapolates to a fm quantum critical point at the critical pressure @xmath15 . superconductivity coexists with ferromagnetism below @xmath15 as blue - yellow hatched area . inset : amplitude of @xmath16 at @xmath17 as a function of pressure . the data follow a linear @xmath14 dependence and extrapolate to @xmath18 gpa . ] the temperature variation of the @xmath19co - nqr spectra shown in fig . [ fig2 ] indicates that the fm region grows continuously around @xmath5 coexisting with the pm region , and that the fm moment in the fm region appears discontinuously , whereas the hysteresis behavior was not observed in the temperature variation of the spectra and @xmath11 in the both states below @xmath5 is almost identical @xcite . since the divergence of @xmath11 and the clear anomaly in the specific heat were observed at @xmath5 @xcite and the fm ordered moment is small ( @xmath20 ) at low temperatures @xcite , the fm phase transition in ucoge is considered to possess weak first - order character with critical fm fluctuations , which is close to second - order one . although we need further investigation of sample and pressure dependence on the fm transition to understand the transition character thoroughly , the presence of the critical fluctuations implies the validity of the scenarios of spin fluctuation - induced superconductivity . + ( a ) @xmath19co - nqr spectra ( @xmath21 ) at 4.2 k in the pm state@xcite . ( b ) temperature dependence of the nqr spectrum from the @xmath22 transitions ( @xmath23)@xcite . the pm and fm signals coexist between 1 and 2.7 k. the blue ( red ) broken lines represent gaussian fits to the pm ( fm ) peaks ; the solid lines are guides to the eye . ] although transport properties show rather three dimensional isotropic characters @xcite , studies of the sc upper critical magnetic field ( @xmath3 ) [ fig . 2 ( a ) ] and angle dependence of @xmath3 [ fig . 2 ( b ) ] have shown that sc properties are highly anisotropic @xcite ; the superconductivity survives in fields exceeding 15 t along the @xmath24 and @xmath25 axes , whereas @xmath3 for fields along the @xmath26 direction ( @xmath27 ) is as small as 0.8 t. + ( a ) temperature dependence of the upper critical fields for @xmath28 and @xmath26-axis @xcite . the temperature is normalized by the sc critical temperature @xmath0 at zero field . ( b ) temperature dependence of the upper critical field @xmath29 close to @xmath30 @xmath24-axis @xcite . the field direction is tilted from the @xmath24 to @xmath26-axis . the inset shows the angular dependence of @xmath29 at 90 mk . the values of @xmath29 greater than 16 t are the results of linear extrapolations to 90 mk . ] the static magnetic susceptibility is strongly anisotropic @xcite ; the magnetic anisotropy is ising like with the @xmath26 axis as a magnetic easy axis [ fig . [ fig4 ] ( a ) ] . in addition , direction - dependent nuclear - spin lattice relaxation rate ( @xmath11 ) measurements on a single crystalline sample have revealed the magnetic fluctuations in ucoge to be ising - type fm ones along the @xmath26 axis ( longitudinal - mode fluctuations ) as shown in fig . [ fig4 ] ( b ) @xcite . the ising - type fluctuations are also suggested from the anisotropic correlation length measured by the inelastic neutron measurements @xcite . + nmr shift @xmath31 measured in field along the @xmath24 , @xmath25 and @xmath26 directions @xcite . remarkable anisotropy was observed . in the inset , @xmath31 is plotted against the bulk susceptibility @xmath32 measured in 2 t. the nearly identified slopes for all the directions indicate that the hyperfine coupling constants are isotropic . ( b ) direction - decomposed dynamic susceptibility @xmath33 and static susceptibility @xmath34 along each direction @xcite . identical ising anisotropy for both quantities found above 8 k suggests that the longitudinal mode dominates these fluctuations . it is noteworthy that in this @xmath35 range , the knight shift scales with @xmath33 , which is predicted by the self - consistent renormalization theory for a 3-dimensional nearly fm metals @xcite . ] 5 . from detailed angle - resolved nmr and meissner measurements @xcite , it was found that magnetic fields along the @xmath26 axis ( @xmath36 ) is a tuning parameter of the longitudinal fm fluctuations and strongly suppresses the fm fluctuations , as shown in fig . 5 ( a ) and 5 ( b ) . this sharp angle dependence of @xmath11 was also confirmed at 600 mk just above @xmath0 @xcite . it was also shown that the superconductivity is observed in the limited magnetic - field region where the longitudinal fm spin fluctuations are active [ fig . 4 ( c ) ] . these results suggest that the superconductivity in ucoge is tightly coupled with the longitudinal fm spin fluctuations along the @xmath26-axis . + ( a ) angle dependence of @xmath11 in the @xmath37 plane measured in three different magnetic fields at 1.7 k @xcite . ( b ) plot of the @xmath11 against @xmath36 . the @xmath11 data collapse onto a single curve when plotted against @xmath36 . ( c ) @xmath36 dependence of magnetic fluctuations along the @xmath26 axis @xmath38 at 1.7 k , extracted using @xmath39 . angle dependence of @xmath29 , determined by the onset of the meissner signal , is plotted against @xmath40 . it is found that the superconductivity is observed in the limited field region ( yellow slashed area ) , where the longitudinal fluctuations are active . inset : plot of @xmath38 against @xmath41 . the relation of @xmath42 is shown by dashed lines in ( c ) and the inset . ] from a theoretical point of view , when an itinerant fm superconductor possesses the large energy splitting between the majority and minority spin fermi surfaces , exotic spin - triplet superconductivity is anticipated , in which pairing is between parallel spins within each spin fermi surface . in addition , it has been argued that critical fm fluctuations near a quantum phase transition could mediate spin - triplet superconductivity @xcite , and actually , ( iv ) in the above indicates that the superconductivity in ucoge is intimately coupled with the ising fm spin fluctuations . however , as far as we know , there have been no reports identifying magnetic fluctuations inducing unconventional superconductivity . here , we show , from model calculations on the basis of the scenario of fm spin fluctuation - mediated superconductivity in ucoge , that unconventional temperature and angle dependence of @xmath3 is consistently understood by adopting the characteristic longitudinal fm fluctuations revealed experimentally . besides , by comparing calculations for possible pairing states , we also discuss candidates of sc gap symmetries in ucoge . in this section , we introduce a simple model to study @xmath3 in ucoge at ambient pressure where the ferromagnetism and superconductivity coexist . although ucoge has a complicated multi - band structure , since the spin fluctuations dominate the low energy physics in ucoge , we can use a simplified model to study its superconductivity with appropriately taking into account the critical fm spin fluctuations observed in the experiments @xcite . the model relevant to ucoge in the fm phase with an applied magnetic field @xmath43 is @xmath44c_k(\tau),\\ s_{\rm ex}&=-({\mbox{\boldmath $ h$}}_{ex}+\mu_b{\mbox{\boldmath $ h$}})\cdot \sum_k\int_0^{1/t}d\tau c^{\dagger}_k(\tau){\mbox{\boldmath $ \sigma$}}c_k(\tau),\\ s_{\rm int}&=-\frac{2g^2}{3}\sum_q \int_0^{1/t}d\tau\int_0^{1/t}d\tau^{\prime } s_q^c(\tau)\chi^c({\mbox{\boldmath $ q$ } } , \tau-\tau^{\prime})s_{-q}^c(\tau^{\prime}),\end{aligned}\ ] ] and effects of the vector potential on electron motions are taken into account in a semiclassical way . @xmath45 is an annihilation operator of low energy quasiparticles which are formed through hybridizations between conduction electrons and @xmath46-electrons . a typical energy scale of the quasiparticles is @xmath47 ( k ) estimated from the specific heat @xcite and the coherence temperature @xcite . since the anisotropy in the resistivity in ucoge is small as discussed in the introduction , we use an isotropic dispersion @xmath48 where @xmath49 is the lattice constant and @xmath6 is the chemical potential . we fix the electron density per site as @xmath50 which leads to a small but nearly spherical fermi surface . as discussed in ref.@xcite , the anisotropic mass model can not explain the anomalous angle dependence of @xmath3 . in our model , the anisotropy in @xmath3 arises from the exchange term and the pairing interaction term in eq.([eq : action ] ) . we use an energy unit @xmath51 and a length unit @xmath52 . the second term of eq.([eq : action ] ) includes the exchange splitting of the fermi surface in the fm phase which is assumed to be large compared to the sc transition temperature at zero - field @xmath53 . the exchange field is parallel to the @xmath26-axis and fixed to be @xmath54 , which would lead to suppression of the pauli depairing effect against in - plane magnetic fields in the equal spin pairing states of the superconductivity @xcite . this is a key assumption to understand the large anisotropy in @xmath3 in the present study , because the equal spin pairing states are not robust against the in - plane zeeman field while the observed @xmath55 are huge in ucoge . however , we put a remark on this assumption . ucoge has a multi - band structure with a strong spin - orbit interaction and 5@xmath56 states are located near the fermi level . in such a system , it is possible that , even if the magnetization in the fm state is small , the exchange splitting for each fermi surface is large . actually , in the band calculations , the orbital magnetizations and the spin polarizations cancel to result in a small total magnetic moment . the present assumption can be checked by nmr knight shift measurements in the sc states . finally , the last term in eq . ( [ eq : action ] ) describes the interaction between the quasiparticles through the ising spin fluctuations , and @xmath57 . the dynamical susceptibility @xmath58 strongly depends on the @xmath26-axis magnetic field as revealed in the nmr experiments @xcite . @xmath59 where @xmath60 and @xmath61 is approximately the fermi velocity . here , we have assumed a simple landau damping in @xmath58 , although existence of non - landau damping was reported @xcite . the coefficient @xmath62 is determined by the nmr experiments , and a normalization factor @xmath63 t is introduced as @xmath64 so that the dimensionless parameter @xmath26 is @xmath65 . although the origin of this magnetic field dependence in the susceptibility is not so clear , we have a possible explanation for it . according to the band calculation , the 5@xmath56 bands in ucoge have some semi - metallic like structures around several @xmath66-points @xcite . for such structures , the density of states would be @xmath67 , which results in @xmath68 near the criticality . in this scenario , detailed structures in the bands are important , and therefore , the spin fluctuations in ucoge under a magnetic field can depend on sample qualities . this problem will be studied elsewhere . we note that the coupling constant @xmath69 should be regarded as a renormalized one including the vertex corrections which enhance the sc transition temperatures . although we have postulated a simple analytic @xmath70-dependence in @xmath71 , there might exist non - analytic corrections @xmath72 in 3-dimensional isotropic systems @xcite . the non - analytic corrections are important in connection with fluctuation - induced first order phase transitions . however , since the first order character of the fm transition is weak , and critical fm fluctuations are well developed in ucoge as discussed in sec.[sec : intro ] , it would be legitimate to neglect the first order character , and assume that @xmath71 is analytic in @xmath70 as long as calculations of the superconducting properties are concerned . in addition , if exist in the ising - like system , the non - analytic corrections would not change the present study qualitatively as discussed in @xcite . in this study , we focus on the anisotropy between the @xmath24 and @xmath26-axes upper critical fields ( @xmath73 and @xmath27 ) . however , it was reported that the @xmath25-axis upper critical field @xmath74 shows a `` s - shaped '' curve at high fields , and relations to the reentrant superconductivity in urhge were discussed @xcite . in urhge , the magnetization flips at a critical @xmath75 and the spin fluctuations at @xmath76 are considered to be different from those of @xmath77 @xcite . similar scenarios are expected to hold also in ucoge where the s - shaped @xmath74 would indicate a crossover of the superconductivities of different mechanisms under magnetic field along the @xmath25 axis . the problem of @xmath74 in ucoge is an another issue and will be studied elsewhere . to determine the upper critical field @xmath3 in the @xmath78-plane , we solve the linearized eliashberg equation in the presence of a vector potential @xcite . the eliashberg equation for an equal - spin pairing state in the fm phase is @xmath79 \delta_{\sigma^{\prime}\sigma^{\prime}}(k^{\prime}),\end{aligned}\ ] ] where @xmath80 and @xmath81 . a short notation @xmath82 is used hereafter . here we have neglected the effect of the magnetization in the orbital motion , because the observed moment is very small in ucoge @xcite . note that the green s function @xmath83 has off - diagonal elements when @xmath84 . the gap function depends both on the momentum in the relative coordinate and the center - of - mass coordinate , and is expanded as @xmath85 , where @xmath86 is the lowest level landau function . we solve the linearized eliashberg equation for two cases corresponding to the classification by the magnetic point group @xmath87 for ucoge which has an orthorhombic crystal structure @xcite . one is the so - called a state which corresponds to point node symmetry with a @xmath88-vector @xmath89 near the @xmath90 point . the other one is the so - called b state corresponding to horizontal line node symmetry with @xmath91 . here , @xmath92 and @xmath93 are real coefficients . both of the gap functions are determined by self consistent calculations , and the resulting sc states are non - unitary because of the presence of the exchange field @xmath94 . we note that , under the applied magnetic field @xmath43 , the calculated gap functions are distorted and has lower symmetry than that of the cubic lattice in the present model . the pairing interaction is evaluated at the first order in @xmath95 , @xmath96 for the selfenergy , we include only the term @xmath97,\end{aligned}\ ] ] where @xmath98 is the non - interacting green s function . other neglected terms in the self energy are much smaller than @xmath99 in magnitude . as noted in the previous section , the coupling constant @xmath69 is a renormalized one . in the present study , the coupling constant is fixed as @xmath100 which is rather large to give a high transition temperature so that the numerical costs are reduced . however , qualitatively , our main results do not depend on the value of @xmath95 . for this value of @xmath95 , the mass enhancement factor @xmath101 is @xmath102 , which is not so large because the density of states at the fermi surface is small in the present parameters . the transition temperature for the a state without magnetic field is @xmath103 which we regard as 1.0 k according to the experiments for ucoge . then , the hopping integral is fixed as @xmath104 k throughout in our study , and the order of this value is consistent with the coefficient of the linear specific heat @xmath105 mj / k@xmath106mol @xcite and the coherence temperature @xmath107 k @xcite . we also fix the lattice constant , @xmath108 which leads to an effective mass for the orbital motion of the cooper pairs @xmath109 where @xmath110 is the bare electron mass . correspondingly , we fix @xmath111 t , which is consistent with the nmr experiments @xcite . before discussing numerical results of @xmath3 of the present model , we note that the orbital limit @xmath112 depends on positions of gap nodes in general @xcite . a simple explanation is as follows . let us consider a superconductor with isotropic fermi surface which has nodes in a gap function . the effective velocity for the cyclotron motion of the orbital depairing effect can be of the form of @xmath113 for the basis function @xmath114 corresponding to the sc gap symmetry , where @xmath115 is a velocity of electrons . for @xmath114 with horizontal line nodes , when the magnetic field is parallel to the @xmath26-axis , the cyclotron motion is suppressed at the nodes where in - plane @xmath116 is zero , resulting in a large orbital limiting field . the cyclotron motion is not suppressed for @xmath117 because @xmath118 is non - zero on the fermi surface , which leads to the anisotropy of the orbital limiting field @xmath119 . on the other hand , for the point node gap function , @xmath116 can not be zero except for the poles of the fermi surface @xmath120 which are realized as cross points of two line nodes @xmath121 and @xmath122 . therefore , for point nodes at the poles of the fermi surface , the order is turned over , @xmath123 . following this consideration , if there is no suppression of the pairing interaction and the pauli depairing effect is negligible , @xmath124 is expected for the a ( b ) state . we now turn to discussions of calculation results . temperature dependence of @xmath125 with @xmath126 for the a state is shown in fig . [ fig : a1 ] . field angle dependence of @xmath3 at @xmath127 in the a state . ] field angle dependence of @xmath3 at @xmath127 in the a state . ] it is found that the qualitative behaviors of @xmath3 are independent of @xmath26 . @xmath73 exhibits a strong coupling behavior and reaches @xmath128 . although the precise value of @xmath129 depends on the choice of @xmath130 and @xmath95 , strong coupling calculations of @xmath3 naturally explain the upward curvature in @xmath73 and its large value observed in the experiments , which is a key to understand the anomalous anisotropic behavior in @xmath3 . the calculated anisotropy ratio @xmath131 becomes @xmath132 in consistent with the experiments , although the fermi surface of our model is merely spherical . we emphasize that , for @xmath3 to have a large anisotropy between the @xmath24 and @xmath26-axes , the experimentally observed @xmath133-dependence in the susceptibility is crucially important . for a comparison , we calculate @xmath27 in the case that @xmath71 has weaker @xmath134-dependence , mean - field like dependence of @xmath135 , for which results are shown with purple symbols in fig . [ fig : a1 ] . in this case , suppression of @xmath3 for the @xmath26-axis is only moderate , and we can not obtain anomalously large anisotropy in @xmath3 . in fig . [ fig : a2 ] , we show field angle dependence of @xmath136 at @xmath127 , where @xmath137 is the angle from the @xmath24-axis to the @xmath26-axis . it is seen that @xmath136 is strongly suppressed when the field direction is slightly tilted from the @xmath24-axis to the @xmath26 axis . this strong angle dependence is again due to the @xmath133 suppression of the pairing interaction . the calculation results well explain the unusual angle dependence of @xmath3 observed experimentally . this good agreement is strong evidence that the superconductivity in ucoge is actually mediated by the critical ising spin fluctuations . we note that the relation between @xmath36 and @xmath3 is considered to be an analog of the isotope effect in conventional superconductors where ion mass @xmath138 controls transition temperature @xmath0 . in both cases , suppression of the pairing interaction by increasing @xmath36 or @xmath138 leads to reduction of @xmath0 . one important reason why we can establish such a relation between @xmath36 and @xmath3 in the @xmath46-electron system ucoge is that the critical spin fluctuations dominate the low energy physics . details of the system and complexity of multi - band are sub - dominant factors for the superconductivity , and therefore , the sc properties are well understood once the spin fluctuations are appropriately taken into account . the transition temperature at zero - field for the b state is @xmath139 which differs from that for the a state , because of the presence of the exchange field along the @xmath26-axis . however , the difference is not important in our study . temperature dependence of @xmath125 for the b state is shown in fig . [ fig : b1 ] . field angle dependence of @xmath3 at @xmath140 in the b state . ] field angle dependence of @xmath3 at @xmath140 in the b state . ] for the b state , strong - coupling behaviors for the @xmath24-axis are not so significant , because the orbital limiting fields depend on the positions of gap nodes as discussed in the previous section , and the enhancement of @xmath73 for the b state is weaker than that for the a state . for the present value of @xmath141 , the pauli depairing effect can not be negligibly small at high magnetic fields , which leads to the suppression of @xmath73 in that region . for the @xmath26-axis , detailed behaviors depend on the value of the parameter @xmath26 . in the case of rather weak suppression of the ising fluctuations , i.e. @xmath142 , @xmath3 is enhanced at low temperatures . if the suppression is sufficiently strong , for example @xmath143 , such enhancement can not be seen . we also show @xmath27 calculated with the mean field @xmath144 by the purple curve as in the a state . in the b state , @xmath27 is not strongly suppressed and @xmath27 is larger than @xmath73 in contrast to the experiments . the relation @xmath145 with the mean - field @xmath144 is directly understood from the general relation @xmath146 for horizontal line node gap functions in simple systems . behaviors of @xmath3 as a function of the field angle also depend on the value of @xmath26 as shown in fig . [ fig : b2 ] . for @xmath142 , @xmath136 shows a non - monotonic behavior . this is because , for the b state , the relation @xmath147 holds in the orbital limit if there is no suppression of the ising fluctuations , while in fact the suppression in @xmath148 gets stronger as @xmath149 . on the other hand , for @xmath143 , @xmath3 monotonically decreases . comparing these results with those for the a state which are robust against the parameter value , one may conclude that the a state is a promising candidate for the sc realized in the fm state in ucoge , although the b state is not ruled out . in this paper , we have studied the upper critical field @xmath3 for the fm superconductor ucoge in the fm phase . in the introduction , we discussed several experimental observations which are key clues to understand relations between the fm and sc . to clarify the relationship , we introduced the simple model which includes the critical fm spin fluctuations with anomalous @xmath133 dependence revealed by the nmr experiments . the linearized eliashberg equations are solved within the first order in the spin fluctuations . in the a state with point nodes , temperature dependence of @xmath73 shows an upward curvature while @xmath27 is suppressed . when the field angle is tilted from the @xmath24 to the @xmath26 axis , @xmath3 is strongly suppressed . these behaviors are totally due to the characteristic suppression of the fm spin fluctuations by @xmath36 , and in nice agreement with the experimentally observed @xmath3 in ucoge . this is strong evidence of the scenario that the spin - triplet superconductivity is actually mediated by the critical ising fm spin fluctuations in ucoge . the relation between @xmath36 and @xmath3 is an analog of the isotope effect in conventional superconductors . in the b state with a horizontal line node , both enhancement of @xmath73 and suppression of @xmath27 are moderate compared to those in the a state . therefore , the a state is a promising candidate for the pairing symmetry in ucoge , although the b state is not ruled out . finally , we put remarks on some remaining problems related to the present study . firstly , in the present study , we have simply adopted the experimentally observed @xmath133 dependence in @xmath148 . although we suggested a possible explanation , the origin of this anomalous dependence is not clear . secondly , we have assumed that the exchange splitting is large enough to suppress the pauli depairing effect . however , this assumption should be checked experimentally by e.g. nmr knight shift measurements . there is also a related problem . we have focused on the coexistence region of the superconductivity and the ferromagnetism where the exchange splitting can be large . under pressure @xcite , @xmath5 is suppressed and the exchange splitting becomes small . in this case and in the paramagnetic phase , the pauli depairing effect would not be suppressed . actually , it was reported that , around the critical pressure where @xmath5 is extrapolated to zero , @xmath150 is suppressed while @xmath151 is increasing @xcite . these behaviors could be understood as a result of two competing effects ; although the pauli depairing effect would not be suppressed by the exchange field near the criticality , the fm spin fluctuations are enhanced . investigations of these issues are needed for a comprehensive understanding of ucoge , and they would be studied in the future . the authors thank s. yonezawa , and y. maeno for experimental support and valuable discussions , and d. aoki , j. flouquet , a. de visser , a. huxley , h. harima , and h. ikeda for valuable discussions . this work was partially supported by kyoto univ . ltm centre , yukawa institute , the `` heavy electrons '' grant - in - aid for scientific research on innovative areas ( no . 20102006 , no . 21102510 , no . 20102008 , no . 23102714 , and no . 23840009 ) from the ministry of education , culture , sports , science , and technology ( mext ) of japan , a grant - in - aid for the global coe program `` the next generation of physics , spun from universality and emergence '' from mext of japan , a grant - in - aid for scientific research from japan society for promotion of science ( jsps ) , kakenhi ( s ) ( no . 20224015 ) , kakenhi ( c ) ( no . 23540406 ) and first program from jsps . 99 saxena s s _ et al . _ 2000 _ nature _ * 406 * 587 belitz d , kirkpatrick t r and vojta t 1997 _ phys . b _ * 55 * 9452 ; belitz d , kirkpatrick t r and vojta t1999 _ phys . lett . _ * 82 * 4707 ; belitz d , kirkpatrick t r and rollbhler j 2005 _ phys . lett . _ * 94 * 247205 ; kirkpatrick t r and belitz d 2012 _ phys . b _ * 85 * 134451 . chubukov a v , ppin c and rech j 2004 _ phys . lett . _ * 92 * 147003 ; efremov d v , betouras j j and a. v. chubukov 2008 _ phys . b _ * 77 * 220401(r ) ; maslov d l , chubukov a v and saha r 2006 _ phys . rev . b _ * 74 * 220402(r ) .	identification of pairing mechanisms leading to the unconventional superconductivity realized in copper - oxide , heavy - fermions , and organic compounds is one of the most challenging issues in condensed - matter physics . clear evidence for an electron - phonon mechanism in conventional superconductors is seen by the isotope effect on the superconducting transition temperatures @xmath0 , since isotopic substitution varies the phonon frequency without affecting the electronic states . in unconventional superconductors , magnetic fluctuations have been proposed to mediate superconductivity , and considerable efforts have been made to unravel relationships between normal - state magnetic fluctuations and superconductivity . here , we show that characteristic experimental results on the ferromagnetic ( fm ) superconductor ucoge ( @xmath1 k and @xmath2 k ) can be understood consistently within a scenario of the spin - triplet superconductivity induced by fm spin fluctuations . temperature and angle dependencies of the upper critical magnetic field of the superconductivity ( @xmath3 ) are calculated on the basis of the above scenario by solving the eliashberg equation . calculated @xmath3 well agrees with the characteristic experimental results observed in ucoge . this is a first example that fm fluctuations are shown to be a pairing glue of superconductivity .
the recent evidence for neutrino masses has brought forward leptogenesis @xcite as a very attractive mechanism to explain the baryon asymmetry of the universe . along this mechanism , the baryon asymmetry of the universe is explained by the same interactions as the ones which can explain the neutrino masses . in the most straightforward seesaw model , which assumes right - handed neutrinos in addition to the standard model particles , both neutrino masses and leptogenesis originate from the yukawa interactions and lepton number violating majorana masses of the right - handed neutrinos @xmath5 where @xmath6 stands for the lepton weak doublets and @xmath7 is related to the standard brout - englert - higgs ( hereafter simply higgs ) doublet @xmath8 by @xmath9 . however , testing this mechanism will be a very difficult task for several reasons . if the right - handed neutrinos have a hierarchical mass spectrum , due to neutrino mass constraints , leptogenesis through @xmath10 decay can lead to the observed amount of baryon asymmetry e.g. only if it involves right - handed neutrinos with masses above @xmath11 gev @xcite . as a result they can not be produced at colliders . moreover there are many more parameters in the yukawa coupling matrices which can play an important role for leptogenesis , than there are ( not too suppressed ) low energy observables which could constrain these parameters . if the right - handed neutrinos have instead a quasi - degenerate spectrum ( for at least 2 of them ) , leptogenesis can be efficient at lower scales @xcite but generically in this case the neutrino mass constraints require suppressed values of yukawa couplings , which hampers their production at colliders . for leptogenesis to be both efficient and tested at low energy , not only is a quasi - degeneracy between 2 right - handed neutrinos required , but also a special flavour structure which allows for larger yukawa couplings while preserving the light neutrino mass constraints , and/or a right - handed neutrino production mechanisms other than through the yukawas and associated neutrino mixings . in this paper we consider the problem of testing leptogenesis mechanisms the other way around . while they can not confirm leptogenesis , could low energy observations at least exclude it ? we propose one particularly clear possibility , namely the observation of a right - handed charged gauge boson @xmath0 . it is known that for high mass right - handed neutrinos and @xmath0 , around @xmath12 gev or higher , the @xmath0 can have suppression effects on leptogenesis through dilution and scattering , but , in the specific case of reheating after inflation , they can also boost the @xmath10 abundances @xcite and hence relax the constraints on yukawa couplings . not surprisingly , with a low scale @xmath0 the suppression effects are dramatically enhanced . actually , see section 2 , they turn out to be so strongly enhanced that , even with a maximal cp asymmetry of order unity , leptogenesis can not be a sufficient cause of the matter excess anymore . right - handed gauge interactions lead in particular to much larger suppression effects at low scale than left - handed interactions do in other contexts ( i.e. than in leptogenesis from scalar @xcite or fermion @xcite triplet decays , whose efficiency have been calculated in refs . this is due to the fact that at the difference of triplets , a single @xmath10 can interact through @xmath0 exchange with fermions which are all in thermal equilibrium , which induces more efficient , and hence dangerous , scatterings and decays . in particular , some of the scatterings involving the @xmath0 turn out to induce a very large suppression due to the fact that they do not decouple through a boltzmann suppression . the production of @xmath10 s through a light @xmath0 , often presented as the easiest way to produce @xmath10 s , is therefore incompatible with successful leptogenesis , and even enhanced @xmath10 production from reheating can not compensate for the large suppression . the lower bounds on the mass of the @xmath0 , required for successful leptogenesis , are given in section 3 . the possible discovery of a low - energy @xmath0 has recently been the object of several analysis by lhc collaborations @xcite . it should be feasible up to @xmath13 - 5 tev ( see more details , and additional possible searches , in section 7 ) . the observation of a @xmath0 is not the only possibility to exclude canonical neutrino decay leptogenesis from current energy data . we give a list of other possibilities in section 5 , considering in particular the implications of the observation of a @xmath1 at lhc . the case of other leptogenesis seesaw models with not only or without right - handed neutrinos is briefly considered in section 6 . as well known the net rate of baryon asymmetry is given in any leptogenesis model by 3 ingredients , the cp asymmetry of the decaying particle , @xmath14 for a right - handed neutrino , the boltzmann equations which determine the efficiency @xmath15 and the @xmath6 to @xmath16 sphaleron conversion rate , which we denote by @xmath17 . let us first discuss and present our results for the case where the lepton asymmetry is created from the decay of a single right - handed neutrino , @xmath10 . later on we will discuss the generalization to more right - handed neutrinos . in this case , from these 3 ingredients the net baryon asymmetry produced by the @xmath10 decays is : @xmath18 with @xmath19 , @xmath20 , @xmath21 , @xmath22 the comoving number density of the species `` i '' , `` eq '' refering to the equilibrium number density , and @xmath23 the comoving entropy density . for a particle previously in thermal equilibrium , the efficiency is unity by definition in absence of any washout effect from inverse decays or scatterings . if all lepton asymmetry has been produced before the sphaleron decoupling at the electroweak phase transition and if the sphalerons have had the time to thermalize completely the @xmath6 abundance , the conversion ratio between lepton and baryon number is given by @xcite @xmath24 where the last equality refers to the sm value , with @xmath25 the number of fermion families and @xmath26 the number of higgs doublets . in the right - handed neutrino decay leptogenesis model without any @xmath0 , the cp - asymmetry is defined by @xmath27 while the evolution of the comoving abundances is given as a function of @xmath28 by the boltzmann equations : @xmath29 -2 \frac{y_{\cal l}}{y_{l}^{\rm eq}}\left(\gamma_{ns}^{\rm sub}+\gamma_{nt}+\gamma_{ht } + \gamma_{hs}\,\frac{y_{n}}{y_{n}^{\rm eq}}\right ) \label{boltzstand2}\end{aligned}\ ] ] where @xmath30 denotes the derivative with respect to @xmath31 . the thermally averaged reaction rate @xmath32 parametrizes the effects of yukawa induced decays and inverse decays with @xmath33 , and @xmath34 bessel functions . the other @xmath35 s take into account the effects of the various scatterings through a @xmath36 or a @xmath10 in the @xmath23 or @xmath37 channels . they are related to the corresponding cross sections in the following way @xmath38 with @xmath39\ , \sigma(s)$ ] the reduced cross section , @xmath40\equiv \sqrt{(a - b - c)^2 -4bc}$ ] and @xmath41 $ ] . the analytic expression of the reduced cross sections can be found in refs . we also neglect as in ref . @xcite the effects of yukawa coupling induced @xmath42 processes which have little effects too . ] @xmath43 in eq . ( [ boltzstand2 ] ) refers to the substracted scattering through a @xmath10 in the @xmath23 channel ( i.e. taking out the contribution of the on - shell propagator in order to avoid double counting with the inverse decay contribution @xcite ) . the above , now traditional approach assumes that @xmath10 are introduced in an isolated way in the model . in many unifying groups ( left - right symmetric @xcite , pati - salam @xcite , @xmath44 @xcite or larger ) the presence of the @xmath10 can be nicely justified as it is precisely the ingredient required to unify all fermions . these groups however do not introduce the @xmath10 in such an isolated way and moreover link the @xmath10 and @xmath0 masses to the same @xmath45 breaking scale @xmath46 . will also contribute to @xmath0 , but the opposite is not necessarily true . ] it is thus a ( generally unwarranted ) assumption to neglect the effect of @xmath45 gauge bosons . if @xmath47 is smaller than @xmath48 gev , these effects must be explicitly incorporated for any @xmath10 whose mass is not several orders of magnitude below the one of the @xmath0 @xcite . their effects for leptogenesis can be incorporated by modifying the boltzmann equations in the following way : @xmath53 with the cp asymmetry unchanged , as given by eq . ( [ cpasym ] ) . in these boltzmann equations there are essentially 2 types of effects induced by the @xmath0 , both suppressing the produced lepton asymmetry : from the presence of alternate decay channels for the heavy neutrinos , @xmath54 , and from scatterings , @xmath55 , see below . \a ) case @xmath56 : in this case the decay of @xmath10 to leptons or antileptons plus higgs particles remains the only possible 2 body decay channels but a series of three body decay channels with a _ virtual _ @xmath0 is now possible : @xmath57 or @xmath58 with @xmath59 , @xmath60 , @xmath61 . we obtain : @xmath62 given the potentially large value of the gauge to yukawa couplings ratio , the three body decays can compete with the yukawa two body decay . since the gauge interactions do not provide any cp - violation and are flavor blind , it can be shown that they do not provide any new relevant source of cp - asymmetry . but still the gauge interaction - induced 3 body decays appear in both boltzmann equations , eqs . ( [ nbogauge])-([lbogauge ] ) , with @xmath63 where @xmath64 is the total three body decay width . unlike in leptogenesis without @xmath0 , not all decays participate in the creation of the asymmetry but only a fraction @xmath65 does . this shows up in the boltzmann equations through the fact that eq . ( [ nbogauge ] ) involves @xmath66 while the cp - asymmetry in eq . ( [ lbogauge ] ) is multiplied only by @xmath67 . ) , we made the choice to keep eq . ( [ cpasym ] ) as definition for the cp - asymmetry . in its denominator , it involves only the yukawa driven decay rather than the total decay width , @xmath68 . therefore this cp asymmetry does nt correspond anymore , as in standard leptogenesis , to the averaged @xmath69 which is created each time a @xmath10 decays . however this definition is convenient for several reasons . it makes explicit the fact that the gauge decay does not induce any lepton asymmetry . moreover in this way , all ( competing ) suppression effects , including the dilution one , are put together in the efficiency , not in the cp - asymmetry . it also allows to take the simple upper bound @xmath70 for any numerical calculations . ] this dilution effect leads automatically to an upper bound on the efficiency . the bound @xmath71 , which applies in standard leptogenesis for thermal @xmath10 s becomes : @xmath72 as a numerical example , for @xmath73 tev , with yukawa couplings of order @xmath74 , so that @xmath75 ev , and with @xmath76 tev we obtain the large suppression factor @xmath77 , consistent with leptogenesis only if the cp - asymmetry is of order unity , which requires maximal enhancement of the asymmetry ( i.e. right handed neutrino mass splittings of order of their decay widths ) . in addition to this dilution effect , the three body decay @xmath54 reaction density also induces a @xmath6 asymmetry washout effect from inverse decays ( proportional to @xmath78 in eq . ( [ lbogauge ] ) ) which can also be large . \b ) case @xmath79 : in this case much heavier than @xmath0 is in general not expected in the left - right symmetric model or extensions given the fact that , as said above , both @xmath0 and @xmath80 have a mass proportional to the @xmath45 breaking scale @xmath46 , and given the fact that @xmath81 with @xmath82 the ordinary gauge coupling which is of order unity . ] the direct 2 body decays @xmath83 are allowed which leads to an even larger dilution and washout effect for low @xmath84 . for example with @xmath85 tev , @xmath86 and @xmath87 gev , we get @xmath88 , which means that the dilution effect makes leptogenesis basically hopeless at this scale , even with the maximum value @xmath89 . in the following we will consider only the case where @xmath90 ( this corresponds to the situation where a discovery of the @xmath0 and @xmath10 at lhc would occur through same sign dilepton channel @xcite , see section 6 ) . to compare eqs . ( [ nbogauge ] , [ lbogauge ] ) and eqs . ( [ n1bogauge_new ] , [ n2bogauge_new ] , [ lbogauge_new ] ) let us first note that the @xmath91 equations differ from the @xmath92 equation only through the @xmath93 and @xmath94 terms . as in the one @xmath10 case it can be checked that the @xmath95 terms have very little effects because their reaction rates are smaller than the @xmath55 ones ( compare for example in fig . 3.a @xmath96 with @xmath97 ) . the @xmath98 terms on the other hand have a size similar to the one of @xmath99 but they are multiplied by @xmath100 . this means that their effect is suppressed because those terms could be important only as long as the @xmath0 effects ( @xmath99 and @xmath54 ) dominate the thermalization of the @xmath101 ( with respect to the yukawa induced processes ) , but these @xmath0 effects equally affect @xmath102 and @xmath103 . similarly it can be checked that the @xmath104 are of little importance . they are relevant only for very large values of both @xmath105 and @xmath106 , beyond the values of interest for our purpose . as a result all these terms can be neglected in eqs . ( [ n1bogauge_new ] , [ n2bogauge_new ] ) and the evolution of @xmath102 and @xmath103 are essentially the same as the one of @xmath92 in eq . ( [ nbogauge ] ) replacing @xmath107 by @xmath108 and @xmath109 respectively . there are no important differences at this level . differences however can come from eq . ( [ lbogauge_new ] ) because this equation involves source and washout terms from both @xmath110 and @xmath111 . to discuss this equation it is useful to split it in two parts as follows @xmath112 with @xmath113 . clearly comparing the @xmath114 ( @xmath115 ) boltzmann equations with the one @xmath10 corresponding equation , eq . ( [ lbogauge ] ) , one observes that these equations are the same except that eqs . ( [ labogauge ] , [ lbbogauge ] ) involve additional washout terms from @xmath111 ( @xmath110 ) . since these terms can only decrease and @xmath116 there is a destructive interference between the contribution of @xmath110 and @xmath111 but even so , from the effects of all other terms , the following inequalities hold ( except for very large @xmath84 close to @xmath117 gev which is not of interest for our purpose ) . ] the absolute value of the lepton asymmetry obtained effects it is a good approximation to start from thermal distributions of @xmath118 , as explained above . therefore there is no change of sign of @xmath78 and the argument applies . ] one consequently gets @xmath119 which gives @xmath120 with @xmath121 which refers to the lepton number asymmetry obtained from eqs . ( [ nbogauge ] , [ lbogauge ] ) . this inequality has several consequences . ( i ) it means that if leptogenesis is ruled out in the one @xmath10 case taking @xmath122 ( as above ) it will be also ruled out in the 2 @xmath10 case if we take @xmath123 ( which is the bound to be considered in this case , see ref . one just need to apply the results of figs . 2 and 5 to both terms of eq . ( [ ineqyl ] ) . ( ii ) as eq . ( [ ineqyl ] ) obviously also holds for the case where we neglect the @xmath0 effects in the lepton number boltzman equation , this conclusion remains true even if we play with flavour ( applying to eq . ( [ ineqyl ] ) the results of fig . ( iii ) if , for a given value of @xmath124 and @xmath47 , both @xmath105 and @xmath106 are outside the allowed range of @xmath107 given in fig . 7.a , the lepton asymmetry produced will be too small . numerically it can be checked also that this figure remains valid to a good approximation for the @xmath125 case . for @xmath47 above @xmath126 tev the allowed region is shrinked by a hardly visible amount . as for the absolute lower bound on @xmath47 it is larger in the 2 @xmath10 case than in the one @xmath10 case ( i.e. than the value @xmath127 tev above ) but not by more than a few tev . with more than 2 right - handed neutrinos these conclusions remain valid . fukugita , yanagida , phys . lett . * b174 * ( 1986 ) 45 ; for recent reviews see e.g. a. strumia , arxiv : hep - ph/0608347 ; s. davidson , e. nardi and y. nir , arxiv:0802.2962 [ hep - ph ] ; t. hambye , arxiv : hep - ph/0412053 ; r. barbieri , p. creminelli , a. strumia and n. tetradis , nucl . * b575 * ( 2000 ) 61 ; k. hamaguchi , h. murayama and t. yanagida , phys . rev . * d65 * ( 2002 ) 043512 ; t. hambye , nucl . phys . * b633 * ( 2002 ) 171 . s. davidson and a. ibarra , phys . * b535 * ( 2002 ) 25 . john r. ellis , junji hisano , martti raidal and yasuhiro shimizu , phys . rev . * d66 * ( 2002)115013 ; s. davidson and a. ibarra , jhep 0109:013 , 2001 . l. covi , e. roulet and f. vissani , phys . b * 384 * ( 1996 ) 169 [ arxiv : hep - ph/9605319 ] ; m. flanz , e. a. paschos and u. sarkar , phys . b * 345 * , 248 ( 1995 ) [ erratum - ibid . b * 382 * , 447 ( 1996 ) ] [ arxiv : hep - ph/9411366 ] . a. pilaftsis , phys . d * 56 * ( 1997 ) 5431 , [ arxiv : hep - ph/9707235 ] ; a. pilaftsis and t. e. j. underwood , nucl . b * 692 * ( 2004 ) 303 [ arxiv : hep - ph/0309342 ] . j. kersten and a. y. smirnov , phys . d * 76 * ( 2007 ) 073005 [ arxiv:0705.3221 [ hep - ph ] ] . m. c. gonzalez - garcia and j. w. f. valle , phys . b * 216 * ( 1989 ) 360 ; s. carlier , j. m. frere and f. s. ling , phys . d * 60 * ( 1999 ) 096003 [ arxiv : hep - ph/9903300 ] ; n. cosme , jhep * 0408 * ( 2004 ) 027 [ arxiv : hep - ph/0403209 ] . e. ma , s. sarkar and u. sarkar , phys . b458 * ( 1999 ) 73 . p. j. odonnell , utpal sarkar , phys . rev . * d49 * ( 1994 ) 2118 ; e. ma and u. sarkar , phys . * 80 * ( 1998 ) 5716 ; t. hambye and g. senjanovic , phys . lett . * b582 * ( 2004 ) 73 - 81 ; g. dambrosio , t. hambye , a. hektor , m. raidal and a. rossi , phys . lett . * b604 * ( 2004 ) 199 . t. hambye , m. raidal and a. strumia , phys . lett . * b632 * ( 2006 ) 667 ; t. hambye , e. ma and u. sarkar , nucl . phys . * b602 * ( 2001 ) 23 - 38 . e. w. kolb and s. wolfram , nucl . b * 172 * ( 1980 ) 224 [ erratum - ibid . b * 195 * ( 1982 ) 542 ] . m. a. luty , phys . d * 45 * ( 1992 ) 455 . m. plumacher , z. phys . c * 74 * ( 1997 ) 549 [ arxiv : hep - ph/9604229 ] . g. f. giudice , a. notari , m. raidal , a. riotto and a. strumia , nucl . b * 685 * ( 2004 ) 89 [ arxiv : hep - ph/0310123 ] . bayatian _ et al . _ [ cms collaboration ] , `` cms technical design report , volume ii : physics performance , '' j. phys . g * 34 * ( 2007 ) 995 . w. m. yao _ et al . _ [ particle data group ] , j. phys . g * 33 * ( 2006 ) 1 ( and 2007 partial update for edition 2008 ) . m. mhlleitner and m. spira , phys . rev . * d68 * ( 2003 ) 117701 ; a. g. akeroyd and m. aoki , phys . rev . d * 72 * ( 2005 ) 035011 e. accomando _ et al . _ , arxiv : hep - ph/0608079 ; d. acosta _ et al . _ [ cdf collaboration ] , phys . rev . * 93 * ( 2004 ) 221802 ; k. huitu , j. maalampi , a. pietila and m. raidal , nucl . b * 487 * ( 1997 ) 27 v.m . abazov et al . 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( cdf collaboration ) , cdf note 9150 , http://www-cdf.fnal.gov/physics/new/top/ 2007/singletop / wprime / public_2fb.html .	it is well known that the leptogenesis mechanism offers an attractive possibility to explain the baryon asymmetry of the universe . its particular robustness however comes with one major difficulty : it will be very hard if not impossible to _ test _ experimentally in a foreseeable future , as most of the mechanics _ typically _ takes place at high energy or _ results _ from suppressed interactions , without unavoidable low - energy implications . an alternate approach is taken by asking : can it be at least falsified ? we show that _ possible _ discoveries at current and future colliders , most notably that of right - handed gauge interactions , would indeed forbid at least the `` canonical '' leptogenesis mechanisms , namely those based on right - handed neutrino decay . general lower bounds for successful leptogenesis on the mass of the right - handed gauge boson @xmath0 are given . other possibilities to falsify leptogenesis , including from the observation of a @xmath1 , are also considered . + 1.5 cm * is leptogenesis falsifiable at lhc ? * 0.5 cm jean - marie frre@xmath2 , thomas hambye@xmath3 and gilles vertongen@xmath3 + .7 cm @xmath3service de physique thorique , + universit libre de bruxelles , 1050 brussels , belgium + @xmath4kitp , university of california , santa barbara ca93106 , usa 0.5 cm 2truecm
so far the entire notion of the existence of dark matter ( dm ) has been based on the manifestations of its gravitational effects at cosmological scales , ranging from the observations of galactic rotation curves to the measurements of the anisotropy of the cmbr @xcite . many different theoretical models of dark matter @xcite suffer from a lack of direct experimental measurements that could reveal the nature of dm in interactions other than gravitational . several direct dark matter search experiments @xcite have created a rather controversial situation , from one side excluding weakly interacting massive particles ( wimps ) in the range of masses @xmath3 6 gev with the weak coupling strength to the target particles of ordinary matter ( om ) , and from another side claiming signals of dark matter detection in a lower mass range than expected in the popular supersymmetric models @xcite . on the other hand , experiments at the large hadron collider so far have not been able to find wimp candidates that could be candidates for a single - component dm @xcite . the general difficulty in identifying the nature of dark matter is the lack of non - controversial and reproducible effects of non - gravitational interactions between the dm and om particles . in this paper we would like to discuss a new , simple particle physics experiment with neutrons that could test one of the dm models of specific interest . in this model , dm ( in whole or at least in some fraction ) consists of mirror matter ( mm ) from a hidden gauge sector that represents an exact copy of the ordinary particle sector . the concept of mirror fermions was introduced in 1956 as a way to restore parity in a seminal paper by lee and yang , while in 1966 kobzarev , okun and pomeranchuk showed that mirror fermions can not have common strong , weak and electromagnetic interactions with ordinary fermions and thus the former can exist in the form of an independent parallel sector which they coined as mirror world @xcite . from today s view , once the ordinary matter and its interactions are described by the standard model ( sm ) , then the mirror particle content and interactions are described by another copy of ( sm@xmath4 ) . it can be equivalently viewed as , if in the context of extra dimensions , the om and mm are localized on two parallel branes with corresponding sm and sm@xmath4 self - interactions , while the gravity propagating through the bulk interacts with both types of matter . thus mm and om interact gravitationally , providing observable manifestations of dm effects , but they have no common sm and sm@xmath4 interactions ; in fact , renormalizable interaction terms between particles of two sectors are forbidden by the particle content of the standard model , in the same way as renormalizable interactions violating baryon or lepton numbers . however , in difference from cold dark matter candidates , mm is self - interacting and dissipative matter in the same sense as om . once mm is adopted as a framework for dm , then the properties of mm with respect to particle content and interactions are well known by the model requirement of sm @xmath5 sm@xmath4 . on the other hand , the cosmological genesis and galactic evolution might be different between mm and om due to differences in abundance and temperature of the two sectors and the obvious lack of observational details in the mirror sector . many aspects of the mm cosmology were discussed and explained in the literature @xcite . although these aspects are not broadly assimilated , the mirror matter hypothesis remains a viable option for dm . a detailed discussion of the viability of the mm model is not in the scope of this paper . the interested reader can learn these ideas e.g. from reviews @xcite . important for this paper are two aspects of mm . first , it was conjectured that the neutral particles of the om and mm sectors , elementary like neutrinos or composite like neutrons , can be mixed , providing an interaction portal between the two sectors . namely , the kinetic mixing of ordinary and mirror photons , first discussed in @xcite , which makes mirror matter effectively mini - charged with respect to the ordinary electromagnetic forces . though the kinetic mixing is severely restricted by cosmological limits @xcite , it can lead to experimentally observable effects , such as positronium oscillation into mirror positronium @xcite , and can also provide an interesting portal for the direct detection of mirror dark matter consisting dominantly of mirror helium and hydrogen and a smaller fraction of heavier mirror nuclei like carbon , oxygen etc . @xcite . in addition , ordinary and mirror particles can also share some hypothetical forces , interacting e.g. with gauge bosons of common family symmetry @xcite or common @xmath6 symmetry @xcite . the lepton and baryon number violating interactions between om and mm are of specific interest since they could lead to the co - genesis of baryon asymmetries in both sectors and shed the light on the near coincidence of the baryon and dark matter fractions in the universe @xcite . such interactions , at low energies , would induce lepton or baryon number violating effects such as the ordinary ( active ) neutrino oscillations into mirror ( sterile ) neutrinos @xcite and neutron oscillations into mirror neutrons @xcite , resembling and perhaps related to the more familiar phenomenon of neutron antineutron oscillation @xcite . particularly interesting for us is the effect of neutron to mirror neutron transformation ( @xmath7 ) first theoretically considered in @xcite . it was noticed in this paper that a @xmath7 oscillation time larger than @xmath8 s can not be excluded by either by the existing at that time experimental data or by cosmological and astrophysical bounds . the reason why such a fast oscillation ( in fact , faster than the neutron decay ) , which leads to the violation of baryon ( neutron ) number could be unnoticed in experiments is the presence of the earth magnetic field , which was never compensated in the experiments measuring the neutron lifetime . in fact , in the absence of matter and with a compensated magnetic field half of the neutrons created in experiments would have to disappear within seconds . however , it was recognized later that a mirror magnetic field @xmath9 , as a natural component of the mm model , might be present but unnoticed in terrestrial experiments , possibly being of galactic origin or due to the accumulation of mm inside the earth @xcite . in particular , the photon mirror photon kinetic mixing @xcite could induce large galactic magnetic fields , both ordinary and mirror , via the electron drag mechanism @xcite . on the other hand , the same interaction portal could lead to the accumulation of some small amount of mirror matter in the earth , which could be sufficient for the induction of a mirror magnetic field up to few gauss via the mirror electron drag due to the rotation of the earth , via the mechanism of ref . @xcite . in this way , the mirror magnetic field at the earth might have a magnitude of the same order as the earth s own magnetic field . in addition , its magnitude and orientation relative to ordinary magnetic field can be arbitrary and might also be variable in time @xcite . the idea of a possible neutron to mirror neutron transformation @xcite and following paper discussing experimental sensitivities for its search @xcite has stimulated several experimental searches @xcite , the results of which were adopted by the particle data group @xcite . these experiments were performed with ultra - cold neutrons ( ucn ) , where the effect of @xmath7 transformation could lead to the unaccounted disappearance of the neutrons stored in the ucn trap . in the presence of earth magnetic field @xmath10 , the neutron with its magnetic moment @xmath11 will change its energy level by @xmath12 , while the mirror neutron will not interact with ordinary magnetic field @xmath10 . thus , the zeeman split of the energy levels would suppress the @xmath7 transformation . the first experiments were thought to require shielding of the earth s magnetic field , due to their assumption that no mirror magnetic field could exist at the earth . in this way , under the hypothesis that the mirror magnetic field at the earth is vanishing , the limit @xmath13 s was obtained at 90 @xmath14 c.l . @xcite . in the later experiments the existence of a non - zero mirror magnetic field @xmath15 was admitted . in this case , the detection of a disappearance signal would require the tuning of the ordinary laboratory magnetic field @xmath16 and would have a resonance character in the transformation probability if the magnitude of the field @xmath10 were the same as that of the mirror field @xmath9 . additionally , the transformation effect would be enhanced when the directions of both fields would be aligned and would be minimal when directions of @xmath16 and @xmath15 were opposite . the probability of @xmath7 transformation in the absence of any fields can be described , as for any two - level system , by the proper oscillation time @xmath17 , where @xmath18 ev @xcite is a small mass mixing @xmath19 . in the presence of two different magnetic fields @xmath16 and @xmath15 , time evolution of the probability of @xmath7 transformation was described in @xcite . since the magnetic moment of neutron ( or mirror neutron ) @xmath20 ev / g , the energy split in milli - gauss magnetic fields @xmath21 and the probability of @xmath22 transition after free flight time @xmath23 can be simplified @xcite to @xmath24 where @xmath25 is the angle between the magnetic field vectors @xmath26 and @xmath27 and @xmath28}{2\tau^2(\omega-\omega')^2 } + \frac{\sin^2[(\omega+\omega')t]}{2\tau^2(\omega+\omega')^2 } \nonumber \\ \\ \mathcal{d}_b(t ) = \frac{\sin^2[(\omega-\omega')t]}{2\tau^2(\omega-\omega')^2 } - \frac{\sin^2[(\omega+\omega')t]}{2\tau^2(\omega+\omega')^2 } \nonumber\end{aligned}\ ] ] with @xmath29 and @xmath30 , where @xmath11 is the magnetic moment of the neutron and mirror neutron , assumed to be identical @xcite . we see that both terms in equation ( 1 ) have a resonance behavior part when the magnitude of the magnetic field @xmath10 is close to the unknown magnitude of mirror magnetic field @xmath9 . resonance occurs even if the directions of the @xmath16 and @xmath15 vectors do not coincide , but the probability is maximum when @xmath31 and minimum when @xmath32 . as was mentioned above , all previously published experiments looking for the disappearance of @xmath7 were performed with ultra - cold neutron ( ucn ) vacuum traps . a relatively small number of ucn were stored in the traps by reflections off the walls coated by material with a high nuclear fermi - potential . the typical velocity of ucn was a few m / s and the time between collisions was typically around 0.1 s. a controlled , intentionally - uniform magnetic field was applied to the ucn volume and was varied during the experiments . neutrons that transformed to the mirror state would have become sterile to interactions with om nuclei and would have escaped the trap , thus leading to an increase of the disappearance rate . in the experiment @xcite , the @xmath7 effect was sought for by studying the variation of the storage time of ucn in the trap with reflecting walls by switching on and off the external magnetic field @xmath10 . in `` off '' mode , the magnetic field in the trap was close to zero ( the earth magnetic field was compensated down to @xmath33 100 nt ) . no disappearance signal was found corresponding to the oscillation time limit @xmath34 103 s ( 95@xmath14cl ) . in a more sophisticated ucn experiment @xcite under similar magnetic field @xmath10 suppression , this limit ( assuming no mirror magnetic field ) was improved to an oscillation time of @xmath35 s ( at 90@xmath14cl ) . these limits , however , are not valid if one assumes that the mirror magnetic fields can be present at the earth . the experimental data of the latter experiment @xcite were made available to and re - analyzed by one of the authors in ref . @xcite with a more advanced theoretical oscillation description from @xcite . additionally , some data runs corresponding to unstable reactor flux conditions , as indicated by the system of two monitoring counters , were excluded . the neutron storage time in the trap was measured multiple times with variation of the laboratory magnetic field that was regularly alternated in the vertical direction with magnitude @xmath36 . from these data the asymmetry parameter for up and down variation of the direction of the magnetic field was calculated and the result has shown a peculiar anomaly with statistical @xmath37 deviation from the expected zero asymmetry for non - polarized neutrons in the be - coated trap . in the absence of consistent alternative interpretations , the asymmetry could be explained as an effect of the variation of the angle between @xmath16 and unknown @xmath15 resulting in a different degree of suppression of the @xmath7 transformation probability . the model @xcite would describe the observed asymmetry by the following parameters : the oscillation time on the order of @xmath38 s and the magnitude of the mirror magnetic field @xmath39 @xmath11 t . these interpretational parameters , however , should be considered as rather approximate for several reasons . first , the angle between @xmath10 and @xmath9 remains unknown ; and second , the effect of magnetic field @xmath10 non - uniformity was not accounted for . in still another experiment @xcite , an attempt was made to scan the magnitude of the vertical magnetic field in the range 0 to @xmath40 12.5 @xmath11 t at several values of @xmath10 with increments of 2.5 @xmath11 t . the analysis here had a statistically insignificant indication of the transformation effect with oscillation time 21.9 s ( other fit parameters were @xmath41 t and the angle between vertical up @xmath10 and @xmath9 @xmath42 degree ) . the data of this experiment also excluded with 95@xmath43 the oscillation time @xmath44 within the range of studied magnitudes of field @xmath10 . a drawback of this experiment was the rather large incremental step in the variation of the magnitude of magnetic field @xmath10 . this step was more than two times wider than the width of the resonance in @xmath45 . thus , the results of all these experiments were controversial . however , they do not exclude unambiguously the possible existence of the effect in the range of oscillation time @xmath46 10 s for a mirror magnetic field with a magnitude similar to or lower than that of the earth magnetic field . the common difficulty in the ucn trap experiments is that the measured small absorption coefficient of the reflecting walls of the trap is not reproduced by the theoretical calculations @xcite , leaving room for the asymmetry anomaly explanation by some other still unidentified phenomena , different than the transformation to a mirror state . thus , the test of the @xmath7 hypothesis require different technique than ucn traps , where multiple collisions of neutrons with @xmath47 walls of the trap would be excluded . in demonstrating the new method we will follow papers @xcite with equations ( 1 ) - ( 2 ) . in paper @xcite the @xmath48 oscillation effect in the presence of two vector magnetic fields @xmath16 and @xmath15 was exactly solved in the schrdinger equation with a @xmath49 hamiltonian describing both the neutron and mirror neutron polarizations and the mixing between the neutron and mirror neutron components . the direction of the @xmath15 field is not known a priori , however , the resonance behavior of the @xmath7 transformation still appears when the directions of both fields do not coincide . thus , the unique characteristic feature of the @xmath7 transformation search will be the observation of a resonance behavior in the counting rate when scanning versus the magnitude of the laboratory field @xmath10 . at least two opposite directions of the vector @xmath16 should be chosen for the magnitude scan to have sensitivity in the whole range of unknown @xmath50 . the idea of the observation of a regeneration transformation is well known , e.g. see @xcite . with a cold neutron beam the regeneration search will attempt to observe the effect of the appearance of neutrons from the mirror state . a beam of slow neutrons would propagate in vacuum through the path where a uniform magnetic field @xmath16 can be tuned in three components . let us assume that the direction of magnetic field @xmath10 can be fixed in 3d space , e.g. along the down - up direction and magnitude of field @xmath10 will be varied in steps from @xmath51 to @xmath52 . the probability of disappearance in the resonance is proportional to the square of the neutron time of flight @xmath53 through the vacuum region where magnetic field is tuned to the resonance : @xmath54 . therefore , a low neutron velocity and a large flight distance will enhance the probability . thick absorbing material at the end of the free flight path should remove all neutrons from the initial beam , leaving generated mirror neutrons propagating through the absorber wall as a beam of non - interacting sterile neutrons . behind the beam - stop absorber there is a second flight path vacuum tube in which identical magnetic field conditions are reproduced . in the second flight path , mirror neutrons with similar probability @xmath55 are transformed back to ordinary neutrons that are detected at the end of the flight path by a @xmath56 detector . this detector , when properly shielded for minimum background , should count no neutrons from the initial beam due to the thick upstream absorber but only the neutrons regenerated from the mirror state in the second tube . the probability of reappearance of regenerated neutrons in the high - efficiency detector in the resonance condition @xmath57 should be @xmath58 if the field @xmath16 field is uniform along the flight path . to demonstrate the sensitivity of such a method , we consider a rather generic configuration of an experiment that could be implemented at many reactors or spallation sources of neutrons where intense cold neutron beams are available ( figure 1 ) . we assume that two down - the - line flight paths , each 15-m long vacuum tubes , made of non - magnetic aluminum 6061-t6 alloy , can be used inside which the magnetic field will be controlled . an effective neutron absorber can be placed between the first and second tube , and a high - efficiency detector of slow neutrons with appropriate shielding can be placed at the end of the second tube . figure 1 also shows a typical collimator defining the divergence of the beam and two additional he-3 counters : one with efficiency @xmath46 1% upstream of the first flight tube serving as the beam intensity monitor and another one with 100% efficiency in front of the beam - stop for a disappearance measurement . as an example of cold neutron beam we are using the parameters of one of the neutron beams at the spallation neutron source ( sns ) @xcite at oak ridge national laboratory . the sns , being the latest in a series of well - characterized modern operating sources , is a suitable candidate for such a conceptual illustration . the sns produces neutrons via the bombardment of high - energy protons into a heavy metal ( liquid mercury ) target . the proton beam is pulsed at 60hz , allowing exploitation of the time structure to resolve thermal neutron energies to @xmath11ev resolution . while this capability is useful for many neutron scattering applications at sns , @xmath59 search experiment would not immediately benefit from this time structure . the high - brilliance of the cryogenic super - critical @xmath60 moderator system provides a very intense source of sub - thermal neutrons @xcite . using the mcstas neutron simulation package @xcite we optimized the layout geometry of the hypothetical regeneration search experiment at one of the typical sns beams . the simulation included the transport with gravity of the neutron beam from the cold moderator through a simple 12.5-m long collimator with opening aperture @xmath61 @xmath62 providing beam divergence of @xmath63 mrad . the simulated normalized cold neutron beam velocity spectrum is shown in figure 2 . cold beam flux can be described by the maxwellian distribution function with equivalent temperature @xmath64k . the integrated neutron intensity was found to be @xmath65 neutrons / sec / mw in the velocity range 200 - 2000 m / s ( wavelength @xmath66 angstroms ) . slow neutrons in the spectrum are more valuable for the @xmath7 observation since they would spend more time in the flight tube ; however , the effect of gravity causes a lower limit on this velocity . we assumed ( see figure 1 ) a @xmath67 detector ( a ) operating in dc integration mode with efficiency 1% upstream of the first flight tube , as the most common equipment for monitoring the intensity of cold beams . with beam intensity @xmath68 n / s , this detector ( a ) will measure the current integrated over 1 second to be proportional to @xmath69 . after the first flight tube , the detector ( b ) with @xmath70 efficiency can be used . this is a @xmath67 dc - integrating counter measuring the full beam intensity . measurements of the integrated currents ratio in ( a ) and ( b ) counters as @xmath71 could detect the @xmath7 `` disappearance '' signal at the end of the first flight tube for the resonance value of magnetic field @xmath10 . beyond the statistical accuracy of the monitoring with counter ( a ) , the stability of the `` counters '' operation and the beam intensity fluctuations might set additional limits on the sensitivity of disappearance detection . at the end of second flight tube , @xmath67 the counter ( c ) , operating in the proportional pulsed mode , is used for counting regenerated neutrons with an efficiency close to 100 % . this counter should be well shielded to provide the lowest background counting rate possible . assuming that counter ( c ) will have a transversal size @xmath72 @xmath62 we found from the comparison with similar existing devices that the achievable background rate can be lower than @xmath73 counts / sec . we will assume this rate for further estimates . we assume that the magnetic field inside the flight tubes can be generated with 3-dimensional coils positioned outside the non - magnetic tube . these coils will compensate the local , uniform earth magnetic field and create controlled 3-d magnetic field inside the flight tube . if the local , environmental magnetic field will be non - uniform , additional coils can be installed to compensate for these local effects . alternatively , an outer layer of @xmath11-metal shielding can be installed around the field - shaping coils and the vacuum tubes . using active feedback of the currents and reference magnetic fields , we assume that the magnetic field can be controlled in any direction with magnitude from 0 to 200 milli - gauss ( 20 @xmath11 t ) and with uniformity better than 2 milli - gauss . the design and control of the magnetic field will probably be the most challenging but doable technical problem in the considered experiment . the effects of magnetic field fluctuations can lead to some deviation from the quasi - free condition as it was shown in ref . @xcite for neutron antineutron oscillation @xcite ; however , these effects can be kept under control . to illustrate the capability of the regeneration method , we performed a simulation of a measurement made with the layout shown in figure 1 , using beam parameters obtained from the sns beam simulations . since the magnitude of mirror magnetic field @xmath9 is unknown , we assumed that a scan will be made with the @xmath10 field magnitude varying e.g. in 100 steps from 0 to 200 mg with a step - size of 2 mg . since the direction of @xmath9 field is also unknown , we anticipate that this 100-point scan should be repeated for the opposite direction of vector @xmath16 . according to above formulas ( 1 ) and ( 2 ) , the transformation probability in the resonance @xmath74 , neglecting non - resonance term in ( 2 ) , will be @xmath75 assuming that the oscillation time is fixed , the probability is maximum for @xmath76 , reduces to @xmath77 at @xmath78 , and further becames zero at @xmath79 according to ( 3 ) . thus , making two 100-point scans with two opposite directions of @xmath16 will cover the possible variation of the disappearance probability magnitude by a factor two due to the unknown direction of @xmath15 . for the regeneration effect at the end of second flight tube , the possible variation of the probability in each of the hemispheres around the two directions of @xmath16 will be a factor 4 . for the regeneration simulation , we have chosen following parameters : the positive direction of vector @xmath16 is vertical up ; @xmath80 mg , @xmath81 ; the background rate is 0.1 @xmath82 . we assumed that 200 counting runs will be made , each run with the fixed magnitude of @xmath10 ranging from @xmath83 mg to @xmath84 mg in steps of @xmath85 mg . the duration of each run will be 3600 seconds . thus , the whole measurement should take @xmath86 days of beam time . to correct for possible beam instabilities , the counting rate of the regeneration counter ( c ) can be normalized to the counting rate of the beam monitor ( a ) . the simulated regeneration counting rate is shown in figure 3 for @xmath87 s. black points represent counts in every run , and the continuous curve is the sum of the simulated effect and assumed background . the counts in the peak of the resonance correspond to a signal of more than 12@xmath88 significance . the resonance in the opposite hemisphere of @xmath16 is statistically insignificant . since scans with two directions @xmath89 are envisaged , for a given oscillation time @xmath90 , at least one resonance peak should appear in counting probability measurements if the magnitude of @xmath9 is in the full range of scan . this peak will have maximum amplitude if two magnetic fields are aligned @xmath57 . in this case second resonance peak will disappear . regeneration probability will be reduced by factor of four if @xmath91 . in this case peaks corresponding to the hemispheres @xmath92 and @xmath93 will have equal height . in the latter case , in the absence of peak the oscillation time @xmath94 s can be excluded at 95% cl . if , at smaller @xmath90 , both peaks are detected , it can give a determination of @xmath50 with a corresponding statistical accuracy . oscillation time @xmath87s . background rate in @xmath67 counter c is assumed as 0.1 counts per second . ] disappearance effect that can be measured at the same time during the hypothetical 10-day magnetic field scan and will provide an independent measurement of the @xmath7 effect . the total beam intensity will need to be measured with total absorption counter ( b ) installed at the end of the first flight tube upstream of the beam stop . due to the high beam intensity , this @xmath67 counter should operate in current - integrating mode . since the disappearance effect is a small fraction @xmath95 of the beam intensity the stability of the neutron flux in the beam needs to be monitored by an independent counter . we assumed that a current - integrating @xmath67 counter ( a ) with 1% efficiency will be installed upstream the first flight tube . statistical fluctuations of the beam intensity monitoring in this counter will be the main contribution to the uncertainty in the measurement of the disappearance effect . thus , with 1% monitoring efficiency , the sensitivity of detection of @xmath7 disappearance will be worse than in the regeneration measurement . however , statistics of 1% beam monitoring by counter ( a ) will practically not contribute to the accuracy of measurement of regeneration effect . the simulated measurement of disappearance effect for the same 10-day scan with the same parameters as above but with an oscillation time of @xmath96 s is shown in figure 4 . the larger disappearance peak in this figure at the resonance value has a significance of more than @xmath97 where we are assuming the contribution of only statistical factors . it is noted that the width of the disappearance resonance is wider than that of the regeneration curve , and the ratio of the heights of @xmath92 and @xmath93 peaks is also higher in accordance with equations ( 1)-(3 ) . to improve the monitoring statistics it is possible to think of an alternative way to monitor the source , such as when the total absorption counter would view the cold source through a different beamline opening . in this case the statistical sensitivity of the disappearance measurement might be essentially better than in the regeneration measurement . if both the disappearance and regeneration measurements are implemented simultaneously , as in our hypothetical case , they can be analyzed together and the combined sensitivity might be improved . with 1% efficiency monitoring , the disappearance method alone can provide statistically 95% cl exclusion limit for @xmath98 s ; with 100% monitor efficiency , 95% cl exclusion limit can be @xmath99 s. an increase of the experiment measurement time by a factor @xmath100 will improve exclusion limits for both disappearance and regeneration by approximately @xmath101 s. oscillation time @xmath96s . ] the hypothetical experiment described above will allow for a resolution of the controversy observed in experiments looking for neutron to mirror neutron disappearance with ucn traps . both beam disappearance and regeneration methods , as described above , will be free of the uncertainties of ucn interactions with material walls . a disappearance oscillation time up to @xmath102 s or higher can be excluded at 95% cl . alternative to exclusion , an observation of the resonance peak in disappearance would demonstrate a new effect of @xmath7 transformation . additionally , the resonance peak observed in regeneration will establish the existence of a beam of sterile neutrons that can be available for further experiments . if the disappearance / regeneration effect would exist , it will allow the determination of the direction of the magnetic field @xmath16 that will maximize the resonance counting . with the resonance magnitude of magnetic field @xmath74 , this can be achieved by varying the direction of magnetic field @xmath16 . thus , the direction of the vector of mirror magnetic field another component of the mm model can be determined . since the effect can be reproduced on rather short time scale of @xmath103 hour , it will be possible to study the stability of magnetic field @xmath15 together with its direction and thus determine cosmological or terrestrial origin of the field @xmath15 . all simulations in this paper were done using the sns / ornl parameters as an example . other sources of cold neutrons and the improved beam line design potentially can provide larger sensitivities . in particular , the european spallation source ( ess ) under construction at lund @xcite with large beam port envisaged in the project for the nnbar experiment , due to higher cold neutron flux and larger length of the neutron flight path can provide sensitivity for an oscillation time of more than 50 seconds . in this way , for the @xmath19 oscillation time of about 10 seconds , the ess would in fact become a factory of mirror neutrons . we are grateful to ornl colleagues leah broussard , franz gallmeier , and erik iverson , and also to christopher crawford from the university of kentucky , chen - 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the damping of quantum effects in a system coupled to external degrees of freedom is a fundamental problem of atomic physics , condensed matter physics and quantum optics . there is great interest in understanding and controlling such damping in well characterised systems . here we study the damping of quantum effects in the transport properties of a two - dimensional electron gas deposited on the surface of a pool of liquid helium @xcite . electrons on the surface of helium are vertically confined by their image charges and an ( optional ) applied holding field . they constitute a two - dimensional electron gas similar to those in semiconductor devices @xcite but with different scattering and damping mechanisms . electrons may scatter off ripples on the surface of the helium pool ( `` ripplons '' ) or off helium vapour atoms above the liquid surface . above 1 k , gas atom scattering dominates @xcite , and we concentrate on this regime . on the electronic time - scale , the helium vapour atoms are almost stationary and hence similar to impurities in a metal film or a semiconductor device . thus there are quantum interference corrections to the resistivity at low temperature familiar from studies of transport in metals and semiconductors @xcite . these corrections result from constructive interference between closed electron paths and their time - reversed counterparts , leading to a small enhancement of the resistance ( `` weak - localization correction '' ) . the slow movement of helium atoms leads to damping of weak - localization . there is an important distinction between the effect of vertical and horizontal motion of helium atoms . roughly , horizontal movement produces damping by scrambling the phase of the interfering paths ; vertical movement , by reducing the weight of contributing paths of long duration . the effect of horizontal movement has been analysed previously @xcite ; it is the purpose of this paper to study the effect of vertical motion . the central result is that due to vertical motion of the helium atoms the interference contribution of paths of duration @xmath0 is reduced by a factor @xmath2 . thus paths of duration greater than the damping time , @xmath3 , are effectively cutoff . an interesting feature is that damping due to both vertical and horizontal movement of helium atoms is not a simple exponential ; it cuts off more sharply as the exponential of @xmath4 . in contrast , electron - electron and electron - phonon interactions in metals and semiconductors are supposed to produce simple exponential damping . damping in atomic physics and nuclear magnetic resonance is also commonly a simple exponential ; this is indicated by the lorentzian shape of spectral and magnetic resonance lines @xcite is a lorentzian . ] . as emphasized by afonin _ @xcite in context of quantum transport , the form of damping can be probed by measuring the magnetic field dependence of the weak - localization correction ( `` weak - localization lineshape '' ) . in section iii we exhibit some lineshapes corresponding to different forms of damping . weak - localization has been observed in a related system , electrons on a surface of solid hydrogen @xcite . in this system helium vapour was deliberately introduced above the solid hydrogen to scatter electrons ; thus gas atom damping is relevant to this type of experiment . more recently , karakurt _ et al . _ have systematically studied the dependence of the damping rate on various experimental parameters ( electron density ; gas vapour pressure , controlled via temperature ; and holding field ) for electrons on helium @xcite . in this way they have obtained quantitative information on the contributions of different mechanisms to the damping rate . it is the experiment of karakurt _ et al . _ that prompted us to carry out the present investigation . for orientation it is useful to recall some typical parameters for the experiment of karakurt _ et al . _ in the absence of a holding field , the electron is bound to the surface by its image . the charge of the image is reduced from the bare charge of the electron by a factor @xmath5 @xcite ; thus the vertical scale of the electronic wavefunction is 76 @xmath6 . the lowest vertical subband wavefunction is of the fang - howard form , @xmath7 , at zero holding field ; this form remains an excellent variational ansatz with @xmath8 an adjustable parameter when a holding field is applied . here @xmath9 denotes the distance of the electron above the helium pool . the subband spacing is 6 k ; hence for sufficiently low temperatures and electron densities below @xmath10 /m@xmath11 the surface electrons behave like a two - dimensional electron gas . much of the data of karakurt _ et al . _ is at temperatures around 2 k and at a typical density of @xmath12 /m@xmath11 corresponding to a fermi temperature of 0.6 mk . note that their two - dimensional electron gas is therefore non - degenerate in contrast to the situation in metal films and typical semiconductor devices . thus transport properties are not determined entirely by mono - energetic electrons on the fermi surface ; instead we must sum the boltzman - weighted contribution of electrons of all energies . the electron - atom collision time inferred from mobility measurements was typically a few ps . the longest relevant electronic time scale is @xmath13 , the time taken by a thermal electron to move a distance @xmath8 ( see eq 5 below ) . at 2 k and zero holding field @xmath14 ps . in comparison , the atom - atom collision time is enormous , of the order of 10 ns . in this section we analyse the damping produced by the vertical motion of helium vapour atoms . first we analyse a simple model ( model i ) that captures some of the essential physics , but leads to the incorrect conclusion that the damping factor goes as the exponential of @xmath15 rather than @xmath4 , a result obtained earlier by stephen @xcite . we then identify a shortcoming of model i and in the next subsection introduce and analyse an improved version ( model ii ) that leads to the correct answer . in this model we assume that the helium atoms are able to scatter electrons only if they are within a certain distance ( denoted @xmath8 ) from the liquid helium surface . it is also assumed that the scattering is independent of the precise height of the atom so long as it lies within the prescribed distance . consider @xmath16 probability that an atom will remain within the scattering distance for a time @xmath0 . at first let us assume the probability decays exponentially , @xmath17 the numerical coefficient in the exponential is for later convenience . since the motion of vapour atoms is essentially independent the probability that @xmath18 atoms will remain within the scattering distance for time @xmath0 is @xmath19^n = \exp \left ( - \frac { n t } { \sqrt{\pi } \tau_z } \right).\ ] ] weak - localization results from constructive interference between the history in which an electron traverses a particular closed path and the history in which it traverses the same path backwards . a path of duration @xmath0 involves @xmath20 collisions . for this path to contribute to weak - localization it is neccessary for all atoms to remain within the electron - scattering region for a duration of order @xmath0 . hence the fraction of paths of duration @xmath0 that contribute to weak - localization is @xmath21 here @xmath22 is the damping factor ; essentially this result is given in ref @xcite . this argument must be improved in two ways . first , it is not neccessary for all the atoms to remain in place for the entire duration of a closed path . in particular , atoms encountered by the electron in the middle of a closed path are encountered by both the forward and backward path at essentially the same time . at the other extreme , atoms encountered early on the forward path are encountered towards the end on the backward path , a time @xmath0 later . thus on average atoms need to remain in place for a time @xmath23 . hence the damping factor is really @xmath24 with @xmath25 . a second improvement is needed because eq ( 1 ) is incorrect . @xmath26 is easily calculated and seen to not be exponential . here we mention only the relevant features of @xmath26 ; the details are relegated to appendix a. ( i ) as expected on dimensional grounds , @xmath26 is a function of @xmath27 alone , where @xmath28 here @xmath29 mass of a helium atom . physically , @xmath13 is the time taken by a thermal atom to move a distance @xmath8 . ( ii ) for short times , @xmath30 , we find @xmath31 ( iii ) for long times , @xmath32 , @xmath26 vanishes in a manner not relevant to our purpose . now the probability that @xmath18 atoms remain near the surface is @xmath33^n & = & \exp [ n \ln p(t ) ] \nonumber \\ & = & \exp \left [ n \ln \left ( 1 - \frac{t}{\sqrt{\pi}\tau_z } + \ldots \right ) \right ] \nonumber \\ & \approx & \exp \left ( - \frac { n t } { \sqrt{\pi } \tau_z } \right).\end{aligned}\ ] ] this shows that for large @xmath18 , @xmath34^n $ ] can be approximated as an exponential only for @xmath35 ; but since it becomes negligible in any case once @xmath36 , there is no significant error in taking @xmath37^n$ ] to be an exponential . the upshot of this discussion is that although @xmath26 is far from exponential , @xmath38^n$ ] is a simple exponential under appropriate circumstances ; eq ( 2 ) is valid , although eq ( 1 ) is not . similarly we see that eq ( 4 ) is also valid provided @xmath39 , a condition needed for weak - localization . in summary , for model i the damping decays as the exponential of @xmath15 . provided @xmath40 , it is given by eq ( 4 ) . the atomic time constant @xmath13 is given by eq ( 5 ) . evidently , the three time scales are arranged in the hierarchy @xmath41 . the shortcoming of model i is the assumption stated in the first paragraph of the previous subsection . it is more realistic to assume that the ability of an atom to scatter electrons turns off smoothly as it moves away from the liquid helium surface . if we treat the atoms as hard core potentials , the contribution of a closed path to the return amplitude is a product of the amplitude for the electron to go to atom 1 , multiplied by the amplitude to scatter off atom 1 , multiplied by the amplitude to go to atom 2 , multiplied by the amplitude to scatter off atom 2 , and so on around the loop . let @xmath42 be the amplitude to scatter from an atom at height @xmath9 above the helium surface . model i can be described as the case in which @xmath42 is a step function . here we choose @xmath43 this is derived by taking the vertical subband wavefunction of the electrons to be of the fang - howard form and treating the helium atom as a short - ranged hard - core potential . if the helium atoms are only allowed to move vertically the forward and backward paths remain in phase ; however the interference contribution to the return probability is still modified because the forward and backward paths have different amplitudes to scatter from each atom . we must consider @xmath44 here @xmath0 is the difference in the times at which the atom is encountered on the forward and return path . the atom is assumed to move ballistically at vertical speed @xmath45 for this time . @xmath46 denotes an average over all possible configurations of the helium atom ( vertical position is assumed to be uniformly distributed and vertical speed is given by the maxwell - boltzmann formula ) . introduce the normalization factor @xmath47 defined by @xmath48 here the average over vertical position is performed as in eq ( 9 ) but the velocity distribution is assumed to be a delta function peaked about zero . @xmath47 is the value of @xmath49 when the atoms do nt move . let @xmath50 the contribution of paths of duration @xmath0 is then reduced roughly by the factor @xmath51 raised to the power @xmath20 , the number of atoms encountered . @xmath51 is analogous to @xmath26 for model i. again on dimensional grounds , @xmath51 depends only on the ratio @xmath27 and again we are interested only in the short time behaviour . this is evaluated in appendix b. the difference from the previous case is that @xmath52 for short times , @xmath53 . the behaviour is quadratic rather than linear ( compare eq 6 ) . quadratic behaviour is generic ; the linear behaviour for model i is an artifact of the discontinuous step in @xmath42 . hence @xmath51 raised to the power of @xmath18 is approximately gaussian rather than exponential @xmath54^n \approx \exp \left ( - \frac{n t^2 } { 3 \tau_z^2 } \right).\ ] ] eq ( 13 ) should be contrasted with eq ( 7 ) above for model i. to obtain the damping factor , roughly we must replace @xmath18 in eq ( 13 ) by @xmath20 , the number of atoms encountered in a path of duration @xmath0 . before that we must replace @xmath15 in eq ( 13 ) by @xmath55 , its value averaged over the interval from 0 to @xmath0 with uniform weight . this is to take into account the range in the difference of times at which an atom is encountered along the forward and reversed histories . the result for the damping factor is @xmath56 where @xmath57 . eq ( 14 ) is the central result of this paper . it is valid provided @xmath58 . karakurt _ et al . _ observed damping by vapour atom motion at low electron density and by electron - electron interaction at high density @xcite . at intermediate densities , damping by both mechanisms was substantial . vapour atom scattering produces cubic exponential damping ; electron - electron interaction is presumably simple exponential . _ have pointed out that the weak - localization lineshape depends on the form of damping and they have given an expression for the lineshape in the extreme cases that the damping is entirely simple exponential or entirely cubic exponential . the purpose of this section is to study the lineshape in the intermediate regime and examine how it crosses over from one extreme form to the other . for simplicity , first let us consider a degenerate electronic system . assuming that the different damping mechanisms are independent the lineshape is given by @xmath59 here @xmath60 is the fermi energy ; @xmath61 , the sample width ; @xmath62 , the sample length ; @xmath63 the electron diffusion constant ; @xmath64 , the simple exponential damping rate ; and @xmath65 , the cubic exponential damping rate . energy dependence enters the integrand in eq ( 15 ) through the diffusion constant @xmath66 and through the energy dependence ( if any ) of the time constants @xmath67 and @xmath68 . the @xmath69 factor in eq ( 15 ) may be recognised as the fourier transform of the directed area distribution for closed random walks on a plane @xcite . it is useful to manipulate eq ( 15 ) into a more revealing form . to this end introduce the dimensionless variable @xmath70 to obtain @xmath71 here @xmath72 . making use of the asymptotic formula @xmath73 we obtain @xmath74 \nonumber \\ & & + \frac{1}{\pi } { \cal f } \left ( \frac{b_1}{b } , \frac{b_3}{b } \right)\end{aligned}\ ] ] where @xmath75 , @xmath76 , @xmath77 is euler s constant and the function @xmath78 . \nonumber \\ & & \end{aligned}\ ] ] eqs ( 18 ) and ( 19 ) constitute the generalisation of the standard weak - localisation lineshape to the case that both @xmath67 and @xmath68 damping are present . for the special case that there is no @xmath68 damping ( hence @xmath79 ) eqs ( 18,19 ) reduce to the familiar expression involving digamma functions by use of the integral representation @xcite @xmath80 a significant feature revealed by eqs ( 18,19 ) is that the lineshape is universal : @xmath81 does not depend on microscopic length scales . note that the magnetic field dependence is entirely in the second term of eq ( 18 ) ; the first term is an additive constant . a practical advantage of eq ( 19 ) over eq ( 15 ) is that the integrand is well behaved for both large and small @xmath82 . in contrast , the integrand in eq ( 15 ) diverges at the lower end . to study the crossover in lineshape we fix the damping rate @xmath83 . equivalently , we fix @xmath84 . @xmath85 is plotted as a function of @xmath86 for several values of the ratio @xmath87 . fig 1 shows that for the same damping rate the lineshape changes noticeably as damping shifts from simple exponential to cubic exponential . fig 2 shows the behaviour of the conductance minimum at @xmath88 for a fixed damping rate . it is given by @xmath89\ ] ] with the crossover function @xmath90 as implied by eqs ( 18 ) and ( 19 ) the crossover depends only on the ratio @xmath91 . @xmath82 has the limiting values @xmath92 and @xmath93 . under experimental conditions @xcite the electron gas is non - degenerate . at finite temperature @xmath94 @xmath18 is the area density of electrons . in the second line of eq ( 23 ) we have approximated the fermi function by a boltzman factor and imposed a lower cutoff @xmath95 . below the cutoff energy the electrons are presumed to be strongly localized and to make an insignificant contribution to the conductance . these finite temperature considerations make it more difficult to extract the form of damping from the lineshape . in summary , we have given a physical argument that due to vertical motion of helium atoms the interference of electron paths of duration @xmath0 is damped by a factor @xmath1 . we have derived a formula for the universal magnetoconductance lineshape for the case that both @xmath67 and @xmath68 damping are present . it should be possible to rederive these results via impurity averaged diagrams ; this is left open for future work . it is a pleasure to acknowledge helpful correspondence with m. stephen . this work was supported in part by nsf grants dmr 98 - 04983 ( dh and hm ) and dmr 97 - 01428 ( ajd ) and by the alfred p. sloan foundation ( hm ) . hm acknowledges the hospitality of the aspen center for physics where this work was completed . we wish to calculate @xmath26 , the probability that a vapour atom will remain within a vertical elevation @xmath8 of the liquid surface for a time @xmath0 . we assume ( i ) the initial elevation of the atom is uniformly distributed between zero and @xmath8 ; ( ii ) the vertical velocity is maxwell - boltzman distributed ; ( iii ) the atom moves ballistically ; and ( iv ) if the atom strikes the liquid surface it sticks and does not reflect @xcite . due to assumption ( iii ) the expression for @xmath26 that we derive is valid only for times short compared to the atom - atom collision time ; however this is not a serious restriction since we are interested only in the short time behaviour of @xmath26 . based on these assumptions we may write @xmath96 the two contributions correspond to the atom moving up and down respectively . by exchanging the order of integration we can perform the @xmath9 integral first to obtain @xmath97 we have rescaled variables so that @xmath98 and @xmath99 . note that @xmath100 and as @xmath101 , @xmath102 . eq ( a2 ) is an exact expression for @xmath26 . the small time , @xmath53 , asymptotic behaviour is @xmath103 to calculate @xmath51 we assume that the initial elevation of the vapour atom is uniformly distributed between the liquid surface and an upper cutoff @xmath62 . ultimately we shall take @xmath104 . aside from this we share the assumptions ( ii ) , ( iii ) and ( iv ) of appendix a. m.w . cole , rev mod phys * 46 * , 451 ( 1974 ) . t. ando , a.b . fowler and f. stern , rev mod phys * 54 * , 437 ( 1982 ) . p.a . lee and t.v . ramakrishnan , rev mod phys * 57 * , 287 ( 1985 ) . afonin , yu . m. galperin , v.l . gurevich and a. schmid , phys rev * a 36 * , 5729 ( 1987 ) ; m.j . stephen , phys rev * b36 * , 5663 ( 1987 ) ; v.v . afonin , yu . m. galperin and v.l . gurevich , phys rev * b40 * , 2598 ( 1989 ) . r. loudon , _ the quantum theory of light _ ( oxford univ press , 1983 ) . slichter , _ principles of magnetic resonance _ ( springer - verlag , 1990 ) . p.w . adams and m.a . paalanen , phys rev lett * 58 * , 2106 ( 1987 ) ; p.w . adams and m.a . paalanen , phys rev * b39 * , 4733 ( 1989 ) ; p.w . adams , phys rev lett * 65 * , 3333 ( 1990 ) . i. karakurt , d. herman , h. mathur and a.j . dahm ( to appear in phys rev lett ) ; cond - mat 9912317 . jackson , _ classical electrodynamics _ , p148 ( second ed , wiley , 1975 ) . e. fermi , z phys * 26 * , 54 ( 1924 ) . stephen , phys rev * b40 * , 2600 ( 1989 ) . for a recent analysis that emphasizes this point of view , see h. baranger and h. mathur ( unpublished ) . morse and h. feshbach , _ methods of theoretical physics _ , vol 1 , p 422 - 423 ( mcgraw - hill , 1953 ) . nayak , n. masuhara and d.o . edwards , phys rev lett * 50 * , 990 ( 1983 ) .	the damping of quantum effects in the transport properties of electrons deposited on a surface of liquid helium is studied . it is found that due to vertical motion of the helium vapour atoms the interference of paths of duration @xmath0 is damped by a factor @xmath1 . an expression is derived for the weak - localization lineshape in the case that damping occurs by a combination of processes with this type of cubic exponential damping and processes with a simple exponential damping factor .
study of exact solutions of einstein s field equations is an important part of the theory of general relativity . this importance is not only from more formal mathematical aspects associated with the theory ( e.g. the classification of space - times ) but also from the growing importance of the application of general relativity to astrophysical phenomena . for example , exact solutions may offer physical insights that numerical solutions can not . present time trend of analyzing different aspects of black hole ( bh ) solutions did lead us to grow our interests in cleaner @xmath1 dimensional gravity . discovery of btz bh @xcite ignited the light first . through this @xmath1 dimensional model if we need to explore the foundations of classical and quantum gravity we would not find any newtonian limit and no propagating degrees of freedom will arise . in literature , very easy - to - find works in this aspect comprise the study of quasi normal modes of charged dilaton bhs in @xmath1 dimensional solutions in low energy string theory with asymptotic anti de - sitter space times @xcite . hawking radiation from covariant anomalies in @xmath1 dimensional bhs @xcite is another beautiful example . lastly , we must also name the study of branes with naked singularities analogous to linear or planar defects in crystals and showing that zero branes in ads space times are negative mass bhs ! " @xcite . taking charged gravastars as an alternative to charged bhs in @xmath1 ads space times is already investigated @xcite . extensions of btz bh solutions with charge are also available in the literature . these are obtained by employing nonlinear born infield electrodynamics to eliminate the inner singularity @xcite . the non - static charged btz like bhs in @xmath2 dimensions have also been studied @xcite which in its static limit , for @xmath3 , reduces to @xmath1 btz bh solutions . study of interior solutions in @xmath1 dimension @xcite shows that even the noncommutative - geometry - inspired btz bh is not free from any singularity . study of interior solutions are farely found in literature . for example , solutions of c. wolf @xcite and s. yazadjiev @xcite , solutions in te framework of brans - dicke theory of gravity by s.m.kozyrev @xcite and new class of solutions corresponding to btz exterior spacetime by sharma _ et . @xcite , which is regular at the centre and it satisfies all the physical requirements except at the boundary where the authors propose a thin ring of matter content with negative energy density so as to prevent collapsing . the discontinuity of the affine connections at the boundary surface provide the above matter confined to the ring . such a stress - energy tensor is not ruled out from the consideration of casimir effect for massless fields . the purpose of the present work is to find exact interior solutions for perfect fluid model both with and without cosmological constant , @xmath0 . the motivation for doing so is provided by the fact that the assumption of equation of state ( eos ) , @xmath4 , which seems to be very reasonable for describing the matter distribution in the study of relativistic objects like stars @xcite , wormholes @xcite and gravastars @xcite . the structure of our work is as follows : in sec ( [ ein_field ] ) , we derive required einstein equations . [ interior ] ) constitutes of different interior solutions for various cases of eos . lastly , in sec . ( [ conclusion ] ) a brief conclusion is provided . we take the static metric to describe the interior region of a @xmath1 dimensional space time as @xmath6 where @xmath7 and @xmath8 are the two unknown metric functions . we take the perfect fluid form of the energy momentum tensor @xmath9 where @xmath10 is energy density and @xmath11 is pressure . einstein s field equations with a cosmological constant , @xmath0 , for the space - time metric ( eq.([metric ] ) ) together with the energy momentum tensor given in eq.([stress ] ) may be written as @xmath12 here superscript ` @xmath13 ' denotes the derivative with respect to @xmath14 . assuming @xmath15 , the generalized tolman - oppenheimer - volkov ( tov ) equation may be written as @xmath16 which represents conservation equations in @xmath1 dimensions . we take the eos of the form @xmath17 where @xmath18 is eos parameter . we first obtain interior solutions without any cosmological constant , thereby taking @xmath0=0 . latter on , we generalize our study to non - zero value of @xmath0 . we choose various cases of eos parameter for both the choices of @xmath0 . for @xmath21 , the field equations ( 3)-(6 ) become @xmath22 the tov equation ( 11 ) takes the form , @xmath23 equation ( [ eos2 ] ) yields @xmath24 solving equations ( 8) and ( 9 ) , we get @xmath25 equating eq.(9 ) with eq.(10 ) , we get @xmath26 now , solving equations ( [ 13 ] ) and ( [ 14 ] ) , we obtain @xmath27,\\ \rho&= & \frac{1}{2 \pi m } e^{-\frac{2a}{1-m}}\left[\frac{1-m}{m}\left ( b-\frac{r^{2}}{2 } \right)\right]^{\frac{1+m}{1-m}}.\end{aligned}\ ] ] here @xmath28 , @xmath29 and @xmath30 are integration constants . for the consistency of solutions , the constants should follow the constraint equation , @xmath31 these solutions are regular at the center . the central density is given by @xmath32^{\frac{1+m}{1-m}}.\ ] ] the interior solution is valid up to the radius @xmath33 . for a physically meaningful solution the radial and tangential pressure should be decreasing function of r. from equation ( 17 ) , we find @xmath34 which gives density and pressure as decreasing functions of r. at @xmath35 , one can get @xmath36 ^{\frac{2m}{1-m } } < 0\ ] ] which support maximality of central density and radial central pressure . here , density and pressure decrease radially outward as shown in fig . 1 . + + ) in the interior region . description of curves are as follows : the red , brown and black colors represent @xmath18=1/3 , @xmath18=1/2 and @xmath18=2/3 , respectively . for all these , solid , dashed and chain lines represent @xmath37 , 0.5 and 0.75 , respectively . thin , thick and thickest lines correspond to b=5 , 10 , and 20 , respectively . ] the above tov ( eq . 11 ) may be re - written as @xmath38 where @xmath39 is the gravitational mass inside a sphere of radius @xmath14 and is given by tolman - whittaker formula , which may be derived from field equations , @xmath40 this modified form of tov equation indicates the equilibrium condition for the fluid sphere subject to the gravitational and hydrostatic forces , @xmath41 where @xmath42^{\frac{2m}{1-m } } , \\ f_h & = & \frac{dp}{dr}=-f_g.\end{aligned}\ ] ] the profiles of @xmath43 and @xmath44 for the specific values of the parameters are shown in fig . 2 which provides the information about the static equilibrium due to the combined effect of gravitational and hydrostatic forces . mass , @xmath45 , within a radius @xmath14 , is calculated as @xmath46^{\frac{2}{1-m } } \nonumber \\ & -&\frac{1}{2}e^{-\frac{2a}{1-m}}\left[\left(b-\frac{r^{2}}{2}\right)\left(\frac{1-m}{m}\right)\right]^{\frac{2}{1-m}}.\end{aligned}\ ] ] + + ) in the interior region . description of curves is the same as in fig . 1 . ] . ] the compactness of the fluid sphere , @xmath47 , is thus defined as be found as @xmath48 this is an increasing function of the radial parameter ( see figure 3 ) . correspondingly , the surface redshift ( @xmath49 ) is given by @xmath50 fig . 4 provides variation of @xmath49 against r for different values of the parameters . in ( 2 + 1 ) dimensional spacetime , the vacuum solution does not exist without cosmological constant . thus it is not possible to match our interior solution with the btz black hole as it is the vacuum solution with non zero @xmath0 . however , if one takes b as large as possible , then the solution is valid for the infinite large fluid sphere . this means that we do nt have the vacuum region left . + . ] for stiff fluid model , @xmath52 , and with @xmath21 , the field equations ( 3)-(6 ) yield following solutions @xmath53 here @xmath54 , @xmath55 and @xmath56 are integration constants . for the consistency of the solutions , the constants should follow the following constraint equation . @xmath57 this ensures that @xmath58 . + the solutions are regular at the center and are valid for infinite large sphere . the central density is @xmath59 . + ) in the interior region for m=1 . description of curves are as follows : @xmath60 and @xmath61 correspond to black , brown , blue , orange and red colors , respectively . @xmath62 and @xmath63 correspond to solid , dotted , dashed and dot - dashed lines , respectively . for @xmath64 values are too small . ] + + + from eq . ( 32 ) , we find at @xmath35 , @xmath65 < 0\ ] ] thus central density is maximum . + the mass , @xmath45 , within a radial distance @xmath14 is given by @xmath66/2~~.\ ] ] the compactness of the fluid sphere is thus , @xmath67 . having @xmath68 , the @xmath49 is determined using eq . ( [ eq34 ] ) . the important physical characteristics such as density , compactness and redshift are shown in figs . + + ) in the interior region . description of curves is the same as in fig . 5 . ] + the tov equation yields @xmath41 where @xmath69 + + the profiles of @xmath43 and @xmath44 for the specific values of parameters are shown in fig . 8 , which provides information about the static equilibrium due to gravitational and hydrostatic forces combined . as before , we can not match our interior solution with btz exterior vacuum solution . + + = 1 . description of curves is the same as in fig . 4 . ] the equation of state of the kind , @xmath71 is related to the @xmath72 , an agent responsible for the second phase of the inflation of hot big bang theory . using the equation of state of the kind , @xmath71 , and with @xmath21 , the field equations ( 3)-(6 ) yield following solutions @xmath73/2,\\ \mu & = & \left[h- \ln ( r^{2}+k)\right]/2.\end{aligned}\ ] ] here @xmath74 , @xmath75 and @xmath76 are integration constants . solutions hold good for the following constraint equation @xmath77 these are regular at the center if @xmath75 is positive and the solution is valid for the infinite large fluid sphere . however , for @xmath78 , solution is valid for @xmath79 up to infinite large radius . for the dust case i.e. when @xmath81 and @xmath82 , the field equations ( 3)-(6 ) reduce to @xmath83 and @xmath84 here , @xmath85 are integration constants . unless specifying the energy density , one can not get exact analytical solution of the field equations . thus dust model in ( 2 + 1 ) dimensional space time is possible for known energy density . as before for the equation of state of the kind @xmath71 with non zero @xmath0 , the field equations ( 3)-(6 ) yield @xmath87 . the metric coefficients may be obtained as @xmath88 and @xmath89 here , @xmath90 , @xmath91 , @xmath92 and @xmath93 are integration constants . these solutions are consistent if @xmath94 these solutions are regular at the center if @xmath92 is positive and the solution is valid for the infinite large fluid sphere . however , for @xmath95 , the solution is valid for @xmath96 up to infinite large radius . the nature of the solutions of the metric potentials is independent of the sign of @xmath0 . however , sign of @xmath0 plays a crucial role to get positive energy density . for positivity of energy density , one should take negative @xmath0 . note that without any loss of generality , we can take @xmath97 as it can be absorbed by re - scaling the coordinates . we match the interior solution to the exterior btz black hole metric @xmath98 at the boundary @xmath99 , which yield @xmath100 @xmath101 solving these two equations , we get @xmath102 the consistency relation assumes the form @xmath103 from the field equations ( 3)-(6 ) after some manipulation , we arrive at @xmath105 one can observe that @xmath106 will be a particular solution of this equation . this yields @xmath107 . equation ( 4 ) implies @xmath108 . finally , we get the following solution for @xmath109 as @xmath110/2~~.\ ] ] here , @xmath111 and @xmath112 is integration constant . for positivity of energy density , one should take positive @xmath0 . the solutions are regular at the center if @xmath113 and valid up to @xmath114 . in this case , we can not match the interior solution to the exterior btz black hole metric which is vacuum solution with negative @xmath0 . for the dust case i.e. when @xmath81 and @xmath82 , one can not obtain the exact analytical solution of the field equations . thus dust model in @xmath1 dimensional spacetime with non zero @xmath0 is not possible . in this paper we have obtained a new class of exact interior solution of einstein field equation in @xmath1 dimensional space time assuming the equation of state @xmath115 ( where @xmath116 is the equation of state parameter ) . the interior solutions obtained without cosmological , @xmath0 , are physically acceptable for the following reasons : + ( i ) the solutions are regular at the origin , + ( ii ) both the pressure ( p ) and energy density ( @xmath10 ) are positive definite at the origin , + ( iii ) the pressure reduce to zero at some finite boundary radius @xmath117 , + ( iv ) both the pressure and energy density are monotonically decreasing to the boundary , + ( v ) the subluminal sound speed ( @xmath118 ) + ( vi ) and ricci scalar is non zero i.e. spacetime is non flat . it is to be noted that at very high densities the adiabatic sound speed may not equal the actual propagation speed of the signal . by studying tov equation , we have shown that equilibrium stage of the interior region without @xmath0 can be achieved due to the combined effect of gravitational and hydrostatic forces . we know btz exterior vacuum solution in ( 2 + 1 ) dimension is valid only for non zero @xmath0 . therefore , it is not possible to match our interior solution ( without @xmath0 ) with btz spacetime at some boundary . we emphasis the following fact that any interior solution in four dimensional space made with a perfect fluid must be glued with an exterior vacuum solution only at a regular surface p = 0 ( this is consequence of the well - known israel matching conditions for the related problem ) . for barotropic equation of state the configurations present p = 0 surfaces at the same location where @xmath119 . for the solutions ( 15)-(17 ) with @xmath120 this occurs at @xmath121 . however , the metric coefficients are singular at the same locus , in fact , there is a curvature singularity at @xmath121 . hence , this is not a regular region where spacetime can be continuously glued with other spacetime . hence , one should take b as large as possible so that the solution is valid for the infinite large fluid sphere and we do nt have the vacuum region left . for m=1 case , there does not exist any radius for which p or @xmath122 , hence , israel matching condition does not occur . while finding interior solution with non zero @xmath0 , we note that density and pressure remain constant . interestingly , we observe that it is not possible to get dust model in @xmath1 dimensional spacetime with non zero @xmath0 . investigation on full collapsing model of a ( 2 + 1 ) dimensional configuration will be a future project . fr , aau and rb would like to thank the inter - university centre for astronomy and astrophysics ( iucaa ) , pune , india , for research facility . fr is also grateful to ugc , govt . of india , for financial support under its research award scheme . pb is thankful to csir , govt . of india for providing jrf . rb thanks csir for awarding research associate fellowship . we are thankful to dr . r. sharma for valuable discussion . banados , m. , teitelboim , c. , zanelli , j. : -p_phys . lett . _ * 69 * , 1849 ( 1992 ) . fernanado , s. : -_gen . _ 36 _ , 71 ( 2004 ) . nam , s.,park , j .- d . : -_classical quantum gravity _ * 26 * , 145015 ( 2009 ) . zanelli , j. : - _ j. phys . ser . _ * 222 * , 012008 ( 2010 ) . rahaman , f. , usmani , a. a. , ray , s. , islam , s. : - _ phys . b _ * 717 * , 1 ( 2012 ) . mazharimousavi , s . h. , halilsoy , m. , tahamtan , t.:-_phys . lett . a _ * 376 * , 893 ( 2012 ) . ghosh , s . g. : -_int . j. * 21 * , 1250022 ( 2012 ) . rahaman , f. et . al . : - _ phys . d _ * 87 * , 084014 ( 2013 ) . c wolf : - _ nouvo cim . b _ * 109 * , 735 ( 1994 ) . s. s. yazadjiev : - _ phys . * 69 * , 127501 ( 2004 ) . s.m.kozyrev : - arxiv ( gr - qc ) 0209026v2 . sharma , r. , rahaman , f. and karar , i. : - _ phys . b. _ * 704 * , 1 ( 2011 ) . banerjee , a. et . al . : - _ gen.rel.grav . _ * 45 * , 717 ( 2013 ) n. cruz and j. zanelli : - _ class . quantum grav . _ * 12 * , 975 ( 1995 ) rahaman , f. et . al . : - _ phys.scripta _ * 76 * , 56 ( 2007 ) won tae kim , john j. oh , myung seok yoon : - _ phys.rev . d _ * 70 * , 044006(2004 ) rahaman , f. et . al . : - _ phys.lett . b _ * 707 * , 319 ( 2012 )	* abstract : * we provide a new class of exact solutions for the interior in ( 2 + 1 ) dimensional spacetime . the solutions obtained for the perfect fluid model both with and without cosmological constant ( @xmath0 ) are found to be regular and singularity free . it assume very simple analytical forms that help us to study the various physical properties of the configuration . solutions without @xmath0 are found to be physically acceptable .
the currently accepted cosmological paradigm , the so - called concordance @xmath1cdm model , i.e. cold dark matter ( cdm ) plus a cosmological constant ( @xmath1 ) , has been overwhelmingly successful in describing the formation and evolution of structures in the universe but there remain many observational discrepancies on galaxy and cluster scales that call for a critical verification . while the flat rotation curves observed since the 1970s in spiral galaxies have been one of the prime factors in support of the existence of dark matter ( einasto et al.1974 ; faber&gallagher 1979 ) , studies of late - type galaxies suggest , in some cases ( e.g. low surface brightness galaxies ) , soft cores and relative low concentration density profiles ( gentile et al . 2004 ; de blok 2005 ) at odd with @xmath1cdm inner density cusps ( navarro et al . 1997 , nfw hereafter ; moore et al . 1999 ) . given the long - standing dark matter puzzles in disk galaxies , it is obviously important to also examine the mass content of the second major galaxy type , the early - types ( etgs : ellipticals and s0 ) . the mass content of these systems , which are generally free of cold gas and have their stars moving in random directions , is more difficult to measure than in disk galaxies . the main difficulty is observational , since the only ubiquitous mass tracers in etgs , i.e. stars , can be investigated through their integrated light ( spectroscopy ) which is hard to be measured with sufficient signal - to - noise outside @xmath2 ( kronawitter et al . 2001 , samurovic & danziger 2005 ) . a suitable option is the x - ray emission from hot gas ( paolillo et al . 2003 , osullivan & ponman 2004 , humphrey et al . 2006 ) , but this turns out to be difficult to be measured around faint galaxies ( fukazawa et al . very promising mass probes are also globular clusters ( grillmair et al . 1994 , richtler et al . 2004 , puzia et al . 2004 ) and more importantly a particular class of stars , planetary nebulae ( ciardullo et al . 1993 , arnaboldi et al . 1998 , napolitano et al . 2001 , peng et al . 2004 ) . none of these techniques is free of limitation , thus after years of investigations there are relatively few ellipticals with a dark matter halo even confirmed and much less shown to have the dark - matter properties expected with @xmath1cdm . + we will concentrate here on the observational tests on etgs allowed by pne which have so far ascertained the presence of both massive ( napolitano et al . 2002 , peng et al . 2004 ) and weak dark haloes ( mendez et al . 2001 , romanowsky et al . 2003 ) which can be either interpreted as a variation of the star formation efficiency or as the reflection of a concentration problem similar to the one found in late - types ( romanowsky et al . 2003 , napolitano et al . yet it is still unclear if the discrepancies can be traced to observational / modeling problems , to a poor knowledge of the baryonic physics ( in n - body simulations predicting the cdm distribution ) , or to a failure of the @xmath1cdm paradigm . planetary nebulae ( pne ) have been proven to be excellent tracers for the dynamics of outer regions of etgs . through their powerful [ oiii ] emission at 5007 , they are easily detectable and their radial velocity measurable also in halo regions of distant galaxies . from 2001 a dedicated instrument to the pn kinematics in galaxy systems , the _ planetary nebula spectrograph _ ( pn.s ) , is operating at the 4 m herschel telescope on la palma . the pn.s is specifically designed for the measurement of kinematics of extragalactic planetary nebulae . it allows this type of observations to be carried out a factor of ten more efficiently than possible so far , mainly thanks to its design which allows the pne to be discovered and their spectra measured in a single observation through the counter - dispersed imaging " technique ( douglas et al . 2002 ) . the primary program of the pn.s is to survey a dozen bright ( @xmath3 12 ) , round ( e0e2 ) , nearby ( @xmath4 mpc ) ellipticals with a large range of stellar light parameters ( luminosity , concentration , shape ) , rotational importance , and environment , with the goal of observing 100400 pn velocities in each . the final reduction pipeline of the instrument data , the velocity and photometric calibration procedures plus comparison of the pn.s velocities of the first complete datasample , the one of ngc 3379 , with external independent pn catalogues has been discussed in douglas et al . ( 2007 ) : we address the reader to this paper for details on the data - reduction and standard kinematical analysis and some basics dynamics of ngc 3379 . .galaxy properties : the pn.s samples and main jeans model parameters [ cols="<,^,^,^,^,^,^,^",options="header " , ] preliminary dataset obtained with the pn.s for three `` ordinary '' elliptical galaxies have been presented in romanowsky et al . 2003 ( r+03 hereafter ) . the velocity dispersion ( vd , herafter ) profile of these three systems showed a pseudo - keplerian decline that was more consistent with constant mass - to - light ratio systems rather than dark matter dominated systems for which a flat vd profile was expected . this unexpected result has produced different interpretations either in the @xmath1cdm framework ( dekel et al . 2005 @xmath5 d+05 ) or in mond theory ( milgrom & sanders 2003 ) . in particular , d+05 address very radial stellar orbits and projection effects in order to explain declining vd profiles . napolitano et al . ( 2005 , n+05 hereafter ) made predictions of gradients of mass - to - light ratios in etgs and have ascertained that `` quasi - constant m / l '' are indeed expected within @xmath1cdm , although the r+03 sample shows @xmath6 gradients which are too low and conflicting with acceptable star formation efficiency and baryon fraction . low m / l gradients mirror a generalised trend of the global dark - to - luminous mass virial ratio ( @xmath0 ) with stellar mass / luminosity : brightest galaxies have a larger fraction of dark- to - luminous matter , @xmath7 with respect to fainter galaxies ( @xmath8 ) , the trend being possibly reversed toward dwarf galaxies . here we characterise the combined vd profiles of the r+03 galaxy sample , i.e. ngc 821 , ngc 3379 and ngc 4494 , for which , after 4 years , we have now the final pn datasets ( see table 1 for the final number of pne observed ) and present very preliminary dynamical estimate of their dark matter content as measured through their @xmath9 . in fig . 1 ( left panel ) , with open symbols , we show the combined vd profile of the three galaxies after having scaled ( with respect to @xmath10 ) and normalised ( with respect to the central velocity dispersion ) the individual galaxy curves . these have been obtained as azimuthally averaged @xmath11 within radial bins following the galaxy isophotes distribution for all galaxies . it is easy to recognise the pseudo - keplerian fall consistent with the one observed in the preliminary datasets as in r+03 . in the same figure we also show a collection of long - slit data from the same galaxies as empty stars which demonstrate that the pn vd profiles lie on the extention of the stellar kinematics of the central regions , thus allowing to continuously map the galaxy kinematics out to many effective radii . beside their declining stellar vds , these galaxies share other common properties like intermediate luminosities ( @xmath12 , see table 1 ) and the fact of being fast rotators ( cappellari et al . 2006 ) , disky / cuspy systems ( n+05 ) . in the same panel of fig . 1 , we also show for comparison two galaxies , ngc 4374 and ngc 5846 ( full symbols ) , for which we have the pn.s observational program lately completed and final pn catalogues done ( see n. pne in table 1 ) . here the velocity dispersion profiles have been normalised to the average central vd value of the faint galaxies in order to mark the kinematical differences between the two samples ; long - slit kinematics are also shown as full stars , accordingly . differently from the r+03 sample , these galaxies show markedly flat vd profiles which well contrasts with the pseudo - keplerian fall of the former ones , suggesting the presence of a significant dark halo . these two `` dark - matter dominated '' galaxies mark differences with respect to the r+03 sample for many other reasons : they are brighter ( @xmath13 ) , slow - rotators ( cappellari et al . 2006 ) and boxy / cored galaxies ( n+05 ) . this dichotomy on many structural galaxy properties is suggestive of some connection with galaxy formation history as discussed elsewhere ( capaccioli et al . 2002 , n+05 ) . basic jeans analysis was performed to substantiate the dark matter content . we assumed spherical symmetry ( well motivated by the round appearence of the galaxy sample discussed here ) and we have fit the @xmath14 by solving the radial jeans equation and projecting the solution along the line - of - sight . the dark halo is modeled as a standard nfw using @xmath15 and @xmath16 ( the dark halo concentration and virial mass respectively defined for @xmath17 ) as free parameters . we assumed either a constant or radial dependence for anisotropy parameter , @xmath18 , in order to investigate the effect of tangential ( @xmath19 ) and radial ( @xmath20 ) orbits on the mass estimates . the main results for four galaxies ( ngc 4374 is not reported here ) are shown in table 1 . as anticipated , ngc 5846 is the only galaxy showing a `` regular '' dark matter halo , having the concentration and virial mass consistent with the @xmath21 relation expected from cosmological simulations ( bullock et al . 2001 , n+05 ) . the faint `` ordinary '' sample instead shows concentrations which are systematically lower than predicted from simulations . another remarkable result is that the virial mass - to - luminous matter ratio , @xmath9 is clearly an increasing function of the galaxy stellar mass , which confirms a smooth trend shown in n+05 and discussed in napolitano et al . this trend can be interpreted in terms of global star formation efficiency , @xmath22 , i.e. the fraction of baryonic mass @xmath23 cooled in stars , assuming baryon conservation , such that @xmath24 ( see n+05 for further details ) . more massive systems are globally less efficient in converting gas to stars than galaxies around @xmath25 , like the ones in the `` ordinary '' pn.s sample , which represent the most efficient ones placing themselves in the maximum ( minimum ) of the @xmath26 ( @xmath9 ) run with @xmath27 . indeed the trend is expected to be reversed moving toward the dwarf galaxies . the fact that in our analysis we are considering appropriate orbital anisotropy , consistent with predictions from cosmological simulations ( see fig . 1 right panel and table 1 ) , allows as to exclude that the trend of the dark matter fraction within the analysed galaxies can be the reflection of an anisotropy sequence with the mass . we have shown that there is a strong indication of a dichotomy between the kinematic behaviour of the first pn complete datasets obtained with the pn.s . faint `` ordinary '' , fast - rotating , discy / cuspy , early - type systems seem to show declining velocity dispersion profiles out to very large distance from the galaxy centres , while bright , slow - rotating , boxy / cored systems have flat velocity dispersion profiles . this corresponds to a different total dark - to - luminous matter virial ratio , which is larger for brighter systems with respect to fainter ones . the only galaxy for which we have a preliminary jeans analysis in the pn.s target sample , ngc 5846 , shows a dm distribution which fairly follows a `` regular '' nfw density profile since the @xmath15 and @xmath16 are consistent with the expectation of n - body simulations ( bullock et al . 2001 , n+05 ) . the faint sample , on the contrary , has @xmath15 which are a factor of two lower than predicted from @xmath28 . all these results seem to support the scenario discussed in napolitano et al . ( 2005 ) where we interpreted the dichotomy in the global structure galaxy parameters as a correlation between dark matter content in galaxies and the mass . this is the consequence of a broader trend of the global efficiency of the star formation with the mass ( benson et al . 2000 , dekel & birnboim 2006 ) which suggests a connection between the observed dark matter properties of early - type galaxies and their formation history . a__pj , 507 , 759 benson a.j . , cole s. , frenk c.s . , baugh c.m . , lacey c.g . , 2000 , _ m__nras , 311 , 793 bullock j.s . , et al . , 2001 , _ m__nras , 321 , 559 2002 , _ _ a__rxiv : astro - ph/0211323 cappellari , m. , et al . 2006 , _ m__nras , 366 , 1126 1993 , _ _ a__pj , 414 , 454 2005 , _ _ a__pj 634 , 237 2006 , _ _ m__nras , 368 , 2 2005 , _ _ n__ature , 437 , 707 2002 , _ _ p__asp , 114 , 1234 2007 , _ _ a__pj , 664 , 257 1974 , _ _ n__ature , 250 , 309 1979 _ _ a__raa , 17 , 135 2004 , _ _ m__nras , 351 , 903 2006 , 1994 , _ _ a__pj , 422 , l9 milgrom , m. , & sanders , r.h . , 2003 , _ _ a__pjl , 599 , 25 2000 , _ a_&as , 144 , 53 2005 , _ _ m__nras , 363 , 705 1999 , _ _ m__nras , 310 , 1147 2001 , _ a_&a , 377 , 784 2002 , _ a_&a , 383 , 791 , 2005 , _ _ m__nras , 357 , 691 ( n+05 ) 2006 , _ _ e__as publications series , 20 , 131 : _ _ a__stro - ph/0510568 navarro j.f . , frenk c.s . , white s.d . , 1997 , _ _ a__pj , 490 , 493 ( nfw ) 2004 , _ _ m__nras , 354 , 935 2003 , _ _ a__pj , 586 , 850 2004 , _ _ a__pj , 602 , 685 2004 , _ a&a _ , 415 , 123 2004 , _ _ a__j , 127 , 2094 romanowsky a.j . et al . , 2003 , _ _ s__cience , 301 , 1696 ( r+03 ) 2005 , _ mnras _ , 363 , 769 _ _ a__pj , 646 , 899	we examine the dark matter properties of nearby early - type galaxies using planetary nebulae ( pne ) as mass probes . we have designed a specialised instrument , the planetary nebula spectrograph ( pn.s ) operating at the william herschel telescope , with the purpose of measuring pn velocities with best efficiency . the primary scientific objective of this custom - built instrument is the study of the pn kinematics in 12 ordinary round galaxies . preliminary results showing a dearth of dark matter in ordinary galaxies ( romanowsky et al . 2003 ) are now confirmed by the first complete pn.s datasets . on the other hand early - type galaxies with a `` regular '' dark matter content are starting to be observed among the brighter pn.s target sample , thus confirming a correlation between the global dark - to - luminous mass virial ratio ( @xmath0 ) and the galaxy luminosity and mass .
the entropy production of the surrounding medium of a system driven out of equilibrium is arguably the most convenient tool for quantifying the irreversiblity . in the long time limit when the steady state value of the entropy production arising due to the boundary contributions could be neglected , the probability distribution function @xmath1 for the change in the medium entropy @xmath2 at time @xmath3 , is known to satisfy certain symmetry relation , generally known as the fluctuation theorem @xcite : @xmath4 this symmetry relation implies that the probability of observing the entropy annihilation over a long time interval becomes negligibly small and can be viewed as the nonequilibrium generalization of the second law of thermodynamics . the derivation of the distribution function in the asymptotic time limit is often performed by finding out the corresponding large deviation function(ldf ) @xmath5 which is related to the distribution function as , @xmath6 the ldf plays the role of the free energy functions in equilibrium systems @xcite and in the case of nonequilibrium systems , its symmetry property , @xmath7 , validates the fluctuation theorem or the gallavotti - cohen symmetry @xcite . for a system described by the continuous time markov dynamics , a microscopic transition from its configuration @xmath8 to @xmath9 with a transition rate , @xmath10 , causes the entropy of the surrounding medium to change @xcite by the amount @xmath11 , where @xmath12 , the boltzmann constant , is assumed to be unity . of the many studies on the entropy production of the medium for systems with microscopic reversibility @xcite , some recent works on the properties of the ldf and its symmetry relation can be found in @xcite where the authors studied the asymptotic distributions of the entropy production by finding out the ldf using a generating function based approach . in reference @xcite , the authors obtained the ldf for partially asymmetric simple exclusion processes and reaction - diffusion processes with microscopic reversibility . the emergence of a kink - like feature in the ldf at zero entropy production was argued to be generic for such processes . this kink could be characterized by the average value of the medium entropy production rate , making the entropy production rate a good candidate for quantifying the irreversibility in nonequilibrium systems . while there is a wide applicability of the above formula for finding the medium entropy production for nonequilibrium systems having bi - directional transitions between its various states , this formula can not be applied directly when we encounter a system with irreversible microscopic transitions between its states @xcite . totally asymmetric simple exclusion processes(tasep ) where particles move in only one direction respecting the exclusion principle is one such example of systems with irreversible microscopic transition . other examples include enzymatic reaction networks modeled by michaelis - menten scheme , directed percolation etc . in these systems when some of the transition rates @xmath10 become zero , this formula predicts an infinite entropy production which have not been observed in realistic situations @xcite . to address this shortcoming , ben - avraham _ et _ _ al . _ @xcite proposed a regularization scheme by sampling the states of the system at small time interval and obtained the modified transition probabilities . using those transition probabilities , they computed the medium entropy production rate and its ldf for a three - state irreversible loop . in reference @xcite , the authors used a slightly different method by introducing small backward transitions and determined the lower bound for the average rate of medium entropy production originating from the predominant irreversible transitions . in this paper , we obtain the effective time - dependent transition rates for systems with irreversible transitions by allowing the systems to undergo all possible allowed transitions over small time interval @xmath0 and , then , deriving the probabilities of transition between any two states at the end of the time interval , @xmath0 . the new transition rates are identical to the original transition rates for small @xmath0 . these transition rates allow us to extend straightforwardly the generating function based approach used earlier @xcite to find out the ldf for the mean entropy production for irreversible systems . for a two - site tasep and a three - state irreversible system , we first obtain the transition probabilities between any two arbitrary states by solving the governing master equations . the transition rates are obtained after keeping the leading order terms in @xmath0 in the taylor expansion of the time - dependent probabilities and then differentiating with respect to @xmath0 . these new rates are used to obtain the average rate of medium entropy production and further to obtain the ldf for the entropy production through a saddle point approximation . at zero entropy production , the ldfs for both the models show a kink which can be characterized by the average entropy production rate @xcite . finally , the ldf is used to find the distribution of the entropy production in the long time limit . in the case of a two - site tasep , the average rate of medium entropy production monotonically increases with the particle injection and withdrawal rates @xmath13 . for large values of the medium entropy production rate , the distribution function for the entropy production appears like a gaussian distribution . for low entropy production rates , the distribution function turns out to be a non - gaussian one . the features of the ldf that are responsible for producing the non - gaussian shape of the entropy distribution function are strongly ( exponentially ) suppressed in the case of large entropy production rates . the rest of the paper is organized as follows . in section [ sec:2 ] , we introduce the entropy production formalisms for markov jump processes and a practical way to apply those for systems having unidirectional transitions . in section [ sec:3 ] , we first compute the rate of medium entropy production in the nonequilibrium steady state for two - site tasep with equal injection and withdrawal rates denoted by @xmath13 . next , we obtain the ldf as well as the distribution function for the medium entropy production for different values of @xmath13 . in section [ sec:4 ] , using the same lines of approach , we obtain the analytical form of the ldf for entropy production for a three - state irreversible loop . the results are summarized in section [ sec:5 ] . in this section , we first present a brief overview of these relations @xcite valid for stochastic dynamics modeled as continuous time markovian dynamics with finite , discrete configuration space . next , we elucidate the feasibility of using the known entropy production formulae for a system having one or more microscopically irreversible transitions between its finite number of discrete states . due to the microscopic irreversibility inherent in the system , some of the transition rates involved in the entropy production formulations are zero . here , without introducing negligibly small backward transition rates as was done in @xcite , we obtain the new set of transition rates by sampling the states of the system at a small time interval @xmath0 . in the limit @xmath14 , these new transition rates approach their original values , thus making the computations of entropy production rate and its ldf more accurate . we consider a continuous - time markov jump process , for the time interval @xmath15 , with finite number of states . the dynamical evolution of the probability @xmath16 , that the system is found in state @xmath8 , is described by the master equation @xmath17 where @xmath18 and @xmath10 are the transition rates for the jump from state @xmath9 to @xmath8 and from state @xmath8 to @xmath9 , respectively . equation([1 ] ) can be written in matrix form as , @xmath19 where @xmath20 is the column matrix and the @xmath21th element of the transition matrix @xmath22 are @xmath23 in order to obtain the expression for the entropy production due to a transition from one state to another , we begin by defining the average gibbs entropy of the system as @xmath24 the expression for the time evolution of the system entropy has the form @xcite @xmath25 where the overdot implies a derivative with respect to time . the first term on the right hand side of the above relation is always positive and is identified as the total entropy production rate due to stochastic transitions . the second term is the medium entropy production rate or the entropy flow into the medium due to these transitions . the total rate of the entropy production is now expressed as @xmath26 with @xmath27 in order to calculate the medium entropy production rate , we first briefly describe the strategy to compute the transition rates employing a matrix - based approach . to begin with , we consider a markov process in which transitions between the discrete states @xmath28 are measured in discrete time steps , @xmath29 , where @xmath30 . the solution of equation([matrix_form ] ) is determined by diagonalizing the matrix @xmath22 as @xmath31 , where matrix @xmath32 is formed of the eigenvectors of @xmath22 arranged column - wise , and @xmath33 is the inverse of @xmath32 . the diagonal matrix @xmath34 has the eigenvalues of @xmath22 as its diagonal elements . the solution of equation([matrix_form ] ) then reads @xmath35 the elements of the @xmath36 matrix are the transition probabilities . for instance , the element @xmath37 , in conventional notation , implies @xmath38 , i.e. , the probability of finding the system at @xmath8-th state at time @xmath3 , provided it was at @xmath9-th state at the initial time . these transition probabilities allow us to obtain the time - dependent transition rates @xcite if the states of the system are sampled at small time interval @xmath39 . to be more specific , let us consider the probability @xmath40 at time @xmath41 of finding the system at state @xmath8 . in the limit @xmath42 , we have , @xmath43 this relation defines @xmath44 which denotes the transition rate from state @xmath9 to @xmath8 . in the subsequent sections , these transition rates are used in relation ( [ medium_entropy ] ) to calculate the average rate of entropy production of the medium , its ldf and the distribution function in the asymptotic time limit . we consider two - site tasep with equal particle injection and withdrawal rates @xmath13 as our first model of a system with irreversible transitions between its four states as shown in figure [ fig : samplefig1 ] . let us consider the time evolution of the probability densities @xmath45 , where the element @xmath46 of the column matrix denotes the probability of finding the system in state @xmath8 . the governing master equation can be written as , @xmath47 where @xmath22 is the @xmath48 matrix having the following form , @xmath49 the eigenvalues of @xmath22 are @xmath50,\ \lambda_4=\frac{1}{2}[-1 - 3\alpha+\sqrt{1 - 6\alpha+\alpha^2}]$ ] with the corresponding eigenvectors @xmath51,-\frac{1}{2\alpha}[1+\alpha + \sqrt{1 - 6\alpha + \alpha^2}],1\right)^t;\\ e_4=\left(1,\frac{1}{2\alpha}[1 - 3\alpha -\sqrt{1 - 6\alpha + \alpha^2}],-\frac{1}{2\alpha}[1+\alpha-\sqrt{1 - 6\alpha+\alpha^2}],1\right)^t.\end{aligned}\ ] ] the matrix @xmath33 has the form @xmath52 with @xmath53 . substituting @xmath32 , @xmath54 and @xmath33 in equation([soln1 ] ) , each element of the column vector @xmath55 is expressed as @xmath56 which can be written in the compact form as @xmath57 in the above equation , the conditional probability @xmath37 implies the probability of finding the system at @xmath8-th state at time @xmath3 , provided it was at @xmath9-th state at initial time . @xmath58 term corresponds to the null transition . if we consider the time interval , @xmath59 , to be small such that the sampling time becomes , @xmath60 , it is then ensured that the transition matrix @xmath61 becomes closer to the original transition matrix ( [ tasep_trmatrix ] ) . in this limit , the matrix @xmath36 has the form , = ( cccc 1-+ & & - & + - & 1-+ & ^2 ^ 2 & - + & -(+ ) ^2 & 1 - 2 + 2 ^ 2 ^ 2 & + & & - & 1-+ + ) . the corresponding transition matrix @xmath61 as defined in ( [ wmatrix ] ) is = ( cccc ^2- & & -3 ^ 2 & + -(1 + ) & -1 & 2 ^ 2 & -(1 + ) + & 1 - 2(+ ) & 4 ^ 2 - 2 & + & & -3 ^ 2 & ^2- + ) . having obtained the time - dependent transition rates in the previous subsection , we now evaluate the average rate of medium entropy production in the nonequilibrium steady state as defined in equation([medium_entropy ] ) . we define the transition rate for a transition from state @xmath9 to @xmath8 as @xmath62 which is related to the corresponding element of the transition matrix as , @xmath63 . the steady state probability densities for the two - site model are obtained as , @xmath64 in the small time limit , the ratio of the forward and the reverse transition rates are approximated as , @xmath65 the average rate of entropy production of the medium is thus obtained as , @xmath66.\end{aligned}\ ] ] the entropy production rate is plotted in the figure [ fig : entropy_prodiction_eigenvalues](a ) with @xmath13 . the positivity of the entropy production suggests that the medium entropy increases as the system undergoes transition from one state to another . with the definition of the transition matrix @xmath61 in the previous section , we calculate the ldf for entropy production@xcite . let @xmath67 be the probability that the system is in the @xmath8-th state at time @xmath3 while the change in the medium entropy is @xmath2 . the probability of finding the system at time @xmath68 after a small time interval @xmath0 , during which the entropy exchange with the medium is @xmath69 due to the jump of the system from state @xmath9 to @xmath8 , is expressed as @xcite , @xmath70.\end{aligned}\ ] ] in the limit @xmath14 , we have , @xmath71.\end{aligned}\ ] ] introducing the generating function @xmath72 we write the time evolution of the generating function as , @xmath73 the above equation can be written in a matrix form as , @xmath74 where & = ( llll -(+^2(-2)/2 ) & ( ) ^1- ( -(1+))^ & ( -3 ^ 2)^1-()^ & + ( -(1+))^1-()^ & -1 & ( 2 ^ 2)^1- [ 1 - 2(+0.5)]^ & ( -(1+))^1-()^ + ( ) ^1-(-3 ^ 2)^ & ( 1 - 2(+0.5))^1-(2 ^ 2)^ & ( 4 ^ 2 - 2 ) & ( ) ^1-(-3 ^ 2)^ + & ( ) ^1-(-(1+))^ & ( -3 ^ 2)^1-()^ & -(+^2(-2)/2 ) + ) , & and @xmath75 . \(a ) ( b ) \(a ) ( b ) in order to find the total probability distribution @xmath76 , we need to introduce the total generating function @xmath77 . in the large time limit , the total generating function can be approximated as , @xmath78 where @xmath79 is the smallest of all the eigenvalues defined through the following equation @xmath80 we evaluate the smallest eigenvalue numerically and it is plotted in figure [ fig : entropy_prodiction_eigenvalues](b ) with @xmath81 . the symmetry of the smallest eigenvalue about @xmath82 validates the fluctuation theorem @xmath83 @xcite . the average rate of entropy production of the medium @xmath84 is related to the smallest eigenvalue as , @xmath85 in order to obtain the probability distribution @xmath67 , one has to invert the relation in equation ( [ genfn ] ) . the final integration is done using a saddle point approximation scheme . the ldf or the rate function @xmath86 with @xmath87 as the normalised rate of entropy production , can be expressed as the legendre transform of @xmath79 @xmath88 here @xmath89 is the saddle point defined through the equation @xmath90 . from figure [ fig : entropy_prodiction_eigenvalues](a ) , it is evident that the average entropy production rate is always positive and it increases as we increase the value of @xmath13 . at zero entropy production , the ldf shows a kink which becomes prominent for larger values of @xmath13 ( see figure [ fig : largedeviation](a ) ) . the symmetry property displayed by the ldf , @xmath91 , as shown in figure [ fig : largedeviation](b ) , is attributed to the symmetry property of the distribution function of entropy production quantified through the fluctuation theorem . . from the left to right panel , the three figures correspond to the rates @xmath92 , @xmath93 and @xmath94 , respectively . for the small values of the entry and exit rates , the distribution is non - gaussian . the distribution tends to be gaussian for larger values of entry and exit rates.,scaledwidth=100.0% ] for @xmath94 . for this value of @xmath13 , @xmath95 . the solid black curve corresponds to the @xmath96 and the red dashed curve is obtained using equation([gaussian_approx ] ) . ] using the expression for the ldf , one may find out the probability distribution function of the normalized entropy production rate @xmath97 in the asymptotic time limit as @xmath98 the ldf provides the detailed information about the distribution function , which is non - gaussian in nature in our case , for large fluctuations . however , for larger values of @xmath13 when the value of the average entropy production rate is also large , the central part of the distribution tends to be gaussian , while for smaller values of @xmath13 it has a non - gaussian form . intuitively , for small @xmath13 , the number of transitions are also small and over the time interval @xmath99 , the smaller number of events cause the distribution function to have a poisson distribution - like form . the distribution functions @xmath97 for three different values of @xmath13 are plotted in figure [ fig : entropy_prodiction_dist ] . to have a qualitative understanding of the distribution function for larger values of @xmath84 , it should be noted from figure [ fig : largedeviation](a ) that the values of the ldf from its minimum are larger if the rate of medium entropy production is increased . these large fluctuations are strongly suppressed in the exponential form of the distribution function . thus , if we perform a taylor expansion of the ldf about its minimum at @xmath100 , we have @xmath101 we determine the second derivative numerically and for @xmath102 , its value is , @xmath103 . taking the form of @xmath86 as in the equation([gaussian_approx ] ) , we obtain the distribution function for large values of medium entropy production ( see figure [ fig : gaussian_approx ] ) . the matching between the original distribution and the approximated one is remarkable and thus , the distribution can be approximated as gaussian for large @xmath84 . however , similar approximation can not be made for smaller values of @xmath84 since in this case , the fluctuations away from the center are not so large . this explains why the distribution function in this case becomes non - gaussian . here we apply the present method to a three - state irreversible loop @xcite where the transitions between the three states denoted as @xmath104 happen in a cyclic way as @xmath105 with rate @xmath106 . in the small time interval @xmath0 , the transition rates @xmath107s have the form , @xmath108 as before , the time evolution of the generating function @xmath109 @xmath110 , as defined in equation([genfn ] ) , is governed by , @xmath111 where @xmath112 and @xmath113 has the form , @xmath114 the smallest eigenvalue of @xmath115 dominates the large time behavior of the total generating function @xmath116 . in this case , the smallest eigenvalue is @xmath117 the rate of medium entropy production is found as , @xmath118.\label{med_entropy_pleimling}\ ] ] \(a ) ( b ) the ldf for the normalized entropy production @xmath119 has the form @xmath120 where the saddle point @xmath89 is defined implicitly as @xmath121 expressing equation ( [ implicit_eq ] ) in terms of @xmath122 , where @xmath123 , we find @xmath124 the saddle point @xmath89 is related to @xmath122 as @xmath125 substituting equation([smallesteigenv ] ) , ( [ med_entropy_pleimling ] ) and ( [ saddle_lambda ] ) into ( [ largedev_2 ] ) , we have @xmath126 the symmetry property of the ldf , @xmath127 , implies that the fluctuation relation for the entropy production in the medium holds for the system in the long time limit . the plot of the ldf for the entropy production ( see figure [ fig : largedeviation_pleimling](a ) ) shows a kink at zero entropy production as a consequence of the fluctuation theorem @xcite . the distribution function for the entropy production , as shown in [ fig : largedeviation_pleimling](b ) , is obtained directly from equation ( [ ld_pleimling ] ) . in summary , we have obtained the ldf and the probability distribution function for the medium entropy production for a two - site tasep and a three - state cyclic process . both tasep and the three - state process involve irreversible transitions due to the hopping of particles in a specific direction and the unicyclic nature of the three - state process . in order to apply the general results of entropy production for stochastic jump processes , we obtained first the time - dependent transition rates by sampling the states of the systems over a short time interval . these new transition rates are incorporated in the subsequent derivations of the ldf for the entropy production which satisfies gallavotti - cohen symmetry . for the two - site tasep , the value of the ldf for large fluctuations becomes higher as the average entropy production rate is increased . as a consequence of this , the distribution function tends to be gaussian . for smaller values of particle injection and withdrawal rates , which in turn makes the average entropy production rate smaller , the distribution function becomes non - gaussian and it resembles poisson distribution because of lesser number of events over the time interval . for the three - state irreversible loop , we have found the analytical forms of the smallest eigenvalue and the ldf . for both the processes , the smallest eigenvalue and the ldf are derived keeping the first order terms in @xmath0 in the new transition rates . the smallest eigenvalue and the ldf for the medium entropy production satisfy the fluctuation theorem . our results for the three - state process differ slightly from the previous study @xcite since we incorporate here the conventional definition of the transition rates in the subsequent derivations of the smallest eigenvalue and the ldf . in reference @xcite , applying bayes theorem to the posterior probabilities , it has been shown that the microscopic reversibility condition is not a necessity to propose a generalize fluctuation theorem for total entropy production . since using time coarse - graining procedure , we obtain nonzero reverse transition probabilities even for processes involving irreversible transitions , it is expected that this procedure leads to holding symmetry relations of certain kinds for the probability distribution of entropy production . using similar coarsening theorem , the authors in reference @xcite have shown the validity of integral fluctuation theorem and crooks relation for hatano - sasa entropy of many - state irreversible processes . finally , the present methodology based on derivation of the time - dependent transition rates seems to be useful for studying a broad range of models relevant to physical and biophysical processes including complex networks involving multiple degrees of freedom @xcite . 99 evans d j , cohen e g d and morriss g p , 1993 _ phys . lett . _ * 71 * 2401 gallavotti g and cohen e g d , 1995 _ phys . rev . lett . _ * 74 * 2694 gallavotti g and cohen e g d , 1995 _ j. stat . phys . _ * 80 * 931 kurchan j , 1998 _ j. phys . a _ * 31 * 3719 lebowitz j l and spohn h , 1999 _ j. stat . phys . _ * 95 * 333 harris r j and schtz g m , 2007 _ j. stat . _ p07020 ellis r s , 1985 entropy , large deviations , and statistical mechanics ( berlin : springer ) touchette h , 2009 _ physics reports _ * 478 * 1 schnakenberg j , 1976 _ rev . phys . _ * 48 * 571 andrieux d and gaspard p , 2004 _ j. chem . phys . _ * 121 * 6167 seifert u , 2005 _ phys . rev . lett . _ * 95 * 040602 seifert u , 2008 _ eur j. b _ * 64 * 423 tom t and de oliveira m j 2012 _ phys . * 108 * 020601 zia r k p and schmittmann b , 2007 _ j. stat . mech . _ p07012 barato a c , chetrite r , hinrichsen h and mukamel d , 2010 _ j. stat . mech . _ p10008 imparato a and peliti l , 2007 _ j. stat . l02001 dorosz s and pleimling m , 2011 _ phys . e _ * 83 * 031107 de oliveira rodrigues j e and dickman r , 2010 _ phys . e _ * 81 * 061108 takeuchi k a , kuroda m , chat h and sano m , 2007 _ phys . * 234503 ben - avraham d , dorosz s and pleimling m , 2011 _ phys . rev . * 84 * 011115 zeraati s , jafarpour f h and hinrichsen h , 2012 _ j. stat . mech . _ l12001 mehl j , speck t and seifert u , 2008 _ phys . e _ * 78 * 011123 speck t , engel a and seifert u , 2012 _ j. stat . _ p12001 reichl l e , 1980 a modern course in statistical physics ( weinheim : wiley - vch ) ohkubo j , 2009 _ j. phys . * 78 * 123001 ben - avraham d , dorosz s and pleimling m , 2011 _ phys . e _ * 83 * 041129 gernert r , emary c and klapp s h l , 2014 _ phys . e _ * 90 * 062115	we obtain the large deviation function for entropy production of the medium and its distribution function for two - site totally asymmetric simple exclusion process ( tasep ) and three - state unicyclic network . since such systems are described through microscopic irreversible transitions , we obtain time - dependent transition rates by sampling the states of these systems at a regular short time interval @xmath0 . these transition rates are used to derive the large deviation function for the entropy production in the nonequilibrium steady state and its asymptotic distribution function . the shapes of the large deviation function and the distribution function depend on the value of the mean entropy production rate which has a non - trivial dependence on the particle injection and withdrawal rates in case of tasep . further , it is argued that in case of a tasep , the distribution function tends to be like a poisson distribution for smaller values of particle injection and withdrawal rates .
it is of great scientific importance to study the dissociation of diatomic molecules on metal surfaces ( dissociative sticking ) , to meet some intrinsic interests on mechanisms for bond breaking and bond formation and origins of activation energy barriers @xcite . the dissociation of the hydrogen molecule ( h@xmath0 ) on metal surfaces are furthermore key events for lots of technological applications such as hydrogen storage for fuels @xcite , hydrogen caused embrittlement @xcite , and heterogeneous catalysis @xcite . among all the metals , the mg(0001 ) surface is one of the most studied prototypes , both theoretically and experimentally @xcite . however , some basic problems concerning the dissociation process and energy barrier still have discrepancies yet . as early as 1981 , nrskov _ et al . _ for the first time studied the dissociation of h@xmath0 on the mg(0001 ) surface , observing a molecularly adsorbed precursor state and an energy barrier of 0.5 ev for dissociation @xcite . later , by using the more precise first - principles methods , bird _ et al . _ discovered that although the most energetically favorable site for dissociation is the bridge site as reported in nrskov s paper , the dissociation energy barrier is 0.37 ev , rather than the reported 0.5 ev , and there is no precursor state @xcite . however , the discrepancy on the dissociation energy barrier remains ever since then . et al . _ reported a value of 1.15 ev for the dissociation , after systematically considered the zero point energy of h@xmath0 within their first - principles calculations @xcite . et al . _ reported a value of 1.05 ev by employing first - principles calculations and transition state theory @xcite . meanwhile , johansson _ recently organized an experiment to study the dissociation energy barrier for h@xmath1 at mg(0001 ) , and the obtained barrier is 0.6 @xmath2 0.9 ev @xcite , which disagrees with all the reported theoretical values . so it is still an open question as to the correct dissociation energy barrier for h@xmath0 at mg(0001 ) . on the other hand , it has been recently suggested that steric effects might be important during the adsorption and dissociation of diatomic molecules on metal surfaces @xcite . for the adsorption of d@xmath0 on cu(111 ) , both experimental and theoretical investigations found that dissociation of d@xmath0 occurs preferentially when the molecule approaches with its bond parallel to the surface @xcite . similar dependence on polar angle has also been theoretically predicted for the dissociative sticking of o@xmath0 on the al(111 ) surface @xcite . when considering the dissociative adsorption of h@xmath0 on metal surfaces , steric effects might be more important because of the low inertia moment of h@xmath0 and correspondingly high possibility to rotate . in fact , it has already been pointed out that during the dissociative adsorption of h@xmath0 on the pd(111 ) surface , the molecular axis orientation has a drastic effect and low activation barriers are only met over a small range of @xmath3 values from parallel the surface @xcite . and for the adsorption on the nial(110 ) surface , h@xmath0 molecules rotate abruptly when they are close to the surface , which allows them to adopt the orientation that is more convenient for dissociation ( i.e. , nearly parallel to the surface ) @xcite . for the adsorption of h@xmath0 on the mg(0001 ) surface , however , this important issue of steric effects has not been considered yet . motivated by this observation , here by using the first - principles calculations , we study the rotation of h@xmath0 during the dissociative adsorption on the mg(0001 ) surface and its corresponding influences on the dissociation energy barrier . we show that the most energetically favorable path for h@xmath0 dissociation at mg(0001 ) is fundamentally determined by the steric effect . our calculations were performed within dft using the vienna _ ab - initio _ simulation package @xcite . the pw91 @xcite generalized gradient approximation and the projector - augmented wave potential @xcite were employed to describe the exchange - correlation energy and the electron - ion interaction , respectively . the cutoff energy for the plane wave expansion was set to 250 ev . the mg(0001 ) surface was modeled by a five - atomic - layer slab plus a vacuum region of 20 . a @xmath4 supercell was adopted in the study of the h@xmath0 adsorption since our test calculations have showed that it is large enough to avoid the interaction between adjacent hydrogen molecules . integration over the brillouin zone was done using the monkhorst - pack scheme @xcite with @xmath5 grid points . the calculated lattice constants of bulk mg ( @xmath6 , @xmath7 ) and the bond length of isolated h@xmath0 are 3.207 , 5.145 and 0.748 , respectively , in good agreement with the experimental values of 3.21 , 1.62 @xcite and 0.74 @xcite . the calculation of the potential energy surface was interpolated to 209 points with different bond length ( @xmath8 ) and height ( @xmath9 ) of h@xmath0 at each surface site . after geometry optimization for the mg(0001 ) surface , we build our model to study the potential energy surface ( pes ) of h@xmath0 on the relaxed mg surface . as shown in fig . 1 , there are four different high - symmetry sites on the mg(0001 ) surface , respectively the top , bridge ( bri ) , hcp and fcc hollow sites . after pes calculations , we find that the dissociation barrier of h@xmath0 at low - symmetry sites is always larger than at high - symmetry sites , proving that high - symmetry sites play crucial roles in the adsorption of diatomic molecules , similar to that has been observed on the adsorption of oxygen molecules on the pb(111 ) surface @xcite . and in the following , we will only give the results at the four high - symmetry sites . at each surface site , an adsorbed h@xmath0 has three different principle orientations , respectively along the @xmath10 ( i.e. , [ @xmath11 ) , @xmath12 ( i.e. , [ @xmath13 ) , and @xmath14 ( i.e. , [ @xmath15 ) directions . herein , we use top-@xmath16 , bri-@xmath16 , hcp-@xmath16 and fcc-@xmath16 respectively to represent the total twelve high - symmetry channels for the adsorption of h@xmath0 on the mg surface . throughout our pes calculations , we find no molecular adsorption precursor states for h@xmath0 at mg(0001 ) , according well with all previous reports , except for the one by nrskov _ _ using jellium model @xcite . our calculated result for the lowest dissociation energy barrier , as well as that in other theoretical and experimental reports , is given in table . i. the minimum energy path ( mep ) for the dissociation of h@xmath0 on the mg(0001 ) surface is found to be along the bri-@xmath12 channel , which is consistent with all previous first - principles studies . the transition state obtained from our pes calculations along the bri-@xmath12 channel is at the point where @xmath8=1.12 and @xmath9=1.16 , which accords with previous results @xcite . however , as we will see later , this transition state needs to be modified after considering the rotational degree of freedom of h@xmath0 . at the bridge site , however , we find that the total energy of the h@xmath0/mg system is not always smaller along the bri-@xmath12 channel than along other channels . this is a key point in this paper . in fact , at large values of h@xmath0 height from mg(0001 ) surface , we find that the total energy is smaller along the bri-@xmath14 channel than along the bri-@xmath12 channel . to show this , we plot in figs . 2(a)-(c ) the two - dimensional cuts of the pes along the bri-@xmath16 channels . correspondingly , the minimum energy paths in figs . 2(a)-(c ) are collected and plotted in fig . it can be clearly seen that there is a prominent crossing point in the minimum energy paths along bri-@xmath12 and bri-@xmath14 channels , at which the distance of the h@xmath0 molecule from the surface takes a value of @xmath17=1.26 . before this crossing , the total energy of the adsorption system along the bri-@xmath14 channel is always lower than along the bri-@xmath12 channel . this finding indicates that h@xmath0 prefers to orient perpendicular to the mg(0001 ) surface until it reaches the height lower than 1.26 . after the crossing point , whereas , the system along the bri-@xmath12 channel has a smaller total energy than along the bri-@xmath14 channel , and h@xmath0 tends to rotate from the bri-@xmath14 channel to the bri-@xmath12 channel . we then further study the influence of the molecular rotation on the dissociation energy barrier of h@xmath0 . for this we have calculated the total energy by fixing the mass center of h@xmath0 at the bridge site with the height of 1.16 and the molecular bond length of 1.12 , while allowing h@xmath0 to rotate around its mass center in the @xmath12-@xmath14 plane ( see the inset in fig . the calculated angle dependence of the total energy is shown in fig . clearly , it can be seen that the transition state ( namely , the saddle point in the pes for h@xmath0 dissociation ) should be the structure where the orientation of h@xmath0 is 21@xmath18 from the @xmath10-@xmath14 plane . this rotation of h@xmath0 results in a 86 mev modification on the dissociation energy barrier . this finding suggests that steric factors that has not been considered in previous theoretical calculations might be ( at least partially ) responsible for their discrepancies with experimental measurement . although it has long been explored for steric effects on the dissociative adsorption of diatomic molecules on metal surfaces , the specific reasoning has seldom been discussed yet . herein we will try to find the underlying mechanisms on the rotation of h@xmath0 during the dissociative adsorption on the mg(0001 ) surface , by analyzing carefully the charge distributions and electronic interactions along the adsorption process of h@xmath1 . figures 4(a ) and ( b ) show the difference electron density for the adsorption system with @xmath19 to be 2.00 and 1.16 along the bri-@xmath12 channel , namely , @xmath20,\ ] ] where @xmath21 , @xmath22 and @xmath23 are respectively the electron density of the adsorption system , the h@xmath1 molecule and the clean mg(0001 ) surface . to calculate @xmath24 , the atomic positions in the last two terms in eq . 1 have been kept at those of the first term . through careful wavefunction analysis , we find that at the beginning of the adsorption process , the molecular orbitals of h@xmath1 orthogonalize with electronic states of mg and thus are broadened . as shown in fig . 4(a ) , the surface electrons of mg are repelled from the region occupied by the h@xmath1 bonding electrons due to these orthogonalizations . this interaction has also been observed during the interacations of h@xmath1 with other metals such as the al(111 ) @xcite and transition metal surfaces @xcite . when the h@xmath1 molecule is close enough to the mg(0001 ) surface and come to the transition state for its dissociation , electrons transfer from electronic states of mg to the antibonding orbital of h@xmath1 , which can be clearly seen from fig . we can also see from fig . 4(b ) that the orthogonalizations between electronic states of mg and the bonding orbital of h@xmath1 still exist at the transition state . in total , the orthogonalizations between molecular orbitals of h@xmath1 and electronic states of mg cause repulsive interactions between electrons of h and mg , and thus will enlarge the total energy of the adsorption system , while the electrons transfer from mg to h@xmath1 causes attractive interactions between h and mg atoms and lowers down the total energy . so during the adsorption process of h@xmath1 , the total energy of the system firstly goes up , then lowers down . this analysis explains why an energy barrier is needed for the dissociation of h@xmath1 on the mg(0001 ) surface . moreover , both the orthogonalizations and electrons transfer are always weaker along the bri-@xmath14 channel than along the bri-@xmath12 channel . therefore , at the beginning of the adsorption process , when no electrons transfer happens , the total energy of the system is smaller along the bri-@xmath14 channel than along the bri-@xmath12 channel . and at around the transition state , when the h@xmath1 molecule is very close to the mg(0001 ) surface , electrons transfer begins to dominate the molecule - metal interaction . so the total energy along the bri-@xmath12 channel is smaller at the transition state . herein , the rotation of h@xmath1 can be seen as the result from the different interactions that respectively favors the bri-@xmath12 and bri-@xmath14 channels . as a result , the minimum energy path for the dissociation of h@xmath1 is neither along the bri-@xmath12 nor along the bri-@xmath14 channels . and the corresponding transition state is the one where h@xmath0 orients 21@xmath18 away from the @xmath10-@xmath14 plane , as shown in fig . in conclusion , we have systematically studied the pess for the dissociative adsorption of the hydrogen molecule on the mg(0001 ) surface . our results accord well with previous reports on the direct dissociative adsorption process . more importantly , we have found that the hydrogen molecule does not always orient parallel to the surface along the dissociation channel with the lowest energy barrier . at large molecular heights , h@xmath0 orients perpendicular to the surface . when getting closer to the surface , h@xmath0 begins to rotate such that at the transition state , h@xmath0 orients 21@xmath18 away from the @xmath10-@xmath14 plane , which causes a @xmath25 mev modification on the dissociation energy barrier . we have revealed that this molecular rotation is because of the two different interactions between h and mg , i.e. , the orthogonalizations between molecular orbitals of h@xmath1 and electronic states of mg and electrons transfer from the mg(0001 ) surface to the antibonding orbital of h@xmath1 . as a final concluding remark , here based on the present results , we would like to point out that steric effects are important to understand the adsorption behaviors of h@xmath0 on metal surfaces . [ c]cccreferences & methods & dissociation barrier ( ev ) + & jellium model ( lda ) & 0.50 + & dft ( lda ) & 0.37 + & dft ( rpbe ) & 0.50 + & dft ( gga ) & 1.15 + & dft ( lda ) & 0.35 + & dft ( pbe - gga ) & 1.05 + & dft ( gga ) & 1.05 + & experiment & [email protected] + this work & dft ( gga ) & 0.85 + * list of captions * + * fig.1 * ( color online ) . ( a ) the @xmath26(@xmath4 ) surface cell of mg(0001 ) and four on - surface adsorption sites . here only the outmost two layers of the surface are shown . ( b ) the sketch map showing that the molecule ( with vertical or parallel orientation ) is initially away from the surface with a hight @xmath27 . + * fig.2 * ( color online ) . contour plots of the two dimensional cuts of the potential energy surfaces ( pess ) for the h@xmath0/mg(0001 ) system as a function of the bond lengths ( @xmath8 ) and the heights ( @xmath17 ) , with h@xmath0 at the bridge site orienting along @xmath10 ( a ) , @xmath12 ( b ) and @xmath14 ( c ) directions . ( d ) minimum energy paths obtained along the three different channels with different heights of h@xmath0 ( @xmath17 ) from the mg surface . + * fig.3 * ( color online ) . the total energy of the adsorption system with different orientations of h@xmath0 at the transition state point along the bri-@xmath12 channel where @xmath8=1.12 and @xmath17=1.16 . the inset depicts the definition of the angle @xmath28 , in which the grey area and blue balls respectively represent the mg surface and two h atoms . + * fig.4 * ( color online ) . the difference electron density for the h@xmath1/mg(0001 ) system with the height of h@xmath1 to be 2.00 ( a ) and 1.16 ( b ) . blue and grey balls respectively represent hydrogen and mg atoms . dark and dashed lines respectively represent plus and negative values , i.e. , electrons accumulation and depletion . +	using first - principles calculations , we systematically study the potential energy surfaces and dissociation processes of the hydrogen molecule on the mg(0001 ) surface . it is found that during the dissociative adsorption process with the minimum energy barrier , the hydrogen molecule firstly orients perpendicular , and then rotates to be parallel to the surface . it is also found that the orientation of the hydrogen molecule at the transition state is neither perpendicular nor parallel to the surface . most importantly , we find that the rotation causes a reduction of the calculated dissociation energy barrier for the hydrogen molecule . the underlying electronic reasons for the rotation of the hydrogen molecule is also discussed in our paper .
this paper describes our computer program speden that reconstructs the density from the diffraction patterns of individual particles . speden is of interest for three reasons . diffractive imaging promises to improve the resolution , sensitivity , and practical wavelength range in x - ray microscopy , for three - dimensional objects that are tolerant to x - rays . a few examples are defects in semiconductor structures , phase separation in alloys , nano - scale machines and laser fusion targets . a long - term vision is the possibility of high - resolution reconstruction of diffraction patterns of single bio - molecules . of broad theoretical interest is speden s unique approach to the reconstruction of scatterers - a difficult mathematical problem . in the rest of this section we expand on these three topics . reconstruction of the electron density from non - uniformly sampled , three - dimensional diffraction patterns is of wide interest and applicability with present - day sources . in radiation - tolerant samples , x - ray diffraction and diffraction tomography are capable of higher resolution than ( straight- or cone - beam ) tomography alone . in tomography , resolution is limited by the quality of the incident beam and by the spatial resolution of the detector ; in diffraction the resolution can be as fine as the wavelength of the incident radiation . experimentally , diffraction imaging has already produced x - ray images at higher resolution than possible with available x - ray optics ( miao et al . , 1999 , and he et al . , 2002 ) . the price to be paid for these benefits is the intrinsic difficulty of the reconstruction . nevertheless , several successful reconstructions from experimental x - ray data , using the iterative hybrid input - output version of the gerchberg - saxton - fienup ( gsf ) algorithm , have been reported recently ( miao et al . , 2002 , and marchesini et al . , 2003 ) . the first successful application of this algorithm to electron diffraction data was reported in 2001 ( weierstall et al . , 2002 ) , and it has been used more recently to produce the first atomic - resolution image of a single carbon nanotube ( zuo et al . , 2003 ) . in biology , the use of the gsf has recently been shown to dramatically reduce the number of images needed for tomographic cryoelectronmicroscopy of protein monolayer crystals , so that phasing can be based mainly on the three - dimensional diffraction data ( spence et al . , 2003 ) . the development of speden was also prompted by the promise of new ways to image bio - molecules . free - electron lasers can , in principle , provide x - ray pulses of tens to hundreds of femtoseconds in length and brightness up to ten orders of magnitude greater than synchrotron radiation . it was predicted that , under such circumstances , it should be possible to dispense with crystals and reconstruct the electron density of single biological particles from their diffraction pattern ( neutze et al . , 2000 ) . in proposed experiments , a large number of single particles will be injected into the x - ray beam in random orientation and their diffraction patterns will be recorded , each in a single shot of the free - electron laser . such diffraction patterns will be very noisy and their resolution will be limited by the signal to noise ( s / n ) ratio . the measured diffraction patterns that correspond to different orientations of the particle will be classified into a number of mutually exclusive classes . the images within each class will then be averaged and the class averages assembled into a three - dimensional diffraction pattern by finding their mutual orientation relationships . finally , the three - dimensional diffraction pattern will be reconstructed to yield the electron density of the molecule . we have worked on the analysis of all three steps of such an experiment . the essence of the first analysis is that the maximum x - ray intensity at a given pulse length is limited by the requirement that the molecule stay intact during the pulse , even though it eventually disintegrates ( hau - riege et al . , the second analysis discusses the division of noisy diffraction patterns into a number of distinct classes . if the images are divided into too few classes , the available resolution is not realized . if the patterns are divided into too many classes , the class averages will be poor and the pattern quality suffers . the individual class averages , each corresponding to a well - defined orientation of the particle , will be assembled into a three - dimensional diffraction pattern . the result will be a three - dimensional diffraction pattern that is measured at a limited number of orientations . it will be , therefore , sparse , irregular and will have a limited signal to noise ratio ( huldt et al . , 2003 ) . the program speden , described in this paper , provides a way to optimally determine the electron density from such a three - dimensional ensemble of continuous diffraction patterns . we first give an analysis of their properties and discuss the methods and the expected difficulties of reconstructing a `` sensible '' electron density from them . we then describe how speden adapts the holographic method ( szoke et al . , 1997a ) in crystallography to deal with continuous diffraction patterns as opposed to discrete bragg spots ; this will be discussed in the next section . we then report quantitative results of preliminary tests for verifying the correctness of our method . these tests use computed and measured diffraction patterns from samples of inorganic particles . the reconstruction of the density of scatterers from its diffraction pattern is an `` inverse problem '' . other , well - studied inverse problems are those of computed tomography , image deblurring , phase recovery in astronomy , and crystallography . in tomography , for example , an inversion algorithm ( e.g. filtered back projection ) is used to recover the density of scatterers from the measured tomograms . it is widely recognized that the reconstructed density is very sensitive to inaccuracies in the measurement . small errors in the diffraction pattern cause large errors in the reconstruction . this property is called ill - posedness or ill conditioning . the reconstruction of the electron density from x - ray diffraction patterns is indeed ill conditioned . it also has two _ additional _ difficulties . first , in contrast to tomography , there are no direct inversion algorithms not even approximate ones . second , the reconstructed electron density at any sample point is influenced strongly by the electron densities of all sample points , as opposed to a limited number of them . therefore , errors in density are non - local and `` propagate '' far . fortunately , very good fundamental discussions of these subjects are provided in the books of daubechies ( daubechies , 1992 ) , bertero and boccacci ( bertero et al . , 1998 , bertero , 1989 ) and natterer ( natterer , 1996 , natterer and wbbeling , 2001 ) . in somewhat simplified terms , the reconstruction of the electron density is similar to finding the inverse of an ill - conditioned non - square matrix , a subject thoroughly discussed in golub and van loan ( golub et al . , 1996 ) . we consider these mathematical properties to be essential for understanding the successes and limitations of reconstruction algorithms ; we will try to be fully cognizant of them in the discussion that follows . the crystallographic phase problem is a good starting point for further discussion . it was first realized by sayre ( sayre , 1952 ) that the number of observable complex structure factors , limited by the bragg condition , is equivalent to a critical sampling of the electron density in the unit cell of the crystal . the sampling theorem of whittaker and shannon teaches us that , if the amplitudes and phases of all the diffraction peaks were accurately measured , the electron density could , in principle , be reconstructed everywhere ( bricogne , 1992 ) . unfortunately , only the amplitudes of the bragg reflections are measured , not their phases . therefore there is not enough information in the diffraction pattern for a unique reconstruction of the electron density . sayre proposed ( sayre , 1980 ) that if we could measure the diffraction amplitudes `` in between '' the bragg peaks , we should have enough information to reconstruct the electron density , or to `` phase '' the diffraction pattern . this is exactly the situation in diffraction from a single particle . nevertheless , it is still difficult to reconstruct the electron density , even from a well `` oversampled '' diffraction pattern . one corollary of critical sampling is that the amplitudes and phases of the bragg reflections of a crystal are independent of one other , but any structure factor in between them depends on the surrounding ones to some extent . therefore , too much oversampling does not help to obtain independent data , although it does improve the s / n ratio by reducing the noise . ill posedness is still with us , although with oversampling , the error propagates less . an additional difficulty with diffraction patterns from a set of discrete orientations of a particle is that at low resolution the diffraction pattern is well oversampled while at high resolution the sampling is sparse . a fundamental property of diffraction is that the position and the handedness of the electron density are undetermined , resulting sometimes in stagnation of the algorithm and drift in the position of the results ( stark , 1987 ) . there are two well - known , necessary remedies for the lack of information and for the ill posedness of the reconstruction problem . the more important one is the need for more information . for example , one way to include _ a priori _ knowledge is to accept reconstructed electron densities only if they are `` reasonable '' . the second remedy is to use `` stabilized '' or pseudo - inversion algorithms . in the next section we introduce our version of a real space reconstruction algorithm ; we will argue that our algorithm deals with all these problems optimally , at least in some sense . we return to the comparison of our algorithm with other methods for phase - recovery in section 2.3 . subsectionspeden , a real - space algorithm in this section we outline the workings of our reconstruction program , speden . speden uses a real - space method for reconstruction ; its acronym stands for single particle electron density . for computational efficiency the particle to be recovered is put into a _ fictitious _ unit cell that is several times larger than the particle itself . all reconstruction algorithms use this artifice in order to be able to calculate structure factors by fast fourier transform techniques . the resulting similarity with crystallography enables the use of many crystallographic concepts . in fact , the recognition of this similarity enabled us to write a program , speden , based on our crystallographic program , eden , with relatively small modifications . the most significant difference between the two programs is that in crystallography the bragg condition restricts the reciprocal lattice vectors to integer values , while the continuous diffraction pattern can be - and usually is - measured at arbitrary , non - integer values of the reciprocal lattice vectors . in speden , in common with eden , the ( unknown ) electron density is represented by a set of gaussian basis functions , with unknown amplitudes , that fill the fictitious unit cell uniformly . this way the recovery is reduced to the solution of a large set of quadratic equations . the program `` solves '' these equations by finding the number of electrons in each basis function so as to agree optimally with the measured diffraction intensities as well as with other `` prior knowledge '' . prior knowledge includes the emptiness of the unit cell outside the molecule , the positivity of the electron density , possibly some low - resolution image of the object , etc . each one of those conditions is described by a cost function that measures the deviation of the calculated data from the observed data . one of the cost functions describes the deviation of the calculated diffraction pattern from the measured one ; others depend on the deviation of the recovered density from prior knowledge . measured data are weighted by their certainty ( inverse uncertainty ) , other prior knowledge is weighted by its `` reliability '' . the mathematical method used is ( constrained ) conjugate gradient optimization of the sum of cost functions . at each step of the optimization , there is a set of amplitudes available that describe the current electron density in the full unit cell . a full set of structure factors is calculated by fourier transforming the current electron density . when the unit cell is larger than the particle , the structure factors can be stably interpolated to compare them with measured structure factor amplitudes . we refer to the cited literature that shows that the procedure we outlined is equivalent to a stabilized ( quasi ) solution of the inverse problem ( daubechies , 1992 ; bertero et al . , 1998 ; and natterer et al . , 2001 ) . as such , it is optimally suited for sparse , irregular , incomplete and noisy data . in the following subsections we describe very briefly the common features of eden and speden as well as their differences . a more complete description of eden can be found in previous papers ( szoke et al . , 1997a , and szoke , 1998 ) . the electron density is represented as a sum of basis functions , adapted to the resolution of the data . specifically , we take little gaussian `` blobs '' of width comparable to the resolution , and put their centers on a regular grid that fills the `` unit cell '' and whose grid spacing is comparable to the resolution . the amplitudes of the gaussians are proportional to the local electron density . in fact , the number of electrons contained in each gaussian constitutes the set of our basic unknowns . the above is identical to the representation of the electron density in eden . some mathematical details follow . the actual formula for the representation of the electron density as a sum of gaussians is @xmath0.\ ] ] the centers of the gaussians are positioned at grid points , @xmath1 , where @xmath2 is a counting index . in our fictitious unit cell , the grid is orthogonal , the grid spacing is @xmath3 and the centers of the gaussians are usually on two intercalating ( body centered ) grids for best representation of the electron density . the number @xmath4 of the order unity , determines the width of the gaussians relative to their spacing , @xmath3 . finally and most importantly , @xmath5 is the _ unknown _ number of electrons in the vicinity of the grid point @xmath6 . the values of @xmath5 are real , and in future may also be complex - valued to allow for photo - absorption in addition to scattering . ( the latter can be significant when diffraction measurements are made at longer x - ray wavelengths . ) given @xmath5 the structure factors can be calculated by @xmath7 } \sum\limits_{p = 1}^{p } n(p)\exp \left [ 2\pi i \bm{h } \cdot { \cal f}\bm{r}(p ) \right ] \,,\ ] ] using a fast fourier transform . the vector @xmath8 , a triplet of integers , denotes the reciprocal lattice vector , the operator f transforms from real space ( cartesian ) coordinates to fractional coordinates , and @xmath9 denotes the dual transformation . the constants appearing in eqs . ( [ eq1 ] , [ eq2 ] ) were discussed in some detail previously ( szoke et al . , 1997b ) . for completeness , we define them here . the crystallographic @xmath10 factor is given by @xmath11 . the `` crystallographic resolution '' , @xmath12 , determines the grid spacing , @xmath3 by the relation , @xmath13 , where @xmath14 is a constant of the order unity . for a body - centered lattice we set @xmath14 = 0.7 and @xmath15 = 0.6 . for a simple lattice , we use @xmath14 = 0.6 and @xmath15 = 0.8 . note that the gaussian basis functions are not used in a one - to - one correspondence with single atoms , but are simply used to describe the 3d electron density at the resolution that is appropriate to the data resolution . in the special case that the resolution was about the size of an atom and an atom happened to be sitting exactly on a grid point , that atom would be represented by a single basis function . if the atom is not on a grid point , or if the atom happens to be fat , because of thermal motion , that same atom would be represented by many basis functions . similarly , at lower resolution , a single basis function represents assemblies of atoms . the measured diffraction pattern of the molecule is proportional to the absolute square of the structure factors . in speden we do account for the curvature of the ewald sphere . there are two subtle points : the diffraction pattern is measured only in a finite number of directions , @xmath16 ; and as a rule , those directions are not along the reciprocal lattice vectors of the ( fictitious ) unit cell for a single particle . in other words , the measurement directions , @xmath16 , are usually not integers and they are not uniformly distributed in reciprocal space . this is the main difference between crystallography and single particle diffraction and , therefore , between eden and speden . the essence of _ any _ reconstruction algorithm is to try to find an electron density distribution such that the calculated diffraction pattern matches the observed one . in our representation , we try to find a set of @xmath5 , such that @xmath17 for each measurement direction , @xmath18 . let us assume for a moment that @xmath18 are integers . when the representation of the unknown density is substituted from ( [ eq2 ] ) , for each measured value of @xmath18 , equation ( [ eq3 ] ) becomes a quadratic equation in the unknowns , @xmath5 . the number of equations is the number of measured diffraction intensities . it is usually not equal to the number of independent unknowns that are the number of grid points in the unit cell . the equations usually contain inconsistent information , due to experimental errors . the equations are also ill conditioned and therefore their solutions are extremely sensitive to noise in the data . under these conditions the equations may have many solutions or , more usually , no solution at all . our way of circumventing these problems is to obtain a `` quasi solution '' of ( [ eq3 ] ) by minimizing the discrepancy , or cost function ( see e.g. stark , 1987 , and bertero et al . , 1998 ) @xmath19 ^ 2\ ] ] the weights , @xmath20 , are usually set to be proportional to the inverse square of the uncertainty of the measured structure factors , @xmath21 . as discussed by szoke ( szoke , 1999 ) , this is equivalent to a maximum likelihood solution of the equations . let us now discuss the first , previously ignored difficulty in the reconstruction . when we try to reconstruct the electron density from real experimental data , we have to compare the set of measured @xmath22 , where @xmath18 are not necessarily integers , with the calculated structure factor amplitudes , @xmath23 , that are on a regular grid , i.e. have integer @xmath24 . in principle , given an electron density of the molecule , one could calculate the structure factors in the experimental directions . nevertheless , for computational efficiency , we put the ( unknown ) molecule or particle into a fictitious unit cell that is larger than the molecule . we will also demand that the gaussians outside the molecular envelope be empty . ( in practice , sizes of molecules are known from their composition ; particle sizes and shapes may be known from lower - resolution imaging . ) as long as the distances of the gaussian basis functions are kept to be the experimental resolution , the number of `` independent '' unknowns neither increases nor decreases , in principle , by this computational device . the structure factors are calculated on an integer grid in the large unit cell , so they are essentially oversampled in each dimension by the same factor of the size of the large cell to the size of the molecule . the oversampling allows stable interpolation of the calculated structure factor amplitudes from integer @xmath24 to the fractional @xmath16 everywhere , independent of the density of the actual measurements . note that interpolation from fractional @xmath16 to integer @xmath24 is not always a stable procedure ! in the present implementation of speden , we get sufficient accuracy with the simplest , tri - linear interpolation in the amplitudes of @xmath25 if we choose the fictitious unit cell to be three times larger than the molecule in each dimension . now , some mathematical details : the reciprocal space vector * h(@xmath26 is within a cube , bounded by eight corners @xmath27 with integer values . let us denote the fractional parts of the components of @xmath18 as @xmath28 . we define weights for the eight corners , @xmath29 , by taking the products of the fractional parts of @xmath30 , or @xmath31 with those of @xmath32 or @xmath33 and @xmath34 or @xmath35 . the cost function to be minimized now becomes @xmath36 ^ 2\,.\end{aligned}\ ] ] a similar approach of applying crystallographic algorithms to continuous diffraction data has been done with direct methods [ spence et al . , 2003 ] . in this case , however , the ewald sphere was approximated by a plane . let us assume that we have some , possibly uncertain , knowledge of the electron density in parts of the unit cell from an independent source , i.e. one that does not come from the x - ray measurement itself . this is the kind of knowledge present when the unknown molecule is placed into a larger unit cell and we demand that the unit cell be empty outside the molecule . this kind of knowledge was also referred to as a `` sensible '' electron density in the introduction . we represent this knowledge by a target electron density @xmath37 and by a real - space weight function @xmath38 . it will be desirable that the actual electron density of the molecule , @xmath39 , as represented by @xmath5 , be close to the target electron density ; the weight function @xmath38 expresses the strength of our belief in the suggested value of the electron density . note that target densities can be assigned in any region of the unit cell independently of those in any other region . the simplest way to express the above statement mathematically is to minimize the value of the cost function @xmath40 ^ 2 \,,\ ] ] where @xmath41 is a scale factor and @xmath42 is a normalizing constant , described in somoza et al . , 1995 . ( in the _ absence _ of information at and around the molecule , weights are generally unity where it is known that there is no molecule and zero elsewhere . ) the same procedure is used in eden . the knowledge of the electron density at low resolution can be expressed by a low - resolution spatial target . crystallographers call this phase extension . the essence is that , during the process of the search for an optimal electron density , we try to keep its low resolution component as close to the known density as possible . a convenient way to accomplish this is the following . given @xmath5 , we smear out its gaussian representation and compare it to the equally smeared out target . actually , it is easier to carry out the computation in reciprocal space . we define @xmath43 where the current `` smeared '' structure factors are calculated using the low resolution , @xmath44 , @xmath45\sum\limits_{p = 1}^{p } { n(p)\exp [ 2\pi i{\rm { \bf h } } \cdot { \cal f}{\rm { \bf r}}(p ) ] } .\ ] ] the original low - resolution target is prepared analogously from the ( presumably ) known electron density , @xmath46 . the same procedure is used in eden . in the presence of a target density , the actual cost function used in the computer program is the sum of @xmath47 ( [ eq5 ] ) , @xmath48 ( [ eq6 ] ) and @xmath49 ( [ eq7 ] ) @xmath50 the fast algorithm described in somoza et al . , 1995 , and szoke et al . , 1997b , is always applicable to the calculation of the full cost function , ( [ eq9 ] ) . there is a clear possibility of defining more target functions . they are all added together to form @xmath51 that is minimized to find the optimum electron density . the minimization of the cost function ( [ eq9 ] ) is carried out in speden ( as in eden ) by d. goodman s conjugate gradient algorithm ( goodman , 1991 ) . it has proven to be very robust and efficient in years of use in eden . the essential properties of the algorithm that make it so advantageous for our application is that the positivity of the electron density , @xmath52 , is always enforced and that the gradient vector in real space can be calculated by fast fourier transform . the gradient calculation needed only a very simple modification for the interpolation in reciprocal space , eq . ( [ eq5 ] ) . the line search algorithm does not use the hessian , so matrices are never calculated . as with any local minimization , global convergence is not achieved . we discussed this problem in our previous papers and came to the conclusion that , usually , the minimum surface of the cost function ( [ eq7 ] ) is so complicated that finding a global minimum would take more computer time than the existence of the universe . reconstruction of the scatterer from a continuous diffraction pattern has a tangled history replete with repeated discoveries . some of the present authors are also guilty of ignorance of prior work . we referred to the pioneering insights in section 2.1.2 . the `` recent '' period of algorithms started with the work of miao , sayre and chapman ( miao et al . 1998 ) who pointed out that the fraction of the unit cell where the density is known is an important parameter for convergence . in somewhat later work , with oversampled structure factors calculated on a regular grid , the crystallographic program eden successfully demonstrated the recovery of the electron density using a simulated data set from the photoactive yellow protein ( szoke , 1999 ) . the protein was put into a fictitious unit cell , twice the size of the original one , and a target with zero density was used outside the original unit cell of the protein . similarly , miao and sayre ( miao et al . , 2000 ) have studied empirically how much oversampling is required in two- and three - dimensional reconstructions of a simulated data set ; using a version of the gerchberg saxton fienup ( gsf ) algorithm . among recent articles we mention robinson et al ( 2001 ) , williams et al ( 2003 ) , marchesini et al . ( 2003 ) and references therein , in addition to those mentioned in the introduction . all reconstruction algorithms of oversampled diffraction patterns use _ a priori _ information on the shape and size of the particle . in our previous studies in crystals we found that such information is very valuable . for example , eden converges surprisingly well for proteins at low resolution where the only information used is that the molecule is a single `` blob '' . eden also converges for synthetic problems with a good knowledge of the solvent volume , which is greater than 50% ( beran et al . , 1995 ) or 60% ( eden ) . a similar conclusion was reached in miao et al . ( 1998 ) . in comparison , when a molecule is embedded in a 3-fold larger fictitious unit cell , the empty `` solvent '' occupies @xmath5396% of the cell volume . as discussed previously , the reconstruction of scatterers from their diffraction pattern is a difficult mathematical problem . in many cases the indeterminacy of the absolute position of the object and of its handedness causes difficulties in convergence . that is definitely the case with speden so , in that sense , speden is not a good algorithm . empirically , the gsf algorithm has a larger radius of convergence and deals better with stagnation ( marchesini et al . 2003 ) another family of difficulties arises when there is _ a priori _ information available , but there is only incomplete and noisy data . under such conditions the main questions are how to find a solution that optimally takes into account the available information and that is the best `` sensible '' one that reproduces the noisy and incomplete data to its limited accuracy . it is this second set of conditions for which speden was written . although , in this paper , we show only its performance for artificial and `` easy '' but incomplete data , speden s older sister , eden has been shown to have those properties on a large range of crystallographic data sets , ranging from cuo@xmath54 to the ribosome . we expect that such properties of eden will be inherited by speden , considering that their fundamental mathematical properties are sufficiently similar . the best known , and successful class of algorithms is the group of iterative transform algorithms that we refer to as gerchberg - saxton ( gs ) ( gerchberg et al . , 1972 ) , and its development , in which support constraints and feedback are added , the gs - fienup ( gsf ) or hybrid input - output algorithm ( fienup et al . , 1982 ; aldroubi et al . , 2001 ; bauschke et al . , 2002 ; bauschke at al . , 2003 ) . the essence of the gsf algorithms is that they iterate the n - pixel data between real and reciprocal spaces via ffts and enforce the known constraints in each of these spaces . depending on the degree of noise in the data , these algorithms usually converge in about a hundred to a thousand iterations . the weak convergence ( non divergence ) of the gs algorithm has been proven in the absence of noise ( fienup et al . , 1982 ) . there is no mathematical proof that these algorithms will converge in general , but it is reasonable that by sequentially projecting onto the set that satisfies the real space constraints and the set that satisfies the reciprocal space constraints , the intersection ( corresponding to a valid solution ) should be approached . this is definitely true for projections onto convex sets , but unfortunately these sets are not convex ( stark , 1987 ) . in practice , despite the fact that the modulus constraint is non - convex , the algorithms often converge even in the presence of noise in about several hundred iterations . the main difference between the gsf algorithms and speden is that speden does not iteratively project onto the sets of solutions that satisfy the real - space or reciprocal space constraints separately , but rather it minimizes a cost function that includes all the constraints of both spaces . it does this by varying quantities in real space only ( the @xmath5 s ) and the cost evaluation only requires a forward transform from real space to reciprocal space . as the cost function never increases , speden reaches only a local minimum . in spite of its `` small '' radius of convergence , there are some expected advantages to speden s algorithm .. in speden we compare the @xmath55 to the @xmath56 by interpolating from the samples of @xmath55(h * ) , calculated on a regular grid , to the measured sample vectors * h. since the gridded @xmath57 are a complete set ( due to the fact they are sampled above the nyquist frequency ) the interpolation is stable and performed with little error . since an inverse transform is required in the gsf algorithms , the measured diffraction data @xmath58 ( recorded on ewald spheres in reciprocal space ) must be interpolated to the gridded data points h. the observed data might not be a complete set , especially at high resolution where the density of samples is sparser : this may lead to error . an additional possible difficulty with the gsf algorithms is that the effective number of unknowns may increase with the size of the fictitious unit cell , while in speden the effective number of unknowns is constant . finally , in speden , weightings can be properly applied to all data and knowledge . the measured data is inversely weighted by its uncertainty ; it is a procedure equivalent to maximum likelihood methods and it should be optimal , at least in theory . as a side effect , when the constraints are inconsistent , speden still converges to a well - defined and correct solution . ( note that for noisy data the constraints are almost always inconsistent . ) weightings are also applied to reflect our confidence in real - space constraints , and these weightings are consistently used in real space . tests * speden has certain built - in limitations . in particular , of course , reconstruction is only as good as the diffraction measurements and derived structure factor amplitudes are reliable . there are also other less obvious limitations . for example , there are inherent inaccuracies due to the gaussian representation of the electron density in real space . ( szoke et al . , 1997a ) also , the trilinear interpolation for representing integer ( hkl ) structure factor amplitudes on a non - integer grid is only approximate . finally , as a fundamental limitation of _ any _ reconstruction method , both the absolute position and the handedness of the molecule are undefined . we performed preliminary tests to verify the capabilities and limitations of our reconstruction method using computed and experimentally obtained diffraction patterns . in this section , we first describe the reconstruction of simple `` molecules '' from synthetic diffraction patterns with speden . specifically , we discuss how the convergence of speden is affected by the errors due to interpolation in reciprocal space , by the quantity of observed structure factors , @xmath59 , by the extent of the `` known '' starting model , and by the uniformity of sampling in reciprocal space . we then describe the reconstruction of simple two - dimensional objects from synthetic and measured diffraction patterns with speden . we created synthetic `` molecules '' in the format of the protein data bank ( pdb ) files ( berman et al . , 2000 ) . each molecule was composed of 15 point - like carbon atoms , placed at random positions within a cube measuring 16.8 in each dimension and `` measured '' to 4 resolution ; these values correspond to crystallographic @xmath10 factors of 185.7 @xmath60 . the molecule was then shifted so that its center - of - mass was at the center of the cube . all our simulations were repeated using molecules with several different random arrangements . initially , the `` unit cell '' coincided with the dimensions of the cube in which the atoms were placed ; later , larger cells were used and the atoms were positioned in their center . we also generated `` starting models '' by removing atoms from the full molecule . both full and partial molecules served to generate structure factors ( @xmath61 ) , using the atomic positions and the b factors . starting models were generated from the @xmath55 , using speden s preprocessor , back , which finds the optimal real - space representation for an input @xmath61 . initially , sets of `` measurements '' ( @xmath62 ) were generated by deleting the phases of the calculated structure factors of the full or partial molecule . the @xmath63 files so generated had all integer @xmath16 . the uncertainty of the measured structure factors , @xmath64 , were chosen to be @xmath65 with @xmath66 for @xmath67 , and @xmath68 for @xmath69 . we used two alternate methods to generate @xmath63 files for non - integer @xmath18 : in the first method , we used a unit cell whose dimensions were incommensurate with one another and with the edge of the cube , but whose volume equaled that of the cube . the resulting @xmath63 file , generated again from @xmath61 files by deleting the phases , then had its indices scaled back appropriately , yielding fractional @xmath18 . a different second method was used to sample the reciprocal @xmath18 space non - uniformly : @xmath70 at fractional @xmath18 were calculated using tri - linear interpolation from @xmath62 data on a regular grid that , in turn , was at four times the regular resolution in reciprocal space . for each of these @xmath63 files , some constraining information is required in order to find the atom positions . we used two types of constraints : one of them identified the ( approximate ) empty region ; the other one used @xmath61 at a considerably lower resolution . we call them the empty target and the low - resolution target , respectively . both are based on the assumption that at a considerably lower resolution , the general position of atoms as one or more `` blobs '' in empty space is known . the low - resolution @xmath71 was prepared by smearing the full @xmath72 file to 10 . the empty target identified the empty points in terms of a mask . then , throughout speden s iteration process , using the solver solve , the program attempted to match the current electron / voxel values in masked - in regions to empty ( @xmath73 ) values . the phase - extension target used the same @xmath71 file ; during the iteration process , at each step , the current real - space solution was smeared to that low resolution in reciprocal space and restrained to agree with that target . besides inspecting the reconstructed electron density visually , we used four quantitative measures to compare the reconstructed image with the electron density from the full molecule at 4 : 1 . real - space rms error : we calculated the real - space electron density from the electron / voxel files , a process we call regridding . we then calculated the root - mean - square ( rms ) error of the electron densities , @xmath74 and @xmath75 , defined as @xmath76 we permitted one file to be shifted with respect to the other file , in order to minimize the distance . phase difference : we calculated the average ( amplitude - weighted ) phase difference between the @xmath61 at the end of the run and the corresponding @xmath61 generated from the full molecule . final @xmath77 factor : we calculated the crystallographic r factor ( giacovazzo et al . , 2002 ) at the end of the run . count error : we compared the integrated real - space electron density generated from a run result with the true number of electron as identified in the pdb file , on an atom - by - atom basis . the integration was performed around each atom over a sphere with a radius that was 1.5 times the grid spacing . the figure - of - merit is the rms error . of all these measures , the final @xmath77 factor was the least useful to assess the quality of the reconstructed image . the @xmath77 factor tends to be lower for a small number of entries in the observation file since there are fewer equations to satisfy during the reconstruction . in such a case , a visual inspection shows that the reconstructed electron density may have little resemblance to the 15 carbon atoms . however , there was a good correlation among the other three measures . the solutions looked correct when the phase difference between solution and full @xmath61 was less than 20 , the count error between solution and pdb model was 0.2 or lower ( out of 6 ) , and the rms distance measure was less than 0.2 . for computational purposes , we placed the ( unknown ) molecule or particle into a fictitious unit cell that is larger than the molecule , and calculated the structure factors on an integer grid in the large unit cell , oversampled by the same ratio : the size of the large cell to the size of the molecule . we then calculated the structure factor at fractional * h * from the structure factor at integer * h * using tri - linear interpolation . the interpolation error becomes smaller when larger unit cells are used ( at the expense of computation time ) . we studied the effect of the unit cell size on convergence of speden . in the initial tests , the unit cell was the same size as the original molecule , and the starting ( known ) part of the molecule consisted of the full molecule . when the molecule consisted of atoms on grid points and the @xmath63 files had integer @xmath78 , unsurprisingly speden converged , as did eden on the same data set however , we found that speden did not converge to a unique solution if either the atoms were not on grid points or when the @xmath63 file had non - integer @xmath18 , or both , since the solution meandered in real space . in subsequent tests , we generated larger unit cells and applied a target over the empty part of the unit cell , in an attempt to restrain the meandering problem . we embedded the molecule in a cell that was 2 or 3 times greater in each dimension . an empty target was used that essentially covered the empty 7/8-th or 26/27-th of the unit cell , respectively . we applied a high relative weight for this empty target , and we still used the full molecule as a starting position . we found that both the larger cell and the empty target are of critical value in enabling speden to converge to the correct solution . comparing the 2-fold larger unit cell versus the 3-fold unit cell , there was a great improvement in the latter case . these results show that for tri - linear interpolation , it is adequate to use a unit cell that is 3 times greater in each dimension . we expect that more sophisticated interpolation algorithms should allow using smaller unit cells . a three - fold enlarged unit cell increases the number of unknown amplitudes of the gaussians , n(p ) , by a factor of 27 . in principle , the emptiness of the volume around the molecule restrains the effective number of independent unknowns . nevertheless , if the number of equations , which is given by the number of entries in the fobs file , is not increased , it is easy for the solver to `` hide '' electrons among the large number of unknowns in the system , even when the empty target constraint is used . in fact we found that when we compared the final fcalc from solve against the starting fcalc , on the one hand , and the correct fcalc on the other , solve s final fcalc was closer to the starting one than to the correct one . in other words , the cost function in reciprocal space was not a sufficiently strong constraint in speden s algorithm , for this synthetic problem . in a similar real case , more experimental diffraction patterns need to be collected in order that speden would be able to find the corresponding image without other information . in the next set of synthetic tests , we attempted to recover missing information by starting from a partial model that contained less than the full complement of 15 atoms . in these simulations , we used @xmath63 files with non - integer @xmath18 , a three - fold enlarged unit cell , an empty target or a phase extension target , and randomly positioned atoms . we found that a low - resolution spatial target significantly helps speden to converge . figure 1 ( a ) shows the results of the comparison of the reconstructed image with the electron density from the full pdb file when a phase extension target is used . the phase extension target was calculated at a resolution of 10 . we found that it was generally possible to recover 5 , 10 , or even all 15 of the atoms . please note that the amount of information in such a phase extension target is ( 4/10)@xmath79 6% of the information in the perfect solution . it was more difficult to reconstruct the original electron density when we used an empty target , as shown in figure 1 ( b ) . speden was able to recover 5 out of the 15 atoms , but did not converge when 10 atoms were unknown . perhaps surprisingly , the case where there was no starting model at all ( 0 atoms known ) did consistently better than those cases where a partial model was given as a starting condition . in this set of synthetic tests , we addressed the question of how difficult it is to recover the molecule from a non - uniform set of samples in reciprocal space , similar to real data sets , and how the results compare with the reconstruction from a uniformly - sampled data set . we generated two - dimensional diffraction patterns of the synthetic carbon molecule for different particle orientations , corresponding to recorded diffraction patterns in a `` real '' experiment . the two - dimensional diffraction patterns were linearly interpolated from a three - dimensional diffraction pattern ; the latter was calculated on an additionally double - fine grid over the already triple - sized unit cells , i.e. using a unit cell that was a total of six times larger than the molecule in each direction . further refining the grid of the three - dimensional diffraction pattern did not alter the results significantly . the three - dimensional diffraction pattern , in turn , was the fourier transform of the gridded electron density of the synthetic molecule . two - dimensional patterns do not sample the diffraction space uniformly . the sampling density near the center of the diffraction space is much larger than the sampling density further away . we used a completeness measure to characterize the sampling uniformity . the reciprocal space is divided into cells that are 4@xmath80/@xmath81 by 4@xmath80/@xmath82 by 4@xmath80/@xmath83 in size , where @xmath81 , @xmath82 , and @xmath83 are the molecule sizes in each dimension . the completeness then is the ratio of cells in reciprocal space that contain at least one measurement over the total number of cells . figure 2 shows the completeness of the input observation files as a function of the number of diffraction patterns . also shown in figure 2 is the number of calculated diffraction intensities . we then used speden to recover 7 out of the 15 atoms . the molecule was embedded in a unit cell that was three times larger in each dimension , and we used an empty solvent target . figure 3 shows the errors of the reconstructed electron density as a function of the number of two - dimensional diffraction patterns . the orientations of the diffraction patterns were chosen at random , and the calculations were repeated for four different molecules . for comparison , also shown in figure 3 are the errors of the electron density of the 8 known atoms ( `` partial model '' ) , and the errors of the reconstructed electron densities when a three - dimensional diffraction pattern is used which was oversampled three times ( `` integer hkl '' ) or six times ( `` fractional hkl '' ) . as discussed above , the @xmath77 factor is not a useful measure to assess the quality of the reconstructed image , but the rms and count errors are better measures for the reconstruction quality . surprisingly , we found that _ four _ two - dimensional patterns are sufficient to reconstruct the electron density as well as in the case the full three - dimensional diffraction pattern is given . four two - dimensional patterns have a remarkably low sampling completeness of only 14% . further increasing the completeness or the number of observations does not improve the quality of the reconstruction . we would like to note , however , that these results could be dependent on the choice of the test model , and that for different test models the number of required two - dimensional patterns may be larger . in the final set of tests , we demonstrate spedens ability to recover missing information for a two - dimensional configuration of 37 au balls in a plane . we reconstructed the au balls using ( i ) a synthetic diffraction pattern and ( ii ) an experimentally obtained diffraction pattern as discussed by he et al . , 2003 . in the following we will discuss both cases , starting with the synthetic diffraction data . the 37 au balls are arranged in a plane as shown in figure 4 . the arrangement of the balls is similar to the experimental case discussed by he et al . , 2003 . the au balls were 50 nm in diameter . we generated an artificial set of `` measurements '' ( @xmath84 ) by calculating the structure factors ( @xmath85 to 30 nm resolution and deleting the phases . the uncertainty of the measured structure factors , @xmath64 , were chosen according to equation ( [ eq10 ] ) . we generated an initial model by smearing the full @xmath72 file to 90 nm , and running back on it . the initial model is shown in figure 5 . in figures 5 7 , we only show one plane . we also used this smeared @xmath61 to generate a low - resolution spatial target as well as an empty target outside the molecule . the corresponding weight function is shown in figure 6 . we then used speden to reconstruct the au balls . as shown in figure 7 , speden reconstructed the electron density successfully . however , we further found that if we use an empty starting model , speden has difficulties converging to the correct electron density . there are two reasons for this behavior . first , without an initial model , the symmetry of the system is not broken , and speden stagnates since the support does not distinguish between the object and its centrosymmetric copy . second , the mask and the reconstructed electron density are possibly shifted with respect to each other . if the initial model is empty , the position of the reconstructed electron density is mostly determined in the early iteration of the solve algorithm and can be partially cut off by the solvent . the algorithm has difficulty shifting the result . it is necessary to provide information about the location of the electron density to a certain degree , for example in the form of a smeared model . note that the gsf algorithms are designed to overcome these problems when there is abundant and accurate data available . we will now discuss the reconstruction of the au balls using experimental data . to generate a starting model , we took the experimental @xmath59 data along with the phases obtained by he at al . , 2003 using a version of the gsf algorithm , and smeared this data to 80 nm . the starting model is shown in figure 8 . we used the same data to generate a real space target with a target fraction of 99.7% , shown in figure 9 . similar to the case of the synthetic test data , we chose @xmath86(h ) according to equation ( [ eq10 ] ) . we then used speden to reconstruct the au balls . figure 10 ( a ) shows the reconstructed electron density , and figure 10 ( b ) shows a scanning electron microscope ( sem ) picture of the sample . we found that speden reconstructed the electron density from the experimental data successfully . in this paper we presented speden , a method to reconstruct the electron density of single particles from their x - ray diffraction patterns , using an adaptation of the holographic method in crystallography . unlike existing gsf algorithms , speden minimizes a cost function that includes all the constraints of both real space and reciprocal space , by varying quantities in real space only , so that the cost evaluation requires only a forward transform from real space to reciprocal space . speden finds a local minimum of the cost function using the conjugate gradient algorithm . we implemented speden as a computer program , and tested it on synthetic and experimental data . our initial results indicate that speden works well . this work was performed under the auspices of the u.s . department of energy by university of california , lawrence livermore national laboratory under contract w-7405-eng-48 and doe contract de - ac03 - 76sf00098 ( lbl ) . sm acknowledges funding from the national science foundation . the center for biophotonics , an nsf science and technology center , is managed by the university of california , davis , under cooperative agreement no .	speden is a computer program that reconstructs the electron density of single particles from their x - ray diffraction patterns , using a single - particle adaptation of the holographic method in crystallography . ( szoke , a. , szoke , h. , and somoza , j.r . , 1997 . acta cryst . a53 , 291 - 313 . ) the method , like its parent , is unique that it does not rely on `` back '' transformation from the diffraction pattern into real space and on interpolation within measured data . it is designed to deal successfully with sparse , irregular , incomplete and noisy data . it is also designed to use prior information for ensuring sensible results and for reliable convergence . this article describes the theoretical basis for the reconstruction algorithm , its implementation and quantitative results of tests on synthetic and experimentally obtained data . the program could be used for determining the structure of radiation tolerant samples and , eventually , of large biological molecular structures without the need for crystallization . szoke chapman szoke marchesini noy he howells weierstall spence speden is a computer program that reconstructs the electron density of single particles from their x - ray diffraction patterns , using an adaptation of the holographic method in crystallography . it is designed to deal successfully with sparse , irregular , incomplete and noisy data .
energy localization due to a nonlinearity in dynamical systems has been observed for more than one century @xcite , and the effect of an exact balance between nonlinearity and linear dispersion of wave packets leading to the appearance of soliton excitations has become a paradigmatic example in nonlinear science which can be found in various textbooks . in the past decade remarkable achievements in the study of localized nonlinear excitations were made with the discovery of stable localized modes in spatially discrete translationally invariant hamiltonian systems - _ discrete breathers _ ( dbs ) @xcite . they have been proved to be generic _ exact _ time - periodic solutions of the corresponding coupled nonlinear ordinary differential equations , eventhough the latter are generally non - integrable . it is worth mentioning that discrete breathers have been observed experimentally in various physical systems including coupled optical waveguide arrays @xcite , coupled josephson junctions @xcite , micromechanical cantilever systems @xcite , anti - ferromagnetic crystals @xcite , high-@xmath0 superconductors @xcite . discrete breathers are predicted also to exist in the dynamics of dusty plasma crystals @xcite . among the most important characteristics of a localized excitation are its localization length and the spatial decay characteristics of its amplitude . although dbs can be localized practically on a single site , in most of the cases they have exponentially decaying tails ( similar to their continuum counterparts - solitons ) . this is true if the interaction potentials are reasonably short ranged ( see @xcite for details ) . however , when an anharmonic interaction between adjacent sites is much stronger then the harmonic one , localized excitations can become even more compact . as it was demonstrated by rosenau and hyman @xcite , in continuous systems nonlinear localized excitations may compactify , i.e. gain strictly zero tails , under nonlinear dispersion . the same was conjectured for discrete systems @xcite , however later it was shown that in discrete systems localized excitations can not have an exact compact structure @xcite , but the tail decay follows a _ super - exponentional _ law @xmath1 , provided that the interaction is purely short - range . this fact was then confirmed numerically @xcite , and the corresponding breather solutions were coined _ compact - like _ @xcite or _ almost - compact _ @xcite dbs . if compact - like breathers are dynamically stable , we may expect that localized perturbations of such solutions will lead to a quasiperiodic in time evolution , which will not induce a radiation of energy away from the breather . this is in contrast to the well - known existence of such a radiation for systems with linear dispersion @xcite . there it appears due to the resonant overlap of combination frequencies of the internal perturbed breather dynamics with the spectrum of small amplitude plane waves . in the case of purely nonlinear dispersion the width of this spectrum is zero , and thus the origin of the radiation is removed . one expects then that perturbed compact - like breathers will not radiate energy away , giving rise to genuine _ quasiperiodic _ compact - like breathers . another important issue which might drastically change the rate of spatial decay in dbs tails is the presence of long - range interactions , essential e.g. in systems with weakly screened coulomb interaction such as ionic crystals , or various biomolecules . usually decaying slower than exponentially in space , long - range interactions introduce a crossover length as a result of competition of the two essentially different length scales @xcite . they can also lead to the appearance of energy thresholds for dbs in some cases , where a pure short - range interaction would not be capable of producing any . in @xcite it was demonstrated , that the effect of length - scale competition with long - range algebraically and exponentially decaying interaction can lead to a new type of multistability of dbs , when in a certain model parameter regime several different types of dbs coexist having the same value of the spectral parameter ( i.e. velocity or frequency ) . it is the purpose of this paper to address the abovelisted issues . the paper is organized as follows . in section [ sec : two ] we introduce the model . we demonstrate , that the specifically chosen nonlinear potentials allow one to completely separate temporal and spatial dependencies and thus significantly simplify the analysis of the problem . we derive the nonlinear coupled algebraic equations for the spatial profile of a solution and in addition an ordinary differential equation ( duffing equation ) for the master function describing uniform oscillations of all the sites with time . in section [ sec : three ] we obtain the different types of discrete breather solutions . we demonstrate , that in general the model supports two classes of discrete breathers with completely different dynamical properties . we then study the linear stability properties of basic types of dbs and observe quasiperiodic localized excitations . in section [ sec : four ] we reveal the effect of long - range interactions along the chain on properties of db solutions . we show that the presence of non - local dispersive terms result in the appearance of several characteristic cross - over lengths . we derive estimations for these cross - over lengths , as well as asymptotes for amplitude distribution in db tails , on the basis of a simple three - site model . in section [ sec : concl ] we conclude . we consider a simple one - dimensional model of ( nonlinearly ) coupled oscillators with the following hamiltonian : @xmath2\right\}\end{aligned}\ ] ] where @xmath3 is the displacement of @xmath4th unit mass oscillator from its equilibrium position , the constant @xmath5 characterizes the rate of spatial decay of long - range interactions between oscillators , and @xmath6 is given by @xmath7 the equation of motion for the displacement of the @xmath4th oscillator from its equilibrium reads : @xmath8 we note that while the interaction decays algebraically for any finite power @xmath9 , in the limit @xmath10 we recover the case of short - range nearest neighbor interaction . the specifically chosen nonlinear potentials allow one to use the time - space separation technique @xcite , so that time - periodic solutions of ( [ mot ] ) can be written in the form @xmath11 with time - independent amplitudes @xmath12 and a master function @xmath13 describing _ uniform oscillations _ of all the sites . after substitution of the ansatz ( [ time - space ] ) into the eqs . ( [ mot ] ) the following equation for the function @xmath13 is obtained : @xmath14 while the amplitudes @xmath12 satisfy algebraic equations : @xmath15\\ \nonumber & & -\phi_n^3,\end{aligned}\ ] ] where @xmath16 is an arbitrary separation constant . its absolute value can be always chosen to be equal to unity . can be removed from equations ( [ ode],[algebra ] ) by rescaling @xmath17 and @xmath18 . note , that this rescaling does not affect the initial variables @xmath19 . ] while the dynamics of all the sites is governed by a unique function @xmath13 , which can be easily found by integrating eq . ( [ ode ] ) , the spatial profile of possible solutions of eqs . ( [ mot ] ) is determined by eqs . ( [ algebra ] ) being of main interest for us . it is important to note , that the dynamics of a db , as well as its spatial profile , depend on the sign of @xmath16 in eqs . ( [ ode],[algebra ] ) . this sign is fixed only for the case of uncoupled oscillators ( @xmath20 ) and for small values of @xmath21 ( @xmath22 ) , while generally it can be arbitrary . as a consequence , for large enough values of @xmath21 eqs . ( [ mot ] ) support _ two classes _ of dbs with different dynamical properties , since the sign of @xmath16 defines the type of nonlinearity ( soft or hard ) in the eq . ( [ ode ] ) for the master function @xmath13 . db solutions of different classes possess essentially different core structures . as a consequence , the impact of long - range interactions on db tail structure is also different for these two classes of dbs . in order to understand how two different classes of solutions of eqs . ( [ mot ] ) appear , it is instructive to start with a simple case of three coupled oscillators . this simple model gives a rather good approximation for the sites of a db core , which are practically not affected by the presence of long - range interactions . here we restrict ourselves in considering only symmetric dbs centered at site @xmath23 ( it is straightforward to modify this approximate model for more complicated types of _ multi - site _ therefore , by putting @xmath24 and @xmath25 , we finally obtain from eqs . ( [ algebra ] ) the following expression for the central site amplitude : @xmath26 while the coefficient @xmath27 is a root of the fourth - order polynomial equation @xmath28 for each given real value of @xmath27 obtained from eq . ( [ polynom ] ) the sign of @xmath16 is fixed and determined by the condition of non - negative valued r.h.s . of eq . ( [ phi0 ] ) . ) . the inset shows the corresponding values of @xmath29 computed from eq . ( [ phi0 ] ) , squares , triangles and circles indicate the same quantity computed for the system of 201 coupled oscillators . , scaledwidth=45.0% ] in fig . [ fig0 ] the real roots of eq . ( [ polynom ] ) are plotted in the range of the coupling constant @xmath30 $ ] . there exist two real non - trivial roots , whose absolute values stay below unity [ and thus corresponding to single - site dbs with amplitudes of the central site @xmath31 _ greater _ than those of the neighboring sites @xmath32 in the whole interval of non - negative values of @xmath21 up to @xmath33 . one of these roots originates from @xmath34 in the uncoupled limit ( i.e. from a single - site excitation ) and remains to be positive in the whole interval of @xmath21 , see dashed line in fig . the corresponding family of dbs have a non - staggered pattern of amplitudes @xmath12 _ in - phase _ oscillations , see fig . [ fig : profs](a ) . the value of @xmath29 computed for this root from eq . ( [ phi0 ] ) is negative for @xmath35 ( see dashed line in the inset in fig . [ fig0 ] ) , therefore one should choose @xmath36 for this type of dbs in eqs . ( [ ode],[algebra ] ) . thus , db solutions of eq . ( [ mot ] ) with a non - staggered pattern should possess _ soft _ nonlinear properties , i.e. their amplitude decays with growing frequency , as it follows from the eq . ( [ ode ] ) with negative @xmath16 . here we mention , that the quantity @xmath29 computed for the 3-site model changes its sign above the value @xmath37 . however , the comparison with numerically obtained solutions of eq . ( [ algebra ] ) for a larger system size ( @xmath38 ) indicates essential discrepancies when @xmath39 ( see triangles in the inset in fig . [ fig0 ] ) . it comes from the fact that a db core of breathers with the non - staggered profile extends while increasing @xmath21 , approaching continuum compacton solutions @xcite as @xmath33 . indeed , as shown in fig . [ fig : profs](a ) , while a db core involves more and more sites with increasing @xmath21 , its characteristic width in terms of the continuum coordinate @xmath40 remains to be fixed and demonstrates rather good agreement with the value @xmath41 reported for dbs in the continuum limit @xcite . another non - trivial real - valued root of eq . ( [ polynom ] ) originates from @xmath42 in the uncoupled limit ( i.e. from a staggered homogeneous excitation ) and remains to be negative in the whole interval of @xmath21 , see solid black line in fig . the corresponding db family is characterized by a staggered profile in the core ( while the tails have more complicated profile , as will be shown below ) , i.e. the central ( @xmath23 ) and the neighboring ( @xmath43 ) sites oscillate in _ anti - phase_. it is remarkable , that the corresponding value of @xmath29 changes its sign at @xmath44 , see solid black lines in the inset in fig . therefore , for small enough values of the coupling constant @xmath45 both staggered and non - staggered types of db solutions of ( [ mot ] ) possess _ soft _ nonlinear properties , in accordance to the chosen type of the on - site nonlinear potential ( [ onsite ] ) . the profile of the corresponding staggered - core db solution at @xmath46 is shown in fig . [ fig : profs](b ) . notably , all the tail sites perform inphase oscillations in this type of dbs , similar to the case of nonstaggered dbs described above . however , the central site oscillates in anti - phase with all the rest of the lattice . the competition between on- and inter - site nonlinearities results in the change of dynamical behavior of the dbs with staggered core profile as the coupling becomes strong enough : the nonlinear term in eq . ( [ ode ] ) for the master function @xmath13 becomes of the _ hard _ type for staggered dbs when @xmath47 . as the result , the amplitude of such a db solution increases with growing frequency . the corresponding profile is shown in fig . [ fig : profs](c ) . in this type of staggered dbs all sites in the core perform anti - phase oscillations . in the case of pure short - range interactions in the system , the staggered pattern persists for the whole spatial profile , including the breather tails . however , long - range interactions destroy the uniform staggered pattern , introducing a complicated domain - like structure in the breather tails , as will be explained below . unlike dbs with non - staggered profile , the staggered core dbs stay localized on a few sites as @xmath21 increases , therefore the 3-site model gives a rather good approximation for larger size systems for arbitrarily large values of @xmath21 ( see squares in the inset in fig . [ fig0 ] ) . simultaneously , the amplitude of these dbs decreases as @xmath33 ( for a fixed value of the frequency ) . however , we note that there is no limit in frequency ( and therefore , in energy ) for this type of dbs , since the master function @xmath13 satisfies the duffing equation ( [ ode ] ) with the _ hard _ nonlinear term ( @xmath48 ) . note also , that as @xmath33 , the influence of the inter - site nonlinear interactions becomes more important than the effect of on - site nonlinearities for the oscillating in anti - phase sites of the db core . therefore , at large values of @xmath21 the staggered - core db asymptotically approaches the high - energy limit of discrete breather solutions in models with purely inter - site nonlinearities ( fermi - pasta - ulam lattices ) @xcite . thus , for @xmath47 eqs . ( [ mot ] ) support _ two different classes _ of db solutions with _ soft _ and _ hard _ nonlinear properties , in what follows we will refer to such breather solutions as _ s - type _ and _ h - type _ dbs , respectively . the above discussed single - site dbs represent only particular ( basic ) members of these two classes of solutions . in general , one can constract more complicated localized s- and h - type solutions - _ multi - site _ dbs . as an example , we mention here _ two - site _ dbs with two sites in the core oscillating with the same ( maximum ) amplitude , see gray lines and symbols in fig . [ fig : profs ] . the center of energy density distribution is located in - between two sites in these dbs , so that they can be viewed as translated half site single - site dbs . finally , we would like to remark , that the existence of the two different classes of db solutions of eqs . ( [ mot ] ) is the result of a competition between soft nonlinearity of the on - site potential and hard nonlinear inter - site interactions . upon change the type of nonlinearity in the on - site potential [ by changing the sign of the quartic term in eq . ( [ onsite ] ) ] , only the h - type dbs with staggered core profile survive . different dynamical properties of dbs with staggered and non - staggered core profiles result , in particular , in different stability properties of these excitations . in this section we perform linear stability analysis of basic types of @xmath49 and @xmath50 type dbs by studying the dynamical behavior of a small perturbation @xmath51 to a given db solution @xmath52 . in order to construct a certain type of db solution of eq . ( [ mot ] ) , we solve numerically eqs . ( [ algebra ] ) for the db profile @xmath53 with a certain sign of the separation constant @xmath16 ( chosen in accordance to the above performed analysis of dbs structure ) and multiply it by a periodic solution @xmath54 of the duffing equation ( [ ode ] ) , according to the ansatz ( [ time - space ] ) . because of the time - space separation ( [ time - space ] ) , the stability properties remain qualitatively the same for the whole family of a given db type with different frequencies @xmath55 [ i.e. with different time - periodic functions @xmath56 . however , they might drastically change by varying the relative strength of the on - site and inter - site nonlinearities controlled by the coupling constant @xmath21 . thus , once a given db solution @xmath52 is obtained numerically , we add a small perturbation to it @xmath57 and linearize equations of motion eqs . ( [ mot ] ) with respect to @xmath51 : @xmath58 in order to simplify the stability analysis , here we restrict ourselves in considering pure short - range interaction terms , thus we keep only the term with @xmath59 in the r.h.s . ( [ linearized ] ) , which corresponds to the limit @xmath10 . being essential for db tail characteristics , long - range interactions practically do not affect the core structure of a db , provided that their decay rate @xmath9 is sufficiently large ( @xmath60 ) . therefore , the impact of long - range interactions on stability properties of dbs is expected to be negligible . the discrete breather acts as a parametic driver for small perturbations @xmath51 with the period @xmath61 being the half - period of the db solution [ i.e. the half period of a given solution @xmath13 in eq . ( [ ode ] ) ] . equations ( [ linearized ] ) define a map @xmath62 which maps the phase space of perturbations onto itself by integrating each point over the period @xmath63 . here we used the abbreviation @xmath64 . the map ( [ map ] ) is characterized by a symplectic floquet matrix @xmath65 , whose complex eigenvalues @xmath66 and eigenvectors @xmath67 provide information about the stability of the db @xcite . here we note that if all eigenvalues @xmath66 are by modulus one , then the db is linearly ( marginally ) _ stable_. otherwise perturbations exist which will grow in time ( typically exponentially ) and correspond to a linearly _ unstable _ db . upon changing a control parameter ( e.g. the coupling constant @xmath21 ) stable dbs can become unstable ( and vice versa ) . such a change of stability is appearing because two ( or more ) floquet eigenvalues collide on the unit circle and depart from it @xcite . with change of the coupling constant @xmath21 for different types of single - site dbs : ( a ) s - type db with the frequency @xmath68 . inset shows the floquet spectrum at @xmath69 , the unit circle in the complex plane is indicated to guide the eye ; ( b ) h - type db with the frequency @xmath70 . inset shows evolution of absolute values @xmath71 with change of @xmath21 . a pair of eigenvalues corresponding to the unstable depinning mode ( see the main body text for the details ) of the two - site db is indicated with gray color . , title="fig:",scaledwidth=45.0% ] with change of the coupling constant @xmath21 for different types of single - site dbs : ( a ) s - type db with the frequency @xmath68 . inset shows the floquet spectrum at @xmath69 , the unit circle in the complex plane is indicated to guide the eye ; ( b ) h - type db with the frequency @xmath70 . inset shows evolution of absolute values @xmath71 with change of @xmath21 . a pair of eigenvalues corresponding to the unstable depinning mode ( see the main body text for the details ) of the two - site db is indicated with gray color . , title="fig:",scaledwidth=45.0% ] in fig . [ fig : stab](a ) the typical floquet spectrum is shown for a _ s - type _ single - site db . all the eigenvalues @xmath66 can be divided into three sub - categories : one pair of eigenvalues is always situated at @xmath72 [ denoted by squares in the inset in fig . [ fig : stab](a ) ] ; it corresponds to perturbations along the db periodic orbit ( phase mode ) and along the corresponding family of db solutions @xcite . the period of these perturbations coincides with the db period . in addition , there are quasi - degenerated bands of eigenvalues at @xmath73 [ filled circles in the inset in fig . [ fig : stab](a ) ] , which correspond to perturbations in breather tails . such perturbations have characteristic frequency @xmath74 defined by the choice of the linear constant in the on - site potential ( [ onsite ] ) , they would correspond to linear phonons if linear coupling between sites were introduced . finally , there is a finite number of eigenvalue pairs bifurcating from the quasi - degenerated bands , which correspond to perturbations of the db core sites [ crosses in the inset in fig . [ fig : stab](a ) ] . the number of such isolated pairs is proportional to the characteristic db core size , it grows as the coupling constant @xmath21 increases , see fig . [ fig : stab](a ) . while increasing the coupling constant @xmath21 , these isolated pairs move on the unit circle and new pairs bifurcate from the quasi - degenerated band , but they do not collide with each other and the non - staggered type of db remains linearly stable up to the continuum limit @xmath33 . notably , the floquet spectrum of the s - type two - site db is qualitatively the same as the one of the single - site db for any @xmath21 . the only principal difference is that it has two degenerated pairs of eigenvalues at @xmath75 , since in the uncoupled limit @xmath20 the corresponding solution has two sites excited with equal amplitude . usually such degeneracy of eigenvalues is lifted for any nonzero @xmath21 , and one pair of eigenvalues is `` pushed out '' from @xmath75 either along the real axis or along the unit circle . however , in the case of purely nonlinear interactions between sites the _ symmetric _ two - site db `` cuts '' the effective linear chain ( [ linearized ] ) into two non - interacting halves . therefore , the additional degeneracy of eigenvalues corresponding to symmetric and anti - symmetric perturbations with respect to the db center remains for any @xmath21 . in contrast to the s - type dbs , the h - type single - site and two - site dbs are linearly stable only within certain windows of the coupling constant @xmath21 values , see fig . [ fig : stab](b ) . close to the critical value of the coupling constant @xmath76 , below which the h - type dbs do not exist , both single- and two - site h - type dbs experience strong instabilities connected with tangent bifurcations of these solutions with other ones , having more complicated spatial structure . in addition , there is another instability of a finite strength , appearing in certain windows of the parameter @xmath21 , see inset in fig . [ fig : stab](b ) . in general , apart from several small intervals in @xmath21 , for any given value of @xmath21 only one of these two db configurations is stable . the corresponding unstable perturbation the `` depinning '' mode `` tilts '' the single - site ( two - site ) db towards the half - site shifted stable two - site ( single - site ) one . changing the coupling constant , the stable configuration varies from the two - site to single - site db and back [ at @xmath77 and @xmath78 , see fig . [ fig : stab](b ) ] , so that the _ exchange of stability _ process @xcite is observed . this exchange of stability process can be connected to an exchange of the dominant roles between inter - site and on - site nonlinearities . indeed , typically for models with purely inter - site nonlinearities ( fermi - pasta - ulam lattices ) the basic stable configuration is the two - site db @xcite . in contrast , for models with weak coupling between sites and a non - linear onsite potential in the form ( [ onsite ] ) ( klein - gordon lattices ) the single - site db configuration is stable @xcite . of principal interest is an influence of the unstable depinning mode on the dynamical behavior of staggered - core dbs . a small perturbation along this mode is generally known to result in depinning of the unstable db from its initial position . depending on the relative hamiltonian energy ( [ hamilt ] ) of the perturbation ( @xmath79 ) and on the strength of instability of the depinning mode , the resulting behavior might vary from quasiperiodic - like oscillations between two neighboring unstable positions ( in a well of the corresponding `` peierls - nabarro potential '' ) to quasi - regular or even chaotic - like motion along the chain ( see e.g. @xcite ) . generally , the depinned db resonates with linear phonons through its excited internal modes and starts to radiate energy . therefore , eventually it will be trapped again at some stable position or even disappear completely transferring totally its energy to excited delocalized phonons . however , in the case of purely nonlinear dispersion there are no linear phonons in the system , and all possible linear resonances are suppressed . as a result , one can observe almost perfect quasi - periodic oscillations of perturbed unstable dbs between two neighboring stable positions , see fig . [ fig : oscil](a ) . similar quasi - periodic behavior is observed in dynamics of a stable db with perturbation along one of its internal modes , see fig . [ fig : oscil](b , c ) . in the latter case no detectable radiation ( within the used double precision ) of energy from the db core was observed during dynamical simulation over @xmath80 breather periods . in this respect , a one - dimensional hamiltonian lattice with purely anharmonic interactions between sites ( [ hamilt ] ) might be an interesting `` toy model '' to study more complicated _ exact _ db solutions like quasi - periodic and moving discrete breathers . let us now fix the value of @xmath69 and study the influence of long - range interactions on the spatial profile of a db . in fig . [ fig1 g ] the profiles of the h - type single - site dbs are shown for various values of the decay constant @xmath9 . they were obtained by solving eqs . ( [ algebra ] ) numerically with use of the standard newton scheme @xcite for the chain of @xmath38 oscillators ( @xmath81 ) with periodic boundary conditions . similarly to the case of nonlinear short - range interaction , the computed dbs have a compact - like structure with mainly three central sites oscillating ( see fig . [ fig : profs ] ) . the other oscillators are almost at rest and can be considered as breather tails . the presence of long - range interactions breaks the uniform super - exponential law of the spatial tail decay known for the case of pure short - range nonlinear interactions @xcite , introducing several cross - over lengths . ) with @xmath48 and @xmath69 for various exponents of the long - range interaction : @xmath82 ( squares ) , @xmath83 ( stars ) , @xmath84 ( crosses ) and @xmath85 ( triangles ) . , scaledwidth=45.0% ] for a few central sites , lying within the breather core , the forces due to the long - range interactions are negligible as compared to those due to the nearest - neighbor interactions . thus , the central part of a breather is practically not affected by the presence of long - range interactions . however , at some distance @xmath86 from the db center interactions with the nearest neighbors ( having small enough amplitudes ) become of the same order as the long - range interactions with the db core ( central three sites having the highest amplitudes ) . this distance is the first cross - over length , where the long - range interactions come into play . it can be roughly estimated by an assumption , that at the distance @xmath86 interactions with the breather core sites are exactly compensated by interactions with the nearest neighbors . thus , keeping only the leading order terms in the sum in the r.h.s . of ( [ algebra ] ) one obtains : @xmath87 since for @xmath88 the relations between the amplitudes @xmath12 are practically the same as in the case of pure nearest - neighbor interactions , i.e. they follow the super - exponential law @xmath89 , one can obtain from ( [ l1_estimate ] ) : latexmath:[\[\label{l1_est2 } limit of extremely large values of @xmath9 the distance @xmath86 will be also large , and satisfy @xmath91 thus the first cross - over length @xmath86 grows approximately logarithmically with @xmath9 . the numerical results in fig . [ fig1 g ] yield @xmath92 , @xmath93 , @xmath94 and @xmath95 . they compare very well with the corresponding solutions of ( [ l1_est2 ] ) : 2.71 , 3.46 , 4.44 , 5.17 . therefore , even extremely fast ( but still algebraically ) decaying in space long - range interactions essentially destroy the concept of compact - like breathers , since only amplitudes of a few sites in the tails obey the super - exponential law of decay , while the rest of the tail amplitudes decay much slower in space ( following a power law , as will be shown below ) . ) with @xmath48 for various exponents of the long - range interaction : ( a ) @xmath85 ; ( b ) @xmath83 . circles : numerical results . solid lines : tail asymptotes ( [ powerlaw ] ) . dashed lines : location of @xmath86 . the inset in ( b ) indicates the change of amplitudes sign around the cross - over point @xmath96.,title="fig:",scaledwidth=40.0% ] ) with @xmath48 for various exponents of the long - range interaction : ( a ) @xmath85 ; ( b ) @xmath83 . circles : numerical results . solid lines : tail asymptotes ( [ powerlaw ] ) . dashed lines : location of @xmath86 . the inset in ( b ) indicates the change of amplitudes sign around the cross - over point @xmath96.,title="fig:",scaledwidth=40.0% ] at large distances from the breather center @xmath97 ( due to the single - site db symmetry around @xmath23 we consider here only non - negative values of @xmath4 ) the impact of short- and long - range interactions is exchanged : now the most powerful contribution comes from the interaction with the breather core , while nearest neighbors , due to their small amplitudes , practically do not affect the dynamics of a tail site . thus , for large @xmath4 one can derive from ( [ algebra ] ) the following asymptote : @xmath98,\ ] ] which in fact gives a rather good approximation for all tail sites starting from the first cross - over point @xmath99 ( see solid lines in fig . [ fig2 g ] ) . note that only amplitudes of the two db core sites and the sign of separation constant @xmath16 are needed to obtain this asymptote for tail amplitude distribution . in this respect we found the simple three - site model , discussed in sec . [ sec : three ] , to be very fruitful : it gives full information not only about the db core sites , but about tail characteristics as well . note that the specific structure of a staggered db core with a central site @xmath23 and two neighboring sites @xmath43 having amplitudes @xmath12 of opposite signs stipulates several other cross - over lengths connected to changes of the sign of the r.h.s . in ( [ powerlaw ] ) which manifest as singularities in the logarithmic plots in fig.[fig2 g ] . the most pronounced cross - over at @xmath100 is associated with the change from a single power law @xmath101 to a more complex one ( [ powerlaw ] ) , see fig . [ fig2 g ] . indeed , in the case @xmath102 the expression ( [ powerlaw ] ) can be re - written as : @xmath103 ( @xmath104 for a staggered core db and @xmath105 for a non - staggered db ) . thus , in leading order , at large enough distance from the db center @xmath106 the tail amplitudes follow the same power law @xmath101 as the decay of long - range interactions . since @xmath96 is defined by the vanishing of the bracket on the right hand side of eq . ( [ powerlaw_as ] ) we obtain in leading order @xmath107 the corresponding values of @xmath96 for @xmath108 and @xmath109 with ( [ l_2 ] ) are @xmath110 . they compare reasonably well with the numerically observed ones @xmath111 , @xmath112 , @xmath113 and @xmath114 . in - between the two characteristic length scales @xmath115 the tail amplitudes decay following a more complicated power law ( [ powerlaw ] ) . to conclude , we revealed the influence of long - range nonlinear interactions on the spatial profile and properties of compact - like discrete breathers in a model of coupled oscillators with pure nonlinear dispersion . as we demonstrate , it is the intriguing property of the model under consideration , that it supports two classes of discrete breathers with _ staggered _ and _ non - staggered _ spatial profiles of a db core having different dynamical properties . the dynamics of a non - staggered db is essentially governed by the _ soft _ nonlinear on - site potential , while dbs with the staggered core have the opposite , _ hard _ , type of nonlinear dynamical behavior caused mainly by the presence of nonlinear interactions in the chain . apart from different dynamical properties , the influence of long - range interactions on spatial profiles of these two types of dbs is also different . with the algebraic spatial decay long - range interactions introduce a new length scale which becomes essential at large enough distances from a db core . we show , that the effect of long- and short - range terms competition results in the appearance of a characteristic cross - over length @xmath86 in both types of dbs , at which the spatial tail decay drastically changes from the super - exponential law to the algebraic one . for large powers @xmath9 of the long - range interactions spatial decay the cross - over length @xmath86 scales logarithmically with @xmath9 . the tail asymptote ( [ powerlaw ] ) demonstrates complex power law spatial decay , which follows essentially the same algebraic decay as the long - range interactions in the system at large enough distances from the db core . while for non - staggered dbs the influence of long - range interactions manifests through the only characteristic cross - over length @xmath86 , the spatial profile of dbs with the staggered core possess several other cross - over lengths associated with sign changes of the asymptote ( [ powerlaw ] ) . thus , the spatial pattern of oscillations in a db with the staggered core becomes rather complicated in the presence of long - range interactions : its core sites perform anti - phase oscillations , while its tails are splitted in several domains of in - phase oscillations . finally , we would like to mention that the discussed case of purely nonlinear coupled oscillators represents a simple model to reveal properties of nonlinear excitations in a system without linear phonons . as we demonstrated in this paper , `` switching off '' the phonons leads not only to the change in characteristic rate of spatial localization of energy , but to appearance of several other intriguing _ dynamical _ properties of nonlinear localized excitations . especially it eliminates the possible source of linear resonances which otherwise would destroy quasiperiodic breathers and possibly also moving breathers . the discussed models thus allow to obtain a better understanding of the general problem of existence / non - existence of quasi - periodic and moving discrete breather solutions . s. aubry , physica d * 103 * , 201 ( 1997 ) ; s. flach , c.r . willis , phys . rep . * 295 * , 181 ( 1998 ) ; _ energy localisation and transfer _ , edited by t. dauxois , a. litvak - hinenzon , r. mackay and a. spanoudaki ( world scientific , singapore , 2004 ) ; d. k. campbell , s. flach and yu . s. kivshar , physics today * 57(1 ) , 43 ( 2004 ) . gaididei , s.f . mingaleev , p.l . christiansen , and k. . rasmussen , phys . e 55 * , 6141 ( 1997 ) ; m. johansson , yu.b . gaididei , p.l . christiansen , and k. . rasmussen , phys . e * 57 * , 4739 ( 1998 ) ; p.l . christiansen , yu.b . gaididei , f. mertens , and s.f . mingaleev , eur . j. b * 19 * , 545 ( 2001 ) . t. cretegny , ph . d. thesis , cole normale suprieure de lyon , france 1998 ; m. ster , m. johansson , and a. eriksson , phys . rev . e * 67 * , 056606 ( 2003 ) ; a.v . gorbach and m. johansson , phys . e * 67 * , 066608 ( 2003 ) .	discrete breathers with purely anharmonic short - range interaction potentials localize _ super - exponentially _ becoming compact - like . we analyze their spatial localization properties and their dynamical stability . several branches of solutions are identified . one of them connects to the well - known page and sievers - takeno lattice modes , another one connects with the compacton solutions of rosenau . the absence of linear dispersion allows for extremely long - lived time - quasiperiodic localized excitations . adding long - range anharmonic interactions leads to an extreme case of competition between length scales defining the spatial breather localization . we show that short- and long - range interaction terms competition results in the appearance of several characteristic cross - over lengths and essentially breaks the concept of _ compactness _ of the corresponding discrete breathers .
the problem of finding variable configurations that minimize the energy of a system with competitive interactions has been and still is a central one in the study of complex systems , like spin glasses in physics , protein folding and regulatory networks in biology , and optimization problems in computer science ( see _ e.g. , _ @xcite ) . among the tools for numerical investigations of complex systems at low temperatures the simulated annealing ( sa ) algorithm @xcite and its variants have played a major role . such stochastic processes satisfy detailed balance and their behavior can be compared with static and dynamical mean - field calculations . however , in problems in which the interest is focused on zero temperature ground states and where the proliferation of metastable states causes an exponential slowdown in the equilibration rate , the applicability of sa - like algorithms is limited to relatively small system sizes . in computer science the field of combinatorial optimization @xcite deals precisely with the general issue of classifying the computational difficulty ( `` hardness '' ) of minimization problems and of designing search algorithms . similarly to statistical physics models , a generic combinatorial optimization problem is composed of many discrete variables_e.g . , _ boolean variables , finite sets of colors or ising spins which interact through constraints typically involving a small number of variables , that in turn sum up to give the global cost - energy function . when the problem instances are extracted at random from nontrivial ensembles ( that is ensembles which contains many instances that are hard to solve ) , computer science meets physics in a very direct way : many of the models considered to be of basic interest for computer science are nothing but spin glasses defined over finite connectivity random graphs , the well studied diluted spin glasses @xcite . their associated energy function counts the number of violated constraints in the original combinatorial problem ( with ground states corresponding to optimal solutions ) . understanding the onset of hardness of such systems is at the same time central to computer science and to @xmath0 statistical physics with surprisingly concrete engineering applications . for instance , among the most effective error correcting codes and data compression methods are the low density parity check algorithms @xcite , which indeed implement an energy minimization of a spin glass energy defined over a sparse random graph . in such problems , the choice of the graph ensemble is a part of the designing techniques , a fact that makes spin glass theory directly applicable . the above example is however far from representing the general scenario for combinatorial problems : in many situations the probabilistic set up is not defined and , consequently , the notion of typical - case analysis does not play any obvious role . the study of the connection ( if any ) between worst - case and typical - case complexity is indeed an open one and very few general results are known @xcite . still , a precise understanding of non - trivial random problem instances promises to be important under many aspects . new algorithmic results as well as many mathematical issues have been put forward by the statistical physics studies , with examples ranging from phase transitions @xcite and out - of - equilibrium analysis of randomized algorithms @xcite to new classes of message - passing algorithms @xcite . the physical scenario for the diluted spin glasses version of hard combinatorial problems predicts a trapping in metastable states for exponentially long times of local search dynamic process satisfying detailed balance . depending on the models and on the details of the process_e.g . , _ cooling rate for sa the long time dynamics is dominated by different types of metastable states at different temperatures @xcite . a common feature is that at zero temperature and for simulation times which are sub - exponential in the size of the problem there exists an extensive gap in energy which separates the blocking states from true ground states . such behavior can be tested on concrete random instances which therefore constitute a computational benchmark for more general algorithms . of particular interest for computer science are randomized search processes which do not properly satisfy detailed balance and that are known ( numerically ) to be more efficient than sa - like algorithms in the search for ground states @xcite . whether the physical blocking scenario applies also to these artificial processes , which are not necessarily characterized by a proper boltzmann distribution at long times , is a difficult open problem . the available numerical results and some approximate analytical calculations @xcite seem to support the existence of a thermodynamical gap , a fact which is of up - most importance for optimization . for this reason ( and independently from physics ) , during the last decade the problem of finding minimal energy configurations of random combinatorial problems similar to diluted spin - glasses_e.g . , _ random k - satisfiability ( k - sat ) or graph coloring has become a very popular algorithmic benchmark in computer science @xcite . in the last few years there has been a great progress in the study of spin glasses over random graphs which has shed new light on mean - field theory and has produced new algorithmic tools for the study of low energy states in large single problem instances . quite surprisingly , problems which were considered to be algorithmically hard for local search algorithms , like for instance random k - sat close to a phase boundary , turned out to be efficiently solved by the survey propagation ( sp ) algorithm arising from the replica symmetry broken ( rsb ) cavity approach to diluted spin glasses . such type of results calls for a rigorous theory of the functioning of sp ( which is a non local process ) and bring new mathematical challenges of potential practical impact . scope of this paper is to display a set of new numerical and algorithmic results which complete previously published results on the sp algorithm . we shall deal only with the random k - sat problem even though we expect the algorithmic outcomes to be applicable to other similar problems like , for instance , the random graph coloring . the paper is organized as follows . in sections [ ksat ] , [ sp ] we briefly review the known results on random k - sat together with the sp equations over single instances at finite pseudo - temperature . we discuss as well in [ sp - y ] how the sp algorithm can be modified in order to study the region of parameters with finite ground state energy ( unsat phase ) , where not all constraints of the underlying random k - sat problem can be satisfied simultaneously . in sec . [ results ] we discuss then the performance of sp as an optimization device . at variance with the sat phase in which many clusters of zero energy configurations coexist and where sp works efficiently without need of correcting variable assignments , in the unsat phase an efficient implementation of sp requires the introduction of at least a very simple form of backtracking procedure ( similar to the one proposed in @xcite ) . we show that a linear time backtrack is enough to reach energies compatible with those predicted by the analytic calculations in the infinite size limit in the relevant region of parameters . we give moreover numerical evidence for the existence of threshold states for one of the most efficient randomized local search algorithms for solving random k - sat , namely walksat @xcite . we display a blocking mechanism at an energy level which is definitely above the lower bound for the dynamical threshold states predicted by the stability analysis of the 1-rsb cavity equations . finally , for the deep unsat phase , we report on numerical data on convergence times for both walksat and sa which are in agreement with the predicted existence of full rsb ( f - rsb ) phases . conclusions and perspectives are briefly discussed in sec . [ conclusions ] . k - sat is a np - complete problem @xcite ( for @xmath1 ) which lies at the root of combinatorial optimization . it is very easy to state : given @xmath2 boolean variables and @xmath3 constraints taking the form of clauses , _ k - sat consists in asking whether it exists an assignment of the variables that satisfies all constraints_. each clause contains exactly @xmath4 variables , either directed or negated , and its truth value is given by the or function . since the same variable may appear directed or negated in different clauses , competitive interactions among clauses may set in . as mentioned in the introduction , in the last decade there has been a lot of interest on the random version of k - sat : for each clause the variables are chosen uniformly at random ( with no repetitions ) and negated with probability @xmath5 . in the large @xmath2 limit , random k - sat displays a very interesting threshold phenomenon . taking as control parameter the ratio of number of clauses to number of variables , @xmath6 , there exists a phase transition at a finite value @xmath7 of this ratio . for @xmath8 the generic problem is satisfiable ( sat ) , for @xmath9 the generic problem is not satisfiable ( unsat ) . this phase transition has been seen numerically @xcite and it is of special interest since extensive experiments @xcite have shown that the instances which are algorithmically hard to solve are exactly those where @xmath10 is close to @xmath11 . therefore , the study of the sat / unsat phase transition is considered of crucial relevance for understanding the onset of computational complexity in typical instances @xcite . a lot of work has been focused on the study of both the decision problem ( _ i.e. , _ determining with a yes / no answer whether a satisfying assignment exists ) , and the optimization version in which one is interested in minimizing the number of violated clauses when the problem is unsat ( random max - k - sat problem ) . on the analytical side , there exists a proof that the threshold phenomenon exists at large @xmath2 @xcite , although the fact that the corresponding @xmath11 has a limit when @xmath12 has not yet been established rigorously . upper bounds @xmath13 on @xmath11 have been found using first moment methods @xcite and variational interpolation methods @xcite , and lower bounds @xmath14 have been found using either explicit analysis of some algorithms @xcite , or some second moment methods @xcite . for random max - k - sat theoretical bounds are also known @xcite , as well as rigorous results on the running times of random walk and approximation algorithms @xcite . recently , the cavity method of statistical physics has been applied to k - sat @xcite and the thresholds have been computed with high accuracy . a lot of work is going on in order to provide a rigorous foundation to the cavity results and we refer to @xcite for a more complete discussion of these aspects . in what follows we shall concentrate on the @xmath15 case and we will be interested in analyzing the behavior of different algorithms in the region of parameter in which the random formulas are expected to be hard to solve or to minimize . the energy function which is used in the zero temperature statistical mechanics studies is taken proportional to the number of violated clauses in a given problem so that a zero energy ground state corresponds to a satisfying assignment . the energy of a single clause is positive ( equals 2 for technical reasons ) if the clause is violated and zero if it is satisfied . the overall energy is obtained by summing over clauses and reads @xmath16 where @xmath17 is the @xmath18-th binary ( spin ) variable appearing in clause @xmath19 and the coupling @xmath20 takes the value 1 ( resp . -1 ) if the corresponding variable appears not negated ( resp . negated ) in clause @xmath19 . for instance the clause @xmath21 has an energy @xmath22 where the boolean variables @xmath23 are connected to the spin variables by the transformation @xmath24 . the phase diagram of the random 3-sat problem as arising from the statistical physics studies can be very briefly summarized as follows . for @xmath25 , the @xmath0 phase is at zero energy ( the problem is sat ) . the entropy density is finite and the phase is replica symmetric ( rs ) and unfrozen . roughly speaking , this means that there exists one giant cluster of nearby solutions and that the effective fields vanish linearly with the temperature . for @xmath26 , there is a full rsb phase . the solution space breaks in clusters and the order parameter becomes a nested probability measure in the space of probability distribution describing cluster to cluster fluctuations . the phase is still sat and unfrozen @xcite . at @xmath27 there is a discontinuous transition toward a clustered frozen phase @xcite . up to @xmath28 the phase is f - rsb while above the 1-rsb solution becomes stable@xcite . the _ complexity _ , that is the normalized logarithm of the number of clusters , is finite in this region . at finite energy there exist even more metastable states which act as dynamical traps . the 1-rsb metastable states become unstable at some energy density @xmath29 which constitutes a lower bound to the true dynamical _ threshold energy _ ( see sec . [ sp ] for more details ) . at @xmath30 the ground state energy becomes positive and therefore the typical random 3-sat problem becomes unsat . at the same point the complexity vanishes . the phase remains 1-rsb up to @xmath31 where an instability toward a zero complexity full rsb phase appears . in the region @xmath32 , the 1-rsb ansatz for the ground state is stable against higher orders of rsb , but the 1-rsb predictions become unstable for energies larger than the _ gardner energy_. the instability line intersects with the 1-rsb ground state extimation at the two extremes of the interval , inside which it provides a lower bound to the true threshold energy ( see ref . @xcite for a comprehensive discussion ) . further ( preliminary ) f - rsb corrections suggest that the true threshold states have energies very close to the lower bound and hence the interval @xmath33 $ ] should be taken as the region where to take really hard benchmarks for algorithm testing . as displayed in fig . [ small_gap_fig ] , the actual value of the energy gap is very small close to the end points of @xmath34 . in order to avoid systematic finite size errors , numerical simulations should be done close to the sat / unsat point , _ i.e. , _ far from the end point of @xmath34 . consistently with the fact that finite size fluctuations are relatively big ( @xmath35 , even close to @xmath11 problem sizes of the order at least of @xmath36 are necessary in order to observe a matching with the analytic predictions . the 1-rsb cavity equations which have been used to study the typical phase diagram of random k - sat become the sp equations once reformulated to run over single problem instance @xcite . this is done by avoiding the averaging process with respect to the underlying random graphs . thanks to the self - averaging property of the random k - sat free energy @xcite , the sp equations can be used both to re - derive the phase diagram of the problem and , more important , to access detailed information of algorithmic relevance about a given problem instance . in particular , the sp equations provide information about the statistical behavior of the single variables in the stable and metastable states of given energy density . the 1-rsb cavity equations are iterative equations ( averaged over the disorder ) for the probability distribution functions ( pdf ) of effective fields that describe their cluster - to - cluster fluctuations . the order parameter is a probability measure in the space of pdf s ; it tells the probability that a randomly chosen variable has a certain associated pdf in states at a given energy density . in sp and more in general in the cavity approach , one assumes to know pdf s of the fields of all variables in the temporary absence of one of them . then one writes the induced pdf of the local field acting on this `` cavity '' variable in absence of some other variable interacting with it ( _ i.e. , _ the so called bethe lattice approximation for the problem ) . these relations define a closed set of equations for the pdf s that can be solved iteratively . the equations are exact if the cavity variables acting as inputs are uncorrelated , _ e.g. , _ over trees , or are conjectured to be an asymptotically exact approximation over locally tree - like structures @xcite where the typical distance between randomly chosen variables diverges in the large @xmath2 limit ( as @xmath37 for diluted random graphs ) . the full list of the cavity fields over the entire underlying graph , in the sp implementation , constitutes the order parameter . from the cavity fields one may determine the total field acting on each variable in all metastable states of given energy density and this information can be used for algorithmic purposes . a clear formalism for the single sample analysis is given by the factor graph representation @xcite of k - sat : variables are represented by @xmath2 circular `` variable nodes '' labeled with letters @xmath38 whereas the k - body interactions are represented by @xmath3 square `` function nodes '' ( carrying the clause energies ) labeled by @xmath39 ( see fig . [ factorgraph ] ) for random 3-sat , function nodes have connectivity @xmath40 , variable nodes have a poisson connectivity of average @xmath41 and the overall graph is bipartite . the total energy is nothing but the sum of energies of all function nodes as given by eq . ( [ energy ] ) . adopting the message - passing notation and strictly following @xcite , we call @xmath42-messages the contribution to the cavity fields coming from the different connected branches of the graph . in sp the messages along the links of the factor graph have a functional nature carrying information about distributions of @xmath42-messages over the states at a given value of the energy , fixed by a lagrange multiplier @xmath43 : we call these distributions of messages @xmath42-surveys . the sp equations can be written at any `` temperature '' ( the inverse of the lagrange multiplier @xmath43 is actually a pseudo - temperature , see @xcite ) . however they acquire a particularly simple form in the limit @xmath44 , which is the limit of interest for optimization purposes , at least in the sat region . in k - sat , the @xmath42-surveys are parameterized by two real numbers and sp can be implemented very efficiently . each edge @xmath45 , from a function node @xmath19 to a variable node @xmath18 , carries a @xmath42-survey @xmath46 . from these @xmath42-surveys one can compute the cavity fields @xmath47 for every @xmath48 , which in turn determine new output @xmath42-surveys ( see fig . [ popdynfig ] ) . very schematically , the sp equations can be implemented as follows . let @xmath49 be the set of function nodes connected to the variable @xmath18 , @xmath50 the set of variables connected to the function node @xmath19 ; let us denote by @xmath51 and @xmath52 the same sets deprived respectively of the clause @xmath19 and of the variable @xmath18 . given then a random initialization of all the @xmath42-surveys @xmath53 , the function nodes are selected sequentially at random and the @xmath42-surveys are updated according to a complete set of coupled functional equations ( see fig . [ popdynfig ] for the notation ) : -messages . the @xmath42-survey for the @xmath42-message @xmath54 depends on the pdf s of the cavity fields @xmath55 and @xmath56 . these are on the other side dependent on the @xmath42-surveys for the @xmath42-messages incoming to the variables @xmath57 and @xmath58 . ] @xmath59 where the @xmath60 s are normalization constants , the function @xmath61 is : @xmath62 and the integration measures are given by : @xmath63 @xmath64 parameterizing the @xmath42-surveys as @xmath65 where @xmath66 , the above set of equations ( [ p],[q2 ] ) defines a non - linear map over the @xmath67 s . once a fixed point is reached , from the list of the @xmath42-surveys one may compute the normalized pdf of the _ local field _ acting on each variable : @xmath68 it should be remarked that @xmath69 is in general different from the family of _ cavity fields _ pdf s @xmath70 computed by mean of ( [ p ] ) . from the knowledge of the cavity and local fields pdf s , one derives the ( bethe ) free energy at the level of 1-rsb : @xmath71 where @xmath72 is the connectivity of the variable @xmath18 and : @xmath73 \sigma_i + \sum_{b\in v(i)\setminus a } \vert u_{b\to i } \vert \right ) \right ] \right\ } , \nonumber \\ \phi^v_i(y)&=&-\frac{1}{y } \ln \left \ { \int \mathcal{d}\widehat{q}_i\,\ , \exp\left[y ( \vert \sum_{a \in v(i ) } u_{a\to i } \vert- \sum_{a \in v(i)}\vert u_{a\to i } \vert ) \right ] \right \}=-\frac{1}{y } \ln ( c_i ) . \label{freeonesamp2}\end{aligned}\ ] ] here , @xmath74 is the energy contribution of the function node @xmath19 . the maximum value of the free - energy functional provides a lower bound estimation of the ground state energy of the hamiltonian ( [ hamiltonian ] ) defined on the sample . in the sat region the free - energy functional @xmath75 is always non positive and it is increasing in the limit @xmath76 ; in the unsat region , on the contrary , it exhibits a positive maximum for @xmath77 ( see @xcite ) . from the free - energy density of a given instance , it is straightforward to compute numerically its complexity @xmath78 and its energy density @xmath79 . we remind that the complexity is linked to the number of pure states ( _ i.e. , _ clusters of configurations ) of energy @xmath80 , by the defining relation @xmath81 . the energy level represented by the largest number of configurations , @xmath82 , is given by : @xmath83 further rsb corrections may be needed to locate the precise value of @xmath82 , which is in any case lower bounded the largest energy of 1-rsb stable states , the so called _ gardner energy _ @xmath84 . it is expected that local search strategies get trapped at energies close , but not necessarily equal , to the threshold energy ( see refs . @xcite for a throrough discussion on the role of the iso - complexity states @xcite ) . more elaborated strategies not properly satisfying detailed balance ( _ e.g. , _ walksat for the k - sat problem ) could in principle overcome this type of barriers ; however , the available numerical and analytical results suggest that also these more sophisticated randomized searches undergo an exponential slowdown , with different layers of states acting as dynamical traps , depending on the details of the heuristics . in the sat phase , where the @xmath85 limit is taken , the convolutions ( [ p ] ) filter out completely any clause - violating truth value assignment . this feature is extremely useful for satisfiable formulas , but it becomes undesired when our sample is presumably unsatisfiable . in the unsat region the sp equations require a finite value of the lagrange multiplier @xmath43 . the filtering action of the exponential re - weighting term in ( [ p ] ) is then weakened and the messages computed by the sp equations can vehicle information pointing to states with a non vanishing number of violated constraints . the sp equations simplify considerably in the @xmath86 limit and lead to extremely efficient algorithmic implementations , as discussed in great detail in @xcite . in the case of finite pseudo - temperature @xmath87 the same simplification can not take place because of the presence of a nontrivial re - weighting factor . still , a relatively fast recursive procedure can be written . let us consider a variable @xmath88 having @xmath89 neighboring function nodes and let us compute the cavity field pdf @xmath90 where @xmath91 . we start by randomly picking up one function node in @xmath92 , denoted as @xmath93 , and we calculate the following `` @xmath94-survey '' : @xmath95 the function @xmath96 would correspond to the true local field pdf of the variable @xmath88 in the case in which @xmath93 was the only neighboring clause ( as denoted by the upper index ) . the following steps of the recursive procedure consist in adding the contributions of all the other function nodes in @xmath92 , clause by clause ( fig . [ popdynrecfig ] ) : , a single clause @xmath93 in @xmath92 is picked up at random and the @xmath42-survey @xmath97 is used to compute equation ( [ pdrecinit ] ) ; ( b ) the contributions of all the other function nodes in @xmath98 are then added , clause by clause ; ( c ) the pdf computed recursively after @xmath99 iterations coincides with @xmath100 . ] @xmath101\nonumber\\ & + & \eta_{b_\gamma\to j}^{\/-}\,\widetilde{p}_{j\to a}^{(\gamma-1)}(h+1)\,\exp\left [ -2y\,\hat{\theta}(h)\right].\nonumber\end{aligned}\ ] ] here @xmath102 is an unnormalized pdf and @xmath103 is a step function equal to @xmath104 for @xmath105 and zero otherwise . the recursion ends after @xmath106 steps , when the influence of every clause in @xmath92 has been taken in account . the final cavity - field pdf @xmath107 can be found straightforwardly by computing the pdf @xmath108 for all values of the field @xmath109 and by normalizing it . as already pointed out in section [ sp ] , the knowledge of @xmath110 input cavity - field pdf s can be used to obtain a single output @xmath42-survey . let us compute for instance the @xmath42-survey @xmath111 ( see always fig . [ popdynfig ] for the notation ) . in order to do that , we need first the cavity field pdf s @xmath107 for every @xmath112 . the parameters @xmath113 are then updated according to the formulas : @xmath114 where we introduced the weight factors : @xmath115 it should be remarked that @xmath116 depends only on one single nontrivial @xmath117 ( from now simply referred to as @xmath118 ) . we could say that a single kind of message can be produced , telling the receiver literal to assume the truth value `` true '' ; this message is transmitted along the edge @xmath119 with a probability @xmath118 , corresponding to the probability that the only way of not violating the constraint @xmath19 is to set appropriately the truth value of @xmath18 . starting from a full collection of @xmath42-surveys at a given time , it is possible to realize a complete update of all the parameters @xmath120 by systematical application of the recursions ( [ pdrecinit ] ) , ( [ pdrec ] ) and of the relation ( [ equpdateeta ] ) ; from the new set of @xmath42-surveys , new cavity field pdf s can be computed and the procedure continues until when self - consistence of @xmath67 s is reached . this procedure can be efficiently implemented numerically and allows us to determine the fixed point of the population - dynamics equations ( [ p ] ) , ( [ q2 ] ) , for a general value of @xmath43 . in the usual sp - inspired decimation @xcite , the computation of the local field pdf s @xmath121 is used to decide a truth value assignment for the most biased variables . indeed , it is reasonable that a spin tends to align itself with the most probable direction of the local field . a ranking can be realized by finding all the probability weights @xmath122 and by sorting the variables according to the values of a bias function : @xmath123 the local field pdf s @xmath124 can be naturally calculated resorting to the iterations ( [ pdrecinit ] ) , ( [ pdrec ] ) : computation is done simply by sweeping over the whole set of neighboring function nodes @xmath125 , including also the contribution of the skipped edge @xmath126 . by fixing in the right direction the spin of the most biased variable , we actually reduce the original @xmath2 variable problem to a new one with @xmath127 variables . new @xmath42-surveys are then computed . doing that we have to take care of fixed variables : if @xmath18 is fixed , its cavity field pdf s must be of the form : @xmath128 regardless of the recursions ( [ pdrecinit ] ) , ( [ pdrec ] ) . the complete polarization reflects the knowledge of the truth value of the literals depending on the spin @xmath129 . the procedure of decimation continues until when a full truth assignment has been generated or until when convergence has been lost or a paramagnetic state has been reached ; in the latter cases the original formula is simplified according to the partial truth assignment already generated and the simplified formula is passed to a specialized heuristic . our choice of preference is the walksat algorithm @xcite , which is by far more efficient than sa in the hard region of the 3-sat problem , as we have checked exhaustively . very briefly , the strategy of walksat is the following one : at each time step the current assignment is changed by randomly alternating greedy moves ( where the variable which maximizes the number of satisfied clauses if fixed ) and random - walk steps ( in which a variable belonging to a randomly chosen unsatisfied clause is selected and flipped ) . walksat stops if either a satisfying assignment is found or if the maximum number of allowed spin flips ( the `` cutoff '' ) is reached ( see ref . @xcite for another recently analyzed and very efficient heuristics ) . when working at finite pseudo - temperature , we have to take in account the possibility that some non optimal fixing is done in presence of thermal `` noise '' . after several updates of the @xmath42-surveys some biases of fixed spins may become smaller than the value they had at the time when the corresponding spin was fixed . certain local fields can even revert their orientation . small or positive values of an index function like : @xmath130 can track the appearance of such dangerous fixed spins and this information can be used to implement some `` error removal '' procedure ; for instance , a simple strategy can be devised where both unfixing and fixing moves are performed at a fixed ratio @xmath131 ( see @xcite for another backtracking implementation ) . the actual sp with finite @xmath43 simplification procedure ( sp - y ) will depend not only on the backtracking fraction @xmath132 , but even more on the choice of the inverse pseudo - temperature @xmath43 . the simplest possibility is to keep it fixed during the simplification , but one may choose to dynamically update it , in order to stay as close as possible to the maximum @xmath133 of the free energy functional @xmath75 ( which corresponds to select the ground state in the 1-rsb framework , as we have seen in section [ sp ] ) . the equations ( [ freeonesamp1 ] ) , ( [ freeonesamp2 ] ) can be rewritten in the following form , suitable for numerical computation : @xmath134,\\ \label{phi_sp - y_var } \phi_{i}^v(y ) = -\frac{1}{y}&\!\!\!\ln&\!\!\!\left(c_i\right).\end{aligned}\ ] ] in fig . [ pseudofig ] we give a summary of the simplification procedure in a standard pseudo - code notation . the first release of the sp - y code can be downloaded from @xcite . as we have already discussed in section [ sp ] , it is expected that , in the thermodynamical limit , any local search algorithm gets trapped in the vicinity of exponentially numerous threshold states with energy @xmath82 and that any local heuristics is in general unable to find the optimal assignment in the thermodynamical limit . to verify this prediction , we conducted various experiments , both in the sat and in the unsat phase , focusing on the comparison between the walksat heuristics performance after and before different kinds of sp - y simplification . in most of the situations , we decided to analyze carefully single large - sized samples instead of a larger number of smaller problems : we verified in fact that the sample - to - sample fluctuations tend to be unrelevant for size of order @xmath135 and larger . the aim of the first set of experiments was to check the actual existence of the threshold effect . we ran walksat over different formulas in the hard - sat region , with fixed @xmath136 and sizes varying between @xmath137 and @xmath36 , reaching a maximum cutoff of @xmath138 spin flips . the obtained results are plotted in fig . [ fig01 ] ; the gardner energy is also reported for comparison with the data . even if for small - size samples the local search algorithm is able to find a sat assignment , for larger formulas ( @xmath139 ) walksat does not succeed in reaching the ground state , its relaxation profile suffers of critical slowdown , and saturates at some well defined level . this is actually expected , because the gardner energy becomes @xmath140 only for @xmath141 or larger , and for a smaller number of variables the threshold effect should be negligible when compared to finite size effects . we remind that walksat can not be considered as an equilibrium stochastic process and that it is not possible to infer that its saturation level coincides with the sample threshold energy ; we can anyway claim that walksat is unable to explore the full energy landscape of the problem , and that the enormous number of non optimal valleys is unavoidably hiding the true ground states . plateaus in the relaxation profiles of walksat have indeed been already discussed in @xcite and ascribed to metastable states acting as dynamical traps . for the @xmath142 formula a trapping effect becomes clearly visible in our experiments , but the saturation plateau is below the gardner lower bound . the finite - size fluctuations are still of the same order of the energy gap between the ground and the threshold states and the experimental conditions are distant from the thermodynamical limit . when the size is increased up to @xmath143 variables , the saturation level moves finally between the full rsb lower bound and the 1-rsb upper bound for @xmath82 . the efficiency of the sp - y simplification strategy against the glass threshold is discussed in fig . [ fig02 ] . we simplified a single randomly generated formula ( @xmath36 , @xmath144 ) at several fixed values of pseudo - temperature . the solid line shows for comparison the walksat results after a standard sp decimation ( _ i.e. , _ @xmath76 ) : the ground state , @xmath145 , is reached as expected , after a rather small number of spin flips . the same happens after sp - y simplifications performed at a large enough inverse pseudo - temperature ( @xmath146 ) ; one should remind indeed that in the sat region the optimal value for @xmath43 would be infinite , and that in that limit the sp - y recursions reduce to the sp equations . after simplification with smaller @xmath43 s , the walksat cooling curves reach again a saturation level , which is nevertheless below the gardner energy , unless @xmath43 is too small : the threshold states of the original formula have not been able to trap the local search , even if the ground state becomes inaccessible . as we have indeed already discussed , working at finite temperature increases the probability of violating a clause when doing a spin fixing , and this is particularly evident in the sat region where every assignment that does not satisfy some constraint should be filtered out . the procedure is intrinsically error prone , and it will allow in general to reach only `` good states '' , but not the true optimal solutions ( the smaller the parameter @xmath43 , the higher the saturation level will be ) . as we shall discuss in the next section , the use of backtracking partially cures the accumulation of errors at finite y : the saturation level can in fact be significantly lowered by keeping the same pseudo - temperature and introducing a small fraction of backtrack moves during the simplification . in fig . [ fig02 ] the data for @xmath147 shows the importance of backtracking . while the run of sp - y without backtracking has led to a plateau above gardner energy , with the introduction of backtrack moves we find energies well below the threshold . when entering the unsat region , the task of looking for the optimal state becomes harder . the expected presence of violated constraints in the optimal assignments really forces us to run the simplification at a finite pseudo - temperature . unfortunately , after many spin fixings , the recursions ( [ pdrecinit ] ) , ( [ pdrec ] ) stop to converge for some finite value of @xmath43 before the maximum of the free energy is reached , most likely because the sub - problem has entered a full rsb phase . at this point one should switch to a 2-rsb version of sp which we did not realize , yet . alternatively , one could try to run directly the final heuristic search ( hoping that the full rsb sub - system is not exponentially hard to optimize ) or more simply one may continue the decimation process by selecting the largest @xmath43 for which the computation converge . we decided to implement the latter choice until either convergence is lost independently from the value of @xmath43 or a paramagnetic state is reached . in our experiments we studied several 3-sat sample problems belonging to the 1-rsb stable unsat phase . we employed walksat as an example of standard well - performing heuristics . although walksat is not optimized for unsatisfiable problems , in the 1-rsb stable unsat region it performs still much better than any basic implementation of sa . we observed anyway that , even after @xmath138 spin flips , the walksat best assignments were still quite distant from the gardner energy , for various samples of different size and @xmath10 . in fig . [ fig03 ] we show the results relative to many different sp - y simplifications with various values of @xmath43 and @xmath132 for a single sample with @xmath36 and @xmath148 . the simplification produced always an improvement in the walksat performance , but , in absence of backtracking , we were unable to go below the gardner lower bound ( although we touched it in some cases : in fig . [ fig03 ] we show the data for a simplification at fixed @xmath149 ; a simplification with runtime optimization of @xmath43 reached the same level ) . the relative inefficiency of these first attempts of simplification was not due to the threshold effect alone , but also to an extreme sensitivity to the choice of @xmath43 , as pointed out by a second set of experiments making use of backtracking . we performed first an extensive analysis of the simultaneous optimization of @xmath43 and @xmath132 , using smaller samples in order to produce more experimental points . after some trials , the fraction @xmath150 appeared to be the optimal one , at least for our implementation , and in the small region under investigation of the k - sat phase diagram . the data in fig . [ fig04 ] refers to a formula with @xmath142 variables and @xmath151 . the dashed horizontal line shows the walksat best energy obtained on the original formula after @xmath152 spin flips . the walksat performance was seriously degraded when simplifying at too small values of @xmath43 , but the introduction of backtracking cured the problem , identifying and repairing most of the wrong assignments . the walksat efficiency became actually almost independent from the choice of pseudo - temperature , whereas in absence of error correction a time consuming parameter tuning was required for optimization . coming back to the analysis of the sample of fig . [ fig03 ] , the backtracking simplifications allowed us to access states definitely below the gardner lower bound . the combination of runtime @xmath43-optimization and of error correction was even more effective : after a rather small number of spin flips , walksat reached a saturation level strikingly closer to the ground state lower bound than to the gardner energy . a further valuable effect of introduction of the backtracking was the increased efficiency of the formula simplification itself : in the backtracking experiments , sp - y was able to determine a truth value for more than 80% of the variables before losing convergence , while without backtracking , the algorithm stopped on average after only 40% of fixings . all the samples analyzed in the previous sections were taken from the 1-rsb stable region of the 3-sat problem , where the equations ( [ p ] ) , ( [ q2 ] ) are considered to be exact . for @xmath153 , the phase becomes full rsb and sp loses convergence before the free energy @xmath75 reaches its maximum from the very first step of the decimation procedure . while a full rsb version of sp would most likely provide very good results , sp - y still can be used in a sub - optimal way by selecting the largest value of @xmath43 for which convergence is reached . numerical experiment show that indeed the performance of sp - y are in good agreement with the analytical expectations . however , it should be noticed that in this region the use of sp is not necessary . although the performance of walksat and sa can be improved by the sp simplification , the sa alone is already able of finding close - to - optimum assignments efficiently ( as expected for a full rsb scenario ) and behaves definitely better than walksat . in this paper , we have displayed the performance of sp as an optimization device and shown that configurations well below the threshold states can be found efficiently . similar results are expected to hold also for random satisfiable instances very close to the critical point for which the combined use of finite pseudo - temperature and backtracking could give access to the sat optima . it would be of some interest to analyze further improvements of the decimation strategies as well as to consider more structured factor graphs within a variational framework , in which some correlations can be put under control . a possible application of sp - y like algorithms can be found in information theory : lossy data compression based on low density parity check schemes leads to optimization problems which are indeed very similar to the one discussed in this paper . we thank a. braunstein , m. mzard , g. parisi and f. ricci - tersenghi for very fruitful discussions . this work has been supported in part by the european community s human potential programme under contract hprn - ct-2002 - 00319 , stipco . w. barthel , a. k. hartmann , and m. weigt , solving satisfiability problems by fluctuations : an approximate description of the dynamics of stochastic local search algorithms , cond - mat/0301271 , preprint ( 2003 ) o. dubois , y. boufkhad , and j. mandler , typical random 3-sat formulae and the satisfiability threshold , in _ proc . 11th acm - siam symp . on discrete algorithms _ , 124 ( san francisco , ca , 2000 ) ; a. kaporis , l. kirousis , and e. lalas , the probabilistic analysis of a greedy satisfiability algorithm , in _ proceedings of the 4th european symposium on algorithms _ ( esa 2002 ) , to appear in series : lecture notes in computer science , springer s. seitz , p. orponen : an efficient local search method for random 3-satisfiability , in _ proc . lics03 workshop on typical case complexity and phase transitions _ ( ottawa , canada , june 2003 ) ; electronic notes in discrete mathematics vol . 16 . ( elsevier , amsterdam , 2003 )	focusing on the optimization version of the random k - satisfiability problem , the max - k - sat problem , we study the performance of the finite energy version of the survey propagation ( sp ) algorithm . we show that a simple ( linear time ) backtrack decimation strategy is sufficient to reach configurations well below the lower bound for the dynamic threshold energy and very close to the analytic prediction for the optimal ground states . a comparative numerical study on one of the most efficient local search procedures is also given .
phenomena related with a biased electrode ( langmuir probe ) imbedded into a plasma have been studied beginning from the rise of plasma physics and are still in progress . the aim of the current investigation is the determination of the link between the measured current - voltage ( c - v ) electrode characteristics and the plasma parameters . for this purpose it is desirable to minimize the disturbances of the actual plasma parameters introduced by the probe . however , there is a number of processes in the surrounding plasma caused by the probe presence that affect significantly the probe c - v characteristics . the probe c - v characteristics can be multi - valued and demonstrate a hysteresis - like behavior . moreover , the current collected by the probe produces a set of instabilities ( sheath - plasma instability @xcite , ionization induced instabilities @xcite , _ etc _ ) . as a result , the self - oscillations of the collected current rise even when the probe potential is kept constant . there are several papers where the physical processes responsible for the excitation of the probe current oscillations are studied . however , the role of a mandatory element the reference electrode for the probe that closes the current circuit through the plasma , has to our knowledge not been adequately explored . in this paper the results of a detailed experimental investigation of the c - v characteristics of a positively charged probe are presented . various probes have been studied whose area @xmath0 is small enough compared with the ion - collected electrode area ( the reference electrode area ) @xmath1 : @xmath2 where @xmath3 and @xmath4 are the electron and ion masses , respectively . it will be shown that probes , whose areas meet the condition ( [ eq00 ] ) , can be separated into three groups : large , intermediate , and small probes . each group is characterized by its own regime of interaction with the plasma . affiliation with one group or another is determined by the relation between the c - v characteristics of the probe and the reference electrode which are connected in series via the plasma . this paper is organized as follows . in section ii , the experimental setup is described . the experimental results are presented in section iii . a simple qualitative theoretical model and comparison with the experimental results are presented in section iv . the paper main results are summarized and discussed in section v. the experiments were mainly performed in a stainless steel vacuum chamber , having an inner diameter of 30 cm and a height of 8 cm ( fig . [ 1]a ) . this chamber was equipped with two insulated tungsten filaments , one of them had 0.3 mm diameter and the other one 0.1 mm . both of them could be heated separately . the first one could be moved just when the chamber was open and the other one could be moved inside the chamber during experiments . the chamber was also equipped with a large surface probe , having an area @xmath5 , with a comparatively small ( @xmath6 ) movable single probe and with a special holder for replaceable platinum probes of various lengths ( 1 25 ) mm and diameters ( 0.05 5 ) mm . the first ( thicker ) filament was used as a hot cathode . it was heated when a voltage pulse @xmath7 of 0.2 s duration was applied per each 5 s. to obtain a hot - cathode discharge and a plasma we applied a dc voltage either between the cathode and the grounded vacuum chamber or between the cathode and the large surface probe . in the latter case the discharge circuit was floating . the plasma density @xmath8 and the electron temperature @xmath9 were derived from the c - v characteristics of the above - mentioned probes . to verify these measurements we could also use the movable probe as a resonance probe @xcite . to measure the plasma potential , the thin filament was used as a hot probe . to heat it , another dc source was used . to bias the small platinum probe we used either a dc power supply or a pulsed saw - tooth or rectangular voltage with pulse duration ( 150 200 ) @xmath10s ( see fig . just a positive bias was used . the pulsed regime was used either to obtain its c - v characteristics in one `` shot '' or to prevent probe overheating if the collected current was too high . a thin glass plate of 3 cm diameter ( not shown in fig . [ 1]a ) could be placed between the hot cathode and the small positively biased probe to prevent direct current of the emitted electrons to this probe . a strong permanent magnet could be placed in various positions in the vicinity of the small probe in such a way that a magnetic field of ( 100 600 ) gauss could be achieved there ( see fig . also a set of spirals having a diameter of ( 5 20 ) mm and a 25 mm length could be placed around this probe ( fig . [ these spirals could also be biased either positively or negatively with respect to the probe . some verifying measurements with just a hot cathode and a small probe could be carried out in a big vacuum chamber having a 66 cm diameter and a 100 cm length . in our experiments we used either ar at a pressure ( 0.5 5 ) mtorr or xe at a pressure ( 0.1 1 ) mtorr . typically we applied an anode voltage @xmath11 v in order to make sure that we work in a regime with the thermal limitation of the emitted current . this is reasonable in order to eliminate dependence of the plasma density @xmath8 on @xmath12 . indeed in the range @xmath13 v , variations of the emitted current @xmath14 did not exceed 10% . on the other hand it was certain that in measurements with a large positive bias @xmath15 on the small probe the value of @xmath12 could be reduced to zero ( the cathode directly connected to the chamber ) . in the investigated pressure range the plasma density @xmath8 was found to be directly proportional to the pressure and the emitted current @xmath14 . the waveform of the emitted current @xmath14 along with the ion saturation current @xmath16 , collected by the large surface probe , are shown in fig . the duration of the @xmath14 growth ( @xmath17 s ) was long enough to consider a steady state plasma at any moment . in our experiments a 100 ma emission current corresponded to @xmath18 , @xmath19 mtorr and @xmath20 ev with xe gas . for the @xmath8 measurements the discrepancy between the probe characteristics method and the resonance probe method did not exceed 15% . the fraction of fast electrons having energy @xmath21 never exceeded ( 1 2)% . they appeared as a step in the ion part of the probe characteristics @xcite . the emitted current @xmath14 as well as the probe current @xmath22 were measured with small current - view resistors . in order to study the influence of the ionization level there was a possibility to insert inside the vacuum chamber a single - core ferromagnetic inductively coupled ( fic ) plasma source , driven by a powerful pulsed rf oscillator @xcite ( see fig . [ 1]b ) . with this source the plasma density increased a few hundred times ( at the same pressure ) and the ionization rate could reach a value of ( 20 30)% . when the positive bias @xmath15 applied to any of the replaceable probes ( fig . [ 1]a ) was relatively low ( @xmath23 v for xe and @xmath24 v for ar ) the various c - v characteristics obtained with our set of probes , along with the measured potential fall @xmath25 between the plasma and the probe , have already been described and explained long ago ( see e.g. ref . 7 and references therein ) . at a higher bias @xmath15 a significant deviation could appear with various combinations of the gas pressure @xmath26 and probe sizes . note , that at each `` shot '' the plasma density @xmath8 increases from a very low value , when the debye length @xmath27 is large and the plasma sheaths near walls and electrodes could be comparable to the chamber sizes , to a value @xmath28 , when @xmath27 becomes small compared to the smallest probes we used . _ large probe . _ thus , with the largest probe we used ( @xmath29 mm ) and at a fixed pressure , say @xmath30 mtorr of xe , the same probe characteristics could be also kept for @xmath31 v just with one exception . namely at very low plasma densities some irregular oscillations appeared in the probe current @xmath22 ( see fig . [ 3]a ) . for higher @xmath8 the behavior of @xmath22 and the potential fall @xmath32 between the probe and the plasma during the pulse are very similar to the previous case when @xmath15 was low ( fig . the maximal electron probe - collected current @xmath22 , saturated at high @xmath15 , was equal to the sum of the emitted current @xmath14 and the total ion saturation current to the vacuum chamber walls ( fig . [ 3]c ) . the total ion saturation current to the walls was derived as the geometrical ratio of the surface probe area ( @xmath33 ) and the total area of the metal vacuum chamber ( @xmath34 ) multiplied by the ion saturation current to this probe @xmath16 ( fig . [ 2 ] ) . in the case of the floating discharge circuit , i.e. when @xmath12 was applied to the large surface probe instead of the chamber wall , @xmath14 was simply subtracted from this sum . it should be noted that at the densities when the irregularities of @xmath22 appear , the measurements showed that the thickness of the plasma sheath near the surfaces become comparable to the chamber size @xcite . in our case it corresponds to @xmath35 , @xmath36 ma for xe , and @xmath37 ma for ar . the pressures were 0.3 mtorr and 1 mtorr for xe and ar respectively . reducing drastically in one order of magnitude the pressure @xmath26 at the same pulse of @xmath14 ( at the same @xmath7 ) it was possible to obtain these @xmath22 irregularities during the whole pulse of @xmath14 . again , if the discharge circuit was floating , the maximal amplitude of the probe current irregular perturbations became smaller at the corresponding @xmath14 value . while these perturbations appeared in the electron probe current @xmath22 , no visible changes were seen neither in the emitted current @xmath14 ( see fig . [ 3 ] ) nor in the ion saturation current collected by a movable or surface probe . the latter means that the plasma density @xmath8 kept constant . _ intermediate probe . _ a very different behavior of @xmath22 could be observed when the probe sizes were reduced to ( 0.6 1.2 ) mm . at very low pressures ( @xmath38 mtorr ) the same irregular oscillations could still be seen . in the pressure range ( 0.1 0.8 ) mtorr for xe , when the probe bias exceeds 25 v , but still below a certain threshold , the probe c - v characteristics appeared as a prolongation of those for the lower voltage . they could be described by the theories mentioned above @xcite . on the other hand , when the bias voltage @xmath15 reached a certain value , a single , few or many current spikes appeared in the probe current @xmath22 . after adding further ( 2 4 ) v , periodic spikes filled the whole plasma pulse , i.e. in the wide range of @xmath39 there appeared a probe - current instability . the corresponding @xmath15 is recognized as the threshold @xmath40 . namely at @xmath41 v and at @xmath30 mtorr of xe in the density range of @xmath42 the periodic spikes in the probe current @xmath22 are shown in fig . [ 4]a , c . their period and duration kept approximately constant in the mentioned above density range and weakly depended on the bias voltage @xmath43 . the waveform of @xmath22 in combination with the potential fall @xmath44 between the plasma and the wall at the same @xmath14 , but at a bit higher @xmath45 v , are shown in fig . it is interesting to note , that these periodic spike oscillations , as a rule , started from a small but finite value of @xmath14 , the stage with irregular oscillations was usually skipped . on the other hand , in the narrow and unstable ranges of the bias @xmath15 and pressure @xmath26 it was possible to obtain irregular oscillations at low @xmath8 which were switched to regular ones at higher @xmath8 . these spikes were quite narrow compared to their period ( see fig . [ 4]c ) . typically the spike duration was @xmath46 for ar and @xmath47 for xe while the spike period was @xmath48 and @xmath49 , respectively . it should be noted that , as seen in fig . [ 4]c , the spikes of @xmath22 correspond to the minimal potential fall between the plasma and the probe : @xmath50 , i.e. @xmath22 and @xmath32 are in opposite phase . also , as seen in fig . [ 4]d , there is a hysteresis in the probe c - v characteristics when this instability exists . the threshold bias @xmath40 definitely decreased with the increase of the pressure @xmath26 : for xe @xmath51 v at @xmath52 mtorr , @xmath53 v at @xmath30 mtorr and @xmath54 v at @xmath55 mtorr . a very similar tendency was obtained with ar . at fixed pressure @xmath26 the spikes amplitude increased monotonically with @xmath15 and eventually reached its maximum which is equal to the total ion saturation current at the chamber wall . the average value of @xmath22 usually did not exceed 30% of the spike amplitude ( see fig . [ 4]b ) . the minimal value of @xmath22 ( see fig . [ 4]c ) usually corresponded to the undisturbed ( with no spikes ) probe current , mentioned above @xcite . a further increase of the probe bias @xmath15 led to qualitative changes of the probe current waveform ( see fig . [ 5 ] ) : the spikes minima increase , the spikes amplitudes decrease , and a visible modulation appears in the ion saturation current @xmath16 collected by the surface probe ( fig . [ 5 ] ) , i.e. there appears a modulation of the plasma density in the whole plasma volume . the above probe current instability could be `` killed '' by an external magnetic field of ( 600 800 ) gauss . such a magnetic field was obtained with a permanent magnet ( fig . [ 1]b ) , placed in the probe vicinity . in this case the ion gyrofrequency crudely corresponded to the oscillation frequency with no magnetic field : 35 khz vs ( 30 40 ) khz ( xe ) . another way to stop this instability was to put a thin spiral around the probe ( fig . when the spiral radius was less than ( 5 7 ) mm , it surely stopped the instability independently whether the spiral was floated , grounded or biased up to @xmath56 v. if the spiral radius exceeded ( 10 12 ) mm , no influence was seen . it means that the processes , determining the instability , were concentrated in the probe vicinity . note , that the ion gyroradius @xmath57 cm corresponds to the ion energy @xmath58 ev , which is about @xmath59 , where @xmath32 is the potential fall between the probe and the plasma corresponding to the moment when @xmath22 is maximal . on the other hand , the existence of this instability was definitely restricted by the ion current collected by the opposite , ion collecting electrode , i.e. by the external circuit . in the case of a small ion collecting area a big fraction of the probe bias @xmath15 is applied to the plasma sheath near this electrode . indeed when this electrode was too small , no instability appeared . this was directly confirmed when the ion collecting electrode we used was either the large surface probe ( @xmath60 ) or the insulated metal chamber cap ( @xmath61 ) . in this case the hot - cathode discharge circuit ( i.e. the plasma creating circuit ) and the probe current circuit were separated and the probe circuit was floating . it is interesting to note that a similar instability may appear not only in four- or three - electrode system , but also in two - electrode system . thus , the same hot cathode and the same probe were placed in the large vacuum chamber ( 66 cm diameter and 100 cm length ) . the cathode was directly connected to the chamber ( actually @xmath62 ) and the probe was biased positively . this scheme is very similar to the one used more than two decades ago in experiments described by stenzel@xcite where , we believe , similar probe - current spikes were noticed and later were recognized and studied @xcite . in our two - electrode experiments , when @xmath15 exceeded a certain threshold @xmath40 , the periodic spikes appeared not only in the probe current but also in the emitted current @xmath14 ( fig . this is contrary to the three- or four - electrode system , where @xmath14 kept constant . to obtain these oscillations the required @xmath15 was approximately the same as in the case described in refs . 1,10 and it was higher than in the case of three- or four- electrode system . again , if the chamber ( which is a big ion collector ) was disconnected from the hot filament ( cathode ) , no oscillation appeared in the probe circuit . _ small probe . _ further decreasing of the probe sizes again led to a drastic change of the probe characteristics . thus , when the probe diameter was below 0.1 mm ( with the same length of 2 mm ) , we could not find any self - consistent oscillations of the probe current @xmath22 . this was correct in the whole investigated ranges of pressure @xmath26 and probe bias @xmath15 for both sort of gases we used . on the other hand , above a certain bias voltage @xmath15 the probe current @xmath22 jumped from its low value to a higher one . it happened almost independently of @xmath63 ( see fig . [ the jump amplitude exceeds one order of magnitude . the threshold bias voltage @xmath40 , when the jump occurs , was close to the threshold bias @xmath40 , corresponding to the probe current instability occurrence in the described above experiments with the larger probe . obviously , this was correct just for the same pressure @xmath26 , sort of gas etc . with further increasing @xmath15 well above @xmath40 , the probe current @xmath22 tended to saturate in the manner shown in fig . actually it also was limited by the total ion saturation current to the chamber wall . the threshold voltage @xmath40 also decreased with the gas pressure rising up : at @xmath30 mtorr of xe , @xmath64 v , and at @xmath65 mtorr , @xmath66 v. for ar it was found that @xmath67 v at @xmath52 mtorr and @xmath68 v at @xmath69 mtorr . unfortunately it was impossible to carry out these measurements in a wide range of pressures because a high @xmath15 at low @xmath26 as well as a high @xmath26 at modest @xmath15 could cause parasite sparking which , in turn , immediately led to the probe evaporation . the `` switching '' time of the probe current jumps was @xmath70 for xe ( fig . [ 7]b ) and @xmath71 for ar . these values exceeded by ( 3 4 ) times the spike duration when the probe current instability occurred . to study the c - v characteristics of such a thin probe we applied a periodic saw - tooth voltage instead of the dc @xmath15 ( see fig . the saw - tooth duration was chosen as @xmath72 . on the one hand this is much longer compared to the switching time , on the other hand this is much shorter compared to the plasma pulse . the latter means that during @xmath73 the plasma may be recognized as stationary . as it is clearly seen in fig . [ 8]b , the current jump starts at a substantially higher bias @xmath15 compared to the bias @xmath15 when the current falls down . it indicates the existence of hysteresis in the probe c - v characteristics ( see fig . [ 8]c ) . a very similar hysteresis loop could be seen in a wide range of plasma densities during the plasma pulse . when the pressure @xmath26 was a bit reduced in order to bring this bias amplitude below @xmath40 , no hysteresis was seen and the probe characteristics appeared as an almost straight line ( see fig . [ 8]d ) , which is in a good agreement with the known theories for small probes @xcite . to complete this study we checked the qualitative influence of the ionization level on the probe characteristics . namely we placed in the vacuum chamber the powerful pulsed single - core fic plasma source @xcite ( see fig . [ 1]b ) . it could work in the same pressure range and in our experiments we achieved ionization level @xmath74% . the plasma density grew up to @xmath75 during a @xmath76 pulse . to prevent the probe damage , we biased the probe with a single variable positive voltage pulse with pulse duration of @xmath77 . typical waveforms of the probe current and plasma potential are shown in fig . [ 9 ] . in the investigated range of @xmath15 the electron probe current could reach a few amperes but the probe characteristics were smooth with no hysteresis , no instability of the probe current was seen and the potential fall @xmath44 between the plasma and wall was always low . the experimental data show that the peculiarities of the probe c - v characteristics ( instability , hysteresis ) are the same either in three or four electrodes schemes . it means that the discharge circuit can be eliminated from the consideration . a positively biased probe collects electron current @xmath22 from the surrounding plasma . the equivalent ion current @xmath78 is collected by a reference electrode ( the chamber wall in the experiments ) . the current loops through an external circuit which maintains the voltage @xmath15 between the electrodes . the probe and the reference electrode ( thereafter cathode ) currents depend on the voltage between them and the plasma : @xmath79 and @xmath80 , where @xmath32 , @xmath44 , and @xmath81 are the probe , plasma , and cathode potentials , respectively ( thereafter @xmath82 . thus , taking into account that @xmath83 , the plasma potential @xmath44 is determined by the condition : @xmath84 in other words , the plasma potential and , consequently , the probe current are determined by both the probe and cathode c - v characteristics . it is clear now that the dependence @xmath85 , which we measure in our experiments , is close to the probe characteristics @xmath86 when the plasma potential is small , @xmath87 . this is possible when the ion - collecting area @xmath1 ( cathode area ) is large enough compared with the probe area @xmath0 and eq . ( [ eq00 ] ) is satisfied . when the bias voltage is large , @xmath88 , the effective collecting area @xmath89 around the small probe can exceed the probe geometric area and the condition in eq . ( [ eq00 ] ) should be written as follows : @xmath90 when the voltage @xmath15 exceeds the ionization potential of the neutral gas , the ionization in the space - charge sheath around the probe can change drastically the collected current even if the electron impact ionization mean free path is large compared with the sheath thickness ( see , e.g. , refs . 11 - 13 ) . the probe c - v characteristics becomes three - valued with unstable middle branch and demonstrates hysteresis under a gradual increase and a subsequent decrease of the probe potential @xcite . the collected current on the stable upper and lower branches of the hysteresis loop can differ by more than an order of magnitude @xcite . when the probe and the cathode c - v characteristics are known , the plasma potential @xmath44 that satisfies eq . ( [ eq0 ] ) can be found graphically as it is depicted in fig . [ fig_1 ] . the cathode c - v characteristics is shown as a function of the potential @xmath91 , @xmath92 , whereas the probe characteristics is shown as a function of @xmath93 , @xmath94 . the intersection of these two curves defines the plasma potential @xmath44 and the circuit current @xmath95 . under the increase / decrease of the bias voltage @xmath15 , the probe c - v characteristics curve is shifted right / left along the horizontal axis as a whole . it is important to bear in mind that the probe characteristics shape depends on the probe dimension and form . further we will consider small probes whose radii are comparable with the plasma debye length . such probes are characterized by the absence of a saturation current @xcite . the characteristic values of the current which are inherent in the @xmath96 dependence ( characteristic scale along the vertical axis in fig . [ fig_1 ] ) increase / decrease monotonically with the increase / decrease of the probe radius . now , using this graphical representation , one can explain the experimental results described above . it is convenient to group the results in accordance with the probe radius : large , intermediate , and small probes . _ large probe . _ when the probe radius is so large that the collected current exceeds the cathode saturation current even on the lower branch of the probe c - v characteristics , the intersection has the form depicted in fig . [ fig_4 ] . independently of the bias voltage , the cathode characteristics intercepts only the lower _ stable _ branch of the probe characteristics . the experiments with the 5 mm radius probe are related to this case ( see fig . [ 3 ] ) . _ intermediate probe . _ under the probe radius decrease its c - v characteristics `` shrinks '' along the vertical axis . the possible types of the intersections between the probe and cathode characteristics are shown in fig . [ fig_3 ] . the distinctive property of the intermediate probe is the following : there is a range of bias voltage values when the cathode characteristics @xmath92 intercepts only the _ unstable branch _ of the probe characteristics @xmath94 ( curve @xmath97 , bias voltage @xmath98 in fig . [ fig_3 ] ) . it means that there are no stable solutions of eq . ( [ eq0 ] ) and self - oscillations of the current in the circuit appear when the bias voltage lies in the above - mentioned range . note that jumps from the lower to the upper branch and backwards occur in different points of the probe c - v characteristics . therefore , the dependence @xmath99 $ ] should demonstrate the hysteresis - like behavior . the experiments with 0.6 - 1.2 mm radius probes demonstrate these peculiarities of the intermediate probe characteristics ( see fig . [ 4 ] ) . _ small probe . _ if the probe radius is so small that the collecting current is small compared with the ion saturation current for all voltages in the range of interest , the intersection between the c - v curves in fig . [ fig_1 ] is located near the cathode floating potential ( potential that corresponds to zero current ) . the intersection types for various bias voltages @xmath15 are shown in fig . [ fig_2 ] . depending on the bias voltage , the cathode c - v characteristics intercepts either the lower branch of the probe characteristics ( curve @xmath100 , bias voltage @xmath101 in fig . [ fig_2 ] ) , or three branches ( curve @xmath97 , bias voltage @xmath98 ) , or the upper branch ( curve @xmath102 , bias voltage @xmath103 ) . it is important to note that independently of the bias voltage the cathode characteristics always intercepts the _ stable branches _ ( upper and lower branches , the intermediate branch is always unstable ) of the probe characteristics . in this case there are stable solutions ( either one or two ) of eq . ( [ eq0 ] ) for all values of the bias potential . the c - v characteristics @xmath95 demonstrates hysteresis under a gradual increase and a subsequent decrease of the potential @xmath15 . this qualitative analysis explains the experimental results related to the small ( less than 0.1 mm diameter ) probe presented in figs . [ period ] , [ cathode_va ] , and [ probe_va ] . the current circuit contains two elements : the plasma - probe sheath and the plasma - cathode sheath . the first one is characterized by a multi - valued c - v characteristics @xmath94 , whereas the c - v characteristics of the second one is a single - valued function @xmath92 . it means that under a gradual increase and a subsequent decrease of the potential @xmath15 , the dependence @xmath92 should be a single - valued function , despite the fact that the dependence @xmath94 demonstrates hysteresis . this conclusion is confirmed by the experimental data shown in figs . [ cathode_va ] and [ probe_va ] . relatively small deviations from the single - valued dependence @xmath92 ( small hysteresis loops ) reflect transient processes of the plasma - cathode sheath reconstruction caused by fast variations of the plasma potential @xmath91 under the `` jumps '' between the lower and upper branches of the probe c - v characteristics @xcite . the key element in the qualitative analysis above is the additional ionization inside the current - collecting sheath around the probe . additional ions can essentially increase the sheath radius if the number of ions @xmath104 in the sheath is comparable with the number of electrons @xmath105 . assuming that the ionization take place in all the sheath volume , it is possible to make the following rough estimate of @xmath104 . any electron entering the sheath produces @xmath106 ions , where @xmath107 is the gas density , @xmath108 is the ionization cross section , and @xmath109 is the sheath radius . thus , @xmath110 , where @xmath111 is the characteristic residence time of the particle in the sheath , and @xmath112 is the characteristic velocity of the particle . because electrons and ions are accelerated by the same potential drop , @xmath113 , where @xmath4 is the ion mass . thus , @xmath114 under typical experimental conditions ( @xmath115 , @xmath116 , @xmath117 ) the number of ions @xmath104 in the sheath is comparable with the number of electrons @xmath105 , @xmath118 , when the sheath radius is of the order of 1 cm . this estimation is in a good agreement with the experimental data . indeed , approximately the same radius @xmath119 cm of the probe collecting area may be obtained directly from the amplitude of the probe current spikes ( fig . [ 4 ] fig . [ 8 ] ) . this is correct in a wide range of plasma densities [ emission currents @xmath120 ma ] . experiments with various spirals placed around the probe showed also that the spiral did not disturb the current - collecting area when the spiral radius exceeds 1 cm . the above estimation of the number of ions @xmath104 in the sheath supposes implicitly that the neutral gas density @xmath107 is large enough so as to provide creation of this amount of ions . however , it does not always happen . in the vicinity of a small probe the electron density exceeds many times the plasma density due to the geometric focusing of the collected particles . the ionization rate in this region can be so large that the ionization degree @xmath121 would reach 100% during the ion residence time or even faster . in this case the number of ions @xmath104 is always smaller than the number of electrons @xmath105 in the sheath and the probe c - v characteristics is single - valued . let us estimate the sheath region where the ionization degree is small , @xmath122 and the condition @xmath123 is satisfied . the ions production rate in a layer of radius @xmath124 is described by the following equation : @xmath125 where @xmath126 and @xmath127 are the collected electrons density and velocity , respectively . during the ion residence time @xmath128 the ionization degree reaches the value : @xmath129 here the particles flux conservation law @xmath130 is used . when the condition ( [ eq4 ] ) is satisfied , the ionization degree is small , and @xmath122 in a sheath region of radius @xmath131 it follows from eq . ( [ eq7 ] ) that the region , where the neutral gas density can not provide production of the required amount of ions , occupies a considerable part of the sheath volume when the plasma density is large enough , all other factors being the same . in the set of experiments where plasma is created by discharge with a hot cathode @xmath132 the ratio @xmath133 is small and the probe c - v characteristics is multi - valued . in contrast , in the experiments with the fic plasma source @xmath134 this ratio is large , @xmath135 , and hence the probe current is stable and self - oscillations are not observed despite that the potential fall between the probe and the plasma could significantly exceed the ionization potential . the characteristics of small positively biased electrodes ( probes ) immersed into a plasma have been studied . these probes were much smaller than the reference electrode size divided by the square root of the ion / electron mass ratio . the electron current branch of the c - v characteristics of such probes is widely used to measure the local plasma parameters . additional ionization of neutral gas by electrons accelerated in the space - charge sheath around the probe could significantly expand the sheath thickness . however , we were focused on the case , when the sheath thickness still remains small compared to the whole plasma and reference electrode sizes . it should be emphasized that in our experiments this additional ionization inside the sheath does not change the plasma parameters in the whole volume . when the probe dimension is large compared to the sheath thickness , the sheath expansion does not affect the current collected by the probe from the plasma . on the other hand , near a very small probe the sheath expansion induced by the ionization leads to considerable increase of the collected current . the probe current loop is closed through the ion - collecting electrode ( cathode ) having its own c - v characteristics . the potential fall between the cathode and the plasma should rise to support the increasing current . the plasma potential growth changes the potential drop between the probe and the plasma , and the probe current should be changed in accordance with the probe c - v characteristics . thus , the current is determined by the relation between the probe and the reference electrode c - v characteristics . it has been shown both experimentally and theoretically that these small probes can be separated into three groups according to the probe dimensions . every group is characterized by its particular dependence of the collected current on the bias voltage . the current collected by the `` large '' probes increases monotonically and continuously with the voltage growth . the probes with an `` intermediate '' size are characterized by excitation of strong periodic spike - like oscillations under a constant bias voltage . the current collected by the `` small '' probes is stable , but changes stepwise under certain bias voltage values and demonstrates hysteresis under a gradual increase and a subsequent decrease of the bias voltage . both oscillations and current steps are caused by additional ionization in the probe vicinity . note , that the same probe can be treated as either large , intermediate , or small depending on the value of the reference electrode area . ignoring the possibility of probe - current instability , neither the plasma density nor the electron temperature could be measured correctly by this probe even if the oscillations are filtered out , averaged etc . in the case of stepwise jumps of the probe current the upper current level does not reflect directly the plasma parameters in the place where the probe is located . in conclusion , we have demonstrated that under certain conditions , phenomena related with langmuir probes can not be correctly interpreted once considered separately from the reference electrode characteristics . bliokh , j. felsteiner , ya.z . slutsker , and p.m. vaisberg , appl . . lett . * 85 * , 1484 ( 2004 ) ; y.p . bliokh , yu.l . brodsky , kh.b . chashka , j. felsteiner , and ya.z . slutsker , j. appl . phys . * 107 * , 015009 ( 2010 ) . mott - smith and i. langmuir , phys . rev . * 28 * , 727 ( 1926 ) ; i.b . bernstein and i.n . rabinowitz , phys . fluids * 2 * , 112 ( 1959 ) ; f. chen , j. nucl . energy , part c * 7 * , * 47 * ( 1965 ) ; j.e . allen , phys . * 45 * , 497 ( 1992 ) .	it is well known that additional ionization in the vicinity of a positively biased electrode immersed into a weakly ionized plasma is responsible for a hysteresis in the electrode current - voltage characteristics and the current self - oscillations rise . here we show both experimentally and theoretically that under certain conditions these phenomena can not be correctly interpreted once considered separately from the reference electrode current - voltage characteristics . it is shown that small electrodes can be separated into three groups according to the relation between the electrode and the reference electrode areas . each group is characterized by its own dependence of the collected current on the bias voltage .
until recently but for a few exceptions the study of quantum dot ( qd ) heterostructures with the staggered ( type - ii ) band alignment had been largely ignored because of the absence of confinement of one of the two types of carriers and their presumed poor radiative efficiency . however , it has come to be recognized that these structures are interesting , especially for their rich physics of excitons.@xcite in contrast to the usual type - i qds ( e.g. inas / gaas ) where the confinement energy scale is far greater than the energy of the coulomb interaction , the role of confinement in type - ii qds is largely limited to defining the geometry of the system . this in itself has interesting consequences . the multiply - connected topology can give rise to an oscillatory ground state energy for the magneto - excitons.@xcite secondly , type - ii qds also act as nanocapacitors@xcite which selectively accommodate only one type of particles ; but once charged they can bind the complementary particle to form an exciton . the strength of the coulomb interaction can be modified by screening or magnetic field.@xcite at higher excitation powers , they can be doubly - charged and form four - particle bound states ( biexcitons ) . the biexcitons in a type - ii qd system are very unlike their counterparts@xcite in type - i qds and quantum wells . they always have negative ` binding energy'.@xcite in the atomic physics language , while the usual biexciton is structurally analogous to a hydrogen molecule , the biexciton@xcite in type - ii qds is more like a helium atom.@xcite in this article we have probed the nature of the ensemble photoluminescence ( pl ) emission from a sample with inp qds in a gaas matrix . although the band offsets@xcite of _ bulk _ inp and gaas and some previous studies suggest that this material combination forms type - ii structures with electrons localized within the inp quantum dots and free holes in the gaas matrix , the energy gap of inp and gaas is within @xmath0mev of each other , and the conduction band offset is relatively small.@xcite thus alloying and anisotropic strain within the qds,@xcite can modify the energy gap and the relative offsets in way that is dependent on the details of the size and shape of the qds . for example , a comprehensive k.p calculation @xcite does not find a type - ii alignment in this system . secondly , thick ( @xmath1 nm ) inp heteroepitaxial layers on gaas and even homoepitaxial inp have shown a broad emission peak at @xmath2 ev@xcite due to donor - acceptor - pair ( da ) recombination , which is rather close to the pl energy of inp / gaas qds . furthermore , the da emission has characteristics@xcite generally used to classify spatially - indirect excitons from type - ii qds excitation power - dependent blue - shift,@xcite spectrally broad emission at a sub - bandgap energy in macro - pl , and narrow emission lines from localized states in micro - pl.@xcite one of the aims of this study is to unambiguously establish the type - ii band alignment in this system and highlight the role of heterostructure boundary conditions on the size and the binding energy of excitons . in this work we will not discuss the aharonov - bohm - type effects associated with the topology of the wave function as these have already been extensively discussed in literature@xcite and manifest on energy scales an order of magnitude smaller than we are interested in here . secondly , by extending the measurements to over six orders of magnitude of excitation powers , we have been able to change the average electronic occupancy within the qds . this leads to significant changes in the diamagnetic shift coefficient that are a result of the interesting physics of biexcitons . the sample was grown by metal - organic vapor phase epitaxy on ( 001 ) gaas substrate.@xcite the qd density was about @xmath3 with an average diameter of @xmath4 nm ( @xmath5 nm ) and average height of about @xmath6 nm ( @xmath7 nm ) as measured by cross - sectional transmission electron microscopy.@xcite but based on experience from other heterostructures , it is possible that the actual size of quantum dots is much smaller.@xcite non - resonant ( excitation wavelength=532 nm ) pl measurements were performed at liquid helium temperature with the excitation powers varied between ( @xmath8w@xmath9 ) . the light from the excitation laser and the sample pl was fibre - optically coupled in and out of ( i ) a variable - temperature cryostat within the bore of a superconducting magnet ( b@xmath10 ) for magneto - pl measurements at low excitation powers and ( ii ) a liquid helium bath cryostat whose tail was within the 18 mm bore of the pulsed field coil ( @xmath11 ) for measurements at higher powers @xmath12 . the pl spectra were recorded by an electron multiplying charged - coupled device after being dispersed by a 30 cm imaging spectrograph . pulsed magnetic fields of up to 50 t were generated using a 5 kv , 500 kj capacitor bank . the field had a duration of about 20ms , during which several pl spectra were recorded . 1(inset ) shows three representative spectra under different conditions . fig.[fig : fig1 ] shows the magnetic field - dependent pl peak - shift measured at @xmath13k , at a very low excitation power of about @xmath14 . using the model of a hydrogenic exciton , one can semi - phenomenologically describe this shift by the following equations:@xcite @xmath15 these equations are derived under the assumption that the magnetic field - induced change in the ground state energy of a hydrogenic exciton from the low - field regime of quadratic ( diamagnetic ) shift to the high field linear shift ( due to transitions between effectively free landau levels ) is adiabatically continuous ( i.e. its functional form is continuous and differentiable ) between two well - defined limits . note that the high field limitis approximate as it assumes that the transitions are between _ free _ landau levels and ignores the weak ( @xmath16 , as @xmath17 ) dependence@xcite of the excitonic binding energy on magnetic field . here @xmath18 is the ground state energy without the magnetic field , @xmath19 corresponds to the magnetic field when @xmath20 , @xmath21 is the excitonic bohr radius , @xmath22 is the magnetic length , @xmath23 is the reduced effective mass , and @xmath24 the magnitude of the electronic charge . the fit to eq.[eqn : fips ] , along with its physical content , is also shown in fig.[fig : fig1 ] . the second term in eq.[eqn : fips](b ) corresponds to the excitonic binding energy and is the extrapolation of the high - field slope [ third term in eq.[eqn : fips](b ) ] to @xmath25.@xcite the crossover from quadratic to linear slope is found at @xmath26 t . the values of the exciton radius , the binding energy and the diamagnetic shift coefficient are given in table 1 . table 1 also compares the effect of the heterostructure boundary conditions on the excitonic parameters in inp . note the systematic trend in the value of the diamagnetic shift coefficient . the effective electron - hole interaction in type - ii dots is the weakest followed by bulk inp , and then finally inp / ga@xmath27in@xmath28p qds which show a much stronger binding [ also see fig.[fig : band diagram ] ] . thus , on physical grounds , the observations are most consistent with type - ii band alignment . .comparison of exciton parameters in inp under different boundary conditions [ cols="^,^,^,^ " , ] next , we will explore the multi - particle states in these qds by excitation - power - dependent pl measurements . these measurements will also help us rule out da recombination , as will be discussed in section iii c. in what follows , we will assume that the qds have type - ii alignment . the pl peak position , measured without the magnetic field [ fig.[fig : combined figure](b ) ] is strongly excitation power - dependent beyond an incident laser intensity of about @xmath29 w-@xmath9 . this marks the point where multiparticle states start to play a role in pl . the observed blue - shift is due to the additional energy associated with the capacitive charging of the qds . the integrated pl intensity ( not shown here for brevity ) also gradually changes its slope from linear to ( slightly ) superlinear beyond @xmath29 w-@xmath9 , indicating that a fraction of emission is from biexcitonic states . the dependence of the pl peak position as a function of the magnetic field at different excitation powers , measured over almost six orders of magnitude , is shown in fig.[fig : combined figure](a ) . notice that the curves are all qualitatively similar and that they all can be fitted to equations [ eqn : fips ] [ solid lines in fig[fig : combined figure](a ) ] . this is because the b - dependent change in the energy of biexcitonic levels is just the sum of single particle energies , but with the important difference that the biexcitonic radius @xmath30 and hence the diamagnetic shift coefficient are significantly different . however , for an ensemble , the analysis is complicated by the facts that ( i ) the emission is from a mixture of excitonic and biexcitonic states with an unknown biexciton fraction @xmath31 , ( ii ) when a photon is emitted by a biexciton , what is measured in the magneto - pl experiment is the difference in the shifts of biexcitonic and excitonic levels , because the emission process involves @xmath32 . the measured change in the pl emission energy at low magnetic fields [ diamagnetic shift regime , equation 1(a ) ] will then be @xmath33b^2 . \;\;\nonumber ( 2)\ ] ] the diamagnetic shift coefficient @xmath34 is thus expected to be strongly excitation - power - dependent . indeed this is observed in fig.[fig : combined figure](c ) . @xmath34 changes by factor of two due to the much smaller biexcitonic radius . recall that in type - i qds , confinement usually renders the diamagnetic shift independent of the charge in the dot.@xcite in general , it depends on the relative extents of the spatial spread of the electron and the hole wave functions . for a rough estimate of the biexciton fraction as a function of excitation power , we use the ratio of the bohr radii for the helium and the hydrogen atom , @xmath35.@xcite fig.[fig : combined figure](d ) is plotted using this value of @xmath36 and we find that the biexciton fraction at the highest excitation power is about 30% . also note that when @xmath37 , equation 2 predicts that the observed diamagnetic shift coefficient would not only reduce but also become negative at very high excitation powers . the strong dependence of the diamagnetic shift coefficient on the excitation power also explains the anomalous results of godoy , et al.@xcite , who found the excitonic diameter to be smaller in inp / gaas compared to the bulk inp . they measured a diamagnetic shift coefficient of between 520@xmath38 , the latter of which is equal to what we measure at the highest excitation power . fig.[fig : combined figure](e ) shows that the exciton mass ( high field slope of the curves in fig.[fig : combined figure](a ) ) stays at approximately @xmath39 , constant within 15% over the whole range of excitation powers . hence the changes in the diamagnetic shift coefficient can be understood as being largely the difference in the excitonic and biexcitonic radii ( equation 2 ) . this provides consistency to the analysis . the value of the mass is between the free electron mass in inp ( @xmath40 ) and the free heavy - hole in gaas ( @xmath41 ) . this is reasonable because the electron is largely immobilized by the quantum dot . strain and non - parabolicity effects may further contribute to the enhancement of the electron mass . the excitation power dependence of the diamagnetic shift coefficient [ fig.3(c ) ] also rules out recombination due to overlap between donor and acceptor wave functions . da recombination has an energy@xcite @xmath42 @xmath43 is the energy gap , @xmath44 and @xmath45 are the binding energies of the donor and acceptor levels , the fourth term is the coulomb repulsion energy of the ionized centres _ after _ recombination , and @xmath46 are the phonon - assisted transitions.@xcite while the da - pair emission shares some characteristics with emission from type - ii qds such as the diamagnetic shifts from da - recombination would be of the same order as for excitons [ fig.1 ] and at higher excitation powers there would be a blue - shift@xcite qualitatively similar to that seen in fig.3(b ) , the diamagnetic shift coefficient itself should not have an excitation power dependence as observed in fig.3(c ) . secondly one should expect a strong quenching of the pl intensity in magnetic field for da recombination the wavefunctions shrink in the magnetic field and since the electron and hole centres are not at the same point in space , their ( exponentially small ) overlap will be strongly reduced . this is also not observed . we studied the pl from inp / gaas qd heterostructures in very high magnetic fields ( @xmath11 ) over almost six orders of magnitude of excitation powers . magneto - pl measurements at very low excitation powers established that the excitons have an average bohr radius of @xmath47 nm and a binding energy of @xmath48mev . these values indicate a much weaker binding for the excitons in comparison with bulk inp and provide strong evidence for type - ii band alignment in these qds . we also studied the evolution of the electron - hole binding as the qd ensemble makes a gradual transition from a regime where the emission is from ( hydrogen - like ) two - particle excitonic states , to a regime where the emission from ( helium - like ) four - particle biexcitonic states also becomes significant . this was demonstrated by a strong variation of the diamagnetic shift with the excitation power . the work at the ku leuven is supported by the fwo and goa programmes and by the methusalem funding of the flemish government . mh acknowledges the support of the research councils uk . e. ribeiro , a. o. govorov , w. carvalho , and g. medeiros - ribeiro , phys . . lett . * 92 * , 126402 ( 2004 ) . i. r. sellers , v. r. whiteside , i. l. kuskovsky , a. o. govorov , and b. d. mccombe , phys . lett . * 100 * , 136405 ( 2008 ) . a. b. kalametsev , v.m . kovalev , and a.o . govorov , jetp lett . * 68 * , 669 ( 1998 ) . m. h. degani , m. z. maialle , g. medeiros - 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body problem in presence of suitable boundary conditions and magnetic field , which is beyond the scope of this work . c. klingshirn , pp353 , _ semiconductor optics _ ( 2nd ed . , springer , berlin , 2005 ) .	we have used magneto - photoluminescence measurements to establish that inp / gaas quantum dots have a type - ii ( staggered ) band alignment . the average excitonic bohr radius and the binding energy are estimated to be 15 nm and 1.5 mev respectively . when compared to bulk inp , the excitonic binding is weaker due to the repulsive ( type - ii ) potential at the hetero - interface . the measurements are extended to over almost six orders of magnitude of laser excitation powers and to magnetic fields of up to 50 tesla . it is shown that the excitation power can be used to tune the average hole occupancy of the quantum dots , and hence the strength of the electron - hole binding . the diamagnetic shift coefficient is observed to drastically reduce as the quantum dot ensemble makes a gradual transition from a regime where the emission is from ( hydrogen - like ) two - particle excitonic states to a regime where the emission from ( helium - like ) four - particle biexcitonic states also become significant .
in recent years , space based infrared observations have permitted the identification of a large sample of _ transition discs _ ( e.g. najita et al 2007,cieza et al 2008,2010 ; espaillat et al 2010 , kim et al 2009 , merin et al 2010 , muzzerolle et al 2010 ) , young stars with spectral evidence for cool circumstellar dust but which lack diagnostics of warm dust . the standard interpretation is that transition discs contain a ( at least partially ) cleared inner cavity and that the temperature at the cavity wall sets the wavelength beyond which a strong spectral excess is detected . the current census of transition discs totals around @xmath1 , which is large enough to permit some examination of trends and correlations within the sample ( e.g alexander & armitage 2009 , kim et al 2009 , owen et al 2011,2012 , owen & clarke 2012 ) . a number of authors have noted that cooler transition discs ( i.e. those with a spectral upturn long - ward of @xmath3 m , corresponding to cavity walls at around @xmath1 k ) are associated with systematically higher accretion rates on to the star and systematically higher millimetre fluxes than warmer ( small hole ) transition discs . owen & clarke ( 2012 ) demonstrated that if one divides the transition disc sample at a mm flux that is equal to the median for young disc bearing stars then the mm bright sub - sample has distinctly different properties i.e. larger hole radii ( @xmath4 a.u . ) and higher accretion rates compared with the mm faint sample ( which is dominated by small holes and a wide range of accretion rates ) . given that there appears to be no correlation between these properties and the mm flux within each of the two sub - samples , owen & clarke ( 2012 ) suggested that these may represent two distinct populations of transition discs . they also noted that the properties of the mm faint sub - sample were consistent with discs that were being cleared at late times as a result of xray photoevaporation . the properties of the mm bright sub - sample are clearly incompatible with disc clearing ( e.g. by photoevaporation ) at a late evolutionary stage . indeed the high mm fluxes imply disc masses that are in several cases @xmath5 of the stellar mass ( andrews et al 2011 ) ; this places them in the regime that is believed to correspond to the early stages of disc evolution and such discs may even be self - gravitating . one possibility for creating a transition disc at this stage is via the formation of a giant planet / brown dwarf at large radius , which then tidally truncates the disc just beyond the orbit of the planet ( kraus & ireland 2012 , nayakshin 2013 ) . although the time - scale for planet formation by core accretion is long in the outer disc , this is a region where massive discs can undergo gravitational fragmentation on account of their low ratio of cooling to dynamical time - scale ( rafikov 2005 , stamatellos & whitworth 2008 , clarke 2009 ) . in order to reproduce the observed high accretion rates in these systems it is necessary that some gas can leak inwards past the planet . on the other hand , in order to produce the spectral signature of a transition disc , this leakage flow has to be depleted in dust . rice et al ( 2006 ) , pinilla et al ( 2012 ) have suggested that such ` transparent accretion ' can be effected via trapping of dust grains at the inner edge of the tidally truncated disc . although many details of the mechanism are still to be quantified , an orbiting companion provides a qualitatively attractive scenario for explaining these objects . in this letter we provisionally assume that the mm bright transition discs are indeed created by embedded planets . we then investigate how such systems ( i.e. planet plus outer disc plus leaky accretion flow ) would evolve over the subsequent lifetime of the disc . for this simple experiment we neglect the possible role of photoevaporation ; see rosotti et al ( 2013 ) for a modeling of combined photoevaporation and planet formation . we also assume that the level of the leaky accretion flow from the outer disc is around @xmath6 of the viscous accretion rate in the outer disc , though - as we discuss in section 2 below - this value , and its influence on planet migration , is not currently well calibrated numerically . we furthermore assume that the dust filtration mechanism works for all companion masses and at all orbital radii . we do not attempt a detailed population synthesis of transition disc properties , due to the large number of model assumptions and degeneracies in fitting the data . moreover , we do not _ require _ that evolution of the mm bright transition discs ( whose properties provide the initial conditions for our experiment ) can necessarily account for all mm faint transition disc objects , since some of these may well have a quite different origin ( e.g. photoevaporation ) . what we _ do _ require is that the model evolution does not populate ` forbidden ' regions of parameter space . specifically we need to a ) avoid the production of large numbers of mm bright sources with small ( warm ) holes and b ) avoid the production of mm faint large holes among conventional transition discs ( i.e. those without stellar mass companions ) . we add this proviso concerning binary companions since discs in wide _ stellar _ binaries have typical mm fluxes that are at least an order of magnitude fainter than those of transition discs without binary companions ( kraus et al 2011,2012 ; see figure 11 , andrews et al 2011 ) . we therefore require that our model generates large mm faint holes only in the limit of high companion masses . we model the evolution of the disc according to the viscous diffusion equation : @xmath7\ ] ] where @xmath8 is the disc surface density and @xmath9 is the kinematic viscosity which we model phenomenologically as a power law of radius following lynden - bell & pringle 1974 , hartmann et al 1998 ; here we adopt @xmath10 ( noting that this implies that in steady state the disc surface density profile scales as @xmath11 - cf hartmann et al . 1998 ; andrews et al 2009 ) . we model the coupled evolution of disc and planet through the disc s inner boundary condition ( see below ) ; a free outflow condition is imposed at the disc s outer edge . the equation is integrated using a standard explicit finite difference method , equispaced in @xmath12 ; we typically employ @xmath13 radial gridpoints over the range @xmath14 a.u .. we have experimented with values of the outer boundary in order to ensure that the disc mass leaving the outer boundary is a small fraction of the initial disc mass ( a few per cent or less ) and that the evolution is independent of the outer boundary location in this case . we first describe the set - up in the absence of leakage from the outer disc . if the planet is located at grid point @xmath15 , the inner edge of the disc is located at grid point @xmath16 , where we impose a zero mass flux boundary condition . we then record the increase in angular momentum of the disc resulting from this boundary condition until the total angular momentum acquired by the disc is equal to the difference in angular momentum of the planet in keplerian orbit at grid points @xmath15 and @xmath17 . ( note that in recording the increase of angular momentum of the disc we also include the angular momentum that is advected through the outer boundary , where a zero torque boundary condition is applied ) . at this point , the planet is moved to grid point @xmath17 , the inner edge of the disc to grid point @xmath15 and the process repeated . this simple approach ensures that the angular momentum of the system is conserved to machine accuracy and does not - as in the approach more usually adopted - rely on a parametrisation of the torque between the disc and planet . we do not expect our method to model the detailed structure of the disc in the region where it is tidally sculpted by the planet and this will have some effect on the mm emission ( although probably not greater in magnitude than the effect of varying the dust to gas ratio in this region , which is an effect that we do explore ) . since we are interested in the orbital evolution of the planet and the global evolution of the disc ( inasfar as this affects the mm flux ) , our simple angular momentum conserving approach is sufficient for our purposes . in addition , we implement a leakage flow from the outer disc . other phenomenological modelling exercises ( e.g. alexander & armitage 2009 , alexander & pascucci 2012 ) have used a prescription in which the leakage flow rises with decreasing planet mass to attain a maximum of around a third of the mass flow rate through the outer disc for planets of around a jupiter mass . the appropriate values are however rather uncertain based on the existing simulation data ( veras & armitage 2004 , lubow & dangelo 2006 ) : as discussed below , in cases where the leakage flow ( and accretion on to the planet ) is significant , there is considerable uncertainty about the consequences for planet orbital migration inasmuch as this would deviate from the type ii planetary migration induced purely by interaction with the outer disc . in order to avoid these uncertainties ( and because the focus of our investigation will end up being in the more massive planetary regime where leakage is expected to be fairly minor ) we simply assume that the leakage flow is around @xmath6 of the flow in the outer disc for all companion masses . we can not rule out that the leakage flow might not become much more significant in the case of planets of much lower planet mass and return to this issue in section 5 . the leakage has three consequences for the system : a ) it implies a finite accretion rate on to the star , b ) it modifies the disc evolution by depleting the outer disc and c ) it affects the planetary migration , both via b ) and via the torques imparted to the planet from the planetary accretion stream and the flow to the inner disc . note that the efficiency factor of the leakage flow ( @xmath18 ; i.e. the ratio of the leakage flow to the accretion rate in the outer disc ) ) critically determines the accretion rate onto the star for all values of @xmath18 ; leakage is also significant in reducing the millimetre flux from the disc ( b ) ) . however c ) is only mildly affected by leakage for low values of @xmath18 such as the value of @xmath19 adopted here . this is fortunate given the uncertainties in c ) . calculation of c ) involves knowledge of the change in specific angular momentum of fluid elements that are either directly accreted onto the planet or are able to cross the planetary orbit into the inner disc . in addition , the finite angular momentum possessed by material in the latter category is eventually passed back to the planet via tidal torques at the outer edge of the inner disc . since we are not modelling either the inner disc nor the detailed trajectories of the material crossing the planet s orbit nor accretion onto the planet , we simply assume that the the entire angular momentum of the material leaving he inner edge of the outer disc is added to the planet . for @xmath20 , relaxation of this assumption makes negligible difference to the orbital migration of the planet which is set almost entirely by the transfer of angular momentum to the outer disc . we use the instantaneous properties of the disc to compute the mm flux , adopting standard opacity values : @xmath21 such that @xmath22 @xmath23 g@xmath24 at @xmath25 mm and compute the luminosity density ( for a face - on disc ) as : @xmath26 ( beckwith et al 1990 ) where @xmath27 is the planck function and @xmath28 ) is the optical depth . we adopt a simple power law parametrisation of the disc temperature : @xmath29 which is motivated by typical parameters that have been found to provide a fit to the spectral energy distributions of circumstellar discs ( andrews & williams 2005,2007 ; andrews et al . 2011 ; beckwith et al 1990 ) . we explore four model discs , in all case adjusting the normalisation of the surface density profile in order that the initial disc has a @xmath25 mm flux ( scaled to the distance of taurus , i.e. @xmath30 pc ) of around @xmath1 mjy ; thus in each case the initial disc has properties that are typical of the mm bright transition discs with large holes . ( we emphasise that throughout we only consider the mm flux from material that is still in the outer disc , assuming that dust filtration suppresses the mm emission from the leakage flow . ) none of the results presented here depend on the normalisation of the viscosity ( since this determines the time - scale of evolution rather than the relationship between millimetre flux and hole size that we explore here ) . it is however worth noting that if we normalise the viscosity such that the initial accretion rate _ onto the star _ is @xmath31 yr@xmath24 , as observed , then the time - scale on which the hole size shrinks to @xmath32 a.u . is a few myr . we also note that the models do not involve the mass of the star except inasmuch as this would , in practice , affect the temperature normalisation of the disc profile ( which we have taken directly from observations ; andrews et al 2009 ) . .initial model parameters ( see text for details ) [ cols=">,<,<,<",options="header " , ] we list the inner and outer disc radii and total disc mass for each model ( designated e , n , p1 and p2 ) disc in table 1 . the extended ( e ) and narrow ( n ) simulations share the same inner ( ` cavity ' ) radius but differ in their outer radii ; the mean emissivity per unit mass is higher in the narrow model ( on account of its higher mean temperature ) and thus the total disc mass required for a fixed mm flux is somewhat lower . in addition , we compute a couple of variant prescriptions for the mm emission , motivated by the results of recent simulations by pinilla et al ( 2012 ) . these relax the assumption of constant gas to dust ratio and follow the evolution of the grain size distribution and spatial variation of the dust in the case of a disc whose gas density profile is sculpted by a planet . dust is concentrated in the resulting structure within a pressure bump located at about twice the orbital radius of the planet , with the disc being strongly depleted in dust at radii interior to this pressure bump ( we however note that these dust calculations are run for a small fraction of a viscous time and thus - since the disc has not evolved into a steady state - the results should be regarded as somewhat provisional ) . we model this situation by two crude approximations that are intended to bracket the simulation results . in model p1 , the planet orbital radius is halved with respect to the default model ( i.e. @xmath33 a.u . compared with @xmath34 a.u . ) and the region between @xmath34 and @xmath33 a.u . is filled with dust - free gas with a surface density profile that is an extrapolation of the power law profile ; this increases the total gas mass by a factor two with respect to the model n ( which shares the same outer radius ) . the gas hydrodynamics and planetary orbital migration is modelled exactly as before ; as the planet migrates , it is assumed that the inner edge of the dusty disc remains at twice the instantaneous planet orbital radius . in model p2 , the disc gas is again extrapolated to the planet location ( @xmath33 a.u . ) ; emission is again only calculated from outward of the cavity radius ( i.e. twice the instantaneous radius of the planet ) and the initial outer radius is again @xmath35 a.u .. in this case , however , the flux contribution from dust that would have been located between the planetary radius and twice this radius is calculated as optically thin emission at the temperature of the cavity radius . the placing of additional emission at the cavity radius increases the mean emissivity per unit gas mass compared with model n and consequently model p2 has a modestly lower total mass in order to reproduce the same mm flux . in summary , each of these models are designed to reproduce observational parameters ( cavity size of 50 a.u . and mm flux of @xmath36 mjy ) that are typical of large hole ( mm bright ) transition discs . we then evolve the coupled disc - planet system for a range of different planet masses , and track the evolution of the system in the plane of mm flux versus cavity radius . we emphasise that at this stage we generically describe the companions as ` planets ' even though we shall include companions with masses up to @xmath1 jupiter masses . we do not extend our calculations to higher masses on the grounds that type ii migration theory ( which places the centre of mass of the system at the primary star ) becomes inapplicable at higher mass ratios . thus we can not directly address the disparity in mm fluxes between large hole transition discs and stellar binaries . as the planet and disc inner edge migrate inwards , the mm flux changes due to three effects : redistribution of material in radius ( and hence temperature ) , optical depth effects and mass loss from the outer disc due to the leakage flow to the inner disc ( which is assumed not to contribute to the mm flux ) . the two latter effects both result in a reduction in mm flux . the former can change the mm flux in either direction since viscous evolution results in material spreading both inwards and out - i.e. into both hotter and cooler regions . in practice , we find that the net effect is either rough constancy of the mm flux or else a gentle fading as the planet migrates inwards . we find that these two outcomes depend on the relative masses of the planet and the disc . in the case of a planet that is comparable to or less massive than the disc , the planet is conveyed inwards as though it were a representative fluid element in the disc ; the disc structure upstream of the planet is not significantly modified by the planet s presence and the mm flux is nearly constant as the planet and associated disc hole moves inwards . this behaviour is seen in models 100 p1 , 40n and 10 n in figure 1 ( where the number refers to the planet mass - in jupiter masses - and the model designation is defined in table 1 ) . on the other hand , in models where the ` planet ' is more massive than the disc ( such as 100 e , 100 n and 100 p2 in figure 1 ) , the behaviour is somewhat different since the finite inertia of the planet impedes the free viscous migration of the disc inner edge ( lin & papaloizou 1986 , syer & clarke 1995 , ivanov , papaloizou & polnarev 1999 ) . the slower migration means that there is time for a significant depletion of the outer disc by the leakage flow and more than half the initial disc mass has leaked past the planet in these models by the time it reaches @xmath32 a.u .. the mm flux declines by more than a factor two over this time , with additional fading resulting from the disc s expansion to large radii where the temperature - and associated mm emission - is low . figure 1 plots observed transition discs in the plane of mm flux ( scaled to a distance of @xmath30pc ) versus hole size ( see owen & clarke 2012 for details of the mm data which is mainly obtained from the mm surveys of andrews & williams 2005,2007 , henning et al 1993 and nuernberger et al 1997 ) . in the minority of objects that lack @xmath25 mm fluxes , this is converted from @xmath37 m data using the prescription of cieza et al 2008 . the open circles denote systems that have been imaged by brown et al . ( 2009 ) & andrews et al ( 2011 ) and which therefore represent the systems with the largest holes and highest mm fluxes . for the remaining unresolved objects the hole radius is either obtained from detailed sed modelling ( andrews et al . 2012 , calvet et al 2002,2005 , espaillat et al 2007,2010 , kim et al 2009 , merin et al 2010 , najita et al 2007 : shown as open symbols ) or , in the case of filled symbols , is simply estimated from the ` turn - off wavelength ' listed in cieza et al ( 2010 ) ; the hole radius is thus more uncertain in these latter systems . the squares and triangles distinguish mm detections from upper limits . figure 1 illustrates that there is a lack of mm bright objects ( with flux @xmath38 mjy at the distance of taurus ) with hole sizes @xmath39 a.u .. although such objects are obviously not the targets of mm imaging studies , they would have been readily picked up in photometric mm surveys and there are likewise no reasons why such objects would not be identifiable as transition discs from their seds ( see owen & clarke 2012 ) . this observational constraint defines the range of models that provide an acceptable fit to observations . evidently it is only models with a rather large planet to disc mass ratio ( a factor two or more ) that avoid evolving into the forbidden region with high mm flux and small hole size . furthermore , we emphasise those models that remain mm bright at @xmath32 a.u . spend a comparable time with hole sizes in the range @xmath40 a.u . and @xmath41 a.u .. we thus can not appeal to rapid inward migration at @xmath39 a.u . in order to explain the observed lack of objects in the forbidden zone . given the requirement that the disc has to be massive enough to generate the observed mm fluxes of large hole , mm bright systems ( @xmath42 mjy ) this in practice rules out systems in which the ` planet ' is less than @xmath36 jupiter masses ( i.e. it excludes all companions in the planetary mass or brown dwarf regime ) . our results above imply that the model in which large mm bright transition discs are associated with a ` planetary ' companion is viable only if the companion is in fact of stellar mass . this is simply because less massive companions are swept to small radii while the system remains mm bright , thus contradicting the observational dearth of small , mm bright holes . our initial conditions are informed by the observed high mm fluxes of large cavity transition discs so that one can not avoid this conclusion by simply invoking lower mass outer discs . we noted above that we do not model the mass ratio regime of most stellar binary companions . however , our results for a @xmath1 jupiter mass companion do not allow us to explain the observed low mm fluxes in young stellar binaries ( kraus et al 2011,2012 ) since they do not show a strong decline of mm flux at large hole radius . this suggests that the low observed mm flux in stellar binaries may more relate to the consumption of the disc when the binary companion is formed rather than to the evolutionary effects explored here . we find that companions of around @xmath1 jupiter masses provide a good fit to the observed distribution of transition discs in the plane of mm flux versus cavity radius , since such systems fade to less than @xmath43 mjy by the stage that the hole size is @xmath44 a.u .. we are not concerned that such systems would not fade to the lowest mm flux levels among transition discs with small inner holes since a separate mechanism - e.g. photoevaporation - could be invoked to explain the faintest objects . nevertheless , there is an unassailable objection to invoking companions of around @xmath1 jupiter mass : such objects would be readily detected by imaging surveys ( whose current sensitivity levels extend to objects of @xmath44 jupiter masses or lower ( kraus et al 2011 , 2012 ) . the absence of such companions in transition discs , combined with the requirement demonstrated here of a rather massive companion , is a serious challenge to the notion that transition discs are associated with companions in _ any _ mass range . ( see also zhu et al 2011,2012 for other arguments against the planetary hypothesis for the origin of transition discs based on difficulties in reproducing the spectral energy distribution ) . we conclude that the popular planet model for large cavity transition discs is faced with a ` planet mobility problem ' . if we set up a system with an outer disc mass that reproduces the mm flux of large cavity transition discs and set a companion within the cavity , then the planet should migrate inwards by type ii migration and the cavity radius thus shrinks with time . we however find that both planets and brown dwarfs ( i.e. objects less than @xmath36 jupiter masses ) are swept to small radii by type ii migration and that the mm flux of the disc does _ not _ fade significantly during this process . thus we would expect to see an associated population of mm bright objects with small holes ( @xmath39 a.u . ) which are _ not _ observed . we can avoid this outcome by instead invoking a more massive companion ( i.e. a low mass star ) . in this case the migration is slow enough for the disc to fade at mm wavelengths before the hole shrinks to 10 a.u .. however , such massive companions in transition discs are clearly ruled out by recent imaging surveys ( kraus et al 2011,2012 ) . in order to ` rescue ' the planet scenario , we need some mechanism that stops the planet migrating inwards and/or suppresses the production of mm flux as the planet migrates . photoevaporation might appear to be an attractive scenario in both respects ( rosotti et al 2013 ) ; however the initial conditions that are required to match the high mm fluxes of large cavity transition discs imply massive discs , so that the photoevaporation time - scale would be long ( @xmath45 a myr ) even for systems with the highest x - ray luminosity . perhaps a more likely explanation is that there is still much to learn about the secular evolution of coupled planet / disc systems . this issue is particularly acute because the low mass planets that would be compatible with the null results from imaging surveys are in the regime where the leakage flow could play an important role in slowing planetary migration . on the other hand , it is not clear whether dust filtration - as is necessary to produce a transition disc signature - would be effective in the limit that the flow past the planet is almost unimpeded . these are issues which can only be assessed by future 2d/3d simulations exploring the secular evolution of coupled planet / disc systems . we are grateful to the referee , richard alexander , for an insightful report which has helped us improve the paper . 99 alexander , r. , armitage , p. , 2009 . apj 704,989 alexander , r. , pascucci , i. , 2012 mnras 422 , l82 andrews , s. m. , & williams , j. p. 2005 , apj 631 , 113 andrews , s. , williams , j. , 2007 . apj 671,1800 andrews , s. , wilner , d. , hughes , a. , qi , c. , dullemond , c. , 2009 , apj 700,1502 andrews , s. , wilner , d. , espaillat , c. , hughes , a. , dullemond , c. , mcclure , m. , qi , c. , brown , j. , 2011 , apj 732,424 armitage , p. , bonnell , i. , 2002 . mnras 330,l11 beckwith , s.v.w . , sargent , a.i . , chini , r.s . , gsten , r. , 1990 , aj , 99 , 924 brown , j. , blake , g.,qi , c. , dullemond , c. , wilner , d. , williams , j.,2009 . apj 704,496 n. calvet , p. dalessio , l. hartmann , d. wilner , a. walsh , m. sitko , 2002 , apj 568,1008 n. calvet et al , 2005 , ap j 630 , l185 cieza , l. a. , swift , j . j. , mathews , g. s. , & williams , j. p. 2008 , apjl 686 , l115 cieza , l. , matthias , r. , romero , g. , mora , m. , merin , b. , swift , j. , orellano , m. , williams , j. , harvey , p. , evans , n. , 2010 apj 712,925 clarke , c. j. 2009 , mnras 396 , 1066 espaillat , c. , et al . 2007 , apjl 664 , l111 espaillat , c. , dalessio , p. , hernandex , j. , nagel , e. , luhman , k. , watson , d. , calvet , n. , muzerolle , j. , mcclure , m. , 2010 apj 717,441 l. hartmann , n. calvet , e. gullbring , p. dalessio , 1998 , ap j 492,323 henning , t.,pfau , w . , zinnecker , h. , prusti , t. , 1993 . a & a 276,129 p. ivanov , j. papaloizou , a. polnarev , 1999 , mnras 307,79 kim , k. h. , et al . 2009,apj 700 , 1017 kraus , a. , ireland , m. , martinache , f. , hillenbrand , l. , 2011 . apj 2011,731 kraus , a. , ireland , m. , 2012 . apj 745 , 5 kraus , a. , ireland , m. , hillenbrand , martinache , f. , 2012 apj 745,19 d. lin , j. papaloizou , 1986 , apj 309,846 s. lubow , g. dangelo , 2006 , apj 641,526 d. lynden - bell , j. pringle , 1974 , mnras 168,603 merin , n. , et al 2010 apj 718,1200 muzerolle , j. , allen , l. , megeath , s. , hernandez , j. , gutermuth , r. , 2010 apj 708,1107 najita , j. r. , strom , s . e. , & muzerolle , j. 2007 , mnras 378 , 369 nayakshin , s. , 2013 . mnras 431,1432 nuernberger , d. , chini , r. , zinnecker , h. , 1997 . a & a 324,1036 owen , j. , clarke , c. , 2012 mnras 426,96 owen , j. , ercolano , b. , clarke , c. , 2011 mnras 412,13 owen , j. , clarke , c. , ercolano , b. , 2012 mnras 422,1880 pinilla , p. , benisty , m. , birnstiel , t. , 2012 a & a 545,81 rafikov , r. , 2005 . apj 631,488 rice , w. , armitage , p. , wood , k. , lodato , g , 2006 , mnras 373,1619 rosotti , g. , ercolano , b. , owen , j. , armitage , p. , 2013 mnras 430,1392 stamatellos , d. , whitworth , a. , 2008 . a & a 480,879 d. syer , c. clarke , 1995 , mnras 277,758 veras , d. , armitage p. 2004 , mnras , 347 , 613 zhu z. , nelson , r. , hartmann , l. , espaillat , c. , calvet , n. , 2011 , apj 729,47 zhu , z. , nelson , r. , dong , r. , espaillat , c. , hartmann , l. , 2012apj 755,18	we assume a scenario in which transition discs ( i.e. discs around young stars that have signatures of cool dust but lack significant near infra - red emission from warm dust ) are associated with the presence of planets ( or brown dwarfs ) . these are assumed to filter the dust content of any gas flow within the planetary orbit and produce an inner ` opacity hole ' . in order to match the properties of transition discs with the largest ( @xmath0 a.u . scale ) holes , we place such ` planets ' at large radii in massive discs and then follow the evolution of the tidally coupled disc - planet system , comparing the system s evolution in the plane of mm flux against hole radius with the properties of observed transition discs . we find that , on account of the high disc masses in these systems , all but the most massive ` planets ' ( @xmath1 jupiter masses ) are conveyed to small radii by type ii migration without significant fading at millimetre wavelengths . such behaviour would contradict the observed lack of mm bright transition discs with small ( @xmath2 a.u . ) holes . on the other hand , imaging surveys clearly rule out the presence of such massive companions in transition discs . we conclude that this is a serious problem for models that seek to explain transition discs in terms of planetary companions unless some mechanism can be found to halt inward migration and/or suppress mm flux production . we suggest that the dynamical effects of substantial accretion on to the planet / through the gap may offer the best prospect for halting such migration but that further long term simulations are required to clarify this issue . 9grs 1915 + 105 [ firstpage ] accretion , accretion discs : circumstellar matter- planetary systems : protoplanetary discs - stars : pre - main sequence
dna microarrays allow the comparison of the expression levels of all genes in an organism in a single experiment , which often involve different conditions ( _ i.e. _ health - illness , normal - stress ) , or different discrete time points ( _ i.e. _ cell cycle ) @xcite . among other applications , they provide clues about how genes interact with each other , which genes are part of the same metabolic pathway or which could be the possible role for those genes without a previously assigned function . dna microarrays also have been used to obtain accurate disease classifications at the molecular level @xcite . however , transforming the huge amount of data produced by microarrays into useful knowledge has proven to be a difficult key step @xcite . on the other hand , clustering techniques have several applications , ranging from bioinformatics to economy @xcite . particularly , data clustering is probably the most popular unsupervised technique for analyzing microarray data sets as a first approach . many algorithms have been proposed , hierarchical clustering , k - means and self - organizing maps being the most known @xcite . clustering consists of grouping items together based on a similarity measure in such a way that elements in a group must be more similar between them than between elements belonging to different groups . the similarity measure definition , which quantifies the affinity between pairs of elements , introduces _ a priori _ information that determines the clustering solution . therefore , this similarity measure could be optimized taking into account additional data acquired , for example , from real experiments . some works with _ a priori _ inclusion of bioinformation in clustering models can be found in @xcite . in the case of gene expression clustering , the behavior of the genes reported by microarray experiments is represented as @xmath0 points in a @xmath1-dimensional space , being @xmath0 the total number of genes , and @xmath1 the number of conditions . each gene behavior ( or point ) is then described by its coordinates ( its expression value for each condition ) . genes whose expression pattern is similar will appear closer in the @xmath1-space , a characteristic that is used to classify data in groups . in our case , we have used the superparamagnetic clustering algorithm ( spc ) @xcite , which was proposed in 1996 by domany and collaborators as a new approach for grouping data sets . however , this methodology has difficulties dealing with different density clusters , and in order to ameliorate this , we report here some modifications of the original algorithm that improve cluster detection . our main contribution consists on increasing the similarity measure between genes by taking advantage of transcription factors , special proteins involved in the regulation of gene expression . the present paper is organized as follows : in section 2 , the spc algorithm is introduced , as well as our proposal to include further biological information and our considerations for the selection of the most natural clusters . results for a real data set , as well as performance comparisons , are presented in section 3 . finally , section 4 is dedicated to a summary of our results and conclusions . a potts model can be used to simulate the collective behavior of a set of interacting sites using a statistical mechanics formalism . in the more general inhomogeneous potts model , the sites are placed on an irregular lattice . next , in the spc idea of domany _ et al . _ @xcite , each gene s expression pattern is represented as a site in an inhomogeneus potts model , whose coordinates are given by the microarray expression values . in this way , a particular lattice arrangement is spanned for the entire data set being analyzed . a spin value @xmath2 , arbitrarily chosen from @xmath3 possibilities , is assigned to each site , where @xmath4 corresponds to the site of the lattice @xmath5 . the main idea is to characterize the resulting spin configuration by the ferromagnetic hamiltonian : @xmath6 where the sum goes over all neighboring pairs , @xmath2 and @xmath7 are spin values of site @xmath4 and site @xmath8 respectively , and @xmath9 is their ferromagnetic interaction strength . each site interacts only with its neighbors , however since the lattice is irregular , it is necessary to assign the set of nearest - neighbors of each site using the so - called @xmath10-mutual - nearest - neighbor criterion @xcite . the original interaction strength is as follows : @xmath11 with @xmath12 the average number of neighbors per site and @xmath13 the average distance between neighbors . the interaction strength between two neighboring sites decreases in a gaussian way with distance @xmath14 and therefore , sites that are separated by a small distance have more probability of sharing the same spin value during the simulation than the distant sites . on the other hand , said probability , @xmath15 , also depends on the temperature @xmath16 , which acts as a control parameter . at low temperatures , the sites tend to have the same spin values , forming a ferromagnetic system . this configuration is preferred over others because it minimizes the total energy . however , the probability of encountering aligned spins diminishes as temperature increases , and the system could experience either a single transition to a totally disordered state ( paramagnetic phase ) , or pass through an intermediate phase in which the system is partially ordered , which is known as the superparamagnetic phase . in the latter case , varios regions of sites sharing the same spin value emerge . sites within these regions interact among them with a stronger force , exhibiting at the same time weak interactions with sites outside the region . these regions could fragment into smaller grains , leading to a chain of transitions within the superparamagnetic phase until the temperature is so high that the system enters the paramagnetic phase , where each spin behaves independently . this hierarchical subdivision in magnetic grains reflects the organization of data into categories and subcategories . regions of aligned spins emerging during simulation correspond to groups of points with similar coordinates , _ i.e. _ , similar gene expression patterns @xcite . this subdivision can be simulated , for example , by using the monte carlo approach , by which one can compute and follow the evolution of system properties such as energy , magnetization and susceptibility , while the temperature is modified . in addition , the temperature ranges in which each phase transition takes place can be localized . rather than thresholding the distances between pairs of sites to decide their assignment to clusters , the pair correlation @xmath17 , indicating a collective aspect of the data distribution , is preferred . it can be calculated as follows @xcite @xmath18 in this way , @xmath17 is the normalized probability for finding two potts spins @xmath2 and @xmath7 sharing the same value for a given temperature step . if both spins belong to the same ordered region , their correlation value would be close to one , otherwise their correlation would be close to zero @xcite . thus , for each temperature step , two sites are assigned to the same cluster if their correlation exceeds a threshold value of @xmath19 . if a site does not have a single correlation value greater than @xmath20 , it is joined with its neighbor showing the highest value . + for our spctf algorithm , we also accept sites whose @xmath17 are larger than @xmath20 in order to build a cluster . however , differently from the traditional spc algorithm @xcite , if two sites do not reach the @xmath17 value greater than @xmath20 they are not connected . this is because with our data we have found that the original condition led to unnatural growth of some clusters when the temperature is increased . as already mentioned , the data are fragmented in various clusters for each temperature value , and for higher temperatures , the number of clusters increases due to finer and finer segmentation . in order to select the more representative clusters through all temperature steps , we assign a stability value to each obtained cluster , based on its evolution . we define @xmath21 as the number of temperature steps until the system reaches the paramagnetic phase and @xmath22 as the number of temperature steps a cluster @xmath23 survives , while @xmath24 and @xmath25 are defined as the total number of sites and the number of elements in a given cluster , respectively . we assign a stability parameter @xmath26 to each cluster , as follows : @xmath27 where @xmath28 is the fraction of temperature steps a cluster @xmath23 survives , while @xmath29 is the fraction of total elements belonging to @xmath23 . the advantage of using the stability parameter @xmath26 is that it gives preference to clusters that survive several temperatures , but also have an acceptable number of elements . we added a small positive real number @xmath30 to the denominator in the expression of @xmath26 for the special case when @xmath31 , where @xmath32 belongs to the range @xmath33 $ ] , leading to @xmath34 instead of the infinity . it has been reported that the main drawback of the spc algorithm consists of dealing with data showing regions of different density @xcite . in this case , either depending on temperature or the number of neighbors selected , some clusters will easily get prominent whereas the detection of others will be hindered . to overcome this problem , at least two techniques have been proposed _ e.g. _ , sequential superparamagnetic clustering @xcite and a modularity approach @xcite . our idea is to take advantage of already available biological information to improve lattice connectivity in such a way that biologically significant clusters have more probability of being detected by the algorithm . indeed , at the transcriptional level , the expression of a gene could be promoted / suppressed by the binding of the proteins named transcription factors to specific sequences on the gene promoter region . then , if a group of genes shows the same expression behavior in a microarray experiment , it is quite possible that they are being regulated by a specific transcription factor , forming a group of coregulated genes @xcite . thus , available information about which genes are targeted by the same transcription factors may be useful in the detection of groups of genes with similar expression profiles . to make effective this idea , we downloaded from _ www.yeastract.com _ a list of yeast transcription factors that are well documented , and whenever two neighboring genes are controlled by the same transcription factor , we increased their interaction strength . it is important to note that the list provided by _ www.yeastract.com _ includes transcription factors associated with several processes and are not only cell cycle related . the formula that takes this into account replaces eq . ( [ eq : js ] ) of the original algorithm , and has the following form : @xmath35 here , @xmath36 is the number of common transcription factors shared by @xmath4 and @xmath8 ( @xmath32 , which varies for each pair of neighboring genes ) , multiplied by a factor @xmath37 which was chosen to be 2.0 after comparing the results obtained with several other values . the selected value has the characteristic of preserving well - defined susceptibility peaks as well as obtaining larger clusters . the objective is to strengthen some connections without preventing the natural fragmentation of clusters caused by the temperature parameter . if two elements do not share a transcription factor , then @xmath38 , recovering the original spc formula therefore , the modified interaction strength between each site and its neighbors is governed by two aspects : the distance between them , which comes from gene expression values generated through microarray experiments , and the number of transcription factors regulating both genes , obtained from documented biological data . any time two genes share a transcription factor , their interaction strength becomes larger , and this favors that the clusters including these sites remain stable for longer temperature ranges , with the corresponding increase of their stability values . we analyzed spellman _ et al . _ @xcite microarray data in which gene expression values from synchronized yeast cultures were obtained at various time moments , aiming to identify cell cycle genes . yeast cultures were synchronized by three methods : adding alpha pheromone , which arrests cells in the g1 phase ; using centrifugal elutration for separating small g1 cells ; and using a mutation that arrests cells late in mitosis at a given temperature . combining the three experiments and using fourier and correlation algorithms , spellman _ @xcite reported @xmath39 cell cycle regulated genes . the goal was to compare the performance of spc and spc with transcription factors ( spctf ) , which are algorithms that do not make assumptions about periodicity . nonetheless , the overall analysis is time consuming and we only selected the data set treated with the alpha pheromone , available at _ http://cellcycle - www.stanford.edu_. genes with missing values were discarded , leaving an input matrix of @xmath40 genes and @xmath41 time courses that included only @xmath42 of the genes reported by spellman _ _ @xcite . furthermore , as we do not include the other two synchronization experiments , we expect to loose some of their cell cycle genes . it is worth mentioning that getz _ _ @xcite also analyzed the spellman alpha synchronized set with the spc algorithm . they took @xmath43 genes which have characterized functions and introduced a fourier transform to take into account the oscillatory nature of the cell cycle . in our case , however , we decided not to introduce any considerations about the periodicity of the data , mainly because the time series cover only two cell cycle periods @xcite . we obtain compact gene clusters implementing spc original algorithm and spctf , both with parameter values @xmath44 and @xmath45 . the cluster with the highest stability value contains an extremely large number of elements without a clear biological linkage between them . it is mainly composed of genes whose expression do not change significantly over time , thus it is possible that they are included here for this very reason . we discard this cluster from our analysis , although it could always be taken apart and analyzed again with spctf by choosing the appropiate number of neighbors to obtain more information . to compare in more detail both approaches , it is necessary to correlate each cluster in the spc method with its equivalent in spctf . in order to do this , we calculate the euclidian distance between the mean position vector of every cluster in each approach , and choose the pairs with the shortest distance between them . ( we recall that the mean position vector of a cluster is obtained by averaging each coordinate between all its elements ) . although different measures could have been used , this one performed adequately , as can be seen in the supplementary information file , where we provide a more detailed comparison between spctf and spc clusters . in table [ tabla1 ] , we present the differences in cluster size as well as the hits , the number of genes reported by spellman _ @xcite , which have been included in the clusters . when going through the spctf approach , one can see that the first largest cluster looses some genes , while the number of the rest of the clusters augments . besides , hits or coincidences with spellman _ et al . _ @xcite cell cycle genes in clusters of six or more elements increase by @xmath46 , from @xmath47 to @xmath48 . therefore , we were able to incorporate several genes to these clusters , mainly from outliers . * comparison between spc and spctf * in the following analysis , we focus on clusters of six or more elements , because we are interested in finding groups of several genes sharing the same expression pattern ( coregulated genes ) . results of the comparison for the first @xmath49 most stable clusters , discarding the first one , are shown in fig . [ fig : todos_cc ] . generally , these clusters incorporate more elements with spctf , including more cell cycle genes as those reported by spellman _ _ @xcite and thus improving the matching . clusters , discarding the first one . gray bars correspond to the clusters obtained with the spc algorithm and black bars to the equivalent clusters in spctf . groups tend to increase in size and also in hits with cell cycle genes reported by spellman _ @xcite , with the exception of cluster @xmath50.,height=211 ] depending on the available information about the genes , we classify the clusters in three groups . the first cluster type , cell cycle genes , cc , corresponds to groups formed in their majority ( @xmath51 ) by already reported cell cycle genes ( fig . [ fig : cc_sp ] ) . the second type , mixed genes , m , contains clusters with non - reported genes as well as already known cell cycle genes ( fig . [ fig : m_n ] ) , and in the third type , no hits , n , we include the clusters that contain only one hit or are entirely composed of non - previously identified cell cycle genes ( fig . [ fig : m_n ] ) . it is worth mentioning that more cell cycle experiments have been done since spellman _ _ @xcite and new genes have been classified meanwhile as cell cycle regulated . some of these newly reported cell cycle genes were obtained by cho _ @xcite , pramila _ et al . _ @xcite , rowicka _ et al . _ @xcite and lichtenberg _ et al . we analize our @xmath49 clusters taking now as hits , genes reported either by spellman _ @xcite or by one of the above mentioned studies . in this way , we gained thirty additional hits in the spc clusters , while in spctf clusters we have fifty - two extra genes . the results including all the aforementioned cell cycle studies are presented in figs . [ fig : all_studies1][fig : m_n_all ] @xcite . most stable clusters . hits are now taken as cell cycle genes reported by all studies . gray bars correspond to the clusters obtained with the spc algorithm and black bars to the equivalent clusters in spctf.,height=211 ] in addition , we analyze the expression profiles of the genes conforming each cluster using the sceptrans tool @xcite , and we notice that all the genes grouped in the same cluster had the same expression pattern . this gives us further confidence that our algorithm is grouping data correctly . the expression profiles for a representative member of each cluster type are shown in fig . [ fig : exp_pro ] . we also find two clusters ( @xmath52 and @xmath49 ) that present an oscillating behaviour that is due to an artifact in the manner the microarray experiment was performed , see @xcite . in the supplementary information file , we include the list of oscillating genes identified in @xcite and the number of these genes inside each of our first @xmath49 clusters . we also include the expression profiles of these clusters as well as those of size @xmath53 and @xmath54 which contain hits with cell cycle genes identified by spellman _ these clusters have also similar expression profiles but were not further analyzed because of their low number of elements . in the case of gene annotation , it is important to have clusters of many elements to effectively assure that an unknown gene shares the biological function already assigned to the other genes in the same cluster . + + + the cc clusters are almost entirely composed of cell cycle regulated genes reported either by spellman _ @xcite or by other authors , besides , their expression patterns are similar , which leaves no doubt on their validity . for the m and n clusters , we know that they are well grouped because their elements share the same expression patterns , but in order to select those of worth for further analysis ( for example in a laboratory experiment ) we analyze them through musa , motif finding using an unsupervised approach algorithm , that can be found at _ www.yeastract.com_. this program searches for the most common sequences ( motifs ) in the regulatory region of a set of genes , and compare them to the transcription factor binding sites already described in yeastract database @xcite . results of this analysis are shown in table [ tabla2 ] , which includes the quorum or percentage of genes containing a motif in each cluster , and the alignment score , which quantifies the level of similarity between the encountered motif and the known transcription factor associated with it . the clusters that probably would give us the best results would be those associated with cell cycle transcription factors with high percentages and scores . we select in this way , the clusters @xmath55 , @xmath53 , @xmath56 , @xmath57 , @xmath58 and @xmath59 because they have percentages higher than @xmath60 and scores higher than @xmath61 . in order to validate the musa analysis , we also constructed various clusters with sizes ranging from six to thirty - seven genes that were composed by genes selected at random from the original data . when analyzing these random clusters in the same way in musa , we obtain at most two cell cycle transcription factor coincidences . * musa analysis * large amounts of biological information are constantly obtained by throughput techniques and clustering algorithms have taken an important place in the unraveling of this information . however , the clustering analyses offer a difficult challenge because any data set can be grouped in numerous ways , depending on the level of resolution asked for and the applied similarity measure . in this work , we propose the use of available biological information in order to strengthen the interaction between genes which share a transcription factor involved in any metabolic process , improving the similarity measure . this information is introduced in the natural evolution of the spc algorithm , and in this way , we are able to enhance the creation and endurance of groups of possible coregulated genes . as the network spanned by the transcription factors information connects all genes , clustering directly _ a posteriori _ using only this information in the present case results into a single massive cluster ( see section iv of the supplementary information ) . however , by having the distance play an important weight in the interaction formula , the far - located clusters will not join , despite sharing transcription factors between their genes . with this in mind , we have modified the spc algorithm , and applied both the original and modified spctf algorithm to one of the three spellman _ @xcite data sets of the yeast cell cycle . the expression profiles of the genes in all resulting clusters show a similar behavior , but we obtain larger clusters with spctf . we classified them in three types , cc , m , and n , depending on the amount of cell cycle reported elements inside each cluster . with spctf , the cc type clusters increase in size including more cell cycle genes , and for the m and n type clusters , we also looked for common sequences in its regulatory regions and selected various groups worth of further research in order to report possible new cell cycle genes . as expected , some of these clusters include already known cell cycle genes sharing a transcription factor , _ but more importantly , at the predictive level , they promote the inclusion of new genes with similar expression patterns_. it is also important to note that the modified algorithm can be applied to any data set , and the followed methodology leads to the selection of the potential gene subsets feasible to be experimentally investigated . our work can serve as an example of how the inclusion of available biological information , such as transcription factors , and bioinformatic tools , such as musa , can lead to better and more confident results , aiding in the analysis of data coming from microarray experiments . the authors thank drs . s. ahnert and g. sherlock for useful discussions and comments . we also thank conacyt for providing support for two of the authors ( m.p.m.a . and j.c.n.m . ) and also the referees for helpful remarks and information . this work was partly supported through the project sep - conacyt-2005 - 49039 . p. t. monteiro , n. d. mendes , m. c. teixeira , s. dorey , s. tenreiro , n. p. mira , h. pais , a. p. francisco , a. m. carvalho , a. b. lourenco , i. sa - correia , a. l. oliveira , and a. t. freitas , nucleic acids res . * 36 * , d132 ( 2008 ) .	in this work , we modify the superparamagnetic clustering algorithm ( spc ) by adding an extra weight to the interaction formula that considers which genes are regulated by the same transcription factor . with this modified algorithm that we call spctf , we analyze spellman _ et al . _ microarray data for cell cycle genes in yeast , and find clusters with a higher number of elements compared with those obtained with the spc algorithm . some of the incorporated genes by using spcft were not detected at first by spellman _ et al . _ but were later identified by other studies , whereas several genes still remain unclassified . the clusters composed by unidentified genes were analyzed with musa , the motif finding using an unsupervised approach algorithm , and this allow us to select the clusters whose elements contain cell cycle transcription factor binding sites as clusters worth of further experimental studies because they would probably lead to new cell cycle genes . finally , our idea of introducing available information about transcription factors to optimize the gene classification could be implemented for other distance - based clustering algorithms . superparamagnetic clustering , similarity measure , microarrays , cell cycle genes , transcription factors . + paper - physa-3 20100901.tex physica a 389(24 ) , 5689 - 5697 ( 2010 ) + doi : 10.1016/j.physa.2010.09.006
cross - layer design of congestion control , routing and scheduling algorithms with quality of service ( qos ) guarantees is one of the most challenging topics in wireless networking . the back - pressure algorithm first proposed in @xcite and its extensions have been widely employed in developing throughput optimal dynamic resource allocation and scheduling algorithms for wireless systems . back - pressure - based scheduling algorithms have also been employed in wireless networks with time - varying channels @xcite@xcite@xcite . congestion controllers at the transport layer have assisted the cross - layer design of scheduling algorithms in @xcite@xcite@xcite , so that the admitted arrival rate is guaranteed to lie within the network capacity region . low - complexity distributed algorithms have been proposed in @xcite@xcite@xcite@xcite . algorithms adapted to clustered networks have been proposed in @xcite to reduce the number of queues maintained in the network . however , delay - related investigations are not included in these works . in this paper , we propose a cross - layer algorithm to achieve _ guaranteed throughput _ while satisfying network qos requirements . specifically , we construct two virtual queues , i.e. , a _ virtual queue at transport layer _ and a _ virtual delay queue _ , to _ guarantee average end - to - end delay bounds_. moreover , we construct a _ virtual service queue _ to _ guarantee the minimum data rate required by individual network flows_. our cross - layer design includes a congestion controller for the input rate to the virtual queue at transport layer , as well as a joint policy for packet admission , routing , and resource scheduling . we show that our algorithm can achieve a throughput arbitrarily close to the optimal . in addition , the algorithm exhibits a tradeoff of @xmath1 in the delay bound , where @xmath0 denotes the distance from the optimal throughput . our main algorithm is further extended : @xmath2 to a set of low - complexity suboptimal algorithms ; @xmath3 from a model with constantly - backlogged sources to a model with sources of arbitrary input rates at transport layer ; @xmath4 to an algorithm employing delayed queue information ; and @xmath5 from a node - exclusive model with constant link capacities to a model with arbitrary link capacities and interference models over fading channels . the rest of the paper is organized as follows : section ii discusses the related work . in section iii , the network model is presented , followed by corresponding approaches for the considered multi - hop wireless networks . in section iv , the optimal cross - layer control and scheduling algorithm is described , and its performance analyzed . in section v , we provide a class of feasible suboptimal algorithms , consider sources with arbitrary arrival rates at transport layer , employ delayed queue information in the scheduling algorithm , and extend the model to arbitrary link capacities and interference models over fading channels . we present numerical results in section vi . finally , we conclude our work in section vii . delay issues in single - hop wireless networks have been addressed in @xcite-@xcite . especially , the scheduling algorithm in @xcite provides a throughput - utility that is inversely proportional to the delay guarantee . authors of @xcite have obtained delay bounds for two classes of scheduling policies . a random access algorithm is proposed in @xcite for lattice and torus interference graphs , which is shown to achieve order - optimal delay in a distributed manner with optimal throughput . but these works are not readily extendable to multi - hop wireless networks , where additional arrivals from neighboring nodes and routing must be considered . delay analysis for multi - hop networks with fixed - routing is provided in @xcite . delay - related scheduling in multi - hop wireless networks have been proposed in @xcite@xcite@xcite@xcite@xcite . however , none of the above - mentioned works provide explicit end - to - end delay guarantees . there are several works aiming to address end - to - end delay or buffer occupancy guarantees in multi - hop wireless networks . worst - case delay is guaranteed in @xcite with a packet dropping mechanism . however , dropped packets are not compensated or retransmitted with the algorithm of @xcite , which may lead to restrictions in its practical implementations . a low - complexity cross - layer fixed - routing algorithm is developed in @xcite to guarantee order - optimal average end - to - end delay , but only for half of the capacity region . a scheduling algorithm for finite - buffer multi - hop wireless networks with fixed routing is proposed in @xcite and is extended to adaptive - routing with congestion controller in @xcite . specifically , the algorithm in @xcite guarantees @xmath1-scaling in buffer size with a @xmath0-loss in throughput - utility , but this is achieved at the expense of the buffer occupancy of the source nodes , where _ an infinite buffer size _ in the network layer is assumed in each source node . this leads to large average end - to - end delay since the network stability is achieved based on queue backlogs at these source nodes . compared to the above works , the algorithm presented in this paper develops and incorporates novel virtual queue structures . different from traditional back - pressure - based algorithms , where the network stability is achieved at the expense of large packet queue backlogs , in our algorithm , `` the burden '' of actual packet queue backlogs is shared by our proposed virtual queues , in an attempt to guarantee specific delay performances . specifically , we design a congestion controller for _ a virtual input rate _ and assign weights in the scheduling policy as a product of actual packet queue backlog and the weighted backlog of a designed virtual queue , which will be introduced in detail in section iv . as such , the network stabilization is achieved with the help of virtual queue structures that do not contribute to delay in the network . since _ all packet queues _ in the network , including those in source nodes , have finite sizes , all average end - to - end delays are bounded independent of length or multiplicity of paths . we consider a time - slotted multi - hop wireless network consisting of @xmath6 nodes and @xmath7 flows . denote by @xmath8 a link from node @xmath9 to node @xmath10 , where @xmath11 is the set of directed links in the network . denoting the set of flows by @xmath12 and the set of nodes by @xmath13 , we formulate the network topology @xmath14 . note that we consider adaptive routing scenario , i.e. , the routes of each flow are not determined _ a priori _ , which is more general than fixed - routing scenario . in addition , we denote the source node and the destination node of a flow @xmath15 as @xmath16 and @xmath17 , respectively . we assume that the source node for flow @xmath18 is always backlogged at the transport layer . let the scheduling parameter @xmath19 denote the link rate assignment of flow @xmath18 for link @xmath20 at time slot @xmath21 according to scheduling decisions and let @xmath22 denote the admitted rate of flow @xmath18 from the transport layer of flow to the source node , where @xmath23 denotes the source at the transport layer of flow @xmath18 . it is clear that in any time slot @xmath21 , @xmath24 @xmath25 . for simplicity of analysis , we assume only one packet can be transmitted over a link in one slot , so @xmath26 takes values in @xmath27 @xmath28 . we also assume that @xmath29 is bounded above by a constant @xmath30 : @xmath31 i.e. , a source node can receive at most @xmath32 packets from the transport layer in any time slot . to simplify the analysis , we prevent looping back to the source , i.e. , we impose the following constraints @xmath33 we employ the node - exclusive model in our analysis , i.e. , each node can communicate with at most one other node in a time slot . note that our model is extended to arbitrary interference models with arbitrary link capacities and fading channels in section v.d . we now specify the qos requirements associated with each flow . the network imposes an _ average end - to - end delay threshold _ @xmath34 for each flow @xmath18 . the end - to - end delay period of a packet starts when the packet is admitted to the source node from the transport layer and ends when it reaches its destination . note that the delay threshold is a time - averaged upper - bound , not a deterministic one . in addition , each flow @xmath18 requires a minimum data rate of @xmath35 packets per time slot . for convenience of analysis , we define @xmath36 , where the pair @xmath37 can be considered as a virtual link from transport layer to the source node . we now model queue dynamics and network constraints in the multi - hop network . let @xmath38 be the backlog of the total amount of flow @xmath18 packets waiting for transmission at node @xmath10 . for a flow @xmath18 , if @xmath39 then @xmath40 @xmath41 ; otherwise , the queue dynamics is as follows : @xmath42^+&\\ + & \sum_{j : ( j , n)\in \mathcal{l}^c}\mu_{jn}^{c}(t ) , \mbox { if } n\in \mathcal{n}\backslash d(c),&\\ \end{aligned}\end{aligned}\ ] ] where the operator @xmath43^+$ ] is defined as @xmath43^+=\max\{x,0\}$ ] . note that in ( [ eq:1 ] ) , we ensure that the actual number of packets transmitted for flow @xmath18 from node @xmath10 does not exceed its queue backlog , since a feasible scheduling algorithm may not depend on the information on queue backlogs . the terms @xmath44 and @xmath45 represent , respectively , the scheduled departure rate from node @xmath10 and the scheduled arrival rate into node @xmath10 by the scheduling algorithm with respect to flow @xmath18 . note that ( [ eq:1 ] ) is an inequality since the arrival rates from neighbor nodes may be less than @xmath46 if some neighbor node does not have sufficient number of packets to transmit . since we employ the node - exclusive model , we have @xmath47\leq 1 \mbox { , } \forall n\in\mathcal{n}.\ ] ] from ( [ e:3])([e:10 ] ) , we also have @xmath48 if it is ensured that no packets will be looped back to the source . now we construct three kinds of virtual queues , namely , virtual queue @xmath49 at transport layer , virtual service queue @xmath50 at sources , and virtual delay queue @xmath51 , to later assist the development of our algorithm : + @xmath2 for each flow @xmath18 at transport layer , we construct a virtual queue @xmath49 which will be employed in the algorithm proposed in the next section . we denote the virtual input rate to the queue as @xmath52 at the end of time slot @xmath21 and we upper - bound @xmath52 by @xmath32 . let @xmath53 denote the time - average of @xmath52 . we update the virtual queue as follows : @xmath54^+ + r_c(t),\end{aligned}\ ] ] where the initial @xmath55 . considering the admitted rate @xmath22 as the service rate , if the virtual queue @xmath49 is stable , then the time - average admitted rate @xmath56 of flow @xmath18 satisfies : @xmath57 + @xmath3 to satisfy the minimum data rate constraints , we construct a virtual queue @xmath50 associated with flow @xmath18 as follows : @xmath58^+ + a_c,\ ] ] where the initial @xmath59 . considering @xmath35 as the arrival rate and @xmath52 as the service rate , if queue @xmath50 is stable , we have : @xmath60 . additionally , if @xmath49 is stable , then according to ( [ e:0 ] ) , the minimum data rate for flow @xmath18 is achieved . + @xmath4 to satisfy the end - to - end delay constraints , we construct a virtual delay queue @xmath51 for any given flow @xmath18 as follows : @xmath61^++\sum_{n\in \mathcal{n}}u_{n}^c(t)\ ] ] where the initial @xmath62 . considering the packets kept in the network in time slot @xmath21 , i.e. , @xmath63 , as the arrival rate and @xmath64 as the service rate , and according to queueing theory , if queue @xmath51 is stable , we have @xmath65 furthermore , if @xmath49 is stable , then according to ( [ e:0 ] ) , we have : @xmath66 in addition , by little s theorem , ( [ eq:3 ] ) ensures that the average end - to - end delay of flow @xmath18 is less than or equal to the threshold @xmath34 with probability ( w.p . ) @xmath67 . from the above description , we know that the network is _ stable _ ( i.e. , each queue at all nodes is stable ) and the average end - to - end delay constraint and minimum data rate requirement are achieved if queues @xmath38 and the three virtual queues are stable for any node and flow , i.e. , @xmath68 @xmath69 @xmath70 now we define the capacity region of the considered multi - hop network . an arrival rate vector @xmath71 is called _ admissible _ if there exists some scheduling algorithm ( without congestion control ) under which the node queue backlogs ( not including virtual queues ) are stable . we denote @xmath72 to be the capacity region consisting of all admissible @xmath71 , i.e. , @xmath72 consists of all feasible rates stabilizable by some scheduling algorithm _ without _ considering qos requirements ( i.e. , delay constraints and minimum data rate constraints ) . to assist the analysis in the following sections , we let @xmath73 denote the solutions to the following optimization problem : @xmath74 @xmath75 where @xmath0 is a positive number which can be chosen arbitrarily small . for simplicity of analysis , we assume that @xmath76 is in the interior of @xmath72 and without loss of generality , we assume that there exists @xmath77 such that @xmath78 @xmath79 . according to @xcite , we have @xmath80 where @xmath81 is the solution to the following optimization : @xmath82 now we propose a control and scheduling algorithm _ * alg * _ for the introduced multi - hop model so that _ * alg * _ stabilizes the network and satisfies the delay constraint and minimum data rate constraint . given @xmath0 , the proposed _ * alg * _ can achieve a throughput arbitrarily close to @xmath83 , under certain conditions related to delay constraints which will be later given in theorem 1 . the optimal algorithm _ * alg * _ consists of two parts : a congestion controller of @xmath52 , and a joint packet admission , routing and scheduling policy . we propose and analyze the algorithm in the following subsections . let @xmath84 be a control parameter for queue length . we first propose a congestion controller for the input rate of virtual queues at transport layer : * 1 ) congestion controller of @xmath52 * : @xmath85 where @xmath86 is a control parameter . specifically , when @xmath87 , @xmath52 is set to zero ; otherwise , @xmath88 . after performing the congestion control , we perform the following joint policy for packet admission , routing and scheduling ( abbreviated as _ scheduling policy _ ) : * 2 ) scheduling policy * : in each time slot , with the constraints of the underlying interference model as described in section iii including ( [ e:3])([e:10])([e:4 ] ) , the network solves the following optimization problem : @xmath89 @xmath90 @xmath91 where @xmath92 and @xmath93 are defined as follows : @xmath94 @xmath95^+,\ ] ] with weight assignment as follows @xmath96,\mbox { if } ( m , n)\in \mathcal{l } , & \\ & \frac{u_{s(c)}^c(t)}{q_m}[q_m-\mu_m - u_{b(c)}^c(t)],&\\ & \qquad\qquad\qquad\mbox { if } ( m , n)=(s(c),b(c ) ) , & \\ & 0 , \qquad\qquad\quad\mbox { otherwise . } & \\ \end{aligned } \right.\end{aligned}\ ] ] in addition , when @xmath97 , without loss of optimality , we set @xmath98 @xmath79 to maximize ( [ eq:11 ] ) . note that @xmath99 forms the @xmath20 pairs in @xmath26 over which the optimization ( [ eq:11 ] ) is performed . thus , the optimization is a typical maximum weight matching ( mwm ) problem . we first decouple flow scheduling from the mwm . specifically , for each pair @xmath20 , the flow @xmath92 is fixed as the candidate for transmission . we then assign the weight as @xmath93 . note also that although similar product form of the weight assignment ( [ e:6 ] ) have been utilized in @xcite@xcite , no virtual queues are involved there . whereas in _ * alg * _ , we assign weights as a product of weighted virtual queue backlog ( @xmath100 ) and the actual back - pressure , in an aim to shift the burden of the actual queue backlog to the virtual backlog . to analyze the performance of the algorithm , we first introduce the following proposition . employing _ * alg * _ , each queue backlog in the network has a deterministic worst - case bound : @xmath101 now we present our main results in theorem 1 . _ remark 1 ( network stability ) _ : the inequalities ( [ eq:26 ] ) from proposition 1 and ( [ eq:14 ] ) from theorem 1 indicate that _ * alg * _ stabilizes the actual and virtual queues . as an immediate result , _ * alg * _ stabilizes the network and satisfies the average end - to - end delay constraint and the minimum data rate requirement . in addition , proposition 1 states that the actual queues are _ deterministically _ bounded by @xmath102 , which ensures finite buffer sizes for all queues in the network , including those in source nodes . _ remark 2 ( optimal utility and delay analysis ) : _ since @xmath103 are stable , the inequality ( [ eq:16 ] ) gives a lower - bound on the throughput that _ * alg * _ can achieve . given some @xmath104 , since @xmath105 is independent of @xmath106 , ( [ eq:16 ] ) also ensures that _ * alg * _ can achieve a throughput arbitrarily close to @xmath83 . when @xmath0 tends to @xmath107 , _ * alg * _ can achieve a throughput arbitrarily close to the optimal value @xmath108 with the tradeoff in queue backlog upper - bound @xmath102 and the delay constraints @xmath109 , both of which are lower - bounded by the reciprocal terms of @xmath0 as shown in ( [ eq:7 ] ) in theorem 1 . _ in other words , the average end - to - end delay bound is of order @xmath1_. we note that in _ * alg * _ , the control parameter @xmath106 , which is typically chosen to be large , does not affect the actual queue backlog upper - bound or the average end - to - end delay bound , but only affects the upper - bound of the virtual queue backlogs ( shown in ( [ eq:14 ] ) ) . in comparison , in the algorithm proposed in @xcite , the authors show that the internal buffer size is deterministically bounded with order @xmath1 , but _ at the expense of _ the buffer occupancy at source nodes which is of order @xmath110 , where @xmath106 has to be large enough for their algorithm to approach @xmath83 . this design assumes an _ infinite buffer size _ at source nodes and typically results in congestion at the source nodes as shown in the simulation results in @xcite , which further induces an unguaranteed and large average end - to - end delay . moreover , one can expect that there are no buffer - size guarantees for single - hop flows by employing the algorithm in @xcite . in contrast , in our proposed _ * alg * _ , we shift `` the burden of @xmath106 '' from actual queues to virtual queues and ensure that the average end - to - end delay constraints are satisfied with finite buffer sizes for all actual _ packet _ queues . _ remark 3 ( implementation issues ) _ : to update the virtual queue @xmath51 and perform the @xmath52 congestion controller at the transport layer , the queue backlog information of flow @xmath18 is crucial . this information can be collected back to the source node by piggy - backing it on ack from each node . in order to account for such delay of queue backlog information , the @xmath52 congestion controller ( [ eq:25 ] ) of the algorithm can employ delayed queue backlog of @xmath51 . similarly , delayed queue backlog information of @xmath49 can be employed at the weight assignment ( [ e:6 ] ) of the scheduling policy . the modified algorithm and its validity are further discussed in section v.c . by employing delayed queue backlog information , we can extend the algorithm to distributed implementation in much the same way as in @xcite@xcite to achieve _ a fraction _ of the optimal throughput . in order to achieve a throughput arbitrarily close to the optimal value with distributed implementation , we can employ random access techniques @xcite@xcite in the scheduling policy with fugacities @xcite chosen as exp@xmath111^+}{q_m}\}$ ] for each link @xmath112 , where @xmath113 is a local estimate ( e.g. , delayed information ) of @xmath49 and @xmath114 a positive weight . it can be shown that the distributed algorithm can still achieve an average end - to - end delay of order @xmath1 with the time - scale separation assumption @xcite@xcite@xcite . a variation of such distributed implementation in single - hop networks can be found in our recent work @xcite . we prove theorem 1 in the following subsection . before we proceed , we present the following lemmas which will assist us in proving theorem 1 . the proof of lemma 1 is trivial and omitted . we will later use lemma 1 to simplify virtual queue dynamics . note that it is not necessary for the randomized algorithm stat to satisfy the average end - to - end delay constraints . similar formulations of stat and their proofs have been given in @xcite and @xcite , so we omit the proof of lemma 2 for brevity . _ remark 4 _ : according to the stat algorithm in lemma 2 , we assign the input rates of the virtual queues at transport layer as @xmath115 . thus , we also have @xmath116 . according to the update equation ( [ eq:4 ] ) , it is easy to show that the virtual queues under stat are bounded above by @xmath32 and the time - average of @xmath117 satisfies : @xmath118 . note that @xmath119 can take values as @xmath73 or @xmath120 or @xmath121 , where we recall @xmath122 and @xmath123 @xmath79 . to prove theorem 1 , we first let @xmath124 and define the lyapunov function @xmath125 as follows : @xmath126 it is obvious that @xmath127 . we denote the lyapunov drift by @xmath128 where we recall that @xmath129 and we employ lemma 1 to deduce the second inequality . from ( [ e:16 ] ) , we have @xmath130 where we employ the fact deduced from ( [ e:4])([e:5 ] ) that @xmath131 and @xmath132 when @xmath133 and @xmath134 when @xmath135 . note that we use the summation index @xmath136 and @xmath137 interchangeably for convenience of analysis . from the queue length dynamics ( [ eq:4 ] ) and by employing lemma 1 , we have : @xmath138 from the virtual queue dynamics ( [ eq:2 ] ) , we have : @xmath139 from the virtual queue dynamics ( [ e:1 ] ) , we have : @xmath140 substituting ( [ e:11])([eq:6])([e:12])([e:13 ] ) into the lyapunov drift ( [ eq:12 ] ) and subtracting @xmath141 from both sides , we then have : @xmath142 we can rewrite the last term of rhs of ( [ e:14 ] ) by simple algebra as @xmath143 then , the second term and the last term of the rhs of ( [ e:14 ] ) are minimized by the congestion controller ( [ eq:25 ] ) and the scheduling policy ( [ eq:11 ] ) , respectively , over a set of feasible algorithms including the stationary randomized algorithm stat introduced in lemma 2 and remark 4 . we can substitute into the second term of rhs of ( [ e:14 ] ) a stationary randomized algorithm with admitted arrival rate vector @xmath73 and into the last term with a stationary randomized algorithm with admitted arrival rate vector @xmath120 . thus , we have : @xmath144 when ( [ eq:7 ] ) holds , we can find @xmath145 such that @xmath146 @xmath147 and @xmath148 . recall that @xmath149 is defined such that @xmath123 @xmath79 . thus , we have : @xmath150 where @xmath151 . we take the expectation with respect to the distribution of @xmath152 on both sides of ( [ eq:10 ] ) and take the time average on @xmath153 , which leads to @xmath154 since @xmath155 is bounded above ( say , by a constant @xmath156 with @xmath157 ) and @xmath158 is nonnegative , by taking limsup of @xmath21 on both sides of ( [ eq:13 ] ) , we have : @xmath159&\\ \leq&\frac{b'}{\delta},&\\ \end{aligned}\end{aligned}\ ] ] which proves ( [ eq:14 ] ) . by taking liminf of @xmath21 on both sides of ( [ eq:13 ] ) , we have @xmath160 which proves ( [ eq:16 ] ) since the first term of the rhs of ( [ eq:17 ] ) is nonnegative . solving mwm optimization problem can be np - hard depending on the underlying interference model as indicated in @xcite . in this section , we introduce a group of suboptimal algorithms that aim to achieve at least a @xmath161 fraction of the optimal throughput . we denote the scheduling parameters of suboptimal algorithms by @xmath162 . for convenience , we also denote the scheduling parameters of _ * alg * _ by @xmath163 . algorithms are called _ suboptimal _ if the scheduling parameters @xmath162 satisfy the following : @xmath164 where @xmath165 is constant and we recall that @xmath92 and @xmath93 are defined in section iv.a . in addition , the congestion controller of suboptimal algorithms is the same as that of _ * alg * _ ( [ eq:25 ] ) . following the same analysis of _ * alg * _ , proposition 1 holds for suboptimal algorithms , i.e. , the queue backlogs are bounded above by @xmath102 , and we derive the following theorem : _ remark 5 _ : from theorem 2 , given an arbitrarily small @xmath0 , we show that a suboptimal algorithm can _ at least _ achieve a throughput arbitrarily close to a fraction @xmath161 of the optimal results @xmath83 . suboptimal algorithms include the well - known greedy maximal matching ( gmm ) algorithm @xcite with @xmath166 as well as the solutions to the maximum weighted independent set ( mwis ) optimization problem such as gwmax and gwmin proposed in @xcite with @xmath167 , where @xmath168 is the maximum degree of the network topology @xmath169 . the delay bound and throughput tradeoff in theorem 1 still hold in theorem 2 . note that in the previous model description , we assumed that the flow sources are constantly backlogged , that is , the congestion controller ( [ eq:25 ] ) can always guarantee @xmath88 when @xmath170 . in this subsection , we present an optimal algorithm when the flows have arbitrary arrival rates at the transport layer . let @xmath171 denote the arrival rate of flow @xmath18 packets at the beginning of the time slot @xmath21 at the transport layer . we assume that @xmath171 is i.i.d . with respect to @xmath21 with mean @xmath172 . for simplicity of analysis , we assume @xmath173 to be in the exterior of the capacity region @xmath72 so that a congestion controller is needed and we assume that @xmath171 is bounded above by @xmath32 @xmath79 . is bounded above by some constant @xmath174 @xmath79 , where @xmath175 . ] let @xmath176 denote the backlog of flow @xmath18 data at the transport layer which is updated as follows : @xmath177^+,l_m\},\ ] ] where @xmath178 is the buffer size for flow @xmath18 at the transport layer . note that we have @xmath179 and @xmath180 if there is no buffer for flow @xmath18 at the transport layer . following the idea introduced in @xcite , we construct a virtual queue @xmath181 and an auxiliary variable @xmath182 for each virtual input rate @xmath52 , with queue dynamics for @xmath181 as follows @xmath183^+ + v_c(t),\ ] ] where initially we have @xmath184 . the intuition is that @xmath182 serves as the function of @xmath52 in congestion controller ( [ eq:25 ] ) and we note that when @xmath181 is stable , we have @xmath185 , where @xmath186 is the time average rate for @xmath182 , recalling that @xmath53 is the time average rate for @xmath52 . thus , when @xmath181 and @xmath49 are stable , if we can ensure the value @xmath187 is arbitrarily close to the optimal value @xmath188 , then so is the throughput @xmath189 since @xmath190 . now we provide the optimal algorithm for arbitrary arrival rates at the transport layer : * 1 ) congestion controller * : @xmath191 @xmath192 @xmath193 where @xmath194 is a weight associated with the virtual queue @xmath181 . note that ( [ eq:22 ] ) and ( [ eq:23 ] ) can be solved independently . specifically , when @xmath195 , @xmath182 is set to zero ; otherwise , @xmath196 . when @xmath197 , @xmath52 is set to zero ; otherwise , @xmath198 . * 2 ) scheduling policy * : the scheduling algorithm is the same as that of _ * alg * _ provided in section iv.b , except for the updated constraints : @xmath199 . since the scheduling policy is not changed , proposition 1 still holds . and we present a theorem below for the performance of the algorithm : theorem 3 shows that optimality is preserved and @xmath1 delay scaling is kept . recall that in _ * alg * _ , congestion controller ( [ eq:25 ] ) is performed at the transport layer and link weight assignment in ( [ e:6 ] ) is performed locally at each link . thus , in order to account for the propagation delay of queue information , we employ delayed queue backlog of @xmath200 in ( [ eq:25 ] ) and employ delayed queue backlog of @xmath103 for links in @xmath11 in ( [ e:6 ] ) . specifically , we rewrite ( [ eq:25 ] ) in _ * alg * _ as : @xmath201 where @xmath202 is an integer number that is larger than the maximum propagation delay from a source to a node , and we rewrite ( [ e:6 ] ) as : @xmath203,&\\ & \qquad\qquad\qquad\mbox { if } ( m , n)\in \mathcal{l } , & \\ & \frac{u_{s(c)}^c(t)}{q_m}[q_m-\mu_m - u_{b(c)}^c(t)],&\\ & \qquad\qquad\qquad\mbox { if } ( m , n)=(s(c),b(c ) ) , & \\ & 0 , \qquad\qquad\quad\mbox { otherwise . } & \\ \end{aligned } \right.\end{aligned}\ ] ] proposition 1 still holds and we present a theorem for the scheduling algorithm using delayed queue backlog information , which maintains the throughput optimality and @xmath1 scaling in delay bound : on employing delayed queue backlogs , we can extend the centralized optimization problem ( [ eq:11 ] ) to distributed implementations with methods introduced in remark 3 . recall that in the model description in section iii , the link capacity is assumed constant ( one packet per slot ) and node - exclusive model is employed . in this subsection , we extend the model to arbitrary link capacities and arbitrary interference models with fading channels of finite channel states . thus , instead of ( [ e:4 ] ) , we have @xmath204 , where @xmath205 is the feasible activation set for time slot @xmath21 determined by the underlying interference model and current channel states , with link capacity constraints @xmath206 , where @xmath207 is the arbitrarily chosen link capacity for a link @xmath112 . we define @xmath208 . note that it is clear that @xmath209 . then we can update the optimization ( [ eq:11 ] ) and weight assignment ( [ e:6 ] ) , respectively , as follows : @xmath210 @xmath211,\mbox { if } ( m , n)\in \mathcal{l } , & \\ & \frac{u_{s(c)}^c(t)}{q_m}[q_m-\mu_m - u_{b(c)}^c(t)],&\\ & \qquad\qquad\qquad\mbox { if } ( m , n)=(s(c),b(c ) ) , & \\ & 0 , \qquad\qquad\quad\mbox { otherwise . } & \\ \end{aligned } \right.\ ] ] it is not difficult to check that proposition 1 still holds with @xmath212 and theorem 1 holds with a different definition of constant @xmath105 . the above modified algorithm can be further extended to solve power allocation problems , where we refer interested readers to our recent work @xcite . in this section , we present the simulation results for the proposed optimal algorithm _ * alg_. simulations are run in matlab 2009a with results averaged over @xmath213 time slots . in the network topology illustrated in figure [ fig1 ] , there are three source - destination pairs @xmath214 , @xmath215 and @xmath216 with same poisson arrival rates and @xmath217 . the required minimum data rate for the three flows are all set to @xmath218 . we denote by _ bp _ the back - pressure scheduling algorithm in @xcite with a congestion controller in @xcite , and denote by _ finite buffer _ the cross - layer algorithm developed in @xcite with buffer size equal to the queue length limit @xmath102 . note that it is shown in simulation results in @xcite that finite buffer algorithm ensures much smaller internal queue length ( of nodes excluding the source node ) than bp algorithm ( and queue length is related to delay performance ) . we set the control parameter @xmath219 , where in simulations we find that a higher @xmath106 can not further improve the throughput . [ cols="^,^,^,^,^,^",options="header " , ] we first illustrate in table [ tab1 ] the throughput optimality of _ alg * _ when the sources are constantly backlogged . we loosen the delay constraint as @xmath220 . as we increase the control parameter @xmath102 , the _ * alg * _ achieves a throughput approaching the throughput of bp algorithm which is known to be optimal . we also note that this approximation in throughput results in worse average end - to - end delay performance , which complies with remark 1 . we then illustrate the throughput and delay tradeoff for both the * _ alg _ * and its corresponding suboptimal gmm algorithm in figure [ fig2 ] for the case of arbitrary arrival rates at transport layer with @xmath179 , where we set @xmath221 and @xmath222 for each flow @xmath18 . note that this pair of @xmath102 and @xmath34 shows that the bound in ( [ eq:7 ] ) is actually quite loose , and thus our algorithm can achieve better delay performance than stated in ( [ eq:7 ] ) . figure [ fig2 ] shows that the average end - to - end delay under _ * alg * _ is well below the constraint ( @xmath222 ) and lower than that under bp and finite buffer algorithms . the throughput of _ * alg * _ is close to ( although lower than ) that of the optimal bp algorithm when arrival rates are small ( @xmath223 ) . specifically , when the arrival rate is @xmath224 , _ * alg * _ achieves a throughput @xmath225 more than the gmm algorithm and @xmath226 less than bp algorithm , with an average end - to - end delay @xmath227 less than the bp algorithm . in the large - input - rate - region ( @xmath228 ) , we also observe that the delay in both the bp and finite buffer algorithm violates the delay constraints . in addition , in the above illustrated scenarios with backlogged and arbitrary arrival rates , the minimum arrival rates and average end - to - end delay requirements are satisfied for _ individual _ flows under _ * alg*_. as a side note , the average end - to - end delay in all four algorithms in figure [ fig2 ] first decreases , which can be explained by the intuition that all the algorithms are based on back - pressure of links ( i.e. , queue backlog difference of links ) and the queue backlog difference tends to be larger for each hop with a larger arrival rate . when arrival rate further increases , congestion level becomes higher since more packets are admitted into the network in this paper , we proposed a cross - layer framework which approaches the optimal throughput arbitrarily close for a general multi - hop wireless network . we show a tradeoff between the throughput and average end - to - end delay bound while satisfying the minimum data rate requirements for individual flows . our work aims at a better understanding of the fundamental properties and performance limits of qos - constrained multi - hop wireless networks . while we show an @xmath1 delay bound with @xmath0-loss in throughput , how small the actual delay can become still remains elusive , which is dependent on specific network topologies . in our future work , we will investigate the capacity region under end - to - end delay constraints applied to different network topologies . our future work will also involve power management in the scheduling policies . 1 l. tassiulas and a. ephremides , `` stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks , '' in _ ieee trans . 1936 - 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agn reverberation mapping ( rm ; * ? ? ? * ; * ? ? ? * ) measures the light travel time ( i.e. , lags ) between different regions of an agn , most commonly the time lag between the uv / optical continuum ( from the accretion disk ) and the broad line ( from the broad - line region , or blr ) emitting regions . rm is a powerful tool for probing the structure and kinematics of agn blrs . rm is used to estimate the masses of agn central supermassive black holes ( smbhs ) , by combining the relation between the blr size and agn luminosity ( the @xmath10@xmath11 relation ) with the assumption of virialized motions of clouds in the blr . through application of this method , rm has been established as the primary _ direct _ smbh mass estimation technique for agn / quasars at @xmath12 . rm campaigns are expensive and time - consuming . they require repeated observations of individual targets with sufficient cadence over durations of a few months to a few years , depending on the source redshift and luminosity . the success rate also relies on other factors , such as whether the variability of the target is significant or not during the rm campaign , which is usually unpredictable . to date rm experiments have been successful for about 60 agn / quasars ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? a more detailed history of agn rm experiments is summarized in a few recent works ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? the majority of the above rm work was done with low - redshift agn at @xmath13 . much higher - redshift ( @xmath14 ) agn / quasars have also been tried ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , yet the number of the successful detections of time lags is still very small . recently the sdss reverberation mapping program ( sdss - rm ; * ? ? ? * ) has enabled a new method of carrying out rm experiments . the sdss - rm program is a dedicated multi - object rm campaign that simultaneously targeted 849 quasars in a single 7 deg@xmath6 field . based on the sdss - rm data , @xcite have reported their first detections of time lags in a sample of quasars at @xmath15 . traditional rm programs use spectroscopic observations to monitor the variability of continuum and line emission . recently , photometric rm has been proposed or performed ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the basic idea is to take photometry in two bandpasses , with one bandpass ` on ' an emission line and the other one ` off ' the line . the combination of the two measurements is used to derive the continuum and line fluxes . the advantage of the photometric rm is that it does not require spectroscopic observations , and can be easily performed with small telescopes . the challenge is the small contribution of the emission line flux to the total bandpass flux within a broad band , so that the photometric uncertainties significantly hamper measurements of variability in the line fluxes . alternatively , one may use a narrow band with a full width at half maximum ( fwhm ) of a few tens ( up to @xmath5120 ) to cover the emission line . this has been done for a few local agn at @xmath16 ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in this case , the line flux contributes a large fraction of the total flux in the narrow band and line variability is more readily detected . in this paper we present our intermediate - band reverberation mapping ( ibrm ) project , which uses the combination of broad and intermediate bands ( with fwhm around 200 ) to perform photometric rm . the usage of intermediate bands has the following two advantages , in addition to the general advantages of photometric rm mentioned above . an intermediate band can usually cover a whole emission line , while the line flux still contributes a significant fraction of the total flux in the band , if the line is selected to have high equivalent width ( ew ) as we do for the ibrm program . secondly , it has a larger ( compared to narrow bands ) dynamic range in wavelength that allows the inclusion of more than one targets per field , which substantially increases observing efficiency . in our ibrm program , we observed 13 quasars within five fields or telescope pointings , and successfully detected time lags in 6 of them . the structure of the paper is as follows . in section 2 we present our quasar sample selection and their optical spectra . in section 3 we introduce our ibrm campaign and the details of observations and data reduction . we derive the light curves and time lags of the quasars in section 4 , and put them in the context of the @xmath10@xmath11 relation in section 5 . in section 6 we summarize the paper . throughout the paper we use a @xmath17-dominated flat cosmology with @xmath18 km s@xmath19 mpc@xmath19 , @xmath20 , and @xmath21 . all magnitudes are on the ab system . in this section we present the selection of our quasar sample . before we go into the detailed steps , we briefly introduce the telescope and instrument that we used for the ibrm project , which is directly related to our sample selection . the telescope that we used is the steward observatory 2.3 m bok telescope , and the instrument is its prime focus imager 90prime . 90prime has a large , square field - of - view ( fov ) of roughly one degree on a side . it uses four 4k thin ccds that were optimized for u - band imaging @xcite . we used two broad - band filters @xmath3 and @xmath4 , and three intermediate - band filters , bact12 , bact13 , and bact14 . these intermediate - band filters were originally designed for the beijing - arizona - taipei - connecticut ( batc ) color survey ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? the broad - band filters are used to measure continuum flux . the effective wavelengths of the three intermediate - band filters are 8045 , 8505 , and 9171 , with the fwhms of 230 , 180 , and 264 , respectively . they cover three wavelength ranges with relatively weak oh sky emission , so imaging in these bands is very efficient . our sample selection began with the sdss dr7 quasar catalog delivered by @xcite and @xcite . the emission lines used for our ibrm project are h@xmath1 and h@xmath2 , two of the strongest lines in quasar spectra . we first selected quasars at certain redshifts so that their h@xmath1 or h@xmath2 emission lines are located in one of the three intermediate bands ( the center of an emission line is roughly within the central 50% of the filter . ) specifically , the redshift ranges considered here are [ 0.216 , 0.236 ] , [ 0.290 , 0.302 ] , [ 0.385 , 0.410 ] , [ 0.642 , 0.668 ] , [ 0.741 , 0.758 ] , and [ 0.870 , 0.904 ] , and there are 4326 quasars in these redshift ranges . we then selected quasars in a certain coordinate range , because the observations of our ibrm project shared the bok nights with the sdss - rm project @xcite , as we will see in the next section . the coordinate range chosen here is [email protected].@xmath2213h and decl.@xmath2325 deg , and 1227 quasars passed this selection . we further selected targets in a certain brightness range ( namely , @xmath24 mag ) with high h@xmath1 or h@xmath2 ew . the h@xmath1 and h@xmath2 ew values were measured from the sdss spectra and taken from @xcite . we required that the observed h@xmath1 ew was greater than 180 , or the observed h@xmath2 ew was greater than 90 . this ensures that the line emission contributes a significant fraction of the total flux in the intermediate bands . this is one of the keys for the success of this program . the choice of @xmath25 mag was to ensure that we can get high snrs in the intermediate bands with 5 min integration time . the choice of @xmath26 mag was to select quasars with expected time lags ( in the observed frame ) shorter than the duration of our observing campaign ( roughly 56 months ) . the expected time lags for most of the selected quasars are between 20 and 60 days . the observed - frame time lags also depends on redshift due to the time dilution of @xmath27 and the strong dependence of the intrinsic luminosity on redshift for a given apparent magnitude . therefore , for very bright quasars at relatively high redshifts ( @xmath28 ) , their expected time lags could be significantly longer and even close to the duration of our rm campaign . we selected 622 quasars in this step . cccccccc f1a & 09:00:45.293 & + 33:54:22.38 & 0.228 & 17.93 & h@xmath1 & 205 & batc12 + f1b & 09:01:56.250 & + 33:33:49.49 & 0.878 & 19.00 & h@xmath2 & 71 & batc14 + f2a & 09:46:59.593 & + 29:32:51.13 & 0.387 & 18.58 & h@xmath1 & 606 & batc14 + f2b & 09:50:46.582 & + 29:38:26.90 & 0.233 & 18.20 & h@xmath1 & 299 & batc12 + f3a & 11:27:59.260 & + 36:02:07.00 & 0.667 & 18.42 & h@xmath2 & 132 & batc12 + f3b & 11:29:56.532 & + 36:49:19.24 & 0.398 & 19.05 & h@xmath1 & 563 & batc14 + f3c & 11:31:14.956 & + 36:02:38.30 & 0.229 & 18.22 & h@xmath1 & 263 & batc12 + f4a & 11:45:53.152 & + 28:13:13.41 & 0.401 & 18.73 & h@xmath1 & 230 & batc14 + f4b & 11:46:34.914 & + 28:26:41.96 & 0.225 & 18.11 & h@xmath1 & 249 & batc12 + f4c & 11:49:36.368 & + 27:44:04.83 & 0.748 & 18.95 & h@xmath2 & 192 & batc13 + f5a & 12:36:35.259 & + 45:02:08.06 & 0.401 & 18.74 & h@xmath1 & 353 & batc14 + f5b & 12:36:58.110 & + 45:53:54.46 & 0.235 & 18.12 & h@xmath1 & 433 & batc12 + f5c & 12:38:42.730 & + 45:18:24.73 & 0.229 & 17.34 & h@xmath1 & 292 & batc12 + after we obtained the list of the quasars from the above steps , we chose the area / fields that have more than one quasar per square degree ( the fov of the 90prime ) . this was to increase the efficiency of the project . meanwhile , we matched the quasars to the pan - starrs1 ( ps1 ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) preliminary catalog , and obtained their variability values as follows . for each quasar , we extracted the standard deviations of the magnitudes from the catalog . there are five standard deviation values for five ps1 bands ( @xmath29 , @xmath30 , @xmath31 , @xmath32 , and @xmath33 ) , and we took the average of the second and third largest standard deviation values as the variability of this quasar . we then eliminated @xmath510% of the sources whose variability was roughly consistent with error bars . finally , from the remaining sources we selected 13 quasars or 5 fields for the ibrm project , by considering the following : 1 ) the fields are roughly evenly distributed between 8h and 13h for the convenience of observations ; 2 ) the quasars have relatively strong h@xmath1 ( or h@xmath2 ) ews ( stronger are better ) ; 3 ) the quasars show relatively strong variability . when the demands for 2 ) and 3 ) are difficult to meet simultaneously , we slightly favored 2 ) , because quasar variability is a stochastic process . table 1 lists the details of the 13 quasars . column 1 shows the i d of the quasars . we use ` f1 ' to ` f5 ' to denote the five fields , and use ` a ' to ` c ' to denote the quasars within a field . the following four columns are the coordinates , redshifts , and @xmath3-band magnitudes drawn from the sdss dr7 catalog . columns 6 and 7 shows the emission lines that we used and their ews in the observed frame . column 8 shows the intermediate filters that cover the lines . these quasars covers a redshift range of @xmath34 . we mainly use the h@xmath1 line ( 10 out of 13 quasars ) , because the h@xmath1 line is much stronger than the h@xmath2 line , and thus quasars with strong h@xmath1 were preferentially selected . note that the ew of f1b is lower than our selection criterion . this is because the line is close to the red end of its sdss spectrum , and the ew value given by the sdss dr7 is not accurate . the value given in the table was measured from an mmt spectrum with much better quality ( see the next subsection ) . the contributions of line emission to the intermediate - band fluxes are roughly between 35% and 70% for all objects except f1b , for which the line contribution is only @xmath520% . single - epoch quasar optical spectra are needed to accurately derive the line contribution to the broad - band photometry . these quasars have optical spectra from sdss i and ii . the sdss spectra do not cover the wavelength range beyond @xmath59100 , which is needed for the batc14 filter . for the quasars observed with the batc14 filter ( see table 1 ) , we obtained new optical spectra using the mmt red channel spectrograph . the observations were carried out as backup targets during other programs in 2014 , when weather conditions were poor . the observations were made in long - slit mode with a spectral resolution of @xmath510 . the integration time was roughly 510 min per object , which is sufficient for our purposes . the spectra were reduced using standard iraf routines . the mmt spectra cover a wavelength range of 700010000 . the final spectra of these quasars are the combination of the sdss and mmt spectra . figure 1 shows the optical spectra of our quasar sample in the observed frame . as we mentioned above , some spectra were directly taken from the sdss , while the others were the combination of the sdss and mmt spectra . in each panel of figure 1 , we also show the transmission curves of the 90prime @xmath3 and @xmath4 filters ( the blue dotted profiles ) and one of the intermediate filters ( the red dotted profile ) that covers h@xmath1 or h@xmath2 . the ccd quantum efficiency has been taken into account . note that the transmission curves of the 90prime @xmath3 and @xmath4 filters are slightly different from those of the sdss @xmath3 and @xmath4 filters . in several cases , the intermediate bands do not entirely cover the emission lines ( e.g. f2a and f5b ) . the effect of missing line wings on the measurement of blr sizes is very small . when we calculate line emission for light curves in section 4 , we only consider the contribution from the part covered by the intermediate bands , which contains more than 90% ( in most cases more than 95% ) of the total line flux . @xcite conducted detailed calculations to estimate the consequences of the above missing line wings on photometric rm results , and concluded that the effect on the measurements of blr sizes is only a few per cent . this is negligible compared to the size measurement uncertainties we will get in section 4 . the ibrm project was carried out in the spring semester , 2014 . it shared the bok nights with the sdss - rm project . as we mentioned earlier , the sdss - rm project was one of the sdss iii ancillary projects . it used the cfht and bok telescopes to do broad - band ( @xmath35 and @xmath3 ) photometry for measurements of continuum light curves in sdss - rm @xcite . our targets and observing time were coordinated with the sdss - rm project . the bok observations of the quasars were conducted in january through june , 2014 . due to the constraints from the telescope scheduling , we obtained one or two long observing blocks each month ( see section 4.1 ) . hence the bok nights for the sdss - rm and ibrm projects were not evenly distributed , and were clustered around the nights with relatively bright moon phase . such an observing schedule does not provide an optimal cadence for rm studies . each of the five fields were observed for between 20 and 30 epochs over the full campaign . because of the large fov of the 90prime , the 13 quasars in our sample were covered by only 5 telescope pointings or fields from f1 to f5 . we usually observed at least 3 fields per epoch / night . the observations were made via observing scripts . each time after we slewed the telescope to a new field , the scripts automatically changed filters , tweaked focus , and took the data , in the order of @xmath3 , @xmath4 , batc12 , batc14 ( and batc13 for field 4 ) . the typical on - source integration time was 150 s in the @xmath3 band , and 300 s in the other bands . the observing conditions were mostly moderate with clear skies , moderate seeing ( @xmath36 ) , but significant moonlight . the 90prime images were reduced in a standard fashion using our own idl routines . the basic procedure was described in @xcite . first , we made a master bias image and a master flat image from bias and flat images taken in the same night . a bad pixel mask was also created from the flat image . science images were then overscan and bias - corrected and flat - fielded . next we identified saturated pixels and bleeding trails , and incorporated them ( along with the bad - pixel mask ) into the weight images . the affected pixels were interpolated over in the science images . we call the science images at this stage ` corrected images ' . the 90prime ccds are thin chips , and thus produce strong fringing in the bands that we used . we subtracted sky background and fringes using two iterations . the first - round of sky subtraction was performed by fitting a low - order 2d polynomial function to the background . a master fringing image ( per filter ) was made by median stacking at least eight sky - subtracted images in the same filter . this fringing image was scaled and subtracted from the original ` corrected ' images ( before sky - subtraction was done ) . we detected objects in the fringe - subtracted images using sextractor @xcite . a better sky image was produced from each fringe - subtracted image with the detected objects masked out . then the second round of sky and fringe subtraction was performed , but with the detected objects masked out . in order to derive astrometry , we detected bright objects using sextractor @xcite , and calculated astrometric solutions using scamp @xcite by matching objects to the sdss . with the new astrometry we re - mapped the images using swarp @xcite . the re - mapped images have a native pixel size of @xmath37 . accurate ( relative ) photometry is another key for the success of this project . in order to achieve accurate photometry , for any given quasar in the whole observing campaign , we used a large number of nearby bright point sources for photometric calibration . these bright sources and our quasar targets was always located in roughly the same part of the same ccd , which minimizes the effect from any large - scale systematics . this allows us to achieve relatively small uncertainties ( @xmath38 mag ) on the magnitude zero points that are usually negligible compared to the uncertainties in the light curves that we derive in section 4 . the details are as follows . the four ccds were read out via 16 amplifiers , with 4 amplifiers per ccd . for any of the 13 quasars in an image , we only performed photometry for the amplifier area in which this quasar was located ( roughly @xmath39 on a side ) . we did not use other parts of the image ( for this quasar ) due to the possible small zero point shift across the amplifiers and ccds @xcite . we first chose a ` standard ' night , which was photometric and relatively dark , and performed photometry for the images taken in this standard night . photometry was measured within an aperture ( diameter ) size of 8 pixels ( @xmath40 ) using sextractor . we then picked up bright ( at least 30@xmath41 detection ) but unsaturated point sources , and matched them to the sdss . the density of the bright stars is about 50100 per amplifier . we used the sdss psf magnitudes , so the measured magnitudes are total magnitudes with aperture corrections automatically taken into account . after we obtained the @xmath3 and @xmath4-band magnitudes for a given object , we calculated its intermediate - band magnitudes as follows . we assumed that its spectrum in the wavelength range of the @xmath3 and @xmath4 bands ( also covers the three intermediate bands ) was a power law , which is determined by its @xmath3 and @xmath4-band magnitudes . then its intermediate - band magnitudes were directly calculated from this power - law spectrum and the intermediate - band transmission curves . the resultant magnitudes have very weak dependence ( usually @xmath42 mag ) on the assumption of the spectrum shape , as long as there are no emission or absorption lines in the intermediate bands . such small uncertainties on the zero magnitude points have no effect on our measurement of light curves , which reply on relative photometry . we calculated ab magnitudes for all these bright stars taken in the standard night . these bright stars were used as standard stars for all other images . we then measured photometry for the images taken in all other nights . the procedure was the same . but we used the bright stars found in the standard night as ` standard stars ' , and used their magnitudes for absolute flux calibration . because of the large numbers of bright standard stars for any given quasar , we achieved high accuracy on relative zero flux points . figure 2 illustrates this point . its horizontal axis shows the magnitude difference of the standard stars taken between the standard night and the first several nights other than the standard night . the vertical axis shows the distributions of the magnitude difference . the sigma values quoted in the figure were estimated by fitting a gaussian profile to the top 80% of the distributions . the bottom 20% deviates from the gaussian distribution , likely due to some unreliable ` standard ' stars , such as variable stars . the tight distributions of the magnitude difference in the four bands suggest that our accurate flux calibration is about 0.010 mag rms . we did not plot the distribution in batc13 , which is similar to those in the other 4 bands . figure 3 shows the measurement uncertainties as a function of total magnitude in 4 bands for all quasars observed in the whole rm campaign . the errors are estimated within an aperture ( diameter ) size of 8 pixels from sextractor . the uncertainties from the absolute flux calibration are not included in this plot . in our images , the noise is completely dominated by sky background , so these errors are quite reliable ( background variance reflects errors ) . the errors are mostly smaller than 0.02 mag . in rare cases errors can be larger than 0.05 mag , mostly caused by low sky transparency . the final photometric errors take into account ( quadratically ) the measurement errors ( figure 3 ) , the errors from the flux calibration ( figure 2 ) , and the uncertainties due to varying seeing . a quasar host galaxy is not a point source , and its radial profile is broader than psf , so the aperture correction derived from point sources does not precisely correct for all light loss . this results in photometric variation with varying seeing . we estimate this variation as follows . in section 5.1 , when we measure the host galaxy contribution for each quasar , we generate a combined image ( from single - epoch images with good seeing ) , build a psf model image , and derive a host - galaxy image using image decomposition . we make use of these ` deep ' and model images , because single - epoch images do not have sufficient snr . we convolve these images with gaussian kernels , which mimics varying seeing . we then perform aperture photometry in the same manner as in single - epoch images . the only difference is that the aperture correction is measured from the corresponding psf images . we find that when psf varies from @xmath43 to @xmath44 ( almost covers our seeing range ) , the photometric variation is about @xmath45 mag . such variation is small for our targets , partly because we used a large aperture for photometry . in many cases , however , it is comparable to the measurement errors shown in figure 3 . thus we carry this error of 0.007 mag to the final photometric errors . in this section we present our main results . for each quasar in table 1 , we first compute the continuum and emission line light curves . this step is straightforward , as we have decent optical spectra and accurate broad- and intermediate - band photometry . we then derive the time lag between continuum and line emission in standard ways . for each quasar we have a single - epoch optical spectrum and a series of @xmath3 , @xmath4 , and intermediate - band photometric measurements . we first calculate the contribution of the line emission ( or equivalently the contribution of the continuum emission ) to the broad - band photometry using the optical spectrum . we select regions with little line emission in the spectrum as continuum windows , and fit a power - law curve ( @xmath46 ) to these windows . this power - law continuum may contain a central agn component and a host galaxy component ( see the next section ) . as long as the host galaxy does not vary ( a constant component ) , the inclusion of the host galaxy component does not affect the determination of time lags . the line emission is obtained by subtracting the power - law continuum from the spectrum . we assume that the contribution of the line emission to the broad - band photometry does not vary with time . the reason is that the line contribution is smaller than 5% , and the line variability is usually smaller than 20% , so the effect of line variability on broad - band photometry is smaller than 1% . as we will see below , the continuum value we derive for a quasar is determined by two broad bands ( @xmath3 and @xmath4 ) , with one band without line contamination , so the effect of line variability on continuum is even smaller by roughly a factor of two , which is much smaller than the uncertainties in the light curves derived below . thus our above assumption has a negligible effect on the measurement of the continuum , but largely simplifies our procedure . we then derive the line flux and continuum flux at the wavelength of the intermediate band from their corresponding @xmath3 , @xmath4 , and intermediate - band photometry at each epoch . the continuum components at @xmath3 and @xmath4 are computed by subtracting the line contribution from these bands . a power - law continuum is derived analytically from the two flux measurements at the effective wavelengths of the @xmath3 and @xmath4 bands . we then determine the continuum value at the effective wavelength of the intermediate - band from the power - law continuum . this continuum value is the continuum that will be used for light curves . finally the line emission is obtained by subtracting the continuum component from the intermediate - band photometry . figure 4 plots the light curves of the 6 quasars that show significant time lags during our ibrm campaign ( see the next subsection ) . for each object , the upper panel shows the light curves in the @xmath3 ( blue circles ) , @xmath4 ( green circles ) , and one of the intermediate bands ( red circles ) . the lower panel shows the light curves of the continuum flux ( upper curves ) and line flux ( lower curves ) . cccccccc f2b & 0.233 & @xmath47 & 43.84 & 0.21 + f3b & 0.398 & @xmath48 & 44.17 & 0.19 + f3c & 0.229 & @xmath49 & 43.84 & 0.37 + f4a & 0.401 & @xmath50 & 44.32 & 0.28 + f5b & 0.235 & @xmath51 & 43.79 & 0.22 + f5c & 0.229 & @xmath52 & 44.25 & 0.36 + we estimate time lags between the line and continuum emission derived above using the javelin package @xcite . as we have seen , the light curves were unevenly ( sometimes sparsely ) sampled , due to the constraints from telescope time scheduling . javelin is able to deal with such an uneven sampling . it assumes that the variability of a quasar / agn can be well described by the damped random walk ( drw ) model ( e.g. * ? ? ? * ) , and its emission line light curve is the lagged and scaled version of its continuum light curve . for a given quasar , we first model its continuum variability using javelin , and find the distribution of the drw parameters we then statistically interpolate the continuum light curve . the light curve is shifted , smoothed , and scaled , before it is compared to the corresponding emission - line curve . the smoothing here refers to the use of a transfer function ( non - delta - function ) in javelin to mimic the realistic line response to continuum light curves . this is due to the fact that the blr clouds are distributed at different radii with different velocities , which results in a transfer function that is broad in lag . for details , see the javelin papers @xcite . this process is performed 10,000 times using the mcmc method . the final results are the best model fits for each try . note that all calculations above are based on flux ( not magnitudes ) . based on the time lag measurements from javelin , we find that 6 ( out of 13 ) quasars in our sample show clear lag detections during our ibrm campaign ( we will discuss the other quasars in the next subsection ) . the lag range allowed in the above calculation is from 100 to + 100 days . we do not consider a wider range , simply because the duration of the ibrm campaign was only 150170 days . figure 5 shows the distributions of the measured lags for the 6 quasars ( their light curves are shown in figure 4 ) . they show clear single distribution peaks . the results are listed in table 2 . the lag errors in the table are calculated by including 16% and 84% of the total distributions around the median distribution ( 50% of the total ) . obviously they depend on the lag range that we consider . on the other hand , they are not sensitive to the lag range , as long as the lag detections are substantial with most distributions clustered around the median values . this applies to all quasars except f2b in figure 5 . the lag measured for f2b is @xmath53 days . it has a large upper error due to the non - negligible fraction of the total distribution beyond 70 days . if we were to reduce the lag range to @xmath54 , the lag becomes @xmath55 days , with a much smaller upper error . we adopt the larger error for consistency in the paper . we further use the discrete correlation function ( dcf ) to validate the above lag detections . the algorithm we adopt is the @xmath4-transformed dcf ( zdcf ) , which was designed to handle unevenly sampled light curves @xcite . the estimated dcfs for the 6 quasars are shown in figure 6 . these quasars also show clear dcf peaks . the lag uncertainties from the zdcf are larger . one reason is that the zdcf uses time - lag bins , and the minimum number required in each bin is roughly 11 for meaningful statistics in the zdcf . given the small numbers of epochs in our ibrm project , the zdcf can only coarsely sample the light curves in the lag time space . nevertheless , the zdcf peaks in figure 6 are consistent with the results from javelin except for f2b . f2b shows a significant lag detection by javelin in figure 5 , but its zdcf peaks at @xmath56 . the lag detection from javelin could be real , or a false positive because of the sparse - sampling in light curves . on the other hand , the zdcf of f2b is broad , which is not against a lag of @xmath57 days . this can be solved with more evenly and finely sampled light curves in the future . we perform a simple experiment to test our results . for any pair of light curves ( continuum and line ) in the 6 quasars , we randomly re - order one curve , and repeat the above processes using javelin to estimate the rate of false positives . this is done a hundred times for each quasar ( each pair of light curves ) . these tests generally show small false positive rates . for the first four quasars that were observed in nearly 30 epochs , the false positive rates are only 34% . the rates increase to 79% for the last two quasars that were observed only @xmath520 times ( see @xcite for a detailed discussion ) . the detected lags in the 6 quasars are all for h@xmath1 , and at relatively low redshifts between 0.22 and 0.40 . it is not surprising , because 10 out of the 13 lines in the original sample are h@xmath1 , and h@xmath1 is much stronger than h@xmath2on average . in addition , higher redshifts usually mean higher intrinsic luminosities and larger time dilution @xmath58 , leading to much larger observed time lags that are likely beyond the detection capability of our ibrm campaign . we did not detect time lags between continuum and line emission in the other 7 quasars in our sample . similar to figure 6 , figure 7 shows the distributions of the lags for these quasars from javelin . unlike those in figure 6 , quasars in figure 7 do not show single strong peaks . they rather show multiple peaks or continuous distributions . there are two main reasons for these non - detection . the first reason is that the expected time lags based on the current @xmath10@xmath11 relation ( e.g. * ? ? ? * ) are comparable to the duration of our campaign . for example , the lags expected for f1b , f3a , and f4c are greater than 120 days , primarily due to their high redshifts . the second reason is that the variability is small , or that the light curves are relatively flat . these quasars show moderate to large variability in the ps1 data . but quasar variability is a stochastic process , and past large variability does not guarantee large variability in the future . in addition , large gaps in light curves can often cause aliasing , and it seems the case for f4c . for these objects , our data were insufficient to detect lags . quasar host galaxies may contribute a significant fraction of the total light in the bands that we measured . there are two general methods to estimate the light from the hosts : image decomposition and spectral decomposition . we do not have the high snr spectra that spectral decomposition requires ( e.g. * ? ? ? * ) , so we rely on image decomposition . image decomposition works better on images with better psfs ( or seeing ) . the site of the bok telescope does not deliver good seeing , and the average psf size of our images is about @xmath59 . in order to construct a deep combined image with a decent psf for each quasar , we choose 50% of the @xmath3-band images with the best psf sizes , and co - add them to a stacked image . the reason to choose the @xmath3-band is twofold . one is that it is the deepest band . the other one is that its effective wavelength is close to the rest - frame 5100 ( the commonly used wavelength ) for our quasars . as for flux calibration , we only consider 1/16 of the image , i.e. , the amplifier that the quasar is located in . this is to avoid any possible large psf variation across the large fov . the psf sizes of the combined images are about @xmath60 , which is good enough for image decomposition on low - redshift quasars ( e.g. * ? ? ? we perform image decomposition on the combined @xmath3-band images . for each quasar / image , the detailed steps are as follows . we first derive a psf model for the quasar using psfex @xcite . psfex finds point sources in the image , and builds psf models as a function of position . we take the psf model that is closest to the quasar position as the psf model for the quasar , although the psf variations are quite small across a single amplifier in our images . the psf image size is 25 pixels on a side , with the peak pixel centered in the middle . we then calculate an accurate central position for the quasar in the image , and re - sample the image to produce a stamp image centered on the quasar . the size of the stamp image is also @xmath61 pixels . following @xcite , we assume that the central pixel ( peak value ) of the quasar image is completely dominated by the quasar / agn component . under this assumption , we scale the psf and subtract it from the quasar image . the residual is referred to as the host galaxy component . figure 8 demonstrates the procedure by showing three quasars . from the above procedure , the fraction of the agn ( or host ) component is simply calculated by doing aperture photometry on the psf image and the quasar image . in our 6 quasars , the host contribution is roughly between 19% and 37% . in order to measure the luminosity of an agn at the rest - frame 5100 , we scale its spectrum in figure 1 to match the mean of the @xmath3-band magnitudes obtained in ibrm . we then calculate the agn luminosity from the spectrum after removing the host contribution . the absolute values of the agn luminosities are listed in table 2 . the relation between the blr size and quasar luminosity provides the basis for determining smbh masses in high - redshift quasars / agn with single - epoch spectroscopy ( for a recent review , see @xcite ) . accurate measurements of smbh masses are particularly important in the context of the smbh and host galaxy co - evolution @xcite . the masses from rm can be calibrated from the local @xmath62 relation ( e.g. * ? ? ? our sample is still small , and would not improve the @xmath10@xmath11 relation . on the other hand , the quasars in this sample are at relatively high redshifts , and thus may test the current @xmath10@xmath11 relation at @xmath7 . figure 9 shows the current @xmath10@xmath11 relation from successful rm campaigns compiled by @xcite . different emission lines have different ionization potentials , so the corresponding blr sizes are different . the relation shown in figure 9 was mostly built from h@xmath2 measurements of local agn . we also plot the recent results on h@xmath2 and mgii lags at @xmath15 ( the blue circles ) from the sdss - rm project @xcite . our results for the 6 quasars with lag detections are shown as the red circles . they are roughly consistent with the @xmath10@xmath11 relation derived from local agn . our sample is primarily based on the h@xmath1 line , which has a shallower ionization potential compared to h@xmath2 , and is thus expected to have a larger blr size . however , the difference between the h@xmath1 lag and h@xmath2 lag is unclear , and may depend on quasar luminosity . several previous studies show that the difference ranges between 20% and 50% ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? given the scatter in the relation , our lag measurements are still consistent with the previous results . we note that our 6 quasars occupy a small part of the parameter space in figure 9 . this is due to the small sample size and the strong target selection bias . our targets were selected to be bright , and they presumably have relatively large blr sizes . if there were fainter quasars in our sample , we would not be able to detect their time lags ( and thus they would not show up in figure 9 ) because of the coarsely sample light curves . on the other hand , much more luminous quasars do not show up in the figure either , since these quasars have much longer time lags and can not be detected in the ibrm duration , as we discussed in section 4.3 . therefore , the consistency of our results with previous studies does not mean that we have validated the current @xmath10@xmath11 relation at @xmath7 . a larger , unbiased sample covering a much larger parameter space is needed . we have presented our ibrm program , a photometric rm program with broad and intermediate - band photometry . the intermediate bands that we chose are centered at 8045 , 8505 , and 9171 . they cover three wavelength ranges with relatively weak oh sky emission , thus imaging in these bands is very efficient . our sample consists of 13 quasars at redshift between 0.2 and 0.9 . these quasars were selected to have strong h@xmath1 or h@xmath2 emission lines that are located in one of the intermediate - bands . the ibrm campaign was carried out with the 90prime camera on the bok telescope . the 90prime has a large fov and covered 13 quasars within five pointings / fields . the five fields were observed in the @xmath3 , @xmath4 , and intermediate - bands in 2030 epochs . these epochs were unevenly distributed in a duration of 56 months , so the cadence is not optimal for rm experiments . by using a large number of standard stars for each quasar , we achieved high accuracy on the photometric measurements . the combination of the broad and intermediate - band photometry allows us to precisely determine the light curves of the optical continuum and emission line . we detected significant time lags between continuum and line emission in 6 ( out of 13 ) quasars in our sample . the time lags are consistent with the @xmath10@xmath11 relation derived from h@xmath2 in low - redshift agn . photometric rm with intermediate - band photometry has two major advantages . first , as with any implementation of photometric rm , it does not require spectroscopic observations , and can be easily performed with small telescopes . second , the bandwidth of an intermediate filter is narrow enough that the line flux still contributes a significant fraction of the total flux in the band . meanwhile , it is wider than narrow bands so that it is possible to include more than one target ( at similar redshifts ) per telescope pointing , which substantially increases observing efficiency . based on our experience from the ibrm program , we may increase our efficiency and improve our success rate in future rm campaigns with intermediate - band photometry . we plan to carry out a larger rm program using the near - earth object survey telescope ( neost ) in xuyu , china . neost is a 1 m telescope with a fov of 9 deg@xmath6 . with such a large fov , we can monitor several ( up to @xmath510 ) quasars per telescope pointing . our current ibrm experiment contains only 2030 unevenly - distributed epochs . we will make more observations ( 4050 epochs ) with better cadence , which will largely increase the success rate and improve time lag measurements . ideally , we can complete this rm campaign for 100 quasars with 60 nights ( 45 epochs in 6 months ) on the neost telescope . we also plan to extend the baseline from 6 months to 18 months , with more sparse sampling after 6 months . this is to explore higher - redshift and higher - luminosity quasars . the maximum redshift that the three intermediate bands can reach for h@xmath2 is roughly 0.9 , which is much higher than the redshifts of the majority quasars shown in figure 9 . we will further extend this method other lines such as mgii , although rm with mgii is significantly more difficult because the line is generally much weaker than h@xmath1 and h@xmath2 . we acknowledge the support from a 985 project at peking university , and the support from a youth qianren program through national science foundation of china . we would like to thank y. alsayyad , y. chen , k.d . denney , s. eftekharzadeh , y. gao , p.b . hall , s. jia , c.m . peters , k. ponder , j.a . rogerson , r.n . smith , and f. wang for their help with the bok observations . the pan - starrs1 surveys ( ps1 ) have been made possible through contributions of the institute for astronomy , the university of hawaii , the pan - starrs project office , the max - planck society and its participating institutes , the max planck 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194 , 45 shen , y. 2013 , bulletin of the astronomical society of india , 41 , 61 shen , y. , brandt , w. n. , dawson , k. s. , et al . 2015 , , 216 , 4 shen , y. , horne , k. , grier , c. j. , et al . 2015 , arxiv:1510.02802 shen , y. , greene , j. e. , ho , l. c. , et al . 2015 , , 805 , 96 tonry , j. l. , stubbs , c. w. , lykke , k. r. , et al . 2012 , , 750 , 99 trevese , d. , paris , d. , stirpe , g. m. , vagnetti , f. , & zitelli , v. 2007 , , 470 , 491 wang , j .- m . , du , p. , hu , c. , et al . 2014 , , 793 , 108 yan , h. , burstein , d. , fan , x. , et al . 2000 , , 112 , 691 zhou , x. , jiang , z .- j . , xue , s .- j . , et al . 2001 , chin . j. astron . astrophys . , 1 , 372 zou , h. , jiang , z. , zhou , xu . , et al . 2015 , , 150 , 104 zu , y. , kochanek , c. s. , & peterson , b. m. 2011 , , 735 , 80 zu , y. , kochanek , c. s. , kozowski , s. , & peterson , b. m. 2013 , arxiv:1310.6774	we present a reverberation mapping ( rm ) experiment that combines broad- and intermediate - band photometry ; it is the first such attempt targeting a sample of 13 quasars at @xmath0 . the quasars were selected to have strong h@xmath1 or h@xmath2 emission lines that are located in one of three intermediate bands ( with fwhm around 200 ) centered at 8045 , 8505 , and 9171 . the imaging observations were carried out in the intermediate bands and the broad @xmath3 and @xmath4 bands using the prime - focus imager 90prime on the 2.3 m bok telescope . because of the large ( @xmath51 deg@xmath6 ) field - of - view ( fov ) of 90prime , we were able to include the 13 quasars within only five telescope pointings or fields . the five fields were repeatedly observed over 2030 epochs that were unevenly distributed over a duration of 56 months . the combination of the broad- and intermediate - band photometry allows us to derive accurate light curves for both optical continuum ( from the accretion disk ) and line ( from the broad - line region , or blr ) emission . we detect h@xmath1 time lags between the continuum and line emission in 6 quasars . these quasars are at a relatively low redshift range @xmath7 . the measured lags are consistent with the current blr size - luminosity relation for h@xmath2 at @xmath8 . while this experiment appears successful in detecting lags of the bright h@xmath1 line , further investigation is required to see if it can also be applied to the fainter h@xmath2 line for quasars at higher redshifts . finally we demonstrate that by using a small telescope with a large fov , intermediate - band photometric rm can be efficiently executed for a large sample of quasars at @xmath9 .
the pauli hamiltonian describes a non - relativistic electron with gyromagnetic constant @xmath6 @xmath7 @xmath8 is the vector of pauli matrices and @xmath9 acts on spinors . we use units where @xmath10 . the electric and magnetic fields are determined by the 4-potential @xmath11 : [ fields ] = , = - + a_0 in 1979 aharonov and casher @xcite observed that the pauli operator for static magnetic field in two dimensions , @xmath12 , so @xmath13 , has ( normalizable ) zero energy modes . the zero modes turn into gapped bound states . ] they are gapless ground states and their number @xmath14 , is determined by the total magnetic flux @xmath15 measured in units of quantum flux , [ zero - modes ] d= ceiling of @xmath16 , i.e. the smallest integer @xmath17 . , the total number of zero modes is 6 . since all the fluxons are critical no mode is localized on any one fluxon . if the two clusters are well separated one can choose two of the zero mode to be localized on the cluster of three fluxons and another three modes to be localized on the cluster of four fluxons . the remaining 6-th mode is supported on the two clusters no matter how far they are separated . , width=6 ] we consider a magnetic field @xmath18 localized on a finite number of disjoint fluxons labeled by @xmath19 . the magnetic flux of the @xmath20-th fluxon , @xmath21 , is localized in a region of radius @xmath22 centered at @xmath23 . we _ do not _ assume that @xmath21 is quantized or that all the fluxes @xmath21 are identical . we shall assume w.l.o.g . that @xmath24 . we say that the @xmath20-th fluxon is super - critical if @xmath25 , subcritical if @xmath3 and critical if @xmath26 . the fluxons are viewed as classical parameters and _ not _ as dynamical degrees of freedom : they _ do not _ have a wave function or an equation of motion . ( the dynamical degree of freedom is the electron wave function . ) when the @xmath20-th fluxon is super - critical it can create @xmath27 zero modes which are _ confined _ to it , in the sense that their wave function decays ( as a power law ) over a typical distance @xmath28 determined by the fluxon radius @xmath22 . more interesting are the zero modes which are bound jointly by a number of separate fluxons . we shall call these solutions _ free zero modes_. these states wave functions live in between the fluxons and have typical size determined by the inter - distance @xmath29 . when _ all _ the fluxons are subcritical , @xmath30 _ all _ the zero modes are free : the probability of finding the charge on any of the fluxons is close to zero ( as @xmath31 ) . in general , confined and free modes coexist . the confined modes behave like the charge - flux composites one encounters in the fractional quantum hall effect @xcite , except that here the charge is quantized but the flux is not whereas in the hall effect it is the flux that is quantized and the charge is not . the free modes are a different kettle of fish as the composite involves a single electron jointly bound by several fluxons . as we shall see , these modes can sometimes turn the fluxons into non - abelian anyons @xcite . these new `` topological '' objects are quite interesting . the distinction between confined and free zero modes is meaningful when the radius of the individual fluxons , @xmath22 , is the smallest length scale in the problem , @xmath32 and is sharp for point - like fluxons . the total number of free modes @xmath33 is , as we shall see , [ extended ] 0d_f= max\{0,_a _ a-1}n-1 , _a=_a - d_a we say that the number of free modes is maximal if @xmath34 . this turns out to be the case where the fluxons become non - abelian anyons . if all the fluxons are identical then to have maximal number of free modes leading to interesting representation of the braid group requires [ con ] 1 - 1 n < _ a < 1 ( the case @xmath35 leads to a trivial representation of the pure braid group . ) to study the holonomy of the zero modes we treat the fluxon coordinates , @xmath23 as ( classical ) adiabatic controls . the adiabatic theory we shall need and describe is of interest in its own right , since the weak electric fields generated by the slow motion of the fluxons are important for the adiabatic transport and since the zero modes are gapless ( see sec . [ sec : ad ] for more details ) . adiabaticity means that the characteristic time scale of the controls is large compared with the characteristic time scale of the system . we shall argue that the characteristic time scale in the case of point - like fluxons is set by their mutual distances . this means that points in control space where fluxons collide must be removed : fluxon collisions is like gap closures in gapped systems . both endow control space with an interesting topology which is sine qua non for interesting topological behaviour . the holonomies of braiding point - like fluxes are summarized in : * the berry phase associated with the confined mode on the @xmath20-th super - critical fluxon braided by the fluxon @xmath36 is the aharonov - bohm phase @xmath37 . * the berry phase for a non - degenerate free mode , ( @xmath38 ) , and two fluxons ( @xmath39 ) is _ topological _ ( path independent ) given by @xmath40 . * the berry phase for a single free mode , ( @xmath38 ) and @xmath41 fluxons is abelian but path _ dependent_. in other words , the adiabatic curvature is non trivial , see fig . [ fig : curvature ] . * for @xmath42 and maximal number of free modes , @xmath34 the holonomy is non - abelian and _ topological_. braiding anyon @xmath20 with anyon @xmath43 is associated with the monodromy matrix , _ a = e^-2i_a the eigenvalues of the holonomy matrix are @xmath44 . this is our most significant result . * if , in addition to @xmath34 , all the fluxons carry identical fluxes , then exchanging them make sense and is described by matrices that give the burau representation of the full braid group @xcite : , the fluxons are identical anyons . like ordinary anyons @xcite they have topological braiding and fusion rules . ( the fusion rules are simply flux addition . ) but , unlike ordinary anyons , they are gapless and hence fragile . * @xmath41 and @xmath45 : the holonomy is non - abelian and , in general , path dependent i.e. not topological . a key to aharonov - casher ( ac ) observation is the fact that when @xmath46 and @xmath6 the pauli hamiltonian is a prefect square @xmath47 since @xmath48 the zero modes , if any , are ground states and are the normalizable solutions of ( -i - e)=0 where @xmath49 is a two component spinor . the second key observation is special to two dimensions . it is convenient then to use complex notation and @xmath50 z = x+iy,2_z=_x -i _ y , 2a_z = a_x - ia_y , one then has ( -i - e ) = -2i ( cc 0 & _ z - iea_z + ) = q+q^ * since @xmath51 the pauli hamiltonian is super - symmetric @xcite : @xmath52 the zero - modes then lie in @xmath53 , i.e. they are either spin up states that lie in the kernel of @xmath54 , or spin down states in the kernel of @xmath55 . for the zero - modes with spin up : [ kernel1 ] = ( c _ + 0 ) , ( \|_z - ie\|a_z)_=0 let us first look for a solution that does not vanish anywhere , so @xmath56 is well defined . we shall call this a fundamental solution and denote it by @xmath57 . it is given by [ alpha ] \|_z _ 0=ie\|a_z _ z\|_z _ 0=ie_z \|a_z using 4_z\|_z= , 4_z\|a_z = ` div ` + i ` curl ` it follows that @xmath58 is a solution of poisson s equation whose source term is determined by @xmath59 [ poisson ] _ 0 = -eb + ie consequently , a unique choice of @xmath58 is made by means of the poisson kernel : ^-1(z , z)= 1 2 \|z - z\| by elliptic regularity @xmath58 is at least as regular as @xmath60 . in the coulomb gauge @xmath61 . consequently @xmath58 is real and the fundamental solution is positive . clearly [ alphapk ] _ 0 -_t\|z\| with @xmath15 the total magnetic flux . since @xmath62 is gauge invariant the fundamental solution ( in any gauge ) decays polynomially : \| _ 0\| \|z\|^-_t the fundamental solution is square integrable iff @xmath63 . similarly , the spin down fundamental solution decays at infinity if @xmath64 and is square integrable iff @xmath65 . we shall assume from now on that @xmath24 and consider only spin up zero modes . now with @xmath66 any holomorphic , [ yr7 ] _ = p(z)_0 _ ker(\|_z - ie\|a_z ) @xmath66 can not have poles , since this conflicts with ( local ) square integrability of @xmath67 ( and the regularity of @xmath58 ) . @xmath66 must therefore be a polynomial . @xmath67 is square integrable provided @xmath68 it follows that there are @xmath14 zero modes with @xmath14 given by eq . ( [ zero - modes ] ) . these results of aharonov and casher ( ac ) @xcite may be viewed as an example of an index theorem for non - compact manifold @xcite . we now turn to the study of static disjoint fluxons with fluxes @xmath21 localized inside discs of radius @xmath22 centered at @xmath23 . we shall denote @xmath69 and @xmath70 . an interesting and very useful feature of the ac zero modes , which follows from eq . ( [ poisson ] ) , is the ` superposition ' property : has no simple relation to the normalization of the factors @xmath71 . ] the fundamental solution for @xmath0 fluxons is the product of the single fluxon fundamental solutions : [ superposition ] _ 0=_a ( _ a)_0 in particular , in the coulomb gauge , @xmath72 consider a single fluxon with uniform @xmath18 localized in a disc of radius @xmath73 centered at the origin . in the coulomb gauge [ buniform ] eb()=\ { lr 1 & r < r + 0 & r > r . e= ( ) ^2 & r < r + 1 & r > r the fundamental solution ( in the coulomb gauge ) is , by eq . ( [ poisson ] ) , [ rnz ] _ 0= \ { lr ( -_t ) & r < r + ( r r)^_t ( -_t ) & r > r . for @xmath74 the fundamental solution is not square integrable near infinity . a sub - critical fluxon can not support zero modes . when the total flux is super - critical @xmath63 the fundamental solution is square integrable for any @xmath75 and it is spread over an area of typical diameter @xmath76 . similar conclusions hold for asymmetric @xmath77 with the same total flux @xmath15 since the asymptotics of eq . ( [ rnz ] ) remains unchanged . it is instructive to compare this result with the formal solution for single point fluxon . the fundamental solution for a delta localized magnetic field is [ point ] _ 0= r^-_t , which is _ never _ square integrable , in contrast with what we found for finite @xmath73 ; the limit @xmath78 must be taken with care . in our discussion of point - like fluxons we shall assume that @xmath73 is smaller than any other length scale in our system but is actually non - zero and can therefore serve as a small distance cutoff . this issue will however not be relevant if all fluxons are sub - critical , in which case one may safely put @xmath79 . consider @xmath0 sub - critical , point - like fluxons . by the superposition property the fundamental solution is just a product of ( translated ) solutions of the form eq . ( [ point ] ) : [ prod ] _ 0()=\|-_a\|^-_a the solution is locally square integrable since , by assumption , the individual fluxons are sub - critical @xmath3 and distinct @xmath80 . it is square integrable at infinity if the total flux is super - critical , @xmath81 . eq.([prod ] ) is the prototype of free zero modes . when the @xmath20-th fluxon collides with the @xmath43-th fluxons the norm of the fundamental solution @xmath82 diverges if @xmath83 . the condition @xmath84 endows the control space of point - like fluxons with a non - trivial topology . for point fluxes , a useful gauge , besides the coulomb gauge , is the _ holomorphic gauge _ which formally has @xmath85 except for cuts where it is delta like . the fundamental solution in this gauge is a holomorphic function in the cut complex plane [ holomorphic ] _ 0(z;)=_a(z-_a)^-_a,_a= _ a ( + i ) , ^n @xmath86 is analytic in @xmath87 . the cut @xmath88 runs from @xmath89 to infinity . is then well defined at infinity . this can be understood as reflecting the dirac flux quantization on compact manifolds . ] the general solution obtained by multiplying by a polynomial @xmath66 is then also holomorphic with the same cuts . [ lijs ] ( z;)=p(z)_a(z-_a)^-_a,(p ) < _ t-1 in general one may allow having both positive and negative fluxons . the solution eq . ( [ lijs ] ) corresponding to @xmath90 is typically not square integrable if some of the fluxons are super - critical . it becomes a legitimate solution only for @xmath91 where @xmath92 . the corresponding modes are then identical to those occurring in a system of ( sub - critical ) fluxons having @xmath93 ( assuming that @xmath94 ) . we call these states free zero modes . as we shall see their behaviour and holonomies are identical to those of a system with fluxes @xmath95 . since in general @xmath96 , the system will also contain another type of zero modes . in these states the electron typically sits in a small @xmath76 neighbourhood of a specific fluxon . we therefore call these states confined . in the special case of integer fluxons certain states may incorporate feature of both confined and free states ( see e.g. fig . [ fig : qubits ] ) . the main focus of this paper is on the holonomies of free states and the reader may assume , for simplicity , that all fluxons are subcritical so that @xmath97 and all states are free . for completeness we give below a brief account of the confined states . confined states localized near the @xmath20th fluxon are typically constructed by taking @xmath98 in eq . ( [ yr7 ] ) . it is convenient to take advantage of our superposition principle and write @xmath49 as a product of ( not necessarily fundamental ) solutions @xmath99 corresponding to all fluxons . a confined state at @xmath20 then takes the form ( z)=_a^j(z)_ba_b(z)_a^j(z)_ba_b(_a).[uigf]here @xmath100 are some wave - functions which depend on the detailed shape and radius @xmath22 of the confining fluxon , while for @xmath101 we have @xmath102 in holomorphic gauge . the approximate equality on the right of eq.([uigf ] ) follows from the fact that @xmath103 is sharply peaked near @xmath104 . the cases when @xmath105 or @xmath106 require a more careful analysis . we shall not delve into this here since our main interest is in the free modes . modes localized on different fluxons are clearly mutually orthogonal . some extra thought also shows that when properly normalized the overlap of free state with a confined state localized at @xmath23 scales as @xmath107 and hence vanishes in the pointlike limit . we are interested in the evolution of the zero modes when the fluxes move adiabatically . control space , parametrized by the fluxon coordinates @xmath108 is @xmath109 dimensional . since the motion of the fluxons generates electric fields we need first to construct corresponding ( time - dependent ) pauli operator , eq . ( [ pauli2 ] ) , with both @xmath110 and @xmath111 . by faraday law a moving magnetic field must be accompanied by a nonzero electric field . if the motion is adiabatic the velocity @xmath112 is small and the acceleration negligible . it follows that radiation and retardation can be neglected . the fields resulting from the motion can be obtained by lorentz transformation to the moving frame [ evb ] = - we therefore need first to determine the full pauli operator , allowing for both scalar and vector potentials , eq . ( [ pauli2 ] ) , due to the motion of fluxons . the main result of this subsection is eq . ( [ phi ] ) which we shall now derive . to determine @xmath113 associated with a moving fluxon we substitute eq . ( [ evb ] ) in the definitions of the potentials , eq . ( [ fields ] ) , -a_0=-== ( ) = ( ) - ( ) ( and on the right we used the fact that @xmath114 is a vector , not a vector field for a discussion of the general case . ] . ) this may be rearranged as [ potentials ] = ( + ) = ( + a_0 ) let the static magnetic field of the @xmath20-th fluxon be described by the vector potential @xmath115 . take @xmath116 to be the rigid transport of @xmath117 , so that @xmath118 , and choose @xmath119 so that eq . ( [ potentials ] ) is satisfied , . ] namely [ rot ] ( , t)=_a(-_a(t)),a_0(,t)=-_a _ a(-_a(t ) ) @xmath120 is the trajectory of the fluxon . this 4-potential generates the fields of a rigidly moving fluxon : [ fields2 ] ( , t)=_a(-_a(t ) ) , ( , t)=-_a(,t ) the generalization to a number of fluxons each moving along its own trajectory is obviously ) may be derived by applying a lorentz boost to the vector @xmath121 and keeping terms only up to first order in @xmath122 . ] [ phi ] = _ a=1^n_a(-_a(t ) ) , a_0=-_a=1^n_a_a(-_a(t ) ) note that @xmath113 _ is not _ necessarily in the coulomb gauge . it has the pleasant feature that a closed path in the space of controls @xmath123 is represented by a closed path of the potential , and hence a closed path of the hamiltonian . we are interested in the evolution of the zero modes due to adiabatic motion of the fluxons . more specifically , we are interested in the holonomy that describes the braiding of fluxons . this adiabatic problem has three subtle points : * gapless zero modes . the zero modes lie at the threshold of the continuous spectrum so the adiabatic evolution is not protected by a gap . one may then appeal to adiabatic theorems that cover the gapless case . for point - like fluxons the only length scale is the distance between them . ] @xcite . these theorems hold provided the space of zero modes changes smoothly . consider the collision of sub - critical point - like fluxons . the norm of the fundamental solution @xmath124 diverges when they form a super - critical fluxon . it follows that the space of zero - modes does not behave smoothly upon flux collisions . flux collisions then play the role of gap closure in gapped adiabatic evolutions . removing points of flux collisions endows the control space with a non - trivial topology . * gauge freedom : adiabatic phases are well defined ( gauge invariant ) for closed cycles of the hamiltonian @xcite . braiding of fluxons is described by a closed cycle in control space @xmath123 , and therefore also a closed cycle in the space of em _ fields _ , but not necessarily in the space of hamiltonian which depends on the potentials . to correctly compute the holonomy , one needs the em _ potentials _ to make a closed cycle . ( one _ is not _ interested in phases that come simply from a change of gauge between the initial and final hamiltonian . ) this is taken care of by the choice of gauge made in eq . ( [ phi ] ) . * parallel transport : in standard adiabatic hamiltonians @xcite the the adiabatic evolution is determined by the frozen hamiltonian . this is not the case here where the weak electric field generates the evolution . it turns out to be instructive @xcite to write the pauli evolution equation as id_t=1 2 m ( ( -i - e ) ) ^2,d_t=_t -iea_0 with @xmath125 replaced by the covariant time derivative @xmath126 . let @xmath127 denote the spectral projection on the zero modes of ( the frozen ) pauli operator . the evolution generated by [ kato ] i= ( _ p + i [ ] ) maps unitarily @xmath128 @xcite . the first term describes the action of the hamiltonian on @xmath129 and the second term guarantees that the states remain within the instantaneous spectral subspace . usually , the first term acts on @xmath129 as a c - number giving it just an overall phase and therefore in spite of being @xmath130 it is usually less important than the second which is only @xmath131 . for the case at hand , both terms act nontrivially on the zero modes space and are @xmath131 . ( [ kato ] ) reduces to i=(i [ , ] -e a_0 ) = ( i [ , ] + e_a=1^n _ a _ a ) and , we have used eqs . ( [ pauli2],[phi ] ) . in particular , if @xmath49 and @xmath132 are zero modes then @xmath133 and @xmath134 and the evolution of zero modes is governed by [ evolution1 ] i\| d = e _ a d_a \| _ a \| = e_a\| v_a a_a+ covariant derivative @xmath126 . ( here @xmath135 . ) given @xmath59 , the instantaneous fundamental solution @xmath136 is uniquely determined as in sec . [ sec : ac ] . the adiabatic evolution can be viewed as a rule for evolving the polynomial @xmath137 [ poly ] = p(z , t)_0 = p(z , t)_a(_a)_0 , p(z , t)=_j=0^d -1 p_j(t ) z^j in the rest of this section we show that the matrix elements of @xmath116 in the evolution equation ( [ evolution1 ] ) can be traded for the derivatives of the zero modes overlaps . the main results are eq . ( [ pt ] ) , or equivalently eq . ( [ gy876 t ] ) below . note first that fundamental mode @xmath71 of each individual fluxon satisfies [ kmode ] 0=(\|_z - ie\|a_a)(_a)_0=-(\|_a+ie\|a_a)(_a)_0 , _ a=__a it follows from this that @xmath138 the evolution equation then takes the form [ pt0 ] 0=\| d + d_a \|_a \| - d\|_a \| \|_a using the fact that the fundamental modes @xmath71 evolve by rigid motion d_0= ( d_a_a)_0=(d_a_a+d\|_a\|_a)_0 we finally arrive at the evolution equation for the polynomial @xmath137 : [ pt ] 0=\|d p\| + d_a _ a\| , = p(z , t)_0,=q(z)_0 which may also be stated as : the evolution of a zero mode under the change of the controls @xmath139 is determined by the evolution of the corresponding polynomial @xmath137 . this is determined by the equations _ 0\|\|qdp\|_0+d_a_a _ 0\|\|q p \|_0=0 [ gy876 t ] when the fluxons are pointlike subcritical @xmath3 and the fundamental mode is chosen in the holomorphic gauge , as in eq . ( [ holomorphic ] ) , the sum on the right of eq . ( [ pt0 ] ) vanishes and the evolution equation simplifies to the statement that the velocity in the manifold of zero modes vanishes : [ parallel ] 0=\| d , = p(z , t)_0,=q(z)_0 note that the scalar product in the holomorphic gauge is well defined even for fractional @xmath21 , independently of the way one chooses the cuts as long as this choice is done consistently . the @xmath14 dimensional space of zero modes can be naturally identified with the space of holomorphic polynomials with @xmath141 . natural coordinates are the coefficient @xmath142 in @xmath143 . let p=(p_0 , , p_d-1)^t , p^d the hilbert space metric induces a metric on @xmath144 [ metric ] ( ) _ jk= _ j \| _ k , _ j = z^j_0 , j , k0 , , d-1 with the properties : * @xmath140 is a positive , hermitian , @xmath145 matrix . * @xmath140 is gauge invariant : it is independent of the choice of gauge for the ( frozen ) pauli operator . * @xmath140 is a smooth function of the fluxes , @xmath21 , provided @xmath146 . it blows up as the total flux @xmath147 reflecting the loss of one mode . * when all the fluxons are finite , @xmath148 , the metric is an everywhere smooth function of @xmath149 , the fluxon coordinates . in the limit of pointlike fluxons , @xmath32 , we can say more : * the metric has an ( approximate ) block structure : all the ( properly normalized ) confined modes are in @xmath150 blocks , and all the free modes are in a single block . ( the terms connecting different blocks scale as positive powers of @xmath151 . ) * the block of the free zero modes is given by [ bkj ] ( ) _ jk= _ j \| _ k , _ j(z ; ) = z^j _ a(z-_a)^-_a , _ a=_a - max\{0,_a } * under scaling the metric of the free modes behaves like : [ scaling ] _ jk()=^k\|^j \|\|^2(1-_t ) _ jk ( ) note that the magnitude of @xmath152 describes dilation and its phase a rigid rotation of the control space . * when two point - like fluxons @xmath20 and @xmath43 collide the metric blows up if @xmath153 . * it is natural to remove from control space ( with coordinates @xmath89 ) the points where @xmath154 . this endows the control space of point fluxons with an interesting topology . consider a path @xmath155 in the space of controls . we are interested in the evolution of @xmath156 along the path . making use of @xmath140 , the transport equation , eq . ( [ gy876 t ] ) , takes the form [ asd ] 0= d p + ( d_a _ a ) p this can be written more compactly using the dolbeault operator stands for dolbeault to be distinguished from @xmath157 and @xmath158 . ] @xmath159 @xcite ( similarly @xmath160 ) : [ yr ] 0= d p + ( ) p the transport equation may then be written in terms of a connection@xmath161 [ gp ] 0= ( d+a)p , = ^-1 ( ) the connection determines the ( adiabatic ) curvature 2-form @xmath162 by a standard formula @xcite : [ rr ] r= da+a =\| ( ^-1 ) in the abelian case @xmath163 this simplifies into [ ra ] r=\| the curvature vanishes , @xmath164 , when the connection @xmath165 is a pure gauge : [ puregauge ] a= ^-1_0d _ 0the connection @xmath166 of eq . ( [ gp ] ) is `` half way '' to be a pure gauge ( @xmath167 is missing ) . it has special properties which we return to below . the study of the connection for point like fluxons is easier in some cases and harder in others : it is relatively easy for the confined modes where arguments based on the aharonov - bohm effect apply . it is more difficult in the more interesting case of free zero modes . this case too splits into cases with different levels of difficulty : the abelian case is simpler than the non - abelian case , and rigid translations and rotations are easier than braiding . this section is organized so that we treat the easier and special cases first . a reader who prefers to start with the most general results may want to read the subsections below in different order . by eqs . ( [ gp],[rr ] ) the connection @xmath166 and curvature @xmath162 inherit the block structure of @xmath140 . as all the confined modes are orthogonal to the free modes ( in the limit @xmath90 ) and to each other , one concludes that the connection @xmath166 and curvature @xmath162 split into a block corresponding to all free states and a number of @xmath168 blocks corresponding to each of the confined modes . each block remains completely unaffected by other blocks and may therefore be discussed separately . braiding the pair @xmath169 one expects is finite due to the power law tails of the state and exact in the limit @xmath31 . ] a confined mode on @xmath20 to acquire aharonov - bohm phase @xmath170 . let us see how this can be understood from the machinery developed in sec . [ ch : adiabatic ] . due to the block structure of the connection , [ sec : block ] , the adiabatic evolution of a confined state has only an abelian phase factor @xmath171 . from sec . [ super ] a mode confined at fluxon @xmath20 is described ( e.g. in holomorphic gauge ) by @xmath172 where @xmath173 is independent of the @xmath174s . the metric ( @xmath168 block ) is the function _ a \| _ a _ ba @xmath175 in particular , if @xmath174 encircles @xmath89 the phase is the aharonov - bohm phase @xmath176 . * under ( rigid ) translations @xmath178 of the entire fluxon configuration independent of @xmath20 but not necessarily of @xmath179 the wave function undergoes simple translation @xmath180 . this fact may be verified by using eqs . ( [ kernel1],[evolution1 ] ) . * a rigid rotation of the entire configuration is described by ( complex ) scaling @xmath181 . from eqs . ( [ scaling ] ) we find @xmath182 from eq . ( [ yr ] ) the @xmath183-th coefficient rotates independently of the others dp_k(t)=-(k+1-_t ) p_k(t ) ( t ) dt in particular a full @xmath184 rotation result in acquiring of the ( abelian ) berry phase [ berry ] 2(_t - k-1 ) as @xmath183 is an integer this phase is identical ( mod @xmath184 ) for all free states . ( this could be anticipated from the fact that changing the origin of rotation mixes different @xmath183s . ) consider @xmath0 subcritical fluxons @xmath3 . the metric @xmath140 is simply the norm of the fundamental mode = _ 0 \| _ 0 , _ 0(z;)=_a(z-_a)^-_a suppose @xmath185 , holding @xmath186 fixed . the adiabatic curvature in the remaining @xmath187 coordinate is given by eq . ( [ ra ] ) [ curva ] r = da=\|_u_u ( u ) , ( u)= there is no reason for @xmath188 to be a harmonic function . in appendix [ 3flux ] we show that @xmath140 and @xmath189 can be evaluated in terms of hyper - geometric functions . indeed as one can see from fig . [ fig : curvature ] for three half fluxes the curvature does not vanish . and @xmath190 . it is symmetric under @xmath191 . ( [ fig : r ] ) and appendix [ 3flux ] for more details on how the figure was drawn.,height=4 ] when @xmath39 and @xmath163 braiding is topological by a special reason that we shall discuss in sec . [ sec : square ] . however , for @xmath41 and @xmath163 , the braiding of two fluxons @xmath169 , keeping all the other fluxons fixed is , in general , path dependent . i.e. @xmath192 . the diagonal blue lines delineate regions without zero modes , with one zero mode and with 2 zero modes . in the triangle marked by @xmath193 the charge is localized on @xmath194 fluxons and berry s phase is @xmath195 as one would expect from the aharonov - bohm effect . the triangle marked @xmath196 describes a zero mode confined on 3 and the phase is @xmath197 . in the triangle marked by @xmath198 the state is free and the berry phase is @xmath199.,height=6 ] for the sake of simplicity of the notation we shall write @xmath200 and @xmath201 in this section . in appendix [ sec : integrals ] we show that the metric @xmath140 for the free zero modes factorizes into a holomorphic and anti - holomorphic factors . by eq . ( [ gexplicit ] ) : [ kjdc ] ( ; ) = ^ * ( ;) ( ) ( ;) where : * @xmath202 is an @xmath203 holomorphic ( matrix ) function whose matrix elements are given in eq . ( [ paj ] ) , as [ paj1 ] _ ak ( ) = _ _ 0^_a d _ k ( ) , a1, ,n , k0 , , d-1 @xmath202 involves a choice , which we call a _ gauge choice _ , and is hidden in the freedom of @xmath204 . alternatively one could add arbitrary integration constant @xmath205 to the r.h.s . * @xmath206 is an @xmath207 hermitian matrix which is independent of the controls @xmath208 . its explicit form is given in appendix [ sec : integrals ] . it has rank @xmath209 with its kernel generated by the vector @xmath210 . note that this guarantees that the metric @xmath211 is not affected by adding arbitrary integration constant to eq.([paj1 ] ) . substituting the expression ( [ kjdc ] ) for the metric into eq . ( [ yr ] ) leads to the transport equation _ dn d(p)=0 [ hjs ] these are @xmath14 equations for the evolution of the @xmath14 coefficients @xmath212 . the solutions of these equations are , in general , path dependent . if one could cancel out the left factor @xmath213 , it would follow that the holonomy is topological . in general however this can not be done since @xmath213 is not an invertible ( or even a square ) matrix . the holonomy is , in general , not topological . when the number of free zero modes is maximal @xmath214 the holonomy is topological . the simplest example of this kind is @xmath39 and @xmath38 . the berry phase associated with taking one fluxon around the other is _ topological _ and is given by eq . ( [ berry ] ) with @xmath215 @xmath216 one way to show that this is the case is by using the ` gauge ' freedom to choose @xmath217 in eq.([paj1 ] ) so that @xmath218 for all @xmath219 . by throwing its last row we can then view @xmath220 as a square @xmath145 matrix which we denote @xmath221 . we may then write for the metric [ square1 ] _ jk = _ a , b=1^d \|_aj g_ab _ bk = ^ * _ _ _ where @xmath222 are all square @xmath223d matrices . repeating the arguments of the previous subsection eq . ( [ hjs ] ) now takes the form ^ * _ _ ( _ p ) = 0 since @xmath211 is positive , all of its ( @xmath145 ) factors are invertible . the equations of parallel transport therefore reduces into @xmath14 conservation laws and @xmath165 is a pure gauge : [ ksg ] d ( _ p)=0 , = ^-1 = ^-1_d _ it follows that the holonomy of adiabatic transport is determined by the monodromy of the ( multivalued function ) @xmath221 . the gauge choice @xmath217 has the disadvantage of treating @xmath224 on different footing than the other @xmath89s . moreover , in braiding operations in which the @xmath0th fluxon is more than a spectator , any dependence of @xmath204 on @xmath224 can lead to extra complication . for this reason we shall prefer in the next section to fix @xmath204 to be a constant independent of @xmath149 . one may avoid the ` gauge ' choice @xmath217 and rewrite equation ( [ ksg ] ) in a gauge independent way as [ dgij ] d ( p ) _ n where @xmath225 is the generator of @xmath226 and the complex coefficient on the r.h.s depends on the arbitrary integration constant chosen in the definition of @xmath220 . we have seen that when @xmath34 there is no curvature . hence , if the unitary holonomy operators are non trivial , and non - abelian , then fluxon braiding can be viewed as ( non abelian ) topological unitary operations on the manifold of zero modes . we start by computing the monodromy of braiding of distinct fluxons . in the special case that the fluxons carry identical fluxes , they may be viewed as identical anyons . in particular , when the fluxes are identical , it obeys the braiding rules of burau representation of the braid group . ( 300,160)(0,0 ) ( 30,40)(1,0)200 ( 30,40 ) ( 80,90)(1,0)150 ( 80,90 ) ( 47,140)(1,0)183 ( 47,140 ) ( 25,25)@xmath227 ( 125,45)@xmath228 ( 220,65)@xmath229 ( 75,75)@xmath230 ( 135,95)@xmath231 ( 220,105)@xmath232 ( 42,125)@xmath233 ( 92,145)@xmath234 ( 220,155)@xmath235 ( 220,20)@xmath236 ( 7,10)(1,0)240 ( 7,10)(0,1)150 ( 230,-4)@xmath16 ( -9,149)@xmath237 we start by computing the monodromy of @xmath238 . in subsection [ uyufc ] we shall relate it to the holonomy of the adiabatic evolution . the components of the matrix @xmath220 are by eq . ( [ paj1 ] ) , [ ppppsi ] _ aj()=__0^_a d ^j _ b=1^n ( -_b)^-_b since @xmath219 is an integer the factor @xmath239 in the integrand has no interesting effect on the monodromy and we can ignore the index @xmath219 ( and henceforth drop it ) without risk . choosing integration paths from @xmath204 to @xmath240 which do not cross any of the cuts shown in fig . [ fig : cuts ] leads to a standard definition of @xmath241 . upon cyclically moving the fluxon positions , we keep fixed @xmath204 independent of @xmath149 . if we do otherwise the monodromy would get extra contribution from possible movement of @xmath204 . ] @xmath89 these paths are deformed as seen in figs . [ fig : braidb1],[fig : braida2 ] into paths which typically do cross the cuts . this leads to another branch @xmath242 of the multivalued function . the monodromy is an @xmath207 matrix @xmath243 : [ m ] = , = ( _ 1, ,_n)^t it is useful to collect properties of @xmath243 before one actually computes it , as they provide tests on the computations . * since the fluxons may be moved backwards , the monodromies @xmath243 must be invertible and generate a group . * since @xmath211 ( of eq . ( [ kjdc ] ) ) must not be affected by the monodromy , @xmath243 and @xmath244 must satisfy a consistency condition [ unitary ] = ^ * ( this may be viewed as a unitarity condition ) . * adding an integration constant ( or equivalently changing @xmath204 ) in eq . ( [ ppppsi ] ) corresponds to @xmath245 and @xmath246 . consistency with eq.([m ] ) thus requires @xmath247 with its ( red dashed ) branch cut @xmath88 going counter clock - wise around fluxon @xmath248 with its ( red dashed ) branch cut @xmath249 . as @xmath20 encircles @xmath43 the integration path ( blue ) from @xmath204 to @xmath174 loops around the branch point of the cut @xmath88 . the integration from @xmath20 to @xmath43 in the old and new paths are related by a factor @xmath250.,width=13 ] a proper definition of the monodromy of @xmath251 requires choosing some definite convention for placements of the cuts , see fig . [ fig : cuts ] . we shall take the cut @xmath88 to run from @xmath89 to @xmath252 , and we order them in such a way that as one goes counter - clockwise along a big circle ( near infinity ) the cuts @xmath88 are traversed successively according to their @xmath20-indexing . let us now compute the monodromy as the flux @xmath253 encircles an adjacent flux @xmath254 counter - clockwise . the integration path associated with @xmath255 loops around the branch point @xmath20 as shown in fig . [ fig : braidb1 ] . the deformed path gives @xmath256 by similar considerations as @xmath20 loops around @xmath43 , the integration path associated with @xmath257 `` stitches '' @xmath43 as shown in fig . [ fig : braida2 ] . it follows that @xmath258 to the fluxon at @xmath89 . as the @xmath20-th fluxons encircle b counter - clockwise it `` stitches '' @xmath43 . the old and new values of @xmath257 differ by integrations from @xmath20 to @xmath43 along two sides of the two cuts . , width=13 ] all other components of @xmath251 remain unaffected . the @xmath259 nontrivial block of the monodromy matrix is therefore ( _ a,_b)= ( cc 1-_a+_a_b & _ a(1-_b ) + 1-_a & _ a ) , = _ a_b the monodromy matrix is not symmetric in @xmath260 due to our convention for ordering the cuts counter clockwise . @xmath261 is related to @xmath262 by @xmath263 by eq.([unitary ] ) the spectrum of @xmath264 should lie on the unit circle . is spanned by @xmath265 known to be an eigenvector of @xmath264 . ] indeed : [ evm ] eigenvalues ( ) = \{1,_a _ b } in the special case where all fluxons are identical ( having the same @xmath21 ) it makes sense to consider also a permutation of two adjacent fluxons . this leads to the burau representation of the braid group @xcite . indeed , inspecting fig . [ fig : ex ] we see @xmath266 the monodromy matrix has the single nontrivial @xmath267 block [ ex ] ( ) = ( cc 1- & + 1 & 0 ) , = -with eigenvalues @xmath268 . note that when @xmath269 this reduces to the standard representation of the symmetric group . this may be understood as due to the fact that in this limit the free zero - modes turn into confined modes which move with the fluxons . the case @xmath270 ( i.e. @xmath271 ) does not occur since it is inconsistent with the assumption @xmath34 , see eq . ( [ con ] ) . the ( non - abelian ) holonomy @xmath272 for a closed path @xmath273 and base point @xmath149 , acts unitarily on the @xmath34 dimensional space of ( free ) zero modes at @xmath149 : @xmath274 one may write @xmath275 or equivalently @xmath276 . since the basis @xmath277 is not orthonormal , the matrix @xmath278 is not unitary . instead it satisfies @xmath279 . this is consistent with unitarity of the holonomy operator @xmath280 the previous sections make it clear that on the @xmath34 dimensional space of free zero modes , the @xmath281 matrix @xmath282 is closely related to the @xmath207 monodromy matrix @xmath264 . the exact relation is however complicated by the gauge freedom of fixing @xmath204 . we would like in this section to state this relation more precisely . for simplicity , consider first only braidings which do not involve the @xmath0-th fluxon . using the gauge choice @xmath217 one writes the conservation law , eq . ( [ ksg ] ) , taking @xmath283 around a closed path ( based at @xmath149 ) : _ p= _ p= _ _ p ( @xmath284 is the @xmath145 matrix obtained from @xmath243 by deleting its last row and column . ) the last identity used the definition of @xmath285 eq . ( [ m ] ) . hence [ u ] p_i=_j u_ijp_j , =( _ ) ^-1 ^-1 _ _ the derivation given above was special to the case where @xmath224 was a spectator . below we give an analysis of the general case . the monodromy matrices @xmath264 acting on @xmath286 preserve the vector @xmath287 i.e. satisfy @xmath288 . it follows that @xmath264 defines a linear transformation @xmath289 on the quotient space @xmath290 . we shall show that the holonomy @xmath272 for a closed path @xmath273 and base point @xmath149 is obtained from @xmath289 by a similarity transformation . the hermitian matrix @xmath291 satisfies @xmath292 . therefore the hermitian form it defines on @xmath286 , projects to a hermitian form @xmath293 on @xmath290 . moreover since @xmath291 has @xmath14 positive eigenvalues the form @xmath293 must give a ( non degenerate ) inner product on @xmath294 . ( [ unitary ] ) shows that @xmath289 are unitary relative to this inner product on @xmath294 . by eq.([dgij ] ) and the definition of the monodromy @xmath264 one has @xmath295 recall that @xmath238 is an @xmath296 matrix i.e. a map @xmath297 . denoting by @xmath298 the corresponding map into @xmath290 we conclude @xmath299 as @xmath298 is clearly invertible ( as follows e.g. from @xmath300 ) , we see that @xmath301 in particular the eigenvalues of the holonomy @xmath302 of fluxon braiding are related to the eigenvalues of the monodromy @xmath264 by [ mu ] eigenvalues ( ) = eigenvalues()\{1}=eigenvalues ( ^-1)\{1 } it follows from the results of the previous sections that when the fluxon @xmath20 goes around the fluxon @xmath43 , the holonomy matrix eigenvalues are [ main2 ] eigenvalues ( ) = \{1 , \|_a\|_b } by considering the @xmath207 matrix @xmath303 obtained by adding @xmath265 as an extra column to @xmath304 and defining @xmath305 we can obtain the simple relation @xmath306 as well as @xmath307 . the set @xmath308 is however an over - spanning set rather than a basis as it satisfies @xmath309 . in this section we show that the metric @xmath140 for the free modes of point - like fluxons factorizes into a product of a holomorphic and anti - holomorphic factors . since we are interested only in the free states , we will , for notational simplicity , assume all fluxons are subcritical . if this is not the case one should replace @xmath21 by its fractional part . let @xmath310 denote the primitive integral of @xmath311 : [ f ] _ j(z;,_0)=__0^z d _ j ( ) , _ j()=^j_a ( -_a)^-_a , j=0, ,d-1 we shall refer to the choice of @xmath204 as a choice of a gauge . for @xmath312 mathematica can evaluate @xmath313 of eq . ( [ f ] ) in terms of known special functions . in general , when @xmath314 the integral form is the best we can do . since @xmath315 one of the two integrations in eq . ( [ bkj ] ) for the metric is for free [ g ] _ jk = i 2 d _ kd\|_j = i 2 _ a _ _ a _ k \|_j d\|z and we have used the generalized stokes theorem . the remaining contour integrals encircle the cuts @xmath88 running from @xmath89 to @xmath252 ( see fig . [ fig : cuts ] ) . the value of @xmath316 above and below the cut are related by _ -=_a ( _ j)_+_a = e^-2i_a to see how @xmath317 behaves across the cut @xmath88 write _ k(z;,_0)= _ ak ( ) + _ k(z;,_a ) , where [ paj ] _ ak ( ) = _ k ( _ a;,_0)= _ _ 0^_a d ^k _ 0 ( ) , the first term is a finite ] constant ( independent of @xmath318 ) . the second term inherits the @xmath250 discontinuity of @xmath311 . it follows that @xmath319 is continuous across the cut @xmath88 and does not contribute to the integral in eq . ( [ g ] ) . the metric reduces to @xmath320 we denote by @xmath321 the value attained by @xmath322 as @xmath318 tends to infinity in the region between the cuts @xmath88 and @xmath323 . ( see fig . [ fig : cuts ] . ) the limit is well defined at infinity provided @xmath324 , which is what we need for the metric . rewrite eq . ( [ eq : g ] ) as a matrix equation [ eq : g2 ] = ( _ -)^ * _ 1 , ( _ 1)_ab= i 2 _ ab(1-\|_a ) @xmath325 is independent of @xmath149 . the @xmath0-tuples @xmath326 and @xmath327 are linearly dependent [ yjf ] _ j(_a-1)-_aj=_a ( _ j(_a)-_aj),a1 , , n where @xmath328 . this comes from integrating @xmath329 along @xmath88 . ( [ yjf ] ) too may be written as a matrix equation [ yjf2 ] _ 2 _ = _ 1 , ( _ 2)_ab = i 2 ( _ ab_a-_a , b+1 ) it follows from eq . ( [ eq : g2 ] ) and eq . ( [ yjf2 ] ) that the @xmath145 matrix @xmath140 can be factored as [ gexplicit ] ( ; ) = ^ * ( ;) ( ) ( ;) , = ( _ 1^(_2 ^ -1)^- ) _ 1 where : * @xmath202 is an @xmath203 holomorphic ( matrix ) function whose matrix elements are given in eq . ( [ paj ] ) . * @xmath206 is an @xmath207 matrix which is independent of the controls @xmath208 . * since @xmath211 is a @xmath145 positive matrix , @xmath244 must be an hermitian matrix . it must have at least @xmath14 positive eigenvalues and the image of @xmath220 must lie in the positive cone " of @xmath244 . in fact one may show that @xmath244 has exactly @xmath14 positive eigenvalues . * the definition of @xmath310 as a primitive integral in eq . ( [ f ] ) allows addition of an arbitrary integration constant ( possibly @xmath219-dependent ) corresponding to a free choice of @xmath204 . change of this choice will change the columns of @xmath220 by constant columns : [ psigauge ] = ( + ccc c_1 & & c_d + & & + c_1 & & c_d + ) since changing @xmath204 must not affect the metric , it follows that the kernel of @xmath244 contains the vector @xmath330 . it is in fact spanned by it . * one convenient ` gauge ' choice is @xmath217 which makes the last row of @xmath331 vanish . as a result eq . ( [ gexplicit ] ) takes the form @xmath332 where @xmath333 is @xmath334 and @xmath335 is @xmath281 . * an explicit expression for @xmath244 is [ ggn ] _ ab= + -((_t-_a ) ) & a = b + ( _ b ) & a < b + the values for @xmath336 may be deduced from hermiticity condition @xmath337 . alternatively the same value may be found from the relation @xmath338 . * when all the fluxes are identical @xmath244 is a tplitz matrix , i.e. constant along the diagonals , _ ab= g_a - b , a , b1, ,n explicitly , if each fluxon carries @xmath339 , then for , @xmath340 , [ gn ] g_0=- ( ) ( ( n-1))(n ) , g_k= e^i ( n-2k ) * away from the threshold for appearance of a new zero - mode , @xmath341 , the elements of @xmath244 are well defined and free of singularities . ( as are the elements of @xmath342 . ) for three fluxons one can find explicit expressions for the metric @xmath140 and the curvature . exploiting translation rotation and dilatation symmetries allow us to fix the location of two fluxons at will . we shall therefore assume the three fluxon are located at @xmath343 . choosing @xmath345 leads in the case @xmath163 to the following [ ucg ] = ( c 0 + u^-_3(1-_1 ) ( 1-_2 ) ( 2-_1-_2 ) _ 2_1(1-_1 , _ 3 ; 2-_1-_2 ; 1u ) + u^1-_1-_3(1-_1 ) ( 1-_3 ) ( 2-_1-_3 ) _ 2_1(1-_1 , _ 2 ; 2-_1-_3;u ) + ) http://functions.wolfram.com/hypergeometricfunctions/hypergeometric2f1/[@xmath346 is a hypergeometric function . for three identical fluxes @xmath339 this reduces into [ psi ] = ( 1-)^2 ( 2 - 2 ) ( c 0 + u^- f(1 u ) + u^1 - 2 f(u ) + ) , f ( u)=_2_1(1-,;2 - 2 ; u ) in particular in the special case @xmath347 it becomes [ hk]= ( cc 0 + 2k(1u ) + 2k(u ) + ) with http://www.wolframalpha.com/input/?i=elliptic+k[@xmath348 the complete elliptic integral of the first kind . using eq . ( [ gn ] ) for @xmath349 , one then finds a simple formula for the metric [ fig : r ] ( u)=8 re ( k(u)k(1-u)^ * ) the associated curvature is plotted in fig . ( [ fig : curvature ] ) . since @xmath220 is defined only up to addition of an arbitrary ( @xmath282-dependent ) multiple of @xmath265 , one may write down various alternative expressions to eq . ( [ ucg ] ) . using the following @xmath350 with @xmath349 the @xmath351 matrix given in eq . ( [ ggn ] ) , leads to expressing the @xmath150 metric @xmath140 as a combination of two squares @xmath352 for @xmath353 this is always positive . one can also get explicit formulas for the non - abelian case . @xmath220 is @xmath355 : [ psig ] = u^-_3 ( cc 0 & 0 + a & b + u^1-_1c & u^2-_1d + ) where @xmath356 when all three fluxes are identical @xmath339 this becomes [ psi2 ] = ( 1-)^22(2 - 2 ) ( cc 0 & 0 + 2u^-f(1 u ) & u^-g(1 u ) + 2u^1 - 2 f(u ) & u^2 - 2 g(u ) + ) where @xmath357 @xmath349 is again @xmath351 given by eq . ( [ ggn ] ) and @xmath140 is @xmath259 . in this appendix we give another ( more abstract ) construction of the connection described in sec . [ sec : connection1 ] and sec . [ sec : square ] corresponding to the free zero modes around point - like fluxons . in particular it shows that in general one may embed our @xmath33-dimensional bundle into a flat @xmath209 bundle . let @xmath294 be the fixed @xmath358-dimensional complex vector space @xmath359 where @xmath225 . since @xmath360 , the @xmath207 hermitian matrix @xmath291 defines a pseudo ( hermitian ) metric on @xmath294 . let @xmath361 be the space of possible positions of @xmath0 fluxons . consider the trivial bundle @xmath362 . for each @xmath219 the vector function @xmath363 defines a ( multivalued ) section of @xmath364 . we shall denote this section by @xmath313 as well although it is actually an equivalence class under quotienting by @xmath365 . at each point @xmath366 the vectors @xmath367 generate a @xmath14-dimensional subspace @xmath368 of @xmath294 . these spaces make up together a @xmath14-dimensional sub - bundle @xmath369 of @xmath364 . the restriction of @xmath291 to @xmath369 is a positive definite hermitian metric . this follows from the fact that @xmath370 is the hilbert space metric on our pauli zero - modes . in particular it follows that @xmath371 the @xmath291-orthogonal complement of @xmath369 is well defined and hence also the @xmath291-orthogonal projection @xmath372 . in fact one may write explicitly @xmath373 where @xmath374 is the inverse of the matrix @xmath375 . as @xmath364 is trivial it is natural to use the trivial connection @xmath376 on it . the projection @xmath372 then defines a connection @xmath377 on @xmath369 . consider a general section @xmath378 of @xmath369 . using the fact that @xmath379 we find that the covariant derivative is given by : @xmath380 the equation @xmath381 for parallel transport thus takes the form @xmath382 which is exactly identical to the transport equation eq . ( [ yr ] ) . consider adiabatically turning one of the flux tubes around itself once . to find the holonomy of zero energy bound states we first need to find the electric and magnetic fields generated by adiabatic rotation at angular rate @xmath383 . to find these , we need a model of a fluxon . consider the following simple model of fluxon , shown in fig . [ box ] : two concentric thin cylinders of radius @xmath73 with charge @xmath384 ( per unit length ) , and charge density @xmath385 , rotating at constant angular velocity @xmath386 . since the overall charge vanishes and the fields are time independent , there is no electric field . the magnetic field is static and it satisfies , ( recall @xmath387 ) @xmath388 leading to a jump in the boundary conditions @xmath389 assuming @xmath390 we then have [ flux ] b ( ) = 4 q & \| \|<r + 0&\| \|>r it follows that the flux , per eq . ( [ zero - modes ] ) , is _ t = 2 e q r^2 and charge density @xmath391 . the red cylinder rotates clockwise and the blue counter - clockwise with the same angular velocity . rotating the fluxon causes the red cylinder to rotate faster and the blue cylinder slower . this creates a voltage difference between the inside and outside of the fluxon . , width=6 ] consider what happens when one adiabatically rotates the whole arrangement by @xmath392 so the two cylinders rotate at different angular velocities . to figure out the addition of angular velocities @xmath393 and @xmath394 , let @xmath395 if @xmath396 then addition gives @xmath397 . but if we allow @xmath398 , the rule follows from additivity of the rapidity @xmath399 . one finds ( assuming @xmath383 small ) @xmath400 if this was all that happened , rotating the fluxon would have no effect on the fields ( to order @xmath383 ) . however , this is not all . relativistic lorentz contraction implies that the geometry of the cylinders must change . the perimeter of the cylinders should contract by the usual rule . as the embedding space remains euclidean the radius needs to adjust to accommodate the contraction . for a cylinder of finite width this would inevitably lead to nontrivial internal stresses , but in the zero width limit we consider here this issue can be ignored . thus if @xmath73 denotes the radius for the cylinder rotating with @xmath393 then the contraction of the radii is given by @xmath401 it follows that , to first order in @xmath383 @xmath402 hence @xmath403 this imply that in the annulus between the two cylinders there is a radial electric field and hence a potential difference between the inside and the outside of the fluxon : [ 2 ] v= 2 q 4 q r=4q r^2 = 2 e_t integrating over the time needed to complete one full rotation gives @xmath404 consider a charged particle having wave function @xmath49 in the presence of the fluxon . the above suggests that fluxon rotation would induce a phase on the part of the wave function which is inside the fluxon . if the evolution is adiabatic this relative phase can be translated into the overall phase @xmath405 @xmath406 is the ( fraction ) of charge inside the fluxon . the phase depends on the total charge inside the fluxon but is independent of how it is distributed there . it is instructive to contrast this result with what one expects from the aharonov - bohm effect . a ( classical , localized ) magnetic flux encircling a localized ( quantum ) charge @xmath406 gives half this phases . not only is the factor 2 intriguing but , even more importantly , the aharonov bohm argument implies that the phase should depend also on the distribution of charge inside the fluxons . in the discussion of moving fluxons , section [ 7tuygd ] , it was assumed for simplicity that @xmath114 stands for a vector rather than a vector field . as a result self rotations of fluxons were not permitted only the position @xmath89 of the fluxons centers were rotated . ] . it is in fact possible to generalize parts of our arguments to arbitrary vector fields . the aim of this appendix is to explain this . it is worthwhile to note that the following does not even require the introduction of a riemann metric as the argument are completely independent of it . by a slowly moving fluxon we shall mean a fluxon whose magnetic field @xmath18 is being dragged along the vector field @xmath112 . more precisely , this is described mathematically by writing @xmath407 where @xmath408 stands for the lie derivative along @xmath112 . in two dimensions @xmath18 is a scalar density and using explicit form of the lie derivative gives @xmath409 the first term describes moving along @xmath112 while the second makes @xmath18 behave as a density in cases where the flow defined by @xmath112 does not preserve volume . in particular this relation guarantees that the total flux is unchanged . in case of translations or rotations the second term vanishes anyway . the corresponding vector potential @xmath59 is of course not determined uniquely , but it is most convenient to assume it is dragged in a similar way . this leads to @xmath410 since @xmath59 is a covariant vector the explicit form in vector notation is @xmath411 the second term represents the required rotation of the vectorial components of @xmath59 . this relation may also be expressed as @xmath412 substituting this into eq.([fields ] ) for the electric field we find @xmath413 as in section [ 7tuygd ] it follows that the relation @xmath414 is consistent with the choice @xmath415 . this holds generally regardless of whether @xmath112 is a rigid motion or a deformation , and of whether it is constant or time dependent . it does not even matter here whether space is flat or curved . ( this may however matter when one considers the spinor @xmath49 . )	aharonov and casher showed that pauli hamiltonians in two dimensions have gapless zero modes . we study the adiabatic evolution of these modes under the slow motion of @xmath0 fluxons with fluxes @xmath1 . the positions , @xmath2 , of the fluxons are viewed as controls . we are interested in the holonomies associated with closed paths in the space of controls . the holonomies can sometimes be abelian , but in general are not . they can sometimes be topological , but in general are not . we analyze some of the special cases and some of the general ones . our most interesting results concern the cases where holonomy turns out to be topological which is the case when all the fluxons are subcritical , @xmath3 , and the number of zero modes is @xmath4 . if @xmath5 it is also non - abelian . in the special case that the fluxons carry identical fluxes the resulting anyons satisfy the burau representations of the braid group .
the historically accepted shape of the heliosphere is that of a comet - like object with a long tail that is dragged downstream by the flow of the local interstellar medium ( lism ) past the sun @xcite . these early pictures , however , were based on a hydrodynamic description of the solar outflow the solar magnetic field was assumed to play a negligible role in the overall structure of the heliosphere and its interaction with the solar wind . computational models based on the magnetohydrodynamic ( mhd ) equations included the solar magnetic field as well as that of the interstellar medium and also produced a heliosphere with a comet - like shape @xcite . on the other hand , the measurements of energetic neutral atoms ( enas ) by ibex and cassini produced some surprises . these enas travel long distances through the heliosphere without being influenced by the ambient magnetic field and therefore yield information about the large - scale structure of the heliosphere . the cassini ena fluxes from the direction of the nose and the tail were comparable , leading the cassini observers to conclude that the heliosphere was `` tailless '' @xcite . the ibex observations from the tail revealed that the hardest spectrum of enas were localized in two lobes at high latitude while the softest spectra were at low latitudes @xcite . recent mhd simulations using a monopole model for the solar magnetic field , designed to reduce the numerical dissipation of magnetic energy that arises from a conventional dipole model , revealed that the solar magnetic field was strong enough to collimate the solar wind into a pair of jets that flow to the north and south @xcite . these jets bend in the direction of the tail , pushed by the flow of the lism . the interstellar rather than the solar wind plasma flows between these jets in the equatorial region downstream . such bent jets have been seen in protostellar systems @xcite and clusters of galaxies @xcite . astrophysical jets around massive black holes are thought to be driven by centrifugal forces that sling the plasma along a rotating helical magnetic field @xcite . however , the jets in the case of the heliosphere are driven in the region downstream of the ts as was proposed for the crab nebula @xcite . in this region of subsonic flow , the magnetic tension ( hoop ) force is strong enough to collimate and drive the wind . mhd models of the global heliosphere are complex and the mechanisms that control the shape of the hp , the thickness of the hs , the structure of the heliospheric jets , including the driver for the outflow , remain uncertain . the voyager 1 observations have revealed that the thickness of the hs is around @xmath1 , which is substantially thinner than expected from the global simulations . we present an analytic model of the heliosphere outside of a spherically symmetric ts where we neglect the ambient flow and magnetic field of the lism . taking the resulting heliosphere as axisymmetric and the flows within the hs as subsonic , we obtain the pressure and magnetic field structure of the hs along with the radius @xmath2 of the hp . the overall shape of the hs takes the classic form of an astrophysical jet : the flows through the ts are accelerated to the north and south by the solar magnetic field . the heliopause radius is determined by continuity : the plasma flow through the ts must balance the outflow through the jets . we present parallel global mhd simulations in the limit of zero magnetic field and flow in the lism which support the analytic model . one reason the influence of the solar magnetic field on the structure of the heliosphere is often neglected in the literature is because the pressure of the ambient plasma is large compared with that of the magnetic field @xmath3 just downstream of the ts . we show , however , that the total plasma pressure does not control either the flows in or the thickness of the hs . the overall pressure in the hs is balanced by the pressure in the lism . it is the tension force of the hs magnetic field that controls the pressure difference between the ts and hp @xcite . to the north and south there is no tension force and this same pressure difference drives the axial flow of the heliospheric jets . thus , it is ultimately the solar magnetic field that controls the large - scale structure of the hs . we consider a simple axisymmetric system in which there is no lism flow or magnetic field and the lism is specified by its ambient pressure @xmath4 . we write down the steady - state mhd equations , including continuity , pressure , momentum and magnetic field , @xmath5 @xmath6 @xmath7 @xmath8 where @xmath9 is the radius in cylindrical coordinates , @xmath10 is the ratio of specific heats and @xmath11 is in the azimuthal direction . these equations are solved in the hs with boundary conditions on the density , pressure , magnetic field and flow given just downstream of the spherical ts which has a spherical coordinate radius @xmath12 . at the ts we assume that the azimuth flow @xmath13 is zero and since there are no forces in the @xmath14 direction ( eq . ( [ momentum ] ) ) , we can take @xmath15 everywhere . thus , from eq . ( [ continuity ] ) we can write @xmath16 where @xmath17 is the stream function for the particle flux . taking @xmath18 with @xmath19 from eq . ( [ nv ] ) , we can reform faraday s law in eq . ( [ faraday ] ) as @xmath20 this equation has components only in the @xmath14 direction so taking the dot product of this equation with @xmath21 yields the constraint that @xmath22 is constant along streamlines or @xmath23 where the function @xmath24 is only a function of @xmath17 . the form of @xmath24 can be determined by the boundary conditions along the ts . downstream of the ts we take the flow to be normal to the shock , in the radial ( in spherical coordinates ) direction with a constant value @xmath25 with the density @xmath26 also constant . so , from eq . ( [ nv ] ) we find @xmath27 or @xmath28 where @xmath29 is the polar angle in spherical coordinates at the shock . thus , the variation of @xmath17 along the shock is known . similarly , we know from the solutions of the parker spiral magnetic field that along the shock @xmath30 so @xmath31 is a constant and so is @xmath32 . throughout the hs we have @xmath33 turning to the pressure equation and using eq . ( [ nv ] ) , we find @xmath34 so @xmath35 is also constant along a streamline and therefore a function only of @xmath17 . as before , we can evaluate it along the ts where it is given by @xmath36 . thus , @xmath37 thus , both @xmath38 and @xmath39 in the hs are linked to @xmath40 and @xmath9 . we now focus on the high @xmath41 limit , which is most relevant to the hs , where the pressure associated with interstellar pickup ions dominates the magnetic pressure . specifically , we take @xmath41 to be a large parameter . in addition , since the flows are subsonic downstream of the ts , inertial forces are also small . we can therefore write the plasma pressure in a series @xmath42 with @xmath43 . thus , to lowest order eq . ( [ momentum ] ) becomes @xmath44 where @xmath45 is an explicit function of @xmath40 and @xmath9 through eq . ( [ p ] ) . equation ( [ p ] ) requires that to lowest order the pressure in the hs is constant everywhere and is given the value @xmath46 at the ts . the density is also a constant , @xmath26 . since @xmath38 is linked to @xmath40 and @xmath9 through eq . ( [ p ] ) the constancy of @xmath45 requires that @xmath47 so that @xmath40 increases with radius outside of the ts @xcite . at first order we include the inertial terms and magnetic field in the momentum equation , which becomes @xmath48 where @xmath39 has been replaced by @xmath26 and @xmath49 . before discussing the flows in the hs , we consider the weak flow limit of eq . ( [ momentum1 ] ) so that the inertial forces in the radial direction can be discarded . in this limit the magnetic tension force in eq . ( [ momentum1 ] ) causes the plasma pressure and total pressure to decrease with radius . this limit is artificial for the hs since the flow @xmath25 downstream of the ts is comparable to the alfvn speed and the associated radial inertial forces are comparable to the magnetic forces . nevertheless , this limit illustrates how the pressure in the hs varies . we do not require zero pressure gradient in @xmath50 . the pressure drop from the ts to the lism is balanced by magnetic tension in the radial direction but the same pressure drop also develops from the equator to the outflow jets to the north and south . this pressure gradient along @xmath50 drives the outward flows associated with the jets . thus , we integrate eq . ( [ momentum1 ] ) from the ts outwards to obtain an explicit expression for @xmath51 , @xmath52 where @xmath53 is the value of @xmath54 at the ts and is dependent on @xmath50 . pressure balance across the hp , which requires that @xmath55 , then yields an explicit expression for @xmath2 , @xmath56 where @xmath57 . at this stage in the calculation the pressure difference @xmath58 remains undetermined . we will show , however , that the requirement that the mass flow into the hs across the ts balance that out the two jets constrains the pressure difference . in fig . [ helio_model ] we show 2-d plots of the plasma pressure , the magnetic pressure and the total pressure in the @xmath9 , @xmath50 plane in the hs . the data is shown for @xmath59 , which as shown later is the upper limit on @xmath58 . the inner boundary of the data shown is the ts and the outer boundary is the hp . the radius of the hp peaks at the midplane and falls off with @xmath50 until @xmath60 , where it remains constant , forming the northward jet . the plasma and total pressure decrease with radius @xmath9 while the magnetic pressure increases with @xmath9 . the total pressure falls off with distance from the equator until it approaches a constant value in the jet . we will show that this pressure difference , which is a consequence of magnetic tension , drives the jet outflow . along the axis @xmath38 remains constant at @xmath46 . this shape was obtained by neglecting the radial plasma inertia , an assumption which breaks down where the straight portion of the hp in fig . [ helio_model ] intersects the curved portion at @xmath61 . the sharp kink in the hp is not real and is not seen in the mhd simulations discussed later . in fig . [ helio_cuts ] cuts along @xmath9 of the total , plasma and magnetic pressures from the data of fig . [ helio_model ] are shown at the equator in ( a ) and across the jet in ( b ) . we now discuss the flows driven in the hs . we derive a bernoulli - like equation by taking the dot product of eq . ( [ momentum1 ] ) with @xmath62 and integrating along the streamline , @xmath63 unfortunately , this equation does not enable us to calculate @xmath64 throughout the hs because it requires that we know the trajectory of a stream line within the hs to link the local radius @xmath9 with its position at the ts where @xmath29 is known . along the axis where @xmath65 eq . ( [ v0 ] ) gives @xmath66 , since the pressure is constant and @xmath40 is zero . similarly , the velocity at the jet radius @xmath67 , can be calculated since at that location @xmath68 . we find @xmath69 , where @xmath70 . thus , the increase in the jet velocity above @xmath25 is linked to the alfvn velocity based on the magnetic field strength @xmath71 at the ts . more generally , we can calculate @xmath72 across the jet radius by noting that within the jet @xmath73 from eq . ( [ psi_s ] ) @xmath74 can be written in terms of @xmath17 so we are left with a single equation for @xmath17 across the jet , @xmath75 where @xmath17 varies from @xmath76 at the jet axis to @xmath77 at the hp . the equality of the particle fluxes through the ts and jets requires that eq . ( [ psi_jet ] ) produce the requisite jump in @xmath17 across the jet . equation ( [ psi_jet ] ) can be simplified by defining an angle variable @xmath78 , @xmath79 where @xmath54 varies from @xmath77 at the jet axis to @xmath80 at the hp . this equation can be integrated directly to obtain the jet radius @xmath67 , @xmath81 the jet radius is a maximum for @xmath82 when the jet outflow velocity is given by @xmath25 . in this limit the conservation of particle flux reduces to the jet cross - sectional area being equal to the ts area or @xmath83 . with increasing @xmath84 the outflow velocity of the jet increases and @xmath67 decreases . for @xmath85 , @xmath86 . an expression for the pressure jump @xmath58 between the ts and the lism can be calculated from eq . ( [ rhp ] ) , which is exact in the jet where @xmath87 , @xmath88 the pressure jump is a maximum when @xmath84 is small and decreases with increasing @xmath84 . the dependence of @xmath67 and @xmath58 are shown as functions of @xmath89 in fig . [ rjet_model ] . from eq . ( [ rhp ] ) @xmath2 therefore also decreases with increasing @xmath89 . we have carried out mhd simulations of the global heliosphere without an interstellar wind and magnetic field . our model is based on the 3d multi - fluid mhd code bats - r - us . it envolves one ionized and four neutral h species as well as the magnetic field of the sun . we used a monopole configuration for the solar magnetic field to eliminate artificial reconnection across the heliospheric current sheet . the basic parameters of the simulation are the same as those described in opher _ et al . the computational grid was @xmath90 in each direction . parameters of the solar wind at the inner boundary at @xmath91 were : @xmath92 , @xmath93 , @xmath94 and the parker spiral magnetic field with a radial component @xmath95 at the equator ( with an azimuthal component @xmath96 ) . the solar wind flow at the inner boundary is assumed to be spherically symmetric and the magnetic axis is aligned with the solar rotation axis . for the lism we assume @xmath97 while the plasma density was raised to @xmath98 to make up for the absence of pressure associated with the interstellar magnetic field . the number density of h atoms in the interstellar medium is @xmath99 and the temperature is the same as for the interstellar plasma . the z - axis is parallel to the solar rotation axis . the grid has cells ranging from @xmath100 at the inner boundary to @xmath101 at the outer boundary . the simulation had a resolution of @xmath102 between @xmath103 and @xmath104 ; @xmath105 , encompassing the entire hs . the run was stepped forward for @xmath106 years . in fig . [ mhd_hp ] we show in yellow the surface of the hp as defined by @xmath107 . the simulation reveals jets to the north and south as in the analytic model . the hp bulges at the equator as in the model . the gray lines are the solar magnetic field . shown in fig . [ mhd_p ] in the @xmath108 plane are the plasma pressure in ( a ) , the magnetic pressure in ( b ) and the speed and streamlines in ( c ) . as in the model , the plasma pressure decreases with cylindrical radius @xmath9 away from the jet axis while the magnetic pressure increases with @xmath9 and the strongest magnetic fields are in the equatorial region just upstream of the hp . the streamlines reveal the north and south directed outflows that make up the jets . the hp boundary does not reveal the sharp indentation seen in the model . finally , in fig . [ helio_cuts](c ) we show cuts of the total pressure ( solid ) , the plasma pressure ( dotted ) , the magnetic pressure ( dashed ) and the magnetic field ( dot - dashed ) in cuts along @xmath9 at the equator from just upstream of the ts to past the hp . the pressures have been normalized to @xmath109 , @xmath40 to @xmath71 and @xmath9 to @xmath12 with @xmath110 taken to be the location of the maximum of @xmath111 . the cuts are in remarkable agreement with the cuts from the model in ( a ) . in the simulation @xmath112 , which from fig . [ rjet_model ] , yields @xmath113 compared with the measured value of @xmath114 from fig . [ helio_cuts](c ) . for @xmath115 ( [ rhp ] ) yields @xmath116 at the equator , essentially identical to the hp radius in the cuts , and the ratio of the hp radius at the equator to that in the jet is @xmath117 ( eq . ( [ rhp ] ) ) compared with the reasured value of @xmath118 . finally the measured particle flux through the ts is the same as that out the jets . we have explored the structure of the hs and hp when the interstellar flow and magnetic field are neglected and the system can be treated as axisymmetric . we show that even in the limit in which @xmath119 in the hs the magnetic field controls the large - scale structure of the hs and drives northward and southward directed jets . to lowest order the pressure in the hs is balanced by the pressure in the interstellar medium . the magnetic field controls the pressure variation within the hs and re - directs and boosts the flow across the ts to the north and south to form heliospheric jets . the radial distance from the ts to the hp and the jet radii are controlled by the requirement that the plasma flowing into the hs across the ts flows outwards in the jets ( see also @xcite ) . for very weak magnetic fields the jet outflow velocity is the same as the velocity @xmath25 downstream of the ts . in this limit the total cross - sectional area of the jets is equal to the area of the ts and @xmath83 . with increasing magnetic field strength the jet outflow velocity increases and the radii of the hp and the outflow jet decrease ( eq . ( [ rjet ] ) and fig . [ rjet_model](a ) . the global mhd models of the heliosphere @xcite produce hs thicknesses that are around @xmath120 , substantially larger than the value of @xmath1 determined from voyager 1 s crossing of the hp in 2012 @xcite . the results here suggest that mechanisms that increase the jet outflows will reduce the hp radius . pressure reductions in the downstream region associated with thermal conduction or other mechanisms might produce such enhanced flows . there is evidence from both the present mhd simulations and those carried out the earlier @xcite that the jets are subject to large - scale instabilities . the resulting turbulence might be a driver of anomalous cosmic rays @xcite . high time - resolution ena measurements might be able to establish the existence of the heliospheric jets and associated turbulence . for a jet radius of around @xmath121 and an alfvn velocity of around @xmath122 , the alfvn transit time is around @xmath123 years . the jet turbulence might , of course , cascade to smaller scales so the relevant time scales could be shorter . this work has been supported by nasa grand challenge nnx14aib0 g and nasa awards nnx14af42 g , nnx13ae04 g and nnx13ae04 g . the mhd simulations were carried out on pleades at the nasa ames research center under the award smd-14 - 4986 . we acknowledge support from the international space science institute for the team `` facing the most pressing challenges to our understanding of the heliosheath and its outer boundaries . '' 2-d images in the hs of the plasma pressure in ( a ) , the magnetic pressure in ( b ) and the total pressure in ( c ) , all normalized to @xmath124 , where @xmath71 is the magnetic field just downstream of the ts at the equator . the images are for @xmath125 and @xmath126 , where @xmath46 is the plasma pressure downstream of the ts and @xmath4 is the pressure of the lism.,width=288 ] cuts through the data of fig . [ helio_model ] of the total pressure ( solid ) , plasma pressure ( dotted ) and magnetic pressure ( dashed ) versus @xmath9 at the equator in ( a ) and within the jet in ( b ) . in ( c ) cuts from the mhd simulation of fig . [ mhd_hp ] along the equator.,width=288 ] the heliopause as defined by @xmath128 from an mhd simulation embedded in an ambient interstellar medium with no mean flow and zero magnetic field . the gray lines are the solar magnetic field with the ts visible as a disc.,width=576 ] the plasma and magnetic pressures ( @xmath129 ) in ( a ) and ( b ) , and the plasma speed ( @xmath130 ) and streamlines in ( c ) . all in the @xmath108 plane through the center of the heliosphere from the simulation in fig . [ mhd_hp ] . , title="fig:",width=288 ] the plasma and magnetic pressures ( @xmath129 ) in ( a ) and ( b ) , and the plasma speed ( @xmath130 ) and streamlines in ( c ) . all in the @xmath108 plane through the center of the heliosphere from the simulation in fig . [ mhd_hp ] . , title="fig:",width=288 ] the plasma and magnetic pressures ( @xmath129 ) in ( a ) and ( b ) , and the plasma speed ( @xmath130 ) and streamlines in ( c ) . all in the @xmath108 plane through the center of the heliosphere from the simulation in fig . [ mhd_hp ] . , title="fig:",width=288 ]	an analytic model of the heliosheath ( hs ) between the termination shock ( ts ) and the heliopause ( hp ) is developed in the limit in which the interstellar flow and magnetic field are neglected . the heliosphere in this limit is axisymmetric and the overall structure of the hs and hp are controlled by the solar magnetic field even in the limit in which the ratio of the plasma to magnetic field pressure , @xmath0 , in the hs is large . the tension of the solar magnetic field produces a drop in the total pressure between the ts and the hp . this same pressure drop accelerates the plasma flow downstream of the ts into the north and south directions to form two collimated jets . the radii of these jets are controlled by the flow through the ts and the acceleration of this flow by the magnetic field a stronger solar magnetic field boosts the velocity of the jets and reduces the radii of the jets and the hp . magnetohydrodynamic ( mhd ) simulations of the global helioshere embedded in a stationary interstellar medium match well with the analytic model . the results suggest that mechanisms that reduce the hs plasma pressure downstream of the ts can enhance the jet outflow velocity and reduce the hp radius to values more consistent with the voyager 1 observations than in current global models .
by the hyperbolic dehn surgery theorem ( * ? ? ? * theorem 5.8.2 ) , all dehn surgeries on a hyperbolic knot give hyperbolic manifolds with only finitely many exceptions . thus a dehn surgery on a hyperbolic knot creating a non - hyperbolic manifold is now called an _ exceptional surgery _ , on which a large mount of studies have been done . see @xcite for a survey on this topic for example . in this paper , we consider exceptional surgeries on pretzel knots @xmath3 in the 3-sphere @xmath4 of type @xmath5 with @xmath6 . here note that @xmath7 and @xmath8 must be odd otherwise @xmath3 has two or more components . also we assume that @xmath9 since @xmath10 and @xmath11 are non - hyperbolic , where @xmath12 denote a torus knot of type @xmath13 . all the other knots @xmath3 are known to be hyperbolic . see @xcite , @xcite , and @xcite . exceptional surgeries on such pretzel knots have been studied extensively , motivated by the fact that the class of the knots includes various interesting examples about exceptional surgeries . see @xcite , @xcite , and @xcite for example . we here recall that exceptional surgeries are classified into the following three types : a reducible surgery ( yielding a reducible manifold ) , a toroidal surgery ( yielding a toroidal 3-manifold ) , a seifert fibered surgery ( yielding a seifert fibered 3-manifold ) , which is a consequence of an affirmative answer to the geometrization conjecture . our first result concerns toroidal surgeries on the knots . it is known that only the @xmath14-surgery on the @xmath5-pretzel knot is toroidal , and the surgered manifold contains the unique embedded incompressible torus up to isotopy . see @xcite for detailed descriptions . [ thm : toroidal_kabaya ] consider the toroidal manifold obtained by the @xmath14-surgery on the hyperbolic @xmath5-pretzel knot with odd integers @xmath15 . let @xmath16 be the one of the two components obtained from the toroidal manifold by cutting along the unique embedded incompressible torus , which is not a twisted @xmath2-bundle over the klein bottle . then @xmath16 is homeomorphic to the manifold obtained by a @xmath17-surgery on the chain - link with three components , where @xmath18 and @xmath19 . the chain - link with three components is depicted in figure [ fig : chain_link ] , whose complement is called the `` magic manifold '' . for its definition and notation , see @xcite in detail . in particular , in @xcite , the exceptional surgeries on the link are completely determined and classified . by referring their classification , we immediately obtain the following : [ cor:(-2,p , p)tor ] under the same setting as in theorem [ thm : toroidal_kabaya ] , the manifold @xmath20 is the seifert fibered space @xmath21 . all the other @xmath16 ( i.e. , @xmath1 ) are hyperbolic . in particular , @xmath22 is homeomorphic to the `` figure-8 knot sister manifold '' . theorem [ thm : toroidal_kabaya ] and the corollary [ cor:(-2,p , p)tor ] will be proved in section [ sec : toroidal ] . our second result concerns seifert fibered surgeries on such pretzel knots . for instance , @xmath23 is well - known for it is the first hyperbolic example , which admits non - trivial seifert fibered surgeries @xcite . on the other hand , in the case where @xmath24 , we obtain the following : [ thm : sf_ij ] a pretzel knot @xmath25 with positive integers @xmath26 admits no seifert fibered surgeries . this will be proved in section 3 by applying a method developed in @xcite by the first two authors . we note that our theorems together with known facts complete the classification of the exceptional surgeries on @xmath25 with @xmath1 . to see this , and also as a background , we recall some of known facts on exceptional surgeries on hyperbolic pretzel knots . actually most of the following results concern montesinos knots . however , for simplicity , we only deal with pretzel knots . see the original references for precise statements . wu showed that there are no reducible surgery on hyperbolic pretzel knots @xcite , and also obtained a complete classification of toroidal surgeries on pretzel knots @xcite . if a pretzel knot contains at most two non - integer tangles , then it is equivalent to a two - bridge knot , and then exceptional surgeries on such knots were completely determined in @xcite . on the other hand , if a pretzel knot contains at least four non - integer tangles , then it was also shown by wu @xcite that such a pretzel knot admits no exceptional surgery . furthermore , on pretzel knots , the first two authors gave a complete classification of surgeries yielding @xmath27-manifolds with cyclic or finite fundamental groups @xcite , and showed that there are no toroidal seifert surgeries on pretzel knots other than the trefoil @xcite . very recently , in @xcite , wu gave several restrictions , in particular , he showed that if a hyperbolic pretzel knot of length three admits an atoroidal seifert fibered surgery , then it is equivalent to @xmath28 with @xmath29 and , up to relabeling , @xmath30 ( * ? ? * theorem 7.2 ) . in this section , we give a proof of theorem [ thm : toroidal_kabaya ] and corollary [ cor:(-2,p , p)tor ] . first of all we set up our definitions and notations . a _ pretzel knot _ of type @xmath31 with integers @xmath32 , denoted by @xmath33 , is defined as a knot admitting a diagram obtained by putting rational tangles of the forms @xmath34 together in a circle . from a given knot @xmath35 in @xmath4 , we obtain a closed orientable 3-manifold by a _ dehn surgery _ on @xmath35 as follows : remove the interior of a tubular neighborhood @xmath36 of @xmath35 , and glue solid torus back . the slope ( i.e. , the isotopy class of an unoriented non - trivial simple closed curve ) on the peripheral torus @xmath37 , which is identified with the meridian of the attached solid torus is called the _ surgery slope_. it is well - known that slopes on the torus @xmath37 are parameterized by @xmath38 by using the standard meridian - longitude system for @xmath35 . thus , when the surgery slope corresponds to @xmath39 , we call the dehn surgery on @xmath35 along the surgery slope the _ @xmath40-dehn surgery _ or _ @xmath40-surgery _ for brevity , and denote the obtained manifold by @xmath41 . let @xmath35 be the @xmath5-pretzel knot . put @xmath35 on a genus two surface @xmath42 which bounds two handlebodies in @xmath4 as shown in figure [ fig : pretzel_on_surface ] , denote the ` outside ' of @xmath42 by @xmath43 and the ` inside ' by @xmath44 . the isotopy class on @xmath37 determined by the intersection @xmath45 is called the _ surface slope of @xmath35 with respect to @xmath42_. now the surface slope is @xmath14 . the manifold obtained from a dehn surgery on @xmath35 along the surface slope is described as follows . let @xmath46 ( resp . @xmath47 ) be the manifold obtained from @xmath43 ( resp . @xmath44 ) by attaching a @xmath48-handle along @xmath35 and @xmath49 . then the manifold obtained from dehn surgery on @xmath35 with surface slope is homeomorphic to @xmath50 . the inside @xmath44 contains a properly embedded one - holed klein bottle whose boundary coincides with @xmath35 , and @xmath44 is homeomorphic to the regular neighborhood of the one - holed klein bottle . therefore @xmath51 is a twisted @xmath2-bundle over the klein bottle and @xmath52 is a torus . we will show that @xmath53 is obtained by a dehn surgery on the chain - link with three components . instead of considering the @xmath5-pretzel knots , we study the @xmath27-component link depicted in figure [ fig : associated_link ] , which is obtained from the @xmath54-pretzel knot @xmath55 by adding two trivial components @xmath56 and @xmath57 encircling the two half - twisted strands respectively . the @xmath5-pretzel knot is obtained from the link by the @xmath58-surgery along @xmath56 and the @xmath59-surgery along @xmath57 , where @xmath60 and @xmath61 respectively . therefore the manifold @xmath46 is obtained from @xmath43 by attaching a @xmath48-handle along @xmath55 then doing the @xmath58-surgery along @xmath56 and the @xmath59-surgery along @xmath57 . let @xmath62 be the manifold obtained from @xmath63 by attaching a @xmath48-handle along @xmath55 . we will show that @xmath62 is homeomorphic to the exterior @xmath64 of the chain - link with three components , and describe the relation between peripheral curves of @xmath62 and @xmath64 . here we fix the standard meridian and longitude on each component of @xmath65 and we identify slopes with @xmath66 . take meridian disks @xmath67 and @xmath68 of the outside handlebody @xmath43 as shown in figure [ fig : associated_link ] and cut @xmath43 along @xmath67 and @xmath68 . then remove the regular neighborhood of the two arcs corresponding to @xmath56 and @xmath57 . cut the resulting manifold along the disks @xmath69 and @xmath70 as indicated in figure [ fig : handle_decomp_1 ] . this gives a handle decomposition of @xmath62 ( figure [ fig : handle_decomp_2 ] ) . in figure [ fig : handle_decomp_2 ] , we modify the regions encircled by dotted curves by the operations described in figure [ fig : simplification_1 ] . then we obtain a simplified handle decomposition of @xmath62 ( figures [ fig : handle_decomp_4 ] and [ fig : handle_decomp_5 ] ) . we regard the diagram of the handle decomposition given in figure [ fig : handle_decomp_5 ] as a trivalent graph , then taking the dual of this trivalent graph , we obtain a triangulation of the boundary of a @xmath27-ball ( figure [ fig : polyhedron ] ) . in this way we regard the handle decomposition of @xmath62 as a topological ideal polyhedral decomposition of @xmath62 . the ideal polyhedron further decomposed into 6 ideal tetrahedra . by using snappea @xcite , we can check that @xmath62 is obtained from gluing 6 positively oriented ideal tetrahedra , therefore has a hyperbolic structure . we can also check that there exists an isometry from @xmath62 to the exterior @xmath64 of the chain - link and we confirm that the isometry maps the slope @xmath58 on @xmath71 to the slope @xmath72 on @xmath65 , and similarly for the slope @xmath59 . this completes the ( computer - aided ) proof , but we also give an explicit homeomorphism between @xmath62 and @xmath64 . we decompose the chain - link exterior into two `` drums '' according to section 6 of @xcite . let @xmath73 , @xmath74 and @xmath75 be the disks bounded by the components of the chain - link in the simplest way ( figure [ fig : chain_link ] ) . slicing the exterior @xmath64 along the disks , then we obtain a solid torus whose boundary is tiled by quadrilaterals ( figure [ fig : sliced_link_complement ] ) . this solid torus is decomposed into two drums and further into 6 tetrahedra as shown in figure [ fig : splited_drums ] . glue together these 6 ideal tetrahedra along the faces which contain the double arrowed edges of figure [ fig : splited_drums ] , we obtain an ideal polyhedron with 12 faces ( figure [ fig : chain_polyhedron ] ) . since the gluing pattern of the ideal polyhedron is equivalent to the one given in figure [ fig : polyhedron ] , @xmath62 and @xmath64 are homeomorphic . finally we observe the correspondence between peripheral curves of @xmath62 and @xmath64 . in the polyhedral decomposition of @xmath62 given in figure [ fig : polyhedron ] , a path @xmath76 from the face @xmath77 to the face @xmath78 and a path @xmath79 from the face @xmath67 to the face @xmath80 form a meridian and longitude pair of @xmath56 . here the path corresponding to @xmath76 in @xmath64 is represented by a word of the form @xmath81 and the path corresponding to @xmath79 by @xmath82 , where @xmath83 ( resp . @xmath81 , @xmath84 ) is an element of the fundamental group of @xmath64 intersecting the disk @xmath73 ( resp . @xmath74 , @xmath75 ) at once ( figure [ fig : chain_link ] ) . on the other hand a meridian and longitude pair of the component corresponding to the disk @xmath74 is represented by the words @xmath81 and @xmath85 . therefore the slope @xmath86 on @xmath56 is mapped to @xmath87 , in particular @xmath58 to @xmath72 . by symmetry , the slope @xmath59 on @xmath57 is mapped to @xmath88 . let @xmath64 be the complement of the chain - link with three components , also known as the _ magic manifold_. we denote the @xmath86- and @xmath89-dehn filling of @xmath64 by @xmath90 . since any two components of @xmath65 can be interchanged by an automorphism which preserves the peripheral structure , this notation makes sense . then , by theorem [ thm : toroidal_kabaya ] , the manifold @xmath16 is homeomorphic to @xmath91 where @xmath60 and @xmath19 with @xmath92 . on the other hand , by the result of martelli and petronio @xcite , we know that @xmath93 is hyperbolic except if one of the following occurs up to permutation : * @xmath94 , * @xmath95 . thus we see that all the manifolds @xmath16 with @xmath1 ( i.e. , @xmath96 ) are hyperbolic . furthermore , when @xmath97 , equivalently @xmath98 , it is shown in @xcite that the manifold @xmath99 is homeomorphic to @xmath21 . in particular , the manifold @xmath22 is homeomorphic to @xmath100 , which is the `` figure-8 knot sister manifold '' . also note that , since @xmath101 is the whitehead sister link ( @xmath102-pretzel link ) , the @xmath103 is obtained from the whitehead sister link . in this section we give a proof of theorem [ thm : sf_ij ] . essentially the proof is on the same line as that for ( * ? ? ? * proposition 3.7 ) . * proof of theorem [ thm : sf_ij ] . * let @xmath35 be a pretzel knot @xmath25 with a positive integer @xmath26 . assume for the contrary that @xmath35 admits a seifert fibered surgery , i.e. , @xmath41 is seifert fibered for some @xmath104 . then by the results in @xcite , @xcite , and @xcite , @xmath41 must be a seifert fibered manifold with a base orbifold @xmath105 having three singular fibers . in particular , @xmath41 is atoroidal . first we see that @xmath40 must be an integer . actually , if @xmath41 is atoroidal seifert fibered , then @xmath107 unless @xmath35 is equivalent to one of the montesinos knots of type @xmath108 or @xmath109 ( * ? ? ? * theorem 8.3 ) . see @xcite for the precise statement . next , we note that @xmath35 is a periodic knot with period two as shown in figure [ period2 ] . the factor knot @xmath110 with respect to this cyclic period is equivalent to a torus knot @xmath111 . then , since the following diagram commutes , by ( * ? ? ? * lemma 3.8 ) , originally observed in @xcite , @xmath112 must be homeomorphic to a lens space . @xmath113_{r\text{-surgery } } \ar[r]^{/f } \ar@{}[dr]\|\circlearrowleft & s^3/f = s^3 \ar@{>}[dr]^{r/2\text{-surgery}}&~ \\ k(r ) \ar[r]^{/\bar{f } } & k(r)/\bar{f } & \hspace{-35pt } \cong k'(r/2 ) } \ ] ] then we have @xmath114 by the classification of dehn surgeries on torus knots due to moser @xcite . since @xmath107 , we have @xmath115 . next we apply the _ montesinos trick _ , originally introduced in @xcite . set an axis which induces a strong inversion of @xmath35 as shown in figure [ kp ] . then , applying the montesinos trick , we see that the surgered manifold @xmath116 is homeomorphic to the double branched cover of @xmath4 branched along the knot @xmath117 depicted in figure [ kp ] . for a strongly invertible hyperbolic knot @xmath35 and a rational number @xmath40 , if @xmath41 is a seifert fibered manifold with the base orbifold @xmath105 , then the link @xmath118 obtained by applying the montesinos trick to @xmath41 , i.e. , the link @xmath118 satisfying that the double branched cover of @xmath4 branched along @xmath118 is homeomorphic to @xmath41 , is equivalent to a montesinos link or a seifert link ( * ? ? ? * proposition 2.1 ) . here a link is said to be _ if its exterior is seifert fibered . also see @xcite and @xcite . note that @xmath117 is a knot since @xmath119 is odd . since seifert links are completely classified in @xcite ( see also ( * ? ? ? * proposition 7.3 ) ) , by this classification , we see that @xmath117 is seifert if and only if @xmath117 is a torus knot . we here apply the following fact : if @xmath120 , then @xmath121 is not a montesinos knot ( * ? ? ? * criterion 2.5 ) . here @xmath122 denotes the _ rasmussen invariant _ for a knot @xmath35 and @xmath123 the _ signature _ of a knot @xmath35 . now we need to calculate or estimate @xmath124 and @xmath125 . first we estimate the rasmussen invariant @xmath124 by using the following inequality obtained in @xcite and @xcite . for a knot @xmath35 and a diagram @xmath70 of @xmath35 , we have @xmath126 where @xmath127 denotes the writhe of @xmath70 and @xmath128 denotes the number of seifert circles of @xmath70 . applying this inequality to the diagram shown in figure [ kp ] , we have @xmath129 next we calculate the signature @xmath125 by using the method due to gordon and litherland @xcite . as shown in figure [ fig : surface ] , @xmath121 bounds a non - orientable surface @xmath130 such that the first betti number of @xmath130 is equal to three . take the loops @xmath79 , @xmath131 , and @xmath132 on @xmath130 , which form a basis of @xmath133 . then a bilinear form @xmath134 introduced in ( * ? ? ? * section 2 ) is presented by the following matrix : @xmath135 since @xmath1 , we see that @xmath136 , where @xmath137 denotes the signature of @xmath138 . furthermore , by considering the boundary slope of @xmath130 , the normal euler number of @xmath139 ( see ( * ? ? ? * section 3 ) ) , denoted by @xmath140 , is shown to be @xmath141 . then by ( * ? ? ? * corollary 5 ) , we have @xmath142 suppose that @xmath121 is a torus knot . as shown in figure [ kp ] , the knot @xmath121 is represented as a closure of a four - braid , the braid index of @xmath121 is at most four . since @xmath144 as in the proof of claim [ clm : montesinos ] , @xmath121 is non - alternating . actually , if a knot @xmath35 is alternating , then we have @xmath145 ( * ? ? * theorem 3 ) . hence the braid index of @xmath121 is three or four . then we see that @xmath146 as follows : for a knot @xmath35 , let @xmath147 be the _ determinant _ of @xmath35 and @xmath148 the _ alexander polynomial _ of @xmath35 . note that we have @xmath149 and it also coincides with the order of the first homology group of the double branched covering space of @xmath4 branched along @xmath35 ( see for example @xcite or @xcite ) . since @xmath41 is the double branched cover of @xmath4 branched along @xmath121 , we have @xmath150 . on the other hand , since @xmath151 , we have @xmath152 or @xmath27 , and @xmath153 . since @xmath154 , @xmath121 is equivalent to @xmath155 and we also have @xmath156 . next we consider the rasmussen invariant of @xmath157 and @xmath158 . for a knot @xmath35 , we denote by @xmath159 the mirror image of @xmath35 . by the inequality , we have @xmath160 since @xmath161 holds for a knot @xmath35 ( * ? ? ? * theorem 2 ) , we have @xmath162 the authors would like to thank professor akira yasuhara for helpful comments on a calculation of the signature of a knot . the first author is partially supported by grant - in - aid for young scientists ( b ) , no . 20740039 , ministry of education , culture , sports , science and technology , japan . the second author is partially supported by grant - in - aid for research activity start - up , no . 22840037 , japan society for the promotion of science . k. ichihara and i. d. jong , _ toroidal seifert fibered surgeries on montesinos knots _ , comm * 18 * ( 2010 ) , no . 3 , 579600 . k. ichihara and i. d. jong , _ seifert fibered surgeries and the rasmussen invariant _ , preprint . j. m. montesinos , _ surgery on links and double branched covers of @xmath4 _ , knots , groups , and @xmath27-manifolds ( papers dedicated to the memory of r. h. fox ) ann . of math . studies , no . 84 ( 1975 ) , 227259 .	we give a complete description of exceptional surgeries on pretzel knots of type @xmath0 with @xmath1 . it is known that such a knot admits a unique toroidal surgery yielding a toroidal manifold with a unique incompressible torus . by cutting along the torus , we obtain two connected components , one of which is a twisted @xmath2-bundle over the klein bottle . we show that the other is homeomorphic to the one obtained by certain dehn filling on the magic manifold . on the other hand , we show that all such pretzel knots admit no seifert fibered surgeries .
more than four decades after deep water cherenkov telescopes for high energy neutrinos have been proposed @xcite , two detectors of this type are successfully taking data : nt-200 in lake baikal , and amanda - ii at the south pole . first components of amanda s follow - up project of cubic - kilometer size , icecube , have been deployed in january 2005 . the science topics of these projects include the search for steady and variable sources of high energy neutrinos like active galactic nuclei ( agn ) , supernova remnants ( snr ) or microquasars , as well as the search for neutrinos from burst - like sources like gamma ray bursts ( grb ) @xcite . underwater / ice telescopes can also be used to tackle a series of questions besides high energy neutrino astronomy . these include the search for neutrinos from the decay of dark matter particles ( wimps ) and the search for magnetic monopoles or other exotic particles - like strange quark matter or q - balls ( see for reviews @xcite ) . all these topics are addressed not only by baikal / nt-200 , amanda and icecube , but also by the mediterranean projects reviewed in the talk of j.j . aubert at this conference @xcite . for deep ice detectors there are two modes of operation which are not or nearly not accessible for detectors in natural water . firstly , due to the low light activity of the surrounding medium , the photomultiplier ( pmt ) count rate in ice is only about 1 khz . this enables the detection of the feeble increase of individual pmt rates as caused , for instance , by multiple interactions of supernova burst neutrinos over time intervals of a few seconds . amanda - ii is actually monitoring about 90% of the galaxy for mev neutrinos from supernova explosions . secondly , deep ice arrays can be operated in coincidence with surface air shower arrays @xcite . this allows one to study questions like the mass composition of cosmic rays up to @xmath0 ev , to calibrate the neutrino telescope , and to use the surface detector as a veto against background events . this paper is organized as follows : in section 2 , the design and performance of the two running large neutrino telescopes are sketched , nt-200 in lake baikal and amanda - ii at south pole . section 3 is a synopsis of the physics results obtained so far by both detectors ( for recent summaries see the talks at the neutrino 2004 conference @xcite . section 4 desribes design and expected performance of icecube . underwater / ice neutrino telescopes consist of a lattice of photomultipliers ( pms ) housed in transparent glass spheres which are spaced over a large volume in the ocean , in lakes ( like the baikal telescope ) or in ice ( as amanda and icecube ) . the pms record arrival time and amplitude of the cherenkov light emitted by muons or particle cascades . the baikal neutrino telescope nt-200 is operated in lake baikal , siberia , at a depth of . an umbrella - like frame carries 8 strings , each with 24 optical modules ( oms ) arranged in pairs . the pressure glass spheres of the oms contain 37-cm diameter pms . the instrumented volume forms a cylinder of 70 m height and 42 m diameter . the angular resolution for through - going muons is about 4@xmath1 . the present , final configuration of the amanda neutrino telescope is named amanda - ii . it consists of 677 oms arranged along 19 vertical strings buried in the glacial ice at the south pole , with most of the oms at depths between 1500 and 2000 m. the geometric shape of the array is a cylinder @xmath2500 m high and @xmath2200 m in diameter . each amanda om houses an 8-inch hamamatsu photomultiplier . the present angular resolution for muons tracks is about 2.5@xmath1 . figure 1 sketches the configuration of both detectors . [ fig : detectors ] nt-200 takes data since april 1998 . it has been deployed in subsequent stages , starting in 1993 with nt-36 , the pioneering first stationary underwater array @xcite . components are deployed from an natural fixed platform : the thick ice layer which covers lake baikal in february and march , when uutside temperatures reach down to -25@xmath1 ( see for more technical details @xcite ) . figure [ fig : nuevents ] ( left ) shows a textbook neutrino event ( an upward moving muon track ) recorded with the early 4-string configuration of 1996 @xcite . the absorption length of deep baikal water varies between 20 and 24 meters . since light scattering is small , nt-200 can monitor a volume exceeding its own geometric volume by an order of magnitude . actually , this is the reason that the sensitivity of nt-200 with respect to high energy , diffuse - flux phenomena is not dwarfed by that of the much larger amanda - ii ( amanda is embedded in ice where light scattering is strong and diffuses light from distant sources ) . in 2005 , nt-200 is going to be upgraded with three sparsely instrumented outer strings ( nt-200 + , see fig.1 ) . the three strings will allow a dramatically improved vertex reconstruction of high energy cascades within this volume and will increase the sensitivity to diffuse fluxes by a factor of four ( see section 3 ) . rather than water , amanda uses ice as detection medium . the glacial ice is extremely transparent for cherenkov light with wavelengths near the peak sensitivity of the oms : at 400 nm , the average absorption length is 110 m. scattering , however , is much stronger than in water the average effective scattering length is only 20 m. below a depth of 1500 m , both scattering and absorption are dominated by dust , and the optical properties vary with dust concentration . to deploy amanda , holes were melted with hot water , and strings with oms frozen into the ice . similar to the baikal telescope , amanda was also deployed step by step . an earlier 10-string stage , called amanda - b10 , was installed between 1997 and 1999 @xcite . first neutrinos have been recorded with the inner four strings deployed in 1996 ( see fig . 2 , right ) . amanda - ii has been taking data since 2000 . for most analysis channels , both amanda and nt-200 use the earth as a filter and separate up - going muons steming from interactions of neutrinos having crossed the earth . the main class of background are down - going atmospheric muons that are misreconstructed as up - going . after their rejection , basically up - going muons from interactions of neutrinos generated in the atmosphere remain . atmospheric neutrinos not only constitute the main background when searching for extraterrestrial neutrinos , but are also a natural calibration source . figure [ fig : amaatmu ] shows the energy spectrum for up - going neutrinos based on amanda - ii data taken in 2000 . it has been obtained by a neural net energy reconstruction , followed by regularized unfolding . this is the first atmospheric neutrino spectrum above a few tev , and it extends up to 300 tev . amanda , with its 60 gev threshold for the standard event selection , is not sensitive to neutrino oscillations . the threshold of the baikal telescope nt-200 is much lower and , for tight selection procedures , can be reduced down to 10 gev . figure [ fig : baiatmu ] shows the angular spectrum for upward tracks close to the opposite zenit . the 20% deficit close to the vertical is well compatible with the the oscillation parameters @xmath3 and @xmath4 . this effect has to be taken into account when using these data as standard signal for detector calibration . corresponds to vertically upward moving tracks . data are compared to simulated distributions including ( full line ) and excluding ( dotted line ) the effect of oscillations . , width=257 ] the primary destination of neutrino telescopes is the identification of individual , point - like sources of high - energy neutrinos . should individual sources be too weak to produce an unambiguous directional signal in the array , the integrated neutrino flux from all sources could still produce a detectable diffuse signal . this flux could be revealed by an high energy excess on top of the omni - present background of atmospheric neutrinos . the data of both experiments have been searched for such a diffuse signal using complementary techniques in different energy regimes . event selection is typically optimized to maximize the sensitivity to an @xmath5 signal spectrum . experimental limits given below include statistical as well as systematic errors ( the latter being typically between 20 and 40 percent ) . * upward moving muons : * the atmospheric neutrino spectrum recorded with amanda - ii ( fig . [ fig : amaatmu ] ) was used to set an upper limit on a diffuse @xmath5 flux of extraterrestrial muon neutrinos for the energy range covered by the highest bin , 100300 tev , by calculating the maximal non - atmospheric contribution to the flux in that bin given its statistical uncertainty . however , the bins in the unfolded spectrum are correlated and the uncertainty in the last bin can not a priori be assumed to be poissonian . the statistics in the bin was therefore determined with many monte carlo samples used to construct confidence belts . given the ( fractional ) unfolded number of experimental events in the bin , a preliminary 90% c.l . upper limit of @xmath6 is derived for @xmath7 . * cascades : * apart from muon tracks , _ cascades _ can be detected . with a typical length of 5 - 10 m and a diameter of 10 cm , cascades can be considered as quasi point - like compared to the spacing of oms . all three neutrino flavors contribute to this signature cascades stem from the leptonic vertex of electron and tau neutrino charged current interactions , and hadronic vertex cascades from all - flavor neutral current interactions . the amanda analysis focuses on contained events ( allowing good energy reconstruction ) , the baikal / nt-200 survey to bright cascades produced at the neutrino interaction vertex in a large volume around the neutrino telescope . ( as pointed out in the introduction , lack of significant light scattering allows to monitor a volume exceeding the geometrical volume of nt-200 by an order of magnitude . ) both analyses did not yield any excess of candidate events over background : 1 observed event ( vs. 0.9 background events ) in the case of the year-2000 amanda data , no event ( vs. 0.4 background events ) in the case of the 1998 - 2003 nt-200 data . the corresponding upper limits with respect to the flux of all three flavors are @xmath8 ( amanda , @xmath9 @xcite ) and @xmath10 ( nt-200 , @xmath11 @xcite ) , calculated under the assumption that neutrinos arrive with a ratio @xmath12 . * ultra high energy neutrinos : * at ultra - high energies ( uhe ) , above 1 pev , the earth is opaque to electron- and muon - neutrinos . the amanda search for extraterrestrial uhe neutrinos is therefore concentrated on events close to the horizon and even from above . the latter is possible since the atmospheric muon background is low at these high energies due to the steeply falling energy spectrum . the search for uhe events in 1997 amanda - b10 data relies on parameters that are sensitive to the expected characteristics of an uhe signal : bright events , long tracks ( for muons ) , low fraction of single photoelectron hits . no excess above background is observed and a 90% c.l . limit on an @xmath5 flux of neutrinos of all flavors is derived , assuming a 1:1:1 flavor ratio at earth @xcite : @xmath13 ( @xmath14 ) . * summary of diffuse searches : * using different analysis techniques , amanda and nt-200 yield limits on the diffuse flux of neutrinos with extraterrestrial origin for neutrino energies from 6 tev up to a few eev . figure [ fig : diffuse ] summarizes the flux predictions , upper bounds derived from observed fluxes of charged cosmic rays and gamma rays , and existing best limits of amanda and nt-200 as well as extrapolations to several years of amanda data and to icecube . with the exception of the limit from the unfolded atmospheric spectrum , which can be seen as a quasi - differential limit , the limits are on the integrated flux over the energy range which contains 90% of the signal . these limits exclude some models like @xcite . note that both experiments enter new territory , being below the gamma bound @xcite where sensitivities are not _ a priori _ too weak to hope for a discovery ( see also section 4.2 ) . searches for neutrino point sources require good pointing resolution and are thus restricted to the @xmath15 channel . both experiments have produced sky - plots based on data taken over several years . figure [ fig : skyplots ] shows 3329 upward moving muons recorded by amanda during four years ( 2000 - 2003 ) , and 372 baikal events recorded during 5 years ( 1998 - 2003 ) . since good pointing resolution is mandatory , and since this can be achieved only for tracks crossing the array , the smaller nt-200 can not compete with amanda . we note , however , that for hard neutrino spectra ( e.g. @xmath5 ) nt-200 is yet the largest telescope at the northern hemisphere and nicely complements amanda in the south . in the following , only amanda data will be discussed in more detail . + amanda events were selected to maximize the model rejection potential for an @xmath5 neutrino spectrum convoluted with the background spectra due to atmospheric neutrinos and misreconstructed atmospheric muons . the _ sensitivity _ of the analysis , defined as the average upper limit one would expect to set on a non - atmospheric neutrino flux if no signal is detected , is about @xmath16 , averaged over the lower hemisphere and for a hypothetical @xmath5 signal spectrum . this is three times lower than the one - year limit for the year 2000 @xcite . it corresponds to a neutrino flux above 1 tev of @xmath171 tev)@xmath18 @xmath19 s@xmath20 about a factor five above the crab gamma flux and close to the gamma flux from markarian-501 in its flaring phase @xcite . the final sample of 3369 neutrino candidates ( 3329 from below , with 3438 expected atmospheric neutrinos ) was searched for point sources with two methods . in the first , the sky is divided into a fine - meshed grid of overlapping bins which are tested for a statistically significant excess over the background expectation ( estimated from all other bins in the same declination band ) . this search yielded no evidence for extraterrestrial point sources . the second method is an unbinned search , in which the sky locations of the events and their uncertainties from reconstruction are used to construct a sky map of significance in terms of fluctuation over background . the hot spots on this map are well within the expectation from a random event distribution . in a search for 33 preselected sources , the strongest excess was observed from the direction of the crab nebula , with 10 events where 5 are expected - again no significant effect given the number of trials . one thus sees no evidence for point sources with an @xmath5 energy spectrum based on the first four years of amanda - ii data , and it seems unlikely that another few years of data will change this result . obviously , a much larger array like icecube is neccessary to detect _ steady _ sources . however , the picture may change principally for _ transient _ sources , where , for searches during known gamma flares ( like e.g. the blazars mkr-421 , mkr-501 , es1959 + 650 ) , the signal - to - noise ratio may improve dramatically . a special case of point source analysis is the search for neutrinos coincident with gamma ray bursts ( grbs ) detected by satellite - borne detectors . here , the timing of the neutrino event serves as an additional selection handle which significantly reduces background . both collaborations have used the grb sample collected by the batse satellite detector which was decomissioned in 2000 . the amanda ( baikal ) and batse data taking periods were overlapping in 1997 - 2000 ( 1998 - 2000 ) . samples containing 312 ( 368 ) bursts triggered by batse from this period have been analyzed by amanda ( baikal ) . data were searched for an excess of events in a 10 min ( 100 second ) window around the grb time . no coincident neutrino event was observed in the case of amanda , and one ( over a background of 0.46 ) for baikal . assuming a broken power - law energy spectrum as proposed by waxmann and bahcall @xcite , the 90% c.l . upper limit on the expected neutrino flux at the earth derived by amanda is @xmath21 . this is approximately a factor 15 above the waxmann - bahcall flux prediction . other classes of bursts are being included in the analysis , like the so - called non - triggered batse bursts and triggers from the third interplanetary network ( ipn3 ) , since 2000 the major source of grb detection . the minimal supersymmetric extension of the standard model ( mssm ) provides a promising dark matter candidate in the neutralino , which could be the lightest supersymmetric particle . neutralinos can be gravitationally trapped in massive bodies , and can then via annihilations and the decay of the resulting particles produce neutrinos . dark matter can therefore indirectly searched for by looking for fluxes of neutrinos from the center of the earth or the sun . both collaborations have searched for vertically up - going tracks from the center of the earth , for amanda - b10 using data from 1997 - 1999 , for nt-200 using data from 1998/1999 . no indication of an excess over atmospheric neutrinos was found . the 90% c.l . upper limit on the muon flux from the center of the earth are compared in fig . [ fig : wimplimits ] ( left ) to limits obtained from underground experiments and to mssm model predictions excluded by direct search . with its larger mass and a higher capture rate due to additional spin - dependent processes , the sun is more effective than the earth in catching wimps . amanda - ii data from 2001 data yield no indication of a wimp signature . the preliminary upper limit on the muon flux from the sun is compared to mssm predictions in fig . [ fig : wimplimits ] ( right ) . for heavier neutralino masses , the limit obtained with less than one year of amanda - ii data is already competitive with limits from indirect searches with detectors that have several years of integrated livetime . it should be noted that the two methods are complementary since they ( a ) probe the wimp distribution in the solar system at different epochs and ( b ) are sensitive to different parts of the velocity distribution . direct searches are sensitive to high - energy recoils and therefore to the high - velocity tail of the wimp flux , indirect searches are more sensitive to low - velocity wimps since those are easier captured by celestial bodies . a magnetic monopole with unit magnetic dirac charge @xmath22 and velocities above the cherenkov threshold in water @xmath23 would emit cherenkov radiation smoothly along its path , exceeding that of a bare relativistic muon by a factor of 8300 . this is a rather unique signature . figure [ fig : monopol ] summarizes the limits obtained until now . a cube kilometer detector could improve the sensitivity of this search by nearly two orders of magnitude . the search could be extended to even lower velocities by detection of the @xmath24 electrons generated along the monopole path . limits on the flux of particles moving with with less than @xmath25 , like gut magnetic monopoles catalyzing baryon decay , q - balls or nuclearites @xcite have been obtained with early stages of the baikal detector @xcite . the trigger system of the future icecube is flexible enough to search effectively for such particles and to lead to much stronger limits than those of @xcite . due to the low external noise rate , amanda is sensitive to the increase of individual counting rates of all pms resulting from a supernova burst @xcite . amanda - ii can detect 90% of supernovae within 9.4 kpc with less than 15 fakes per year . this is sufficiently robust for amanda to contribute to the supernova early warning system ( snews ) with neutrino detectors in the northern hemisphere . icecube will monitor the full galaxy . an alarm would confirm other records within snews but also provide directional information : if several detectors spread around the world would measure the signal front with an accuracy of a few ms , one might determine the supernova direction by triangulation . the high statistics of hits recorded by icecube would allow a precise measurement of the set - in of the burst @xcite . with 4800 optical modules on 80 strings , horizontally spaced by 125 m , icecube @xcite covers an area of approximately 1 km@xmath26 , with the oms at depths of 1.4 to 2.4 km below surface . each string carries 60 oms , vertically spaced by 17 m. the strings are arranged in a triangular pattern . the configuration of icecube is shown in fig.[fig : icecube ] . at each hole , one station of the icetop air shower array @xcite will be positioned . an icetop station consists of two ice tanks of total area 7 m@xmath26 . a first icecube string has been successfully deployed on january 27 , 2005 . for icecube , a new drilling system has been constructed . the power for heaters and pumps of the ehwd is now @xmath2 5 mw , compared to 2 mw for amanda . this , and the larger diameter and length of the water transporting hoses , results in only 40 hours needed to drill a 2400 m deep hole ( three times faster than with the old amanda drill ) . mounting , testing and drop of a string with 60 doms takes about 20 hours , and deployment of ultimately 18 strings per season seems feasible . the present amanda - ii detector will be integrated into icecube . icecube will deliver efficient veto information for low energy cascade - like events or short horizontal tracks recorded in amanda . horizontal tracks could be related to neutrinos steming from wimp annihilations in the sun . the icecube om contains a 10-inch diameter pm hamamatsu r-7081 . different to amanda , the pm anode signal is digitized within the om and sent to the surface via electrical twisted - pair cables . waveforms of the signals are recorded with 250 mhz over the first 0.5 @xmath27s and 40 mhz over 5 @xmath27s . the fine sampling is done with the analog transient waveform recorder ( atwr ) , an asic with four channels , each capable to capture 128 samples with 200 - 800 hz . the 40 mhz sampling is performed by a commercial fadc . each pulse time stamped with 7 ns r.m.s .. see @xcite for more details . figure [ fig : area ] shows the icecube effective area for muons after @xmath28 reduction of events from downward muons , as a function of the muon zenith angle @xcite . whereas at tev energies icecube is blind towards the upper hemisphere , at pev and beyond the aperture extends above the horizon and allows observation of the southern sky . denotes vertically upward moving muons , @xmath29 marks the horizontal direction . , width=283 ] the icecube sensitivity to diffuse fluxes after three years of data taking is shown in fig . [ fig : diffuse ] ( section 3.2 ) . the dashed - dotted lines indicate the stecker and salamon model for photo - hadronic interactions in agn cores @xcite and of the model of mannheim , protheroe and rachen on neutrino emission from photo - hadronic interactions in agn jets @xcite . in case of no signal observed , these models could be rejected with model rejection factor of @xmath30 and @xmath31 respectively . also shown is the grb estimate by waxman and bahcall @xcite which would yield of the order of ten events coinciding with a grb , for 1000 monitored grbs . for not too steep angles the icecube pointing resolution is 0.6 to 0.8 degrees , improving with energy . we expect that evaluation of waveform information will improve these numbers significantly , at least at high energies , and increase the potential for point source identification . figure [ fig : pointfuture ] sketches a possible scenario for the point source search over the next decade . best present limits are from macro , super - kamiokande ( southern sky ) and amanda ( northern sky ) . baikal limits for the southern sky will appear soon . this picture will not change until the medium stage mediterranean detectors come into operation . the ultimate sensitivity for the tev - pev range is likely reached by the cubic kilometer arrays . this scale is set by many model predictions for neutrinos from cosmic accelerators or from dark matter decay . a discovery with amanda is not yet excluded be it a flaring blazar or a supernova ; for icecube , hundred times larger than amanda and thousand times larger than underground detectors , it seems extremely likely . however , irrespective of any specific model prediction , icecube will hopefully also keep the promise for any detector opening a new window to the universe : to detect _ unexpected _ phenomena . i thank my colleagues in the collaborations baikal , amanda and icecube for the long , fruitful cooperation and for helpful discussions . 9 m. markov , proc . 1960 int . conf . on high energy physics , rochester 1960 , 578 . f. halzen , these proceedings , and f. halzen and d. hooper , rept . phys . * 65 * ( 2002 ) 1065 . j. edsj , proc . neutrino-2004 , paris , and references therein . d.bakari et al . , hep - ex/0004019 , s. wick et al . , astropart . phys . * 18 * ( 2003 ) 663 . aubert , these proceedings . j. ahrens et al . , astropart . * 21 * ( 2004 ) 565 . dzilkibaev for the baikal collaboration , nucl phys b(proc . suppl . ) 143 ( 2005 ) 335 . k. woschnagg for the amanda collaboration , ibid . o. botner for the icecube collaboration , ibid . i. belolaptikov et al . , astropart . * 7 * ( 1997 ) 263 . i. belolaptikov et al . , astropart . phys.12 ( 1999 ) 75 . c. spiering for the baikal coll . vlvnt workshop , amsterdam 2003 , 26 , and astro - ph/0404096 . e. andrs et al . , nature * 410 * ( 2001 ) 441 . k. daum et al . , z. phys . c * 66 * ( 1995 ) 417 . l.v.volkova , sov.j.nucl.phys . * 31 * ( 1980)784 . j. ahrens et al . lett . * 90 * ( 2003 ) 251101 . v.s.berezinsky and a.yu . smirnov , astrop . * 32 * ( 1975 ) 461 . m. ackermann et al . , astropart . * 22 * ( 2004 ) 127 , astro - ph/0405218 . v. aynutdinov et al . , to be published in proc . 19th ecrs , florence ( 2004 ) . m. ackermann et al . , astropart . * 22 * ( 2005 ) 339 . stecker and m.h . salamon , space sci . * 75 * ( 1996 ) 341 . k. mannheim , r. protheroe and j. rachen , phys . d63 ( 2001 ) 023003 . e. waxmann and j. bahcall , phys . d59 ( 1999 ) 023002 . j. ahrens et al . * 92 * ( 2004 ) 071102 . see e.g. d. horan , d.c . weekes , new astron . * 48 * ( 2004 ) 527 , and astro - ph/0310391 . j. ahrens et al . d66 ( 2002 ) 032006 . akerib et al . , astro - ph/0405033 . bezrukov et al . , sov . j. nucl . * 52 * ( 1990 ) 54 . j. ahrens et al . , astropart . * 16 * ( 2002 ) 345 . t. neunhffer for the amanda collaboration , proc . 27th icrc , hamburg 2001 , 1125 . a. karle for the icecube coll . neutrino-2002 , munich 2002 , 388 , and c. spiering for the icecube coll . vlvnt workshop , amsterdam 2003 , 21 , and astro - ph/0404090 . t. stanev and r. ulrich for the icecube collaboration , astro - ph/0501046 . j.ahrens et al . , astropart . phys * 20 * ( 2004 ) 507 , astro - ph/0305196 . c.spiering , journ . g , nucl . part . * 29 * ( 2003 ) , astro - ph/030368 . c. distefano et al . , phys . d66 ( 2002 ) 063004 .	this talk reviews status and results from the two presently operating underwater / ice neutrino telescopes , nt-200 in lake baikal and amanda - ii at the south pole . it also gives a description of the design and the expected performance of icecube , the next - generation neutrino telescope at south pole . christian spiering + desy , platanenallee 6 , d-15738 zeuthen , germany
the behavior of an excess eletron in a wide variety of fluids has been an interesting topic for many years @xcitein the gas phase or the dilute liquid phase , the electron behaves almost like a free particle . as the solvent density increases , the electron exihibits different properties depending upon the nature of the solvent and electron - solvent interactions . at liquid density , the electron may become self - trapped in a cavity of solvent particles or remain _ quasifree _ depending upon the nature of electron - solvent interaction . observed properties such as the electron mobility@xcite and the absorption spectra @xcite probe the nature of the electronic states in the fluid and phenomena of localization@xcite . there are a broad range of problems in condensed matter physics that are intimately related to the problem of excess electrons in deformable medium . these include charge transfer kinetics in biological reactions , metal - insulator transitions in fluids , polarons , phonon - assisted hopping of charge carriers in semiconductors and insulators , quantum - tunneling , etc . while the excess electron problem belongs to the general problem of electrons in disordered materials , the liquid environment is in many ways different from the solid medium . in liquids , the constituent particles can diffuse , and local environment around the solute electron can be substantially different from that in solid . when an electron is solvated in a polar liquid such as water or ammonia , the strong anisotropic electron - solvent interaction causes significant local modification of the equilibrium fluid structure @xcite . the electron becomes localized in a small cavity because molecules in a solvation shell orient to create a potential minimum . even simple fluids are found to exhibit electron mobilities that change by many orders of magnitude as the density of the fluid is altered slightly . in super critical helium , for example , the electron mobility drops by over 4 orders of magnitude as fluid density increased by a factor of 2 in low density regime@xcite . the reason for this behavior is strong repulsion between electron and solvent atom . this causes the the depletion of the solvent atoms from the region of the electron and forms a highly localized state of the electron . in many other nonpolar fluids such as ar , ch@xmath1 , etc . the electron always remains in a state of high mobility@xcite comparable to many semiconducting materials . an interesting density dependence of the mobility has been observed in them . it shows a minimum near critical fluid density and a maximum at liquid density . electronic states in reduced dimension are of considerable interest . for example , for a system less than two spatial dimension , electrons are localized with an infinitesimal amount of disorder @xcite . the interest in problems of electron or phonon propagation in a one dimensional random potential stems from the discovery and extensive experimental study of a certain class of organic or metallo - organic materials @xcite . these matarials exhibit strongly anisotropic , quasi - one - dimensional behavior attributed to the fact that they consist of long chains , weakly interacting with each other . in many of these , the presense of a random potential has been proposed in order to explain their behavior . electronic surface states play an important role in a wide variety of physical problems . for example , surface electrons on liquid helium has shown many interesting properties and led to important theoretical advances such as the spectrum of bound electronic states , electron transport on the he surface , effects due to deformation of the he surface , and the possibility that the electrons may form a two - dimensional crystal@xcite in the field of low - dimensionality physics . the theory for the excess electrons in fluids developed by chandler , singh , and richardson@xcite ( csr ) is based on the path integral formulation of quantum theory which maps the behavior of the electron on to that of a classical isomorphic polymer@xcite . the solvent - induced potential surface for the self - interaction of the isomorphic polymer is evaluated using an integral equation ( e.g. , reference interaction site model ) . with known potential surface , the polymer statistics is solved using variational approach@xcite that allows the determination of electronic properties and the structure of the liquid near the electron . the input of the theory is the pure solvent structure factor and the electron - solvent particle interaction potential . the csr theory in its formulation is applicable to an adiabatic solvent ( _ i.e. , solvent particles are treated classically _ ) , but has been extended to treat the effect of the quantum mechanical charge density fluctuation in the solvent particles @xcite . the calculated electron - absorption line shape and mobility are in good agreement with the simulations@xcite and experiments @xcite . the predictions of csr theory were verified by computer simulations @xcite . for a one - dimensional system we @xcite have shown recently that the repulsive and attractive parts of electron - solvent interaction potential lead separately to localization of electron , respectively , by creating cavity or forming a cluster of the fluids atoms around it . in two dimensional system we @xcite have shown that dispersion of the wavepacket associated with the solvated electron is very sensitive to the interaction between the electron and the fluid atoms , and exhibits complicated behavior in its density dependence . csr theory has been extended to calculate the effective mass @xcite as well as the density matrix of the excess electron in fluid @xcite . the csr theory involves three or more characteristic lengths , depending upon the nature of electron - solvent interaction . these lengths are the thermal wavelength of excess electron @xmath2 ( where @xmath3 is the inverse of temperature in the unit of the boltzmann constant @xmath4 , @xmath5 is the mass of a bare electron , and @xmath6 is the planck s constant divided by 2@xmath7 ) , characteristic length associated with the electron - solvent pseudopotential , and a length associated with the mean volume occupied by each solvent atom , which is related to @xmath8 , where @xmath9 being the number density of the solvent and @xmath10 is the diameter of the solvent atom , and @xmath11 is the spatial dimensionality of the system . the behavior of the excess electron is expected to depend sensitively on these lengths . laria and chandler @xcite have attempted to explain the contrasting behaviors of the electron in super critical helium and xenon on the basis of different ranges of the electron - solvent repulsive interactions . in the present work we examine in detail the role played by different lengths and the spatial dimensionality of the system ( in which we have considered the same solvent - solvent and electron - solvent model potential ) to study the self - trapping behavior of the electron . the organization of the rest of the paper is as follows . in sec . ii we briefly review the csr theory . in sec . iii we have presented the results and their discussions . finally , section iv presents concluding remarks . appendix a provide some mathematical material for @xmath11- dimensional integration the system we consider is a single electron dissolved in a single component classical solvent . in the csr theory , an excess electron is mapped , using a discretized version of the path integral formulation of quantum mechanics , onto a polymer of p interaction sites or beads . under this isomorphism @xcite , the electron can be viewed as a classical ring polymer . the total potential energy can be written as @xmath12 with @xmath13 and @xmath14 here , * r * denotes the position of the excess electron , @xmath15 is the collection of the coordinates for a solvent atom , and n is the number of solvent atoms and @xmath16 and @xmath17 are , respectively , electron - solvent atom and solvent atom - solvent atom interaction potentian . we consider a @xmath11-dimensional system of spheres of diameter @xmath10 in which the pair interaction between the solvent atoms is taken to be @xmath18 where @xmath19 is the @xmath11-dimensional distance ( for notational convenience the @xmath11 dependence will not always be explicitly indicated ) . the electron - solvent atom interaction in a real system consists of a strong repulsion at short distance due to orthogonality requirements between wavefunctions of core electrons in the solvent particle and that of the excess electron and attraction at large distances due to dispersion interaction . however , in a system of neutral atoms the electronic states are determined primarily by the short range repulsive interaction or excluded - volume effect . the attractive interaction becomes important only at low densities . the interaction between the electron and solvent atom is taken to be @xmath20 here , @xmath21 is the distance of closest approach between electron and solvent atom . in csr theory@xcite the partition function z for an electron in a bath of classical particles is written as the functional integral @xmath22\end{aligned}\ ] ] where * r(u ) is the electronic path in imaginary time which is periodic in time interval @xmath23 , i.e. , r(0)=r(@xmath24 ) . to concentrate our attention on the electron degrees of freedom , the partition function given by eq.(6 ) can be written as @xmath25\bigr\}\end{aligned}\ ] ] where @xmath26 denotes the partition function of the solvent , @xmath27 $ ] is the excess chemical potential for the fixed electronic path , @xmath28 = { 1\over \hbar}\int_0^{\beta\hbar } du { 1\over 2}m\|\dot{\bf r}(u)\|^2\end{aligned}\ ] ] and @xmath29\}$ ] is called the influence functional which represents the solvent effects on the electon . in the continuum limit @xcite , @xmath30 = \rho_s\hat{c}_{es}(0 ) + { 1\over 2}(\beta\hbar)^{-2 } \int_0^{\beta\hbar } du\int_0^{\beta\hbar } du^\prime v(\|{\bf r}(u ) - { \bf r}(u^\prime)\| ) , \end{aligned}\ ] ] where @xmath31 is the k = 0 spatial fourier transform of @xmath32 . @xmath33 here r * and @xmath34 appear as independent coordinates , * r * is the distance between two sites and u measures the length along the contour of the polymer , and @xmath35 is the density - density correlation function of the unpurturbed bath . in eqs . ( 9 ) and ( 10 ) , @xmath36 is the direct correlation function . its value is determined from the equation @xcite @xmath37 where @xmath38 is the intrapolymer correlation function . ( 12 ) is solved for @xmath36 and @xmath39 using suitable closure relation @xcite . since all sites of a ring polymer on the average are equivalent , the site dependence disappears from eq.(12 ) and only the zero - frequency component @xmath40 of the equilibrium response function is required in eq.(12 ) . to complete the evaluation of excess chemical potential , the electronic path integral still has to be performed . following feynmam @xcite and chandler _ et . al . _ , @xcite the excess chemical potential for the fixed electronic path is mimicked by a gaussian functional , @xmath41 = -\gamma_\circ + { 1\over 2}(\beta\hbar)^{-2 } \int_0^{\beta\hbar } du \int_0^{\beta\hbar } du^\prime \gamma(u - u^\prime ) \times \|{\bf r}(u ) - { \bf r}(u^\prime)\|^2\end{aligned}\ ] ] where @xmath42 is a solvent - induced force constant between different sites on the electron polymer and @xmath43 merely determines the zero of energy . the bogoliubov inequality provides a upper bound for the excess chemical potential , @xmath44 - \delta\mu_{\rm { ref}}[{\bf r}(u)]>_{\rm { ref}}\end{aligned}\ ] ] here , @xmath45 is the electronic partion function for the gaussian reference system and @xmath46 means the average over the reference system weight determined by @xmath47 $ ] . minimizing the right hand side of eq.(14 ) provides the optimal gaussian reference system . this procedure leads to the following equations . the correlation function for the intrapolymer correlation in k - space : @xmath48\end{aligned}\ ] ] where @xmath49\end{aligned}\ ] ] is the mean square displacement between two points on the electron path separated by a imaginary time increament @xmath50 with @xmath51 where @xmath52 , and @xmath53 \int { d^dk k^2\over ( 2\pi)^d } v(k ) \exp ( -k^2r^2(u)/2d).\end{aligned}\ ] ] we solve eq.(12 ) for @xmath39 and @xmath36 using closure relation @xmath54 @xmath55 we can express eq.(12 ) and the closure ( 20 ) in the variational form @xmath56 where @xmath57 where @xmath31 is the k = 0 spatial fourier transform of @xmath32 , @xmath11 is the dimensionality of the space , @xmath58 is the dimensionality dependence volume element , @xmath34 labels the beads in the polymer ring @xmath59 , @xmath5 is the bare electron mass , and @xmath60 is the fourier transform of the potential between beads , which is found in eq.(10 ) . the information about the electron - solvent atom interation is contained in the closure relation . in eq.(11 ) , @xmath61(r ) is the intra polymer distribution function averaged over all beads of the ring polymer . in writing eq.(11 ) it has been assumed that for each polymer configuration , the solvent sees only average polymer rather than individual beads . the intra polymer distribution function @xmath62 is determined in the _ polaron approximation _ @xcite,@xmath63 is the fourier transform of @xmath64 and is given by @xmath65 . \end{aligned}\ ] ] eq.(22 ) is solved self - consistently @xcite for a given solvent and model potential representing the electron - fluid particle interaction . this solution gives information about @xmath66 , @xmath67 , @xmath68 , @xmath69 and @xmath70(r ) [ = 1+@xmath39(r ) ] . note that the quantity _ r_(u ) is the root mean square ( rms ) value of the displacement between two points on the electron path separated by a time increment @xmath71 . the characteristic size or breadth of the polymer is measured by @xmath72 . this is a measure of the spread of the wave packet associated with the particle . since in the csr theory a periodic boundary condition , @xmath73 , has been imposed on the path of the electron , @xmath68 is found to be symmetric about @xmath74 , _ i.e. _ it starts from zero at u = 0 , attain a maximum value at @xmath75 and decreases for @xmath76 reaching zero at @xmath77 . @xmath78 gives information about the average packing of solvent particles around the electron . the variational parameter @xmath69 measures the strength of the electron fluid coupling . quantities such as average kinetic energy , potential energy and effective mass etc . , can be expressed in terms of @xmath69 @xcite as @xmath79 . \end{aligned}\ ] ] @xmath80 @xmath81 in this equation @xmath82 is the effective mass of the solvated electron . as mentioned earlier , we need two input for this theory . one is electron - solvent and another is the density - density correlation function of solvent which is related to the structure factor of the solvent . for @xmath11-dimensional hard sphere solvent under cocsideration the percus - yevic ( py ) equation can be solved analytically for @xmath11 = 1 @xcite and for @xmath11 = 3 @xcite . for @xmath83 excellent results of thermodynamic and structural properties have been obtained by baus and colot @xcite . in the above @xmath84 , @xmath85 , and @xmath86 , are the spatial fourier transform of @xmath61 , @xmath36 , and @xmath87 . in presenting our results we mainly focus on the imaginary time correlation function @xmath88 and the electron - solvent radial distribution function . note that @xmath89 gives a measure of the physical size of the electron chain or the spread of the wave packet associated with the electron . for a free particle , @xmath90 in fig.1 we plot the reduced correlation length , _ s _ @xmath91 which is the dispersion of the wavepacket associated with the solvated electron relative to the free particle in one , two and three dimension as a function of density for @xmath92 , @xmath93 = @xmath94 and @xmath95 = 0.29 for several values of the attractive interaction , @xmath96 . from these figures we find that when the electron - solvent interaction is solely repulsive ( @xmath97 ) , the electron is always gets trapped inside a solvent cage as solvent density is increased in one and two dimensions . in three dimension [ see fig . ( 1c ) ] the repulsive interaction is not strong enough ( because @xmath98 ) to localize the electron due to ordered structure formed when the solvent density is high . when the attractive interaction between electron and solvent atom is large ( @xmath99 ) , the reduced correlation length of the electron is very small at very low solvent density and it stays almost constant upto @xmath100 . this behavior strongly indicates the electron is localized to a single atom irrespective of space dimensionality . as the space dimensionality increases , the range of the constant value of the solvent density decreases for example for @xmath11=1 , _ s _ is constant upto @xmath101 . one interesting feature we have noted from figs . 1a , 1b , and 1c is that as the attractive interaction increases , the reduced correlation length of the electron decreases as we increase the solvent density in the low - density regime , but the correlation turns upwardto have a maximum in the middle density regime , and finally the electron is trapped in the high density regime . if we carefully look at figs.1 , we observe that as we increase the space dimensionality , we find these effect are pronounsed for large attractive interaction . the similar type of behavior has been observed for the other values of @xmath95 . when @xmath95 is less than @xmath102 we found the reduced correlation length of the electron decreases as we increase the attractive interaction _ i.e. _ @xmath103 and is self trapped at lower density . opposite trend is found for increasing @xmath95 . from figs . 1[a ] , 1[b ] and 1[c ] it is clear that at the begining when we start increasing the attractive interaction ( @xmath104 ) the wave packet associated with the electron increases then further increase of electron solvent attractive interaction i.e. when @xmath105 the electron is localized on a single solvent atom . the reason behind the delocalized state at small vales of attractive interaction ( @xmath96 ) is the cancellation between the repulsive and attractive electron - solvent interaction . in fig . 2[a ] to fig.2[f ] we plot the electron - solvent radial distribution function for one , two and three dimension at various solvent density and attractive iteraction ( @xmath103 ) . 2 can be explained in consistent manner with the same physical picture given for figs . [ 1 ] . when the attractive interaction is weak , the electron pushes the solvent atoms makes space to self - trap . when the attractive interaction is very large ( @xmath106 ) the electron is trapped on a single atom irrespective of space dimensionality , as evidenced by a large peak in @xmath78 [ see figs . 2[f ] . in fig . 3[a ] , 3[b ] , and 3[c ] we plot the reduced correlation length as a function of @xmath107 at @xmath108 for @xmath11=1 , 2 , and 3 respectively . in one and two dimension ( fig.3[a ] and fig.3[b ] ) we observed as we increase @xmath0 the electron is trapped in the _ solvent cage_. if we compare fig . 3a with fig . 3b as the space dimensionality increases , the electron is strongly trapped in the _ solvent cage_. one interesting feature we found in fig . 3[c ] at @xmath109 there is sharp transition from delocalized to self - trapped state . in less than two dimension when there is no attractive interaction , the electron will always be in a localized state@xcite but in three dimension there is always a tendendency the electron will be in delocalized state . on the other hand , as the temperature is decreased , the electron has tendency to be in the localized state . this competition between vaious length scales probably make the sharp transition around @xmath110 . in fig . [ 4a ] , [ 4b ] and [ 4c ] we plot the reduced correlation length as a function of density ( @xmath111 ) for various values of @xmath95 . we found for low @xmath95 ( for example @xmath95 = 0.15 ) the attractive interaction dominates and the electron is self - trapped in the low density regime . on the other hand , when @xmath95 is large the self - trapping of the electron is dominated by repulsive interaction or self - trapping is by _ cage effect_. in the case of electron in helium and xenon , the self - trapping of electron in helium and delocalized state of the electron in xenon has been explained on the basis of @xmath95 value @xcite when elecron - solvent interaction is repulsive _ i.e. _ @xmath112 . the csr theory for the excess electrons in simple fluid ( consisting of spherical atoms ) has been studied in one , two , and three dimension . the detailed study led us to the following conclusions : 1 . the reduced correlation length , _ s _ , is very sensitive to the nature of the electron - fluid atom interaction , thermal wavelength of the excess electron ( @xmath0 ) and a length associated with the mean volume occupied by each fluid atom which is related to @xmath113 , where @xmath114 , @xmath115 being the number density of the fluid atoms and @xmath10 is the effective diameter of a fluid atom . the behavior is interpreted in terms of an interplay among the length scales noted above . when the electron - solvent attractive interaction is very large , the density dependence of the size of the electron polymer relative to the free particle is dominated essentially by the pair attractive interaction , and the electron is trapped in a single solvent atom irrespective of the dimensionality ( @xmath11 ) . on the contrary , if the attractive interaction is absent , the electron is trapped in a cage formed by the solvent atoms at higher density . when the attractive interaction is low ( @xmath116 ) due to cancellation between repulsive and attractive interaction , the electron is delocalized irrespective of the dimensionality . 2 . in the one dimensional case the reduced correlation length is almost independent of @xmath95 when electron - solvent attractive interaction is absent . on the other hand , in two and three dimensions , @xmath117 , the reduced correlation length of the electron is sensitive on the value of the @xmath95 . 3 . in three dimension we found when @xmath98 and @xmath118 , the temperature dependence of the reduced correlation length , @xmath117 , shows a sharp transition ( metal - insulator type ) around @xmath119 at @xmath109 . 4 . the long range electron - solvent attraction dominates at low density , while hard core repulsion dominates at high densities . when both interactions are present , they can counterbalance each other . consequently , at some intermediate density , the effective electron - solvent interaction can be quite small , resulting in the electron delocalization . in the present work we have explored the electron self - trapping in simple liquid for various space dimension ( @xmath11= 1,2 , and 3 ) which have same model potential ( solvent - solvent interaction potential and electron - solvent interation potential ) in every space dimension . in every space dimension ( i.e. @xmath11=1 , 2 , and 3 ) we have in the model system considered here there is repulsive as well as attractive electron solvent interation potential which is not considered in refs . 25 , 27 , and 28 . we have presented the dimensionality dependent result for self trapping behavior of electron in simple liquid . this work was supported in part by national science foundation , the robert welch foundation , and the institute for molecular science , ( japan ) . in @xmath11-space we have for any function f(x ) depending only on the distance @xmath120 , @xmath121 or on @xmath122 and one integration angle @xmath123 , @xmath124 where @xmath125 is the surface area of the unit sphere of volume @xmath126,where @xmath127 is the @xmath128 function . g. r. freeman , annu . phys . chem . * 34 * 463 ( 1983 ) ; r. a. holroyd , in _ radiation chemistry : principles and applications _ , edited by x. farhataziz and m. a. j. rodgers ( vch , new york , 1987 ) ; r. a. holroyd and w. f. schmidt , annu . phys . chem . * 40 * , 439 ( 1989 ) . n. e. cipollini and a. o. allen , j. chem . phys . * 67 * , 131 ( 1977 ) ; s. s. s. huang and g. r. freeman , _ ibid _ * 68 * , 1355 ( 1978 ) ; s. s. s. huang and g. r. freeman , phys . a * 24 * , 714 ( 1984 ) ; f. m. jacobson , n. gee , and g. r. freeman , phys . a * 34 * , 2329 ( 1986 ) . e. abrahams , p. w. anderson , d. c. licciardello , and t. v. ramakrishnan , phys . lett . * 42 * , 673 ( 1979 ) ; p. a. lee and t. v. ramakrishnan , rev . * 57 * , 287 ( 1985 ) and relatedreferences therein . a. n. bloch , r. b. weisman , and c. m. varma , phys . * 28 * , 753 ( 1972 ) ; p. f. williams , and a. n. bloch , phys . b 10 * , 1097 ( 1974 ) ; c. papatriantafillou , e. n. economou , and t. p. eggarter,_ibid_. * 13,910 ( 1976 ) ; r. johnston , and b. kramer , z. phys . b63 * , 273 ( 1986 ) . m. w. cole in e. y. andrei ( ed . ) , two - dimensional electron system , kluwer academic publ . , 1997 ; m. w. cole , rev . phys . * 46 * , 451 ( 1974 ) ; c. c. grimes , * 73 * , 379 ( 1978 ) ; v. b. shikin , * 73 * , 396 ( 1978 ) . ) dependence of reduced correlation length @xmath117 for various values of attractive interaction ( @xmath96 ) with @xmath92 , @xmath93 = @xmath94 and @xmath95 = 0.29 ( a ) for one dimension ( @xmath11 = 1 ) , ( b ) for two dimension ( @xmath11 = 2 ) , and ( c ) for three dimension ( @xmath11 = 3 ) . ] , relative to the free paricle value at @xmath129 and @xmath95 = 0.29 for various values of density [ a ] for one dimension ( upper panel ) ( @xmath11 = 1 ) , [ b ] for two dimension ( @xmath11 = 2 ) ( middle panel ) , and [ c ] for three dimension ( @xmath11 = 3 ) ( lower panel ) . ]	the behavior of an excess electron in a one , two and three dimensional classical liquid has been studied with the aid of chandler , singh and richardson ( csr ) theory [ j. chem . phys . * 81 * 1975 ( 1984 ) ] . the size or dispersion of the wavepacket associated with the solvated electron is very sensitive to the interaction between the electron and fluid atoms , and exhibits complicated behavior in its density dependence . the behavior is interpreted in terms of an interplay among four causes : the excluded volume effect due to solvent , the pair attractive interaction between the electron and a solvent atom , the thermal wavelength of the electron ( @xmath0 ) , a balance of the attractive interactions from different solvent atoms and the range of repulsive interaction between electron and solvent atom . electron self - trapping behavior in all the dimensions has been studied for the same solvent - solvent and electron - solvent interaction potential and the results are presented for the same parameter in every dimension to show the comparison between the various dimensions . 2g^(2 )
the top quark ( @xmath0 ) is the heaviest known elementary particle , with a mass that is much higher than those of all the other quarks , and close to the masses of the @xmath1 , @xmath2 and higgs bosons . its large mass means that it decays quickly without forming hadrons , offering the unique opportunity to study the properties of a ` bare ' quark . with its @xmath3 yukawa coupling to the higgs boson , it may also be closely connected to electroweak symmetry breaking , or offer a window to physics beyond the standard model ( sm ) . studies of the properties of the top quark are therefore central to both the lhc and tevatron physics programs . this review focuses on measurements of the top - pair ( ) production charge asymmetry , and measurements of spin correlation and top polarisation in events , performed by atlas @xcite and cms @xcite in @xmath48tev @xmath5 collisions at the cern large hadron collider . other properties of the top quark are covered in other reviews presented to the top2014 conference . one of the most intriguing legacy results from the tevatron is the forward - backward asymmetry . analyses of the angular distributions of top quarks and anti - quarks by cdf and d0 indicate that the top quarks ( antiquarks ) are produced preferentially following the direction of the proton ( antiproton ) beam . the latest measurements from cdf suggest that the size of this asymmetry is slightly larger than expected in the sm , whilst the measurements from d0 are in agreement with the sm @xcite . this asymmetry can not be measured at the lhc , as it collides @xmath5 and not , but an analogous charge asymmetry @xmath6 can be defined by looking at the difference in absolute rapidity values of the produced top quark and antiquark . the rapidity difference @xmath7 is positive when the top quark is produced at a smaller angle to the beam direction ( large @xmath8 ) than the antiquark , and negative otherwise . the asymmetry is defined from event counts @xmath9 as : @xmath10 in the sm , this asymmetry is slightly positive for pairs produced via @xmath11 , where interference effects generate a correlation between the direction of the incoming quark and the outgoing top quark , but zero for @xmath12 . the total resulting asymmetry is an order of magnitude smaller than the tevatron forward - backward asymmetry , _ e.g. _ an nlo qcd calculation including electroweak corrections gives @xmath13 at @xmath4tev @xcite . the most precise measurements of @xmath6 come from the semileptonic final state , in which the @xmath1 boson from one top quark decays to an electron or muon and a neutrino , and the other to a pair . by selecting events with an isolated electron or muon , missing transverse momentum ( ) and at least four jets , atlas and cms both isolate samples of about 60000 events with about 20% non- background , dominated by @xmath1+jets and single top production . kinematic fits are used to fully reconstruct the system and determine @xmath14 on an event - by - event basis . the @xmath14 distribution is then unfolded to correct for background , efficiency and resolution effects , and the inclusive @xmath6 corrected back to the parton level extracted . the measurements from atlas @xcite and cms @xcite at @xmath4tev are shown in table [ t : asym ] , and have been combined @xcite to give a value of @xmath15 , consistent with both zero and the sm prediction . cms has also measured @xmath6 at @xmath16tev using a very similar analysis @xcite . .[t : asym ] measurements of the inclusive asymmetry @xmath6 and leptonic asymmetry from atlas @xcite and cms @xcite at @xmath4tev and @xmath16tev with their statistical and systematic uncertainties , together with the lhc combination @xcite and corresponding theoretical predictions @xcite . [ cols="<,<,<,<,<,<",options="header " , ] both collaborations have also measured @xmath6 differentially , as a function of the rapidity , transverse momentum and invariant mass of the system . the latter in particular increases the sensitivity to new physics scenarios , which are expected to be more prominent at high as shown in figure [ f : acljets ] . atlas has also measured the asymmetry for @xmath17 , this cut on the longitudinal velocity of the system increasing the fraction of @xmath11 events and the potential new physics contributions , but no significant deviations from the sm have been seen . measurements of the charge asymmetry @xmath6 as a function of the invariant mass of the system in semileptonic events from atlas at @xmath4tev @xcite and cms at @xmath16tev @xcite . the measurements are compared to sm nlo qcd predictions with electroweak corrections from ref . @xcite ( cms nlo1 ) and ref . @xcite ( cms nlo2 and atlas ) , and to various beyond - standard model scenarios ( see @xcite for details).,title="fig:",width=264 ] measurements of the charge asymmetry @xmath6 as a function of the invariant mass of the system in semileptonic events from atlas at @xmath4tev @xcite and cms at @xmath16tev @xcite . the measurements are compared to sm nlo qcd predictions with electroweak corrections from ref . @xcite ( cms nlo1 ) and ref . @xcite ( cms nlo2 and atlas ) , and to various beyond - standard model scenarios ( see @xcite for details).,title="fig:",width=245 ] the charge asymmetry has also been measured in the dilepton channel at @xmath4tev @xcite . in dileptonic events , the @xmath1 bosons from both top quarks decay to leptons and neutrinos , giving an under - constrained system since the measurement can not resolve the separate contributions of each neutrino . extra assumptions , such as the expected distribution of neutrino rapidities , are used to find the most probable kinematic configuration for each event , allowing @xmath14 to be extracted and @xmath6 to be measured . in dileptonic events , a complementary asymmetry observable can be defined , based on the difference in @xmath18 between the positive and negatively charged leptons : @xmath19 . the distributions of both variables for the updated atlas dilepton analysis @xcite ( new for this conference ) are shown in figure [ f : aclldist ] , and the @xmath6 and values from both atlas and cms are shown in table [ t : asym ] . unfolded normalised @xmath20 distribution ( left ) and dilepton @xmath21 distribution ( right ) from the atlas dilepton charge asymmetry analysis at @xmath4tev @xcite . the data is compared to the simulation predictions based on powheg+pythia.,title="fig:",width=230 ] unfolded normalised @xmath20 distribution ( left ) and dilepton @xmath21 distribution ( right ) from the atlas dilepton charge asymmetry analysis at @xmath4tev @xcite . the data is compared to the simulation predictions based on powheg+pythia.,title="fig:",width=230 ] the inclusive charge asymmetry measurements are summarised in table [ t : asym ] . the semileptonic measurements have a precision of around 1% , not yet precise enough to distinguish between zero and the small non - zero asymmetry expected in the sm . the dilepton measurements have lower precision , due to the smaller event samples and ambiguities inherent in the dileptonic final state . the top quark lifetime of about @xmath22s is much shorter than the time required to form hadrons , so the top quark decays as a ` bare ' quark , transferring information on its spin to the decay products . in pair production , the polarisation is expected to be negligible , but the spins of the @xmath0 and @xmath23 are correlated , such that the asymmetry in the numbers of events where the @xmath0 and @xmath23 have like and unlike spins , @xmath24 , is non - zero . the double - differential cross - section in the decay angles of the two top quarks , and , is given by : @xmath25 where the asymmetry parameter @xmath26 is scaled by the spin - analysing power of the chosen top quark decay products , and and are the cosines of the angles between the top quark decay product and the chosen polarisation axis . normally the helicity basis is used , in which the polarisation is measured using the direction of the top quark momentum in the rest frame . the spin - analysing power @xmath27 is @xmath28 for positively - charged leptons , @xmath29 for down quarks from the @xmath1 decay , and -0.393 for @xmath30 quarks from the top decay . measurement of the spin correlations using the above formalism requires the full reconstruction of the system , using techniques similar to those used in the charge asymmetry measurement . they have been measured at @xmath4tev by cms in dilepton events @xcite ( see figure [ f : cmspol ] ( left ) ) , and by atlas in both dileptonic and semileptonic events @xcite , using both the helicity basis and the so - called maximal basis which is particularly sensitive to correlations in the @xmath12 subprocess . all measurements are consistent with the spin correlations predicted by the standard model . unfolded distributions of @xmath31 ( left ) and top quark decay angle @xmath32 ( right ) measured by cms in dilepton events at @xmath4tev @xcite , and compared to predictions with and without spin correlations and from the mc@nlo event generator.,title="fig:",width=207 ] unfolded distributions of @xmath31 ( left ) and top quark decay angle @xmath32 ( right ) measured by cms in dilepton events at @xmath4tev @xcite , and compared to predictions with and without spin correlations and from the mc@nlo event generator.,title="fig:",width=207 ] the spin correlation also affects the distribution of , the difference in azimuthal angles ( transverse to the beamline ) of the two leptons in dileptonic events . both collaborations exploited this variable to demonstrate the existence of spin correlations in @xmath4tev data . the cms analysis @xcite unfolded the distribution to parton level , correcting for background , acceptance and resolution effects , and compared the result to simulation predictions with and without spin correlations as predicted by the sm ( see figure [ f : spdphi ] ( left ) ) . the level of spin correlation was quantified from the asymmetry of the parton - level distribution about @xmath33 as @xmath34 , in agreement with the nlo prediction of 0.115 . atlas @xcite instead compared the detector - level distribution to fully - simulated events with and without spin correlation , quantifying the spin correlation strength with a fit to sm - like correlated and uncorrelated simulation - derived templates . the fitted fraction of correlated template was measured to be @xmath35 , compatible with unity . at the conference , atlas presented a new analysis using the distribution at @xmath16tev @xcite with an optimised dilepton event selection having higher efficiency ; as shown in figure [ f : spdphi ] ( right ) , this is also compatible with the sm expectation , with a fitted @xmath36 , corresponding to a spin correlation strength in the helicity basis of @xmath37 . the distribution can also be used to set limits on new physics contributions within the selected event sample , for example top squark pair production as shown by atlas @xcite , or an anomalous -gluon interaction parameterised as a chromomagnetic dipole moment as explored by cms @xcite . measurements of spin correlation using the distribution in dilepton events from cms at @xmath4tev @xcite and atlas at @xmath16tev @xcite , compared to simulation predictions with and without spin correlation at parton level ( cms ) or detector level ( atlas).,title="fig:",width=226 ] measurements of spin correlation using the distribution in dilepton events from cms at @xmath4tev @xcite and atlas at @xmath16tev @xcite , compared to simulation predictions with and without spin correlation at parton level ( cms ) or detector level ( atlas).,title="fig:",width=226 ] the cms collaboration also used the same @xmath4tev dilepton sample to measure the top quark polarisation in events , via the unfolded distribution of the decay angle @xmath32 @xcite . the results are shown in figure [ f : cmspol ] ( right ) , and are consistent with zero polarisation . atlas also measured the top polarisation in both dilepton and semileptonic events , looking for polarisation effects which polarise the @xmath0 and @xmath23 quarks with the same ( cp - conserving ) or opposite ( cp - violating ) signs , and again found results compatible with zero @xcite . the lhc data have already produced a wealth of top quark property measurements , so far mainly at @xmath4tev . top - pair charge asymmetry measurements are reaching 1% precision , and the inclusion of the full @xmath16tev dataset may allow the small non - zero expected asymmetry to be observed . no hints of deviations in either the inclusive or differential asymmetries have been seen , strongly constraining various new physics scenarios . the expected spin correlations have been clearly observed at both @xmath4tev and @xmath16tev , and the top quark polarisation in events has been seen to be consistent with the expectation of zero within about 2% . many of these measurements are limited by systematics , and advances in modelling or new analysis techniques will be needed to fully exploit the @xmath16tev dataset . this will be even more important in the upcoming @xmath3814tev run , which in particular will allow top charge asymmetry measurements to be pushed to sub - percent precision and extended to higher invariant masses .	measurements of top quark properties performed at the large hadron collider are reviewed , with a particular emphasis on top - pair charge asymmetries , spin correlations and polarisation measurements performed by the atlas and cms collaborations . the measurements are generally in good agreement with predictions from next - to - leading - order qcd calculations , and no deviations from standard model expectations have been seen .
water is a very special molecule in the interstellar medium for its crucial role in cooling the molecular gas . however , because of its abundance in the earth s atmosphere , observations of water have to be made from space . notable exceptions are the observations of its isotopes ( h@xmath1o , hdo and d@xmath0o ) , whose rotational transitions can be observed with ground - based telescopes ( e.g. , @xcite for the detection of d@xmath0o , or some maser lines @xcite ) . similarly , water is of particular importance in protoplanetary disks , the sites where planets are formed . more difficult to detect than in other astrophysical environments , hdo has so far been detected in only one protoplanetary disk @xcite , and only very recently the spitzer space telescope detected warm h@xmath0o from the innermost regions of protoplanetary disks @xcite . these observations rose the question of the origin of the observed warm water as h@xmath0o was predicted to be absent in the innermost disk . on the contrary , water is expected to be present in the outer regions of the disk , where the bulk of the gas resides . but water has never been detected in those colder regions . the situation will hopefully soon change thanks to the herschel space observatory , and , specifically , the hifi instrument which will allow to observe the water lines with unprecedented sensitivity , spectral and spatial resolution . in this letter , we report the predictions of the h@xmath0o spectrum toward a typical disk surrounding a herbig ae star , similar to the one of hd 97048 . we first describe the adopted physical and chemical structures , based on previously published models , and the method to compute the emerging line spectrum ( [ sec : model - description ] ) . we then show the results of our computations ( [ sec : results ] ) and discuss how they depend on the distribution of the grain sizes across the disk , a key parameter in the evolution of the protoplanetary disks and the formation of planets . we also show the impact of the adopted set of collisional rates on the predictions and the importance of having a full modeling of the spectrum to derive reliable physical and chemical parameters from the water lines . the disk structure used in the present calculations is inspired by the disk model derived for hd 97048 by @xcite and @xcite . the dust mass in the disk is 10@xmath2m@xmath3 , distributed following power - laws for the scale height and surface density , and located between [email protected] au and r@xmath5 400 au . the flaring exponent is @xmath6 1.26 and the exponent for the surface density is @xmath7 . the gas scale height is 51 au at the reference radius of 135 au . the continuum radiative transfer and dust temperature calculations are performed with mcfost , a monte carlo based code @xcite . the central star has t@xmath8=10000k and a radius of 2 r@xmath3 . the dust opacity is calculated with the mie theory and the dust composition is the mixture given by @xmath9 of @xcite . the grains size distribution follows a power - law ranging from 0.03@xmath10 m to 1 mm in radius and the usual ism slope of -3.5 . two models are calculated , one where dust is fully mixed throughout the disk , and one where vertical settling has occurred and large particles have been removed from the disk surface and progressively settled to the disk midplane as a function of their size , the larger the grain the more complete the settling . in the following we refer to the latter as stratified disk model . details about the parameterization of the dust settling are given in @xcite . the exponent @xmath11 describing the settling has been fixed to 0.25 , a value that fits the observations of gg tau @xcite . the continuum calculations made with mcfost provided the density and temperature profiles , as well as the uv radiation field fully propagated including multiple scattering , to further compute the h@xmath0o abundance and line spectrum with other tools described below . the h@xmath0o abundance profile has been computed following the model described in @xcite . the model has been verified against the more extended chemical model by @xcite to give similar results . briefly , at the conditions prevailing in the protoplanetary disks described above , the major reservoir of water is that frozen on the dust grain mantles . the latter inject water into the gas phase because of two major mechanisms : i ) the sublimation of the h@xmath0o - rich ices when the dust temperature exceeds about 100 k , and ii ) the photo - desorption of the h@xmath0o - rich ices due to the fuv photons from the interstellar field and the star itself @xcite . these two mechanisms completely dominate the abundance profile of the water across the disk . the results for the disk described above are shown in fig . [ fig : chemi ] which shows the case of a standard grain size distribution and the case where a stratification is present ( see above ) . in both cases , three regions can be identified : 1 ) at radii less than about 20 au , water is abundant at any height of the disk because of the warm dust temperature ( @xmath12 k ) and water ice sublimation ; 2 ) between 20 au and about 100 au ( or 250 au in the case of dust stratification ) , water is mostly frozen onto the grain mantles on the equatorial plane ( where the density is larger than @xmath13 @xmath14 and the dust temperature lower than @xmath15 k ) but it is very abundant ( @xmath16 with respect to h@xmath0 ) in the regions just above the plane because of ice sublimation ; 3 ) at radii larger than about 100 au ( or 250 au in the case of dust stratification ) , water is frozen in the equatorial plane and abundant ( @xmath17 ) above it because of the photo - desorption of the ices by the fuv photons . the major difference in the water abundance distribution between the non stratified and stratified case is , therefore , a much larger region where ices sublimate and water is abundant ( @xmath1810@xmath2 ) . we will show that this has important consequences on the emerging h@xmath0o line spectrum . another important difference between the two cases is the continuum emission from the dust grains which also largely affects the emerging water line profiles . , distribution in the disk surrounding hd 97048 , in the case of no stratification ( left panel ) and stratification ( right panel ) of the dust grains ( see text ) . the right bar shows the value of the grey contours . the lines mark the h@xmath0 logarithmic density , increased by steps of 100 from 10@xmath19 to 10@xmath20 @xmath14 ( upper & lower contours respectively ) . [ fig : chemi ] ] given the intrinsic geometry of the problem , a full modeling of the water line spectrum would require a 2-d treatment . however , in the specific case of water , the lines are greatly optically thick and , because of the disk geometry , the line optical depths in the horizontal direction are by far larger than in the vertical one . we therefore developed a pseudo 2-d code where we computed the radiative transfer along the vertical direction at 50 and 70 different disk radii for the non stratified and the stratified case respectively . other authors have discussed and verified the validity of this approach @xcite . the radial distances for each set of models have been selected in order to sample the variations of density and water abundance given by the physical structure of the disk . we have assumed an intrinsic turbulent velocity of 0.5 kms@xmath21 ( see , e.g. , the case for mwc758 and cq tau published by @xcite ) . the sampling in height has been adjusted to trace narrowly the upper layers of the disk where the line opacity of the thicker lines is @xmath221 and photons do escape . the equatorial zone is treated as a continuum source ( water abundance in these regions is @xmath2310@xmath24 ) with a temperature identical to that of the dust grains and a spectral dependence on the dust opacity of 0.65 . the code used for the 1-d radiative transfer models has been described in @xcite . we used the collisional rates with the h@xmath0 by @xcite , assuming that all h@xmath0 is in the ortho form . we will discuss the impact of this assumption on the computed emerging line spectrum by comparing the predicted emerging spectrum obtained by considering different sets of collisional coefficients . finally , in all models the orto - to - para h@xmath0o ratio was assumed equal to 3:1 . figure [ fig : h2o - lines ] shows the profile of a selected sample of lines , namely the brightest lines observable with the hifi instrument onboard the herschel observatory . in order to identify the regions contributing to the emerging profiles we present the results for a face - on disk convolved with the beam of herschel s telescope . hence , the line profiles are totally dominated by the opacity of the lines and the variation of excitation temperature with radius and height above the disk . several kind of profiles are predicted , depending on the line and whether dust grains are stratified or not . in the case of dust stratification , several lines are predicted to be in emission with a central dip caused by self - absorption . other lines show a combination of emission and absorption profiles . for example , the para line at 1.111 thz is in absorption with two `` emission peaks '' at velocities at about 1 km / s arising from the outer ( r@xmath25200 au ) disk , where the water abundance is lower . another important example is represented by the 557 ghz line , which shows two emission peaks again at about 1 km / s produced by the regions with r@xmath26180 au , and a deep absorption in the central velocities of the continuum produced at all radii but particularly strong at r@xmath23 120 au ( see fig . [ fig : tex ] ) . the situation dramatically changes for the case with no dust stratification . almost all lines go entirely in absorption against the dust continuum , because of the brighter continuum . the only exception is the 1153 ghz line , thanks to its relatively low spontaneous emission coefficient ( 10 times lower than , for example , the line at 1163 ghz ) . in general , the line profiles are dominated by the line optical depth and the variation of excitation temperature with height and radius . besides , due to the relatively large telescope beam and to the large line opacities the water lines of fig . [ fig : h2o - lines ] largely probe the upper layers of the outer disk where ices are photo - desorbed . the wiggles presented by some water lines in the no - stratified model are due to the strong decrease of water abundance for r@xmath26120 au compared to the stratified case . o lines observable with the herschel hifi spectrometer . the frequency of the line is reported in each panel . the ordinates report the t@xmath27 in mk , and the abscissas refer to the velocity in kms@xmath21 . the emerging profiles result from the convolution of the herschel hifi beam ( 40@xmath28 at 557 ghz and 13@xmath28 at 1.7 thz respectively ) with the computed brightness temperatures of h@xmath0o . solid lines refer to the stratified case , while dashed lines refer to the case with no stratification ( see text ) . [ fig : h2o - lines ] ] several h@xmath0o lines show masering effects in the inner disk equatorial plane because of the high densities and temperatures prevailing there . figure [ fig : maser ] shows the brightness temperature t@xmath29 of the four brightest maser lines as a function of disk radius . all of them probe the inner ( @xmath30 au ) and denser regions of the disk , but each of them peaks at a different radius , probing , therefore , a different part of the disk . in principle , observations of these maser lines would allow to constrain the physical and chemical structure of the inner disk , even if the spatial resolution is not large enough to resolve the emission . of all the shown maser lines , the 183.3 ghz seems to be the most promising . for example , we predict a signal of 3 and 1 k at the iram-30 m telescope for the non stratified and the stratified case respectively . such signals can easily be detected under good weather conditions @xcite . this is particularly true for the alma interferometer . ( in k ) of the four brightest maser lines at 22.2 , 183.3 , 380.2 and 440 ghz respectively ( as marked on the curves ) . dashed lines refer to the case with no dust grain stratification , solid lines to the case with stratification . the lower x - axis reports the linear distance from the star in au , while the upper x - axis reports the angular distance in arcseconds . [ fig : maser ] ] we have also computed the emerging profiles for the isotope h@xmath1o of water assumed to be 500 times less abundant than h@xmath0o . in spite of the large abundance ratio the intensities predicted for the fundamental line of the rare isotope are not very different from those of the main isotope . of course , the h@xmath1o fundamental line penetrates deeper in the disk probing a different region than the main isotope . finally , in order to help understanding the role of collisional pumping on the emerging water line spectrum , fig . [ fig : tex ] shows the excitation temperature , t@xmath31 , of the fundamental line of water 1@xmath32 - 1@xmath33 as function of the height above the disk at radii 20 , 120 and 350 au . three different sets of collisional coefficients are used : i ) collisions with he , scaled by the h@xmath0 mass , as computed by @xcite ; ii ) collisions with the ortho - h@xmath0 , as computed by @xcite ; iii ) collisions with para - h@xmath0 , computed by the same authors . we notice three effects . first , t@xmath31 is always larger for computations obtained with the ortho - h@xmath0 collision coefficients set and smaller with the he set . this directly reflects the value of the collisional coefficients , larger for the former set of coefficients . second , at the three radii the difference in the t@xmath34 obtained with the three sets of collisional coefficients is larger going towards the equatorial plane , and can be as high as a factor two . this reflects the different population mechanisms at work in the different regions . at larger heights , the difference diminishes because the lines are more and more radiatively populated , an effect nicely seen at radii 20 au where the three sets of collisional coefficients produce the same t@xmath31 . third , the lines , integrated along the disk height , at r=120 and 350 au , show slightly different profiles and very different intensities , reflecting the different excitation conditions and line optical depths . at the smaller radius , 120 au , the line is in absorption against the continuum : the absorption is larger in the case of the he collisional coefficients set and smaller in the case of ortho - h@xmath0 set . at the larger radius , 350 au , the line goes in emission with a self - absorption dip in the central velocities : the line is weaker in the case of the he collisional coefficients set and brighter in the case of ortho - h@xmath0 set . -1@xmath33 intensity for each set of collisional rates . the second inset in the top panel shows the excitation temperatures and the temperature of the gas . for larger radii the lines are almost subthermally populated at all heights . [ fig : tex ] ] the emerging line profile is obtained by integrating these profiles over the radii and will , therefore , depend on the balance of the absorption / emission in the different parts of the disks . in the specific case of the 556 ghz line reported in fig . [ fig : h2o - lines ] , the line can be substantially in absorption or in emission , depending on the choice of the collisional coefficients . for small radii ( r@xmath23120 au ) , the line profiles do not depend on the set of collisional rates while differences of up to a factor of 2 can be found at larger radii . we have performed the full calculations for the case in which h@xmath0 is in the para form , rather than ortho as assumed in the calculations of fig . [ fig : h2o - lines ] , using again the collisional coefficient calculated by @xcite . the emerging line profiles dramatically change : lines in emission become weaker and the absorption becomes stronger . we note that we obtain similar results ( with differences below 5% ) when we used the more recent collisional coefficients by @xcite . in summary , the emerging profile of the water spectrum strongly depends on the o - h@xmath0/p - h@xmath0 abundance ratio , which likely depends on the chemical history prior to the formation of the protoplanetary disk and its subsequent chemical evolution . the interpretation of the water line profiles will remain , therefore , challenging in spite of the accurate collisional rates available in the litterature . in summary , the major results of the present work are the following : * a ) * first , and foremost , several h@xmath0o lines are predicted to be detectable with the newly launched space - born telescope herschel . some maser lines are also observable with ground telescopes . * b ) * even though we did not study the case of a different chemistry ( for example the absence of water ices photodesorption , as in several previous published models ; e.g. @xcite , or @xcite , the comparison between the stratified versus no stratified case shows that the assumed water abundance profile is crucial in the emerging line spectrum . the comparison of our predictions with those recently published by woitke et al . ( 2009 ) , who used a different chemical structure , strengthens this statement . the differences between our results and theirs are mainly due to the different water abundance profile , in particular in the upper layers , to the dust settling we have introduced in our models , and to the inclination assumed for their disk . besides , the dust continuum emission is also crucial . in fact , if not for other reasons , line fluxes and profiles depend on the position of the water molecules with respect to the dust absorbing continuum . therefore , predictions based on simplistic models based on constant abundance across the disk can not be trusted . * c ) * the transition where a line goes from emission to absorption not only depends on the dust and gas temperature and density profiles , but also on the collisional coefficients . this dependence is indeed critical . in practice , the emerging line spectrum strongly depends on the h@xmath0 para - to - orto ratio , a poorly known quantity . to conclude , observations of water lines will be a very powerful diagnostic tool to understand the structure of proto - planetary disks . the analysis of water line profiles will be a challenge because the levels are sub - thermally populated , the dust photons play a crucial role in the pumping , and the huge line opacities favour the vertical diffusion of photons . on the other hand , the information which can be extracted by h@xmath0o observations warrants the effort : amongst others , the amount of water present in the first phases of a planetary system birth . 0.25 cm this work has been supported by spanish micinn through grant aya2006 - 14876 , by dgu of the cm under iv - pricit project s-0505/esp-0237 ( astrocam ) . we also thank the french anr ( contracts anr-08-blan-0225 , anr-07-blan-0221 ) and pnps of cnrs / insu for support . we thank j.r . goicoechea for useful comments . 58 aikawa y. , nomura h. , 2006 , , 642 , 115 asensio - ramos a. , ceccarelli c. , elitzur m. , 2007 , , 471 , 187 butner h. , charnley s. , ceccarelli c. , et al . , 2007 , , 659 , l137 carr j.s . , najita j.r . , 2008 , science , 319 , 1504 ceccarelli c. , dominik c. , caux e. , et al . , 2005 , , 631 , l81 cernicharo j. , thum c. , hein h. , et al . , 1990 , , 231 , l15 cernicharo j. , gonzlez - alfonso e. , et al . , 1994 , , 432 , l59 cernicharo j. , bachiller r. , gonzlez - alfonso e. , 1996 , , 305 , l5 cernicharo , j. , goicoechea j. , pardo j.r , asensio - ramons a. , 2006 , , 642 , 940 chapillon e. , guilloteau s. , dutrey a. , pitu v. , 2008 , , 488 , 565 dominik c. , ceccarelli c. , hollenbach d. , kaufman m. , 2005 , , 635 , l85 doucet c. , habart e. , pantin e. , et al . , 2007 , , 470 , 625 dubernet m.l . , daniel f. , et al . , 2009 , , 497 , 911 faure a. , crimier n. , ceccarelli c. , et al . , 2007 , , 472 , 1029 gonzlez - alfonso e. , cernicharo j. , 1993 , , 279 , 506 gonzlez - alfonso e. , cernicharo j. , 1997 , , 322 , 938 green s. , maluendes s. , mclean a.d . , 1993 , , 85 , 181 lagage p.o . , doucet c. , pantin e. , et al . , 2006 , science , 314 , 621 mathis j.s . , whiffen g. , 1989 , , 342 , 808 pavlyuchenkov ya , henning th . , wiebe d. , 2007 , , 669 , l101 pinte c. , mnard f. , duchene g. , bastien p. , 2006 , , 459 , 797 pinte c. , fouchet l. , mnard f. , et al . 2007 , , 469 , 963 pinte c. , padgett d.l . , mnard f. , et al . 2008 , , 489 , 633 pinte c. , harries t.j . , min , et al . , 2009 , , 498 , 967 salyk c. , pontoppidan k. , blake g.a . , et al . , 2008 , , 676 , l49 willacy k. , 2007 , , 660 , 441 woitke p. , thi w.f . , kamp i. , et al . , 2009 , 501 , l5	water is a key species in many astrophysical environments , but it is particularly important in proto - planetary disks . so far , observations of water in these objects have been scarce , but the situation should soon change thanks to the herschel satellite . we report here a theoretical study of the water line spectrum of a proto - planetary disk surrounding ae stars . we show that several lines will be observable with the hifi instrument onboard the herschel space observatory . we predict that some maser lines could also be observable with ground telescopes and we discuss how the predictions depend not only on the adopted physical and chemical model but also on the set of collisional coefficients used and on the h@xmath0 ortho to para ratio through its effect on collisional excitation . this makes the water lines observations a powerful , but dangerous -if misused- diagnostic tool .
the idea of a shape - space @xmath2 , whose elements are the shapes of @xmath0 labelled points in @xmath3 , at least two being distinct , was introduced in a statistical context by kendall ( 1984 ) . here , it is natural to identify shapes differing only by translations , rotations , and dilations in @xmath3 ( although there are situations of interest , not to be considered here , in which the scale is also relevant ) . however , this identification will not apply to reflections . thus , the resulting shape space is the quotient @xmath4 of the sphere by the rotation group . this is the base space of a fibre bundle with total space @xmath5 and fibre @xmath6 . the former is naturally endowed with a uniform riemannian metric , while the latter , being a compact lie group , possesses an invariant metric . the key observation of kendall is that the metrical geometry of these quotient spaces , long studied by geometers , is precisely the required tool for the introduction of measures appropriate for the systematic comparison and classification of various shapes . now , for a planar distribution of points , with @xmath7 , the shape space @xmath8 of @xmath0 points is simply a complex projective space @xmath9 of dimension @xmath1 , with the fubini - study metric defining the geodesic distances between pairs of shapes . the vertices of a shape in @xmath10 ( viewed as a complex plane ) , with the centroid at the origin , determine the homogeneous coordinates of a point in @xmath9 . the permutation group @xmath11 of order @xmath0 interchanges homogeneous coordinates , and is therefore represented by projective unitary transformations of @xmath9 . for three - point configurations , i.e. @xmath12 , the shape space is a complex projective line @xmath13 , which , viewed as a real manifold , is a two - sphere @xmath14 . hence , the natural shape space for triangles is essentially a riemann sphere ( kendall 1985 , watson 1986 ) . a quantum state @xmath15 of a physical system is represented by a vector in a complex hilbert space @xmath16 of dimension , say , @xmath17 . quantum states determine expectations of physical observables , that is , if @xmath18 is a random variable , then its expectation in a given state @xmath15 is determined , in the dirac notation , by @xmath19 where @xmath20 denotes the complex conjugate of the vector @xmath15 . notice that this expression is invariant under the complex scale change @xmath21 , where @xmath22 . thus , a pure quantum state is an equivalence class of states , i.e. a ray through the origin of @xmath16 . this is just the complex projective space @xmath23 , with the fubini - study metric that determines transition probabilities . to see this , we recall that the transition probability between a pair of states @xmath15 and @xmath24 is given by @xmath25 to recover the fubini - study metric we set @xmath26 hughston 2001 ) . in particular , the maximum separation between a pair of states is given by @xmath27 , whereas if the two states are equivalent in @xmath23 , then @xmath28 . therefore , a shape space of planar @xmath0-point configurations may be viewed as a pure quantum state space with @xmath29 generic energy levels . however , this correspondence applies only to planar configurations , and not those in higher dimensions . nonetheless , all this suggests the possibility of applying quantum theoretical methods to the statistical theory of shapes , or conversely . i shall briefly outline some relevant ideas which might prove fruitful . as noted above , the space of planar configurations of labelled triangles is @xmath31 , or in quantum theory , the state space of a spin-@xmath30 particle . specification of the hamiltonian determines a pair of energy eigenstates , identifiable with the poles of the sphere . as in quantum mechanics where any orthogonal pair of states can be the energy eigenstates , any pair of orthogonal triangles can form the poles of @xmath14 . a natural symmetrical choice of the ` triangular eigenstates ' would be as follows . let @xmath32 be a nontrivial cubic root of unity ; this satisfies the cyclotomic equation @xmath33 , and the complex conjugate of @xmath32 is @xmath34 . then the vectors @xmath35 could form two eigenstates . thus , the corresponding shapes are two origin - centred equilateral triangles , differing only by two - vertex interchanges . see fig . 1 ( p. 118 ) in kendall s rejoinder ( 1989 ) , displaying the triangles on @xmath14 corresponding to this choice of basis . the shape of an arbitrary triad of points can be expressed as a complex linear combination @xmath36 of these two states such that @xmath37 . in many statistical problems , the labelling of the vertices has no significance , and we can work modulo permutations of points . as regards the representation of the permutation group @xmath38 by transformations of @xmath14 , we let the two - sphere be appropriately subdivided into six regions ( ` lunes ' in kendall s term ) in a manner topologically equivalent to the subdivision of the surface of a cube into six faces . the projective unitary transformations induced by @xmath38 then act in a manner that corresponds to the the permutations of the three oriented cartesian coordinate axes . the total group of rigid isomorphisms of the cube has 48 elements , i.e. the six permutations of the oriented coordinate axes , followed by any of the eight possible combinations of the reflections . in cases where the labelling of points is of no significance , the relevant state space is thus reduced to the ` spherical blackboard ' of kendall ( 1984 ) . if @xmath32 is a nontrivial quartic root of unity , so that the complex conjugate of @xmath32 is @xmath39 , then the most symmetrical choice for the square eigenstates consists of the two orthogonal states @xmath40 eigenstates . the corresponding shapes are two regular origin - centred squares , differing by vertex interchanges . in this case , the third eigenstate ( for spin-0 ) is the degenerate square , i.e. a line , given by @xmath41 and these three states form an orthonormal basis , so that any configuration of four points is linearly expressible in terms of these three states . for pentagons , on the other hand , if @xmath32 denotes a nontrivial fifth root of unity , satisfying the cyclotomic equation @xmath42 , with the complex conjugate of @xmath32 given by @xmath43 , then a symmetric basis consists of the four orthogonal states @xmath44 of origin - centred regular pentagons . these states are identifiable with the eigenstates of a spin-@xmath45 particle in quantum mechanics . in general , the projective space @xmath23 corresponds to configurations of @xmath46 points . in this case , the cyclotomic equation @xmath47 indeed determines an orthonormal set of shape eigenstates . however , as i shall indicate below , if the number @xmath48 of points is not a prime , then the most symmetrical choice of shape eigenstates , to be referred to as eigenshapes , will be degenerate in a sense analogous to that of the square eigenstates considered in the previous section . for example , in the case of six - point configurations , two of the eigenshapes are regular hexagons , two of the eigenshapes are equilateral triangles , and the fifth eigenshape is just a line . in particular , any sextet of points can be expressed as a linear combination of these five eigenshapes , and we do not require five hexagonal shapes to express an arbitrary configuration . in order to construct general eigenshapes for an arbitrary number @xmath49 of points , we let @xmath50 denote a nontrivial root of the cyclotomic equation . then a generic eigenshape can be expressed in the form @xmath51 where @xmath52 ( @xmath53 ) is a set of integers between 0 and @xmath54 , not all of which need be distinct . when they are distinct for a given @xmath49 , the corresponding eigenshape @xmath55 is nondegenerate . for a general @xmath49-point configuration , there are @xmath54 eigenshapes @xmath56 for an arbitrary @xmath49 : [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] note that the numbers @xmath57 successively appear in a cycle . therefore , for example , if @xmath58 , then the value of @xmath59 in the eigenshape @xmath60 , which in this table is given by @xmath61 , indicates the number following @xmath62 in the cycle . in the present example , this is the number 0 rather than 9 , so that the sequence represented by @xmath59 is @xmath63 . this cyclic property clearly implies that if @xmath49 is a prime number , then by following the sequence of numbers along any given row or column in this table , one never encounters any repetitions . thus , the corresponding eigenshapes are all nondegenerate . conversely , if @xmath49 factors , then , by definition , repetitions will appear in some rows and columns , and consequently the corresponding shape becomes degenerate . therefore , only if the number of points is a prime can regular polygons be chosen for all the corresponding eigenshapes . although all orthonormal bases are unitarily equivalent , the choice given here is preferred not only for aesthetic reasons but also for the practical purpose of systematically generating bases that are readily visualised . in a quantum mechanical problem , one begins by specifying the hamiltonian . the eigenstates of the hamiltonian constitute a naturally preferred basis . on the other hand , in a problem concerning statistical analysis of shape , there is no _ a priori _ preferred basis unless further conditions are specified . consequently , for a given configuration with a large number of points , it has hitherto been unclear how one can systematically ` decompose ' the configuration into a set of simple orthogonal components . the present scheme offers one possibility of achieving this task . given an arbitrary pair of @xmath64-point configurations in @xmath10 we can determine the state vectors in the projective space @xmath23 that correspond to these two configurations . then , the separation of these two shapes ( what kendall calls the distance between two shapes ) is determined by the transition probability between the two representative elements of @xmath23 . the superposition of different shapes is also useful from the quantum mechanical point of view . in particular , the set of eigenshapes constructed in the previous section is complete in the sense that any shape can be expressed uniquely as a linear superposition of these eigenshapes . for example , superposition of a pair of orthogonal equilateral triangles , given by @xmath65 will generate all possible shapes associated with three points . in the special case where @xmath66 , all the collinear configurations of three points are obtained by varying the phase variable @xmath67 . similarly , an arbitrary four - point shape can be expressed in the form @xmath68 thus , the decomposition scheme considered here allows us to recover simple parametric families of states that represent the totality of possible point configurations . it should be evident how these examples can be generalised to shapes with larger number of points . for a given hamiltonian , the unitary dynamics of a spin-@xmath30 quantum state corresponds to a rigid rotation of @xmath14 around the poles specified by the energy eigenstates . thus , to study the unitary evolution of triangles , we must determine the infinitesimal deformations of a normalised triangle representing the direction orthogonal to that of a rotation about the origin . after diagonalisation , the unitary evolution generated by the hamiltonian consists in rotating the vertices of the triangle about the origin , but at generally different angular velocities . in general , if the angular velocities are commensurable , then the trajectory is closed in the projective space , but all the vertices of the triangle are hereby rotated through the same angle about the origin . this angle is just the geometric phase associated with the corresponding quantum state . another interesting question in the present context is whether quantal notions such as entanglement could be relevant to statistical shape analysis . the entanglement concept arises when two or more physical systems are combined . if a system with @xmath69 energy eigenstates ( corresponding to an @xmath70-point shape space ) and another with @xmath71 energy eigenstates ( an @xmath72-point shape space ) are combined , one obtains the state space of an @xmath73-eigenstate system ( an @xmath74-point shape space ) . in particular , if the constituent subsystems are disentangled , then such states form an @xmath75-dimensional subspace @xmath76 of the total space @xmath77 . thus , for example , if we combine a pair of triangles , we obtain a disentangled pair equivalent to a four - point configuration , while entangled states represent generic configurations of five points . more generally , when a pair of shapes are combined , then a generic shape that corresponds to a disentangled state will have four points that are collinear . the possible statistical significance of this situation constitutes an intriguing open question . some examples of disentangled and entangled shape combinations are shown in the figure below . what would be quite relevant to the statistical analysis of shape is the notion of mixed states , i.e. distributions over the shape space @xmath23 . often , in cases of interest , certain interactions or dynamics are associated with the points of the configurations under consideration . this permits one to define distributions of shapes . an example of this is the notion of a two - dimensional froth , which has important applications in biology ( e.g. , epithelial tissue growth ) . each froth can be viewed as an irregular polygon , and the froth vertices where three polygonal edgesmeet ( a vertex with higher incidence number is unstable ) are the points of interest . for a given polygon , the number of edges can vary , although its expectation value , assuming we have a flat surface ( i.e. zero curvature ) and a large number of cells , must be 6 by virtue of euler s relation ( cf . et al_. 1996 ) . this is closely related to the fact that , for a wide range of interaction energies between point particles on a plane , the minimum energy configuration is typically given by a regular hexagonal lattice . in the case of biological cells , for which divisions and disappearances occur , statistical analysis of the distribution and the dynamics of point configurations is important in understanding the properties of biological processes . more specifically , the statistical theory of shape is relevant here because the ability of a damaged tissue to restore its stable configuration can be explained by shape - dependent information stored in the cells , no further information being required ( dubertre , _ et al_. 1998 ) . the problem of shape diffusion ( cf . kendall 1988 ) , which would characterise the dynamical evolution of froths , can be formulated as a diffusion process for a single point in @xmath23 . in the quantum mechanical context this is known as the problem of quantum state diffusion . one special class of processes that has been studied extensively ( when phrased in the shape - theoretic context ) is as follows : given a set of eigenshapes that represent energy extremals , ( that is , they are eigenstates of an energy operator ) , an arbitrary initial configuration will diffuse into one of the extremal eigenshapes in such a manner that the probability of terminating in such an eigenshape is given by the transition probability between the initial and final shapes . for such a process , an explicit solution to the diffusion equation is known ( brody @xmath78 hughston 2002 ) for an arbitrary number of points . these notions may be applicable to study diffusive dynamics of complex planar systems . finally , i cite some recent developments in the study of point configurations ( atiyah @xmath78 sutcliffe 2002 , battye , _ et al_. 2003 ) . these investigations are motivated by the physical question of determining the minimum classical energy ( i.e. most stable ) configurations of point particles in two or three dimensions . note , however , that their analysis does not exclude the possibility of all the points coinciding . hence , the relevant shape spaces are slightly distinct from those investigated by kendall . these studies , along with the above - mentioned quantum interpretation of shapes , might shed new light on statistical shape theory . i am grateful to michael atiyah for stimulating discussions , and to the royal society for support .	the shape space of @xmath0 labelled points on a plane can be identified with the space of pure quantum states of dimension @xmath1 . hence , the machinery of quantum mechanics can be applied to the statistical analysis of planar configurations of points . various correspondences between point configurations and quantum states , such as linear superposition as well as unitary and stochastic evolution of shapes , are illustrated . in particular , a complete characterisation of shape eigenstates for an arbitrary number of points is given in terms of cyclotomic equations .
due to their large sizes and short orbital periods , hot jupiters ( roughly jupiter - mass planets with periods between 0.8 and 6.3 days ; * ? ? ? * ) are among the easiest exoplanets to detect . both the first exoplanet discovered around a main sequence star @xcite and the first known transiting exoplanet @xcite were hot jupiters . until the launch of the _ kepler _ space telescope in 2009 , the majority of known transiting exoplanets were hot jupiters . hot jupiters allow for the determination of many planetary properties , including their core masses @xcite and atmospheres @xcite . for these reasons , transiting hot jupiters were and continue to be the subject of many follow - up studies @xcite . one such follow - up study is the search for additional planets in the system revealed by small departures from perfect periodicity in the hot jupiter transit times ( called transit timing variations or ttvs ) . ttvs were predicted @xcite and searched for @xcite , but very little evidence for ttvs was found until the _ kepler _ mission discovered smaller transiting planets on longer period orbits than the hot jupiters detected from the ground @xcite . the lack of transit timing variations for hot jupiters implies a dearth of nearby planets in these systems . while systems exist with a known hot jupiter and a distant ( @xmath0-day period ) companion @xcite or a warm jupiter ( orbital period 6.3 - 15.8 days ) and a close - in planet ( for example , koi 191 : * ? ? ? * ; * ? ? ? * ) , searches for close - in , companions to hot jupiters ( as in * ? ? ? * ) have not yet been successful . this apparent scarcity supports the idea that hot jupiters form beyond the ice line and migrate inwards via high eccentricity migration ( hem ) , a process which would destabilize the orbits of short - period companions @xcite . studies of the rossiter - mcaughlin effect have also found the fingerprints of high eccentricity migration @xcite . however , statistical work has shown that not all hot jupiters can form in this way @xcite , so some hot jupiters may have close - in planets . additionally , hem may not exclude nearby , small planets @xcite . in this paper , we present an analysis of the wasp-47 system ( originally announced by * ? ? ? * ) that was recently observed by the _ kepler _ space telescope in its new k2 operating mode @xcite . in addition to the previously known hot jupiter in a 4.16-day orbit , the k2 data reveal two more transiting planets : a super - earth in a 19-hour orbit , and a neptune - sized planet in a 9-day orbit . we process the k2 data , determine the planetary properties , and measure the transit times of the three planets . we find that the measured ttvs are consistent with the theoretical ttvs expected from this system and measure or place limits on the planets masses . finally , we perform many dynamical simulations of the wasp-47 system to assess its stability . _ kepler _ observed k2 field 3 for 69 days between 14 november 2014 and 23 january 2015 . after the data were publicly released , one of us ( hms ) identified additional transits by visual inspection of the pre - search data conditioned ( pdc ) light curve of wasp-47 ( designated epic 206103150 ) produced by the _ kepler_/k2 pipeline . we confirmed the additional transits by analyzing the k2 pixel level data following @xcite . a box - least - squares ( bls ; * ) periodogram search of the processed long cadence light curve identified the 4.16-day period hot jupiter ( wasp-47b ) , a neptune sized planet in a 9.03-day period ( wasp-47d ) , and a super - earth in a 0.79-day period ( wasp-47 ) . because of the previously known hot jupiter , wasp-47 was observed in k2 s `` short cadence '' mode , which consists of 58.3 second integrations in addition to the standard 29.41 minute `` long cadence '' integrations . k2 data are dominated by systematic effects caused by the spacecraft s unstable pointing which must be removed in order to produce high quality photometry . we began processing the short cadence data following @xcite to estimate the correlation between k2 s pointing and the measured flux ( which we refer to as the k2 flat field ) . we used the resulting light curve and measured flat field as starting points in a simultaneous fit of the three transit signals , the flat field , and long term photometric variations ( following * ? ? ? * ) . the three planetary transits were fit with @xcite transit models , the flat field was modeled with a spline in _ kepler _ s pointing position with knots placed roughly every 0.25 arcseconds , and the long term variations were modeled with a spline in time with knots placed roughly every 0.75 days . we performed the fit using the levenberg - marquardt least squares minimization algorithm @xcite . the resulting light curve shows no evidence for k2 pointing systematics , and yielded a photometric precision of 350 parts per million ( ppm ) per 1 minute exposure . for comparison , during its original mission , _ kepler _ also achieved 350 ppm per 1 minute exposure on the equally bright ( @xmath1 ) koi 279 . we measured planetary and orbital properties by fitting the short cadence transit light curves of all three planets with @xcite transit models using markov chain monte carlo ( mcmc ) algorithm with affine invariant sampling @xcite . and for each planet , we fit for the orbital period , time of transit , orbital inclination , scaled semimajor axis @xmath2 , and @xmath3 . our best - fit model is shown in figure [ transitfits ] and our best - fit parameters are given in table [ bigtable ] . our measured planetary parameters for wasp-47b are consistent with those reported in @xcite . we also fitted for the transit times and transit shapes of each transit event in the short cadence light curve simultaneously ( due to the relatively short orbital periods sometimes causing two transits to overlap ) using mcmc . our measured transit times[multiblock footnote omitted ] are shown in figure [ ttvs ] . we find that the ttvs of wasp-47b and wasp-47d are detected at high significance . the two ttv curves are anti - correlated and show variations on a timescale of roughly 50 days . this is consistent with the ttv super - period we expect for planets orbiting in this configuration , which we calculate to be @xmath4 days using equation 7 of @xcite . transiting planet signals like those we find for wasp-47 and wasp-47d can be mimicked by a variety of astrophysical false positive scenarios . in this section , we argue that this is unlikely in the case of the wasp-47 system . the hot jupiter , wasp-47b , was discovered by @xcite and confirmed with radial velocity ( rv ) follow - up , which showed no evidence for stellar mass companions or spectral line shape variations , and detected the spectroscopic orbit of the planet . in the k2 data , we detect transit timing variations of wasp-47b which are anti - correlated with the transit timing variations of wasp-47d , and which have a super - period consistent with what we expect if both of these objects are planets . the ttvs therefore confirm that wasp-47d is a planet in the same system as wasp-47b . the light curve is not of sufficiently high quality to detect ttvs for the smaller wasp-47 , so we validate its planetary status statistically . we do this using ` vespa ` @xcite , an implementation of the procedure described in @xcite . given constraints on background sources which could be the source of the transits , a constraint on the depth of any secondary eclipse , the host star s parameters and location in the sky , and the shape of the transit light curve , ` vespa ` calculates the probability of a given transit signal being an astrophysical false positive . both visual inspection of archival imaging and a lucky imaging search @xcite found no close companions near wasp-47 , but the lucky imaging is not deep enough to rule out background objects that could cause the shallow transits of wasp-47 . following @xcite , we define a conservative radius inside of which background sources could cause the transits . we adopt a radius of 12 arcseconds ; we detect the transits with photometric apertures as small as 6 arcseconds in radius and allow for the possibility that stars outside of the aperture could contribute flux due to _ kepler _ s 6 arcsecond point spread function . we find that wasp-47 has a false positive probability ( fpp ) of roughly @xmath5 . we consider wasp-47 to be validated as a _ bona fide _ planet . we test the dynamical stability of the wasp-47 planetary system with a large ensemble of numerical simulations . the k2 data determine the orbital periods of the three bodies to high precision and place constraints on the other orbital elements . we sample the allowed ranges of the orbital elements for all three planets , randomizing the orbital phases of the three bodies . we assigned masses by sampling the distribution of @xcite for the measured planet radii . we chose eccentricities from a uniform distribution that extends up to @xmath6 = 0.3 . we discard systems that do not satisfy the stability criteria enumerated in @xcite . given a set of 1000 such initial conditions , we numerically integrate the systems using the ` mercury6 ` integration package @xcite . we use a bulirsch - stoer ( b - s ) integrator , requiring that system energy be conserved to 1 part in @xmath7 . we integrate the system for a total simulation length of 10 myr , unless the system goes unstable on a shorter time scale due to ejection of a planet , planetary collisions , or accretion of a planet by the central star . to perform these computationally intensive simulations , we use the open science grid ( osg ; * ? ? ? * ) accessed through xsede @xcite . the results from this numerical survey are shown in figure [ fig : stability ] . the left panel shows the fraction of systems remaining stable as a function of time . about 30% of the systems are unstable over short time scales , and almost 90% of the systems are unstable over long time scales . once the systems reach ages of @xmath8 yr , they tend to survive over the next three orders of magnitude in integration time . the remaining three panels show the mass and initial eccentricity of the three planets , sampled from the distributions specified above . one clear trend is that low eccentricity systems tend to survive , whereas systems with @xmath9 are generally unstable . a second trend that emerges from this study is that stability does not depend sensitively on the planet masses ( provided that the orbits are nearly circular ) . stable systems arise over a wide range of planet masses , essentially the entire range of masses allowed given the measured planetary radii . wasp-47b and wasp-47d orbit within about 20% of the 2:1 mean motion resonance ( mmr ) . for completeness , we carried out a series of numerical integrations where the system parameters varied over the allowed , stable range described above . in all trials considered , the resonance angles were found to be circulating rather than librating , so there is no indication that the system resides in mmr . each of the numerical integrations considered here spans 10 myr , which corresponds to nearly one billion orbits of the inner planet . tidal interactions occur on longer timescales than this and should be considered in future work . in particular , the survival of the inner super - earth planet over the estimated lifetime of the wasp-47 system could place limits on the values of the tidal quality parameters @xmath10 for the bodies in the system . we performed a second ensemble of numerical simulations to estimate the magnitude of transit timing variations in the wasp-47 system . we used initial conditions similar to those adopted in the previous section , but with starting eccentricities @xmath11 . we integrated each realization of the planetary system for 10 years using the ` mercury6 ` b - s integrator with time - steps @xmath12 seconds . we extracted transit times from each integration for each planet , resulting in theoretical ttv curves . the resulting distributions of predicted ttv amplitudes are shown in figure [ fig : ttvhist ] . the three distributions have approximately the same shape and exhibit well defined peaks . the ttv amplitudes we measured in section [ k2data ] are consistent with the distributions we produced theoretically . we measure the ttvs with high enough precision that dynamical fits can give estimates of the planetary masses . we use ` ttvfast ` @xcite to generate model transit times for each observed epoch for each planet , and use _ emcee _ @xcite , an mcmc algorithm with affine invariant sampling , to minimize residuals between the observed ttvs and these model ttvs . in these fits , we allow each planet s mass , eccentricity , argument of pericenter , and to float . we imposed a uniform prior on eccentricity between 0 and ( as required for long - term stability ) . we initialized the chains with random arguments of pericenter and masses drawn from the @xcite mass posterior for wasp-47b and the distribution of @xcite for wasp-47 and wasp-47d . we used walkers and iterations to explore the parameter space , and discarded the first 2500 iterations as ` burn - in ' . we confirmed that the mcmc chains had converged using the test of @xcite . we find that we are able to measure the masses of wasp-47b and wasp-47d and place an upper limit on the mass of wasp-47 . we additionally provide limits on the quantities @xmath13 . these masses and limits are summarized in table [ bigtable ] . we measure a mass of m@xmath14 for wasp-47b , which is consistent with the mass measured by @xcite of @xmath15 m@xmath14 at the 1@xmath16 level . the mass of wasp-47d is @xmath17 . only an upper limit can be placed on wasp-47 of @xmath18 . wasp-47 is unusual : it is the first hot jupiter discovered to have additional , close - in companion planets . using the exoplanet orbit database @xcite , we found that among the 224 systems containing a planet with mass greater than 80 m@xmath14 and orbital period less than 10 days , only six contain additional planets , and none of them have additional planets in orbital periods shorter than 100 days . that the additional planets in the wasp-47 system are coplanar with the hot jupiter and that the planets are unstable with @xmath19 implies that the wasp-47 planets either migrated in a disk or some damping near the end of migration took place to bring them into their present compact architecture . wasp-47 is a rare system for which planet masses can be determined using ttvs measured from the k2 data set . this is because ( a ) the planets are far enough away from resonance that the super - period ( 52.7 days ) is shorter than the k2 observing baseline ( 69 days ) , and ( b ) the planets are massive enough that the ttvs are large enough to be detectable . the detection of ttvs was also aided by the fact that wasp-47 was observed in short cadence mode , which is unusual for k2 . finally , wasp-47 is a favorable target for future follow - up observations . the v - band magnitude is 11.9 , bright enough for precision rv follow - up studies . the k2 light curve shows no evidence for rotational modulation , indicating that wasp-47 is photometrically quiet and should have little rv jitter . measuring the mass of the two planetary companions with rvs could both improve the precision of the inferred masses and test the consistency of ttv and rv masses , between which there is some tension ( e.g. koi 94 : * ? ? ? * ; * ? ? ? the 1.3% depth of the transits of wasp-47b makes it easily detectable from the ground . previous ground based searches for transit timing variations of hot jupiters have attained timing uncertainties of @xmath20 seconds , lower than the measured ttv amplitude for wasp-47b @xcite . follow - up transit observations could place additional constraints on the masses of the wasp-47 planets . in this work we have studied the wasp-47 planetary system by using data from the _ kepler_/k2 mission along with supporting theoretical calculations . our main results can be summarized as follows : 1 . in addition to the previously known hot jupiter companion wasp-47b , the system contains two additional planets that are observed in transit . the inner planet has a ultra - short period of only 0.789597 days , and radius of 1.829 r@xmath14 . the outer planet has a period of 9.03081 days and a radius of 3.63 r@xmath14 , comparable to neptune . the system is dynamically stable . we have run 1000 10 myr numerical integrations of the system . the planetary system remains stable for the 10 percent of the simulations that start with the lowest orbital eccentricities . the particular planetary system architecture of wasp-47 results in measurable ttvs , which are in good agreement with the ttvs we find from numerical integrations of the system . we use the ttvs to measure the masses of wasp-47b ( consistent with rv measurements ) and wasp-47d . this compact set of planets in nearly circular , coplanar orbits demonstrates that at least a subset of jupiter - size planets can migrate in close to their host star in a dynamically quiet manner , . the wasp-47 planetary system provides a rare opportunity where planets can be both inferred from ttvs and seen in transit . future observations comparing the system parameters inferred from ttvs with those inferred from rvs will qualitatively test ttvs as a general technique . lcccc _ stellar parameters _ & & & + right ascension & 22:04:48.7 & & & + declination & -12:01:08 & & & + @xmath21 [ @xmath22 & 1.04 & @xmath23&@xmath24 & a + @xmath25 [ @xmath26 & 1.15 & @xmath23&@xmath27 & a + & & & & b , d + & & & & b , d + @xmath28 [ cgs ] & 4.34 & @xmath23&@xmath29 & a + [ m / h ] & @xmath30 & @xmath23&@xmath31 & a + @xmath32 [ k ] & 5576 & @xmath23&@xmath33 & a + & & @xmath23 & & a , b , c + & & @xmath23 & & a , b , c + & & + _ wasp-47b _ & & & + orbital period , @xmath34 [ days ] & 4.1591287 & @xmath23&@xmath35 & b + radius ratio , @xmath36 & 0.10186 & @xmath23&@xmath37 & b + scaled semimajor axis , @xmath2 & 9.715 & @xmath23&@xmath38 & b + orbital inclination , @xmath39 [ deg ] & 89.03 & @xmath23&@xmath40 & b + transit impact parameter , @xmath41 & 0.164 & @xmath23&@xmath42 & b + time of transit @xmath43 [ bjd ] & 2457007.932131 & @xmath23 & 0.000023 & b + [ min ] & 0.63 & @xmath23 & 0.10 & b + @xmath44 [ m@xmath14 ] & & & & a , b , c + @xmath46 [ r@xmath14 ] & 12.77 & @xmath23&@xmath47 & a , b + & & + _ wasp-47 _ & & & + orbital period , @xmath34 [ days ] & 0.789597 & @xmath23&@xmath48 & b + radius ratio , @xmath36 & 0.01456 & @xmath23&@xmath49 & b + scaled semimajor axis , @xmath2 & 3.24 & @xmath23&@xmath50 & b + orbital inclination , @xmath39 [ deg ] & 87.0 & @xmath23&@xmath51 & b + transit impact parameter , @xmath41 & 0.17 & @xmath23&@xmath52 & b + time of transit @xmath43 [ bjd ] & 2457011.34849 & @xmath23 & 0.00038 & b + [ min ] & & b + @xmath44 [ m@xmath14 ] & & & & c + @xmath46 [ r@xmath14 ] & 1.829 & @xmath23&@xmath53 & a , b + & & + _ wasp-47d _ & & & + orbital period , @xmath34 [ days ] & 9.03081 & @xmath23&@xmath54 & b + radius ratio , @xmath36 & 0.02886 & @xmath23&@xmath55 & b + scaled semimajor axis , @xmath2 & 16.33 & @xmath23&@xmath56 & b + orbital inclination , @xmath39 [ deg ] & 89.36 & @xmath23&@xmath57 & b + transit impact parameter , @xmath41 & 0.18 & @xmath23&@xmath58 & b + time of transit @xmath43 [ bjd ] & 2457006.36927 & @xmath23 & 0.00044 & b + [ min ] & 7.3 & @xmath23 & 1.9 & b + @xmath44 [ m@xmath14 ] & & & & c + @xmath46 [ r@xmath14 ] & 3.63 & @xmath23&@xmath50 & a , b + & & +	using new data from the k2 mission , we show that wasp-47 , a previously known hot jupiter host , also hosts two additional transiting planets : a neptune - sized outer planet and a super - earth inner companion . we measure planetary properties from the k2 light curve and detect transit timing variations , confirming the planetary nature of the outer planet . we performed a large number of numerical simulations to study the dynamical stability of the system and to find the theoretically expected transit timing variations ( ttvs ) . the theoretically predicted ttvs are in good agreement with those observed , and we use the ttvs to determine the masses of two planets , and place a limit on the third . the wasp-47 planetary system is important because companion planets can both be inferred by ttvs and are also detected directly through transit observations . the depth of the hot jupiter s transits make ground - based ttv measurements possible , and the brightness of the host star makes it amenable for precise radial velocity measurements . the system serves as a rosetta stone for understanding ttvs as a planet detection technique .
= 10000 = 10000 searches for supermassive black holes ( bhs ) based on spatially resolved kinematics have found @xmath035 candidates ( see kormendy et al . 2000 for a review ) . almost all are in weakly active or inactive galaxies . the reason is that bright active galactic nuclei ( agns ) swamp the light from the surrounding stars and gas , and complicate the kinematic observations . in addition , agns are rare , so most are distant . even with the _ hubble space telescope _ ( _ hst _ ) , the central kinematics of galaxies are well enough resolved to reveal bhs only in nearby galaxies . the ironic result ( kormendy & richstone 1995 ) is that the bright seyfert nuclei and quasars that motivate the bh search are conspicuously rare in the dynamical bh census . reverberation mapping ( blandford & mckee 1982 ; netzer & peterson 1997 ) avoids this problem . in this technique , time delays between brightness variations in the continuum and in the broad emission lines are interpreted as the light travel time between the bh and the line - emitting region farther out . this provides an estimate of the radius @xmath1 of the broad - line region ( blr ) . we also have a velocity @xmath2 from the fwhm of the emission lines . together , these measure a mass @xmath3 , where @xmath4 is the gravitational constant . an important advantage is that the blr is @xmath0@xmath5 times closer to the bh than the stars and gas that are used in _ hst _ spectroscopy . however , several authors have pointed out that reverberation mapping yields smaller bh masses at a given bulge luminosity than do dynamical models of spatially resolved kinematics ( e.g. , wandel 1999 ; ho 1999 ) . that is , in the observed correlation between bh mass and bulge luminosity @xmath6 ( kormendy 1993 ; kormendy & richstone 1995 ) , @xmath7values from reverberation mapping are systematically low ( see figure 1a which shows reverberation masses that are as much as a factor of 40 smaller than predicted by the correlation ) . this discrepancy is overstated when using the @xmath7 @xmath6 correlation from magorrian et al . those bh masses are based on two - integral models applied to low - resolution data . comparison with _ hst _ data and three - integral models shows that the magorrian et al . ( 1998 ) bh masses are high by about a factor of three , mainly due to radially - biased anisotropy in the stellar orbits ( gebhardt et al . 2000c ) that was not modeled in magorrian et al . ( 1998 ) mass estimates . nevertheless , even using the best kinematic data , ho ( 1999 ) finds that @xmath7values from reverberation mapping are still low by a factor of @xmath8 compared with masses based on spatially resolved kinematics of different galaxies but similar bulge luminosities . it is important to resolve this discrepancy . gebhardt et al . ( 2000b ) and ferrarese & merritt ( 2000 ) find a new correlation between @xmath7and the effective velocity dispersion @xmath9 of the host galaxy . this relation is significantly tighter than the @xmath7 @xmath6 correlation , consistent with zero intrinsic scatter . in this _ letter _ , we add reverberation mapping masses to the new correlation and find that the systematic offset between the two mass estimators is no longer significant . ho ( 1999 ) and wandel , peterson , & malkan ( 1999 ) measure @xmath7values for 22 seyfert 1 galaxies using reverberation mapping . unfortunately , the absorption - line kinematics of these galaxies are not well studied , so we are unable to obtain velocity dispersions for the whole sample . only seven galaxies have usable published dispersions . the three sources for these dispersions are nelson & whittle ( 1995 ) , di nella et al . ( 1995 ) , and smith , heckman , & illingworth ( 1990 ) . for most of these galaxies , the velocity dispersions are difficult to measure . some are late - type galaxies , so template matching is difficult because of the presence of young stars . in many cases , dilution of the stellar absorption lines by the nonstellar continuum of the agn is a problem . dilution does not alter the velocity dispersion of the lines , but it does make them hard to detect . ideally , we should use spectral regions that are minimally sensitive to template mismatch and to line dilution . the calcium infrared triplet near 8500 is preferable to the traditional mg b @xmath105170 region ( dressler 1984 ) . in the present paper , we adopt velocity dispersions derived from the calcium triplet region whenever possible . the study of terlevich , daz , & terlevich ( 1990 ) contains three galaxies with reverberation masses ; however , the dispersions measured for many of their other galaxies do not compare well with those from other groups . for example , their dispersions for m33 , m32 , and m31 are significantly different than the accepted values in their apertures : 77 @xmath11 compared with 21 @xmath11 ( kormendy & mcclure 1993 ) for m33 , 56 @xmath11 compared with @xmath0 80 @xmath11 ( van der marel et al . 1994 ) for m32 , and 137 @xmath11 compared with 195 @xmath11 ( van der marel et al . 1994 ) for m31 . therefore we exclude their measurements from our analysis . for their dispersion estimate , gebhardt et al . ( 2000b ) use the projected , luminosity - weighted value inside the half - light or effective radius of the bulge , which we call the effective dispersion and denote by @xmath9 . for the agn sample , we do not have dispersion profiles and can not perform the same calculation . consequently , we must use _ central _ dispersions . however , based on their sizes , these galaxies have bulge half - light radii of only a few arcseconds ( kotilainen , ward , & williger 1993 ; baggett , baggett , & anderson 1998 ) . these sizes are similar to the typical seeing and extraction window used ( @xmath02 ) . gebhardt et al . ( 2000b ) find that central aperture dispersions measured at this resolution are similar on average to effective dispersions , with a scatter of at most 10% . thus , the reported dispersion should be a good approximation to the effective dispersion , although a systematic study using the dispersion profile would be worthwhile . when using central dispersions , the most crucial concern is whether the black hole affects the measured dispersion . assuming typical stellar mass - to - light ratios , the spheres of influence for these black holes are a few tenths of an arcsecond . they should have little effect on the dispersions . the effective dispersion used in gebhardt et al . ( 2000b ) assumes edge - on configuration , and since the projected dispersion varies with orientation we must consider whether it needs correction for inclination . this effect may be more important in agn disk galaxies , where we might expect significant rotation ( a rotating galaxy will have a larger projected dispersion edge - on than face - on ) . four of the seven galaxies are inclined greater than 45@xmath12 and are likely to have corrections smaller than their uncertainties . the three galaxies more face - on than 45@xmath12 are mrk590 , ngc4151 , and ngc4593 . based on systems with large bulge fractions , even these would have corrections less than 10% ( gebhardt et al . 2000b ) ; however , better kinematic data on the bulge rotation profiles for a larger sample of agns is needed before we fully understand inclination corrections . ho ( 1999 ) compiled bulge luminosities and bulge - to - total light ratios ( @xmath13 ) from three sources . kotilainen et al . ( 1993 ) provide surface photometry and disk bulge decompositions for 3c120 , mrk590 , ngc3227 , ngc4151 , and ngc4593 ; granato et al . ( 1993 ) provide disk bulge decompositions for mrk590 and ngc3516 ; while baggett et al . ( 1998 ) give profiles and decompositions for ngc3227 , ngc4051 , ngc4151 , and ngc4593 . for the galaxies that overlap among the various groups , we find consistent @xmath13 values . however , each study uses a de vaucouleurs profile for the bulge component . if these bulges are more nearly exponential or if the agn contributes significant light unaccounted for , then the bulge light will have been overestimated . table 1 lists the data we have discussed , and figure 1 plots @xmath7versus the bulge luminosity and effective dispersion @xmath9 . the masses from resolved kinematics and the associated least - squares fits come from gebhardt et al . ( 2000b ) . the relation fitted only to the galaxies with spatially resolved kinematics for the @xmath7 @xmath9 correlation is @xmath7@xmath14 . the reverberation masses lie a factor of 510 too low in the luminosity plot ( black dots ) , but are much more consistent with the correlation in the @xmath9 plot . in the @xmath9 relation , the reverberation mapping masses have an average offset of @xmath15 ( @xmath16 ) dex and a dispersion of 0.34 dex relative to that average . the scatter ( 0.30 dex in log@xmath7 at fixed dispersion ) is the same regardless of whether or not we include the reverberation mapping masses in the fit , but the slope changes from 3.75 to 3.90 if we include them . the average uncertainties in m@xmath17 , @xmath18 , and @xmath9 for the reverberation mapping estimates are shown in the bottom corner in fig . 1 . unfortunately , the uncertainties in reverberation mapping are dominated by systematics that are uncertain or unknown ( wandel et al . 1999 ) and can be quite large . since we have only seven reverberation mapping masses , we do not attempt a rigorous statistical analysis including the measurement uncertainties . llcccccc 3c120 & s0 : & 132 & 2.29 & 0.24 & @xmath19 & 162 & smith et al . 1990 mrk590 & sa : & 105 & 2.78 & 0.47 & @xmath20 & 169 & nelson & whittle 1995 ngc3227 & saba & 21 & 1.16 & 0.52 & @xmath21 & 128 & nelson & whittle 1995 ngc3516 & sb0 : & 39 & 2.00 & 0.61 & @xmath22 & 124 & di nella et al . 1995 ngc4051 & sabbc & 9 & 0.13 & 0.20 & @xmath23 & 88 & nelson & whittle 1995 ngc4151 & sabab & 20 & 1.20 & 0.36 & @xmath20 & 119 & nelson & whittle 1995 ngc4593 & sbb & 40 & 3.13 & 0.48 & @xmath24 & 124 & nelson & whittle 1995 the apparent discrepancy between reverberation mapping and dynamical modeling of spatially resolved kinematics arose from a comparison of @xmath7with bulge luminosities . since the reverberation mapping masses are consistent with the correlation and not with the @xmath7 @xmath6 correlation , the discrepancy in the latter is likely due to problems with the use of bulge luminosities , not with estimation of the bh masses . velocity dispersions are more difficult to measure in agns , but we have little reason to suspect that they have systematic errors . however , it is important to examine the potential complications of both techniques . sections 3.1 and 3.2 below suggest that the bh masses from resolved stellar kinematics have only small systematic errors but that the bh masses from reverberation mapping may be biased slightly low . \(1 ) model limitations were once a concern but are now under control . the current state of the art is to use schwarzschild s method ( schwarzschild 1979 ; richstone & tremaine 1988 ) to construct three - integral models that include galaxy flattening and velocity anisotropy ( van der marel et al . 1998 ; gebhardt et al . 2000a ; richstone et al . the galaxies with stellar kinematical masses in figure 1a , b all have three - integral models . when such models are fitted to _ hst _ data , the errors in @xmath7 are small . however , there is still some concern about whether non - axisymmetric structure affects the masses , and thorough comparisons of the different modeling codes have not yet been carried out . \(2 ) selection effects may be present since early bh searches were biased toward objects with unusually high bh masses ( the first _ targets were galaxies that showed high central dispersions at ground - based resolution ) . this bias still persists in the current overall bh census based on stellar - dynamical measurements , but the present sample is large enough to overcome effects from a few galaxies with high bh masses . \(3 ) it is possible that galaxies contain central concentrations of ordinary dark matter ( e.g. , stellar remnants ) that are included in most bh mass measurements . this concern is prompted by the fact that the radii that we resolve with _ hst _ spectroscopy are @xmath25 larger than the blr that is used in reverberation mapping . however , measurements in our galaxy ( genzel et al . 1997 , 2000 ; ghez et al . 1998 ) and in ngc4258 ( greenhill et al . 1996 ) probe a small region comparable to that probed by reverberation mapping in other galaxies , and find no suggestion of any dark mass in addition to a bh . \(1 ) the geometry and orbital distribution of the blr are poorly known . if , as is often assumed in the agn unification model ( antonucci 1993 ) , seyfert 1 nuclei are viewed preferentially face on and if the blr and the obscuring tori are roughly coplanar , then the inclination correction for reverberation masses would be significant . however , because the thickness of the blr disk is unknown , the actual correction is uncertain . nonetheless , if the corrections are factors around 2 , then the comparison of these masses in fig . 1b will improve . \(2 ) the measured `` lag '' in reverberation studies is a peculiar moment over the distribution of distances between the central nucleus and the line - emitting gas , and the measured line width is a different peculiar moment over the velocity distribution . these are affected by the adopted weightings , the shape of the continuum fluctuation power spectrum , and the sampling of the monitoring . \(3 ) selection effects restrict the bh masses that are currently measurable by reverberation mapping . the timescales of agn variability scale with luminosity ( e.g. , netzer & peterson 1997 ) and presumably with mass for eddington - limited systems . ongoing studies of slowly - varying , high - luminosity agns ( kaspi et al . 2000 ) and of rapidly - varying , low - luminosity agns ( peterson et al . 2000 ) should remedy this situation in the future . ( 4 ) it is important to consider non - gravitational effects acting on blr gas . they include radial motions caused by radiation pressure or by mechanical energy from jets . the resulting mass measurement errors could have either sign , but it is most likely that we would overestimate @xmath7 ( krolik 1997 ) . the following are additional complications that arise when estimating bulge luminosities . \(5 ) agns are commonly associated with starbursts ( e.g. , heckman 1999 ; sanders 1999 ) . this may cause @xmath7 to look too small in the @xmath7 @xmath6 correlation . although @xmath7 is plotted against blue luminosity , the physical correlation is presumably with bulge mass , and star formation can easily reduce @xmath26 by a factor of 2 4 compared to its value in bulges that are made of old stars . then the bulge would look too bright for the given @xmath7 . \(6 ) it is possible that the light from the agn biases the estimate of the bulge light . first , the agn makes the center of the galaxy look exceptionally bright in poor - quality images , so there is a tendency to assign a hubble type that is too early if using qualitative visual inspection . if one then uses the loose correlation between hubble type and bulge - to - disk ratio ( simien & de vaucouleurs 1986 ) to estimate bulge luminosities , they will be overestimated . second , even if one uses disk / bulge decompositions , unless the agn is modeled separately , it will likely cause an overestimate of the bulge light as well . we have shown that masses derived from reverberation mapping are consistent with the relation between bh mass and galaxy velocity dispersion derived from spatially resolved kinematics . based on a sample of seven seyfert galaxies , we find that the systematic and random errors in bh masses determined from reverberation mapping are around 0.21 dex and 0.34 dex , respectively . it is remarkable that , despite the large number of possible systematic biases ( especially in reverberation mapping ) , both methods appear to provide consistent and reliable estimators of bh masses . peterson & wandel ( 2000 ) provide further support of the reliability for the agn mass estimates by showing a keplerian relation between line width and time lag . one could even use the @xmath27 correlation to infer properties of the broad - line region ; for example , any differences in the agn masses compared with the correlation may provide insight into the blr geometry . the fact that reverberation mapping successfully delivers bh masses offers tremendous hope of getting bh masses in objects that otherwise would not be accessible , namely bright agns , including qsos ( e.g. , kaspi et al . 2000 ) , and high - redshift agns . the latter hold some hope of probing the time evolution ( growth history ) of bh mass ( e.g. , wandel 1999 ) . furthermore , the correlation between photoionization and reverberation models ( wandel et al . 1999 ) offers the possibility of wholesale agn mass estimates . future studies aimed at comparing the two mass estimators on the _ same _ galaxies are required to confirm both techniques , but the present results are encouraging . we are grateful for comments from b. peterson , a. wandel , j. krolik , and the referee , k. anderson . this work was supported by grants to the nukers , go02600.0187a , g06099 , and g07388 , and by nasa grant nag5 - 8238 . a.v.f . acknowledges nasa grant nag5 - 3556 . k.g . is supported by nasa through hubble fellowship grant hf-01090.01 - 97a awarded by the space telescope science institute , which is operated by the association of the universities for research in astronomy , inc . , for nasa under contract nas 5 - 26555 . antonucci , r. r. j. 1993 , , 31 , 473 baggett , w. e. , baggett , s. m. , & anderson , k. s. j. 1998 , 116 , 1626 blandford , r. d. , & mckee , c. f. 1982 , , 255 , 419 di nella , h. , garcia , a. m. , garnier , r. , & paturel , g. 1995 , , 113 , 151 dressler , a. 1984 , , 286 , 97 ferrarese , l. , & merritt , d. 2000 , , 539 , l9 gebhardt , k. et al . 2000a , , 119 , 1157 gebhardt , k. et al . 2000b , , 539 , l13 gebhardt , k. et al . 2000c , in preparation genzel , r. , eckart , a. , ott , t. , & eisenhauer , f. 1997 , , 291 , 219 genzel , r. , pichon , c. , eckart , a. , gerhard , o. e. , & ott , t. 2000 , , submitted ghez , a. m. , klein , b. l. , morris , m. , & becklin , e. e. 1998 , , 509 , 678 granato , g. l. , zitelli , v. , bonoli , f. , danese , l. , bonoli , c. , & delpino , f. 1993 , , 89 , 35 greenhill , l.j . , gwinn , c.r . , antonucci , r. , & barvainis , r. , 1996 , , 472 , l21 heckman , t. m. 1999 , wolf - rayet phenomena in massive stars and starburst galaxies , ed . k. a. van der hucht , g. koenigsberger , & p. r. j. eenens ( san francisco : asp ) , 703 ho , l. c. 1999 , in observational evidence for black holes in the universe , ed . s. k. chakrabarti ( dordrecht : kluwer ) , 157 kaspi , s. , smith , p. s. , netzer , h. , maoz , d. , jannuzi , b. t. , & giveon , u. 2000 , , 533 , 631 kormendy , j. 1993 , in the nearest active galaxies , ed . j. beckman , l. colina , & h. netzer ( madrid : consejo superior de investigaciones cientficas ) , 197 kormendy , j. , & mcclure , r. d. 1993 , , 105 , 1793 kormendy , j. , & richstone , d. 1995 , , 33 , 581 kormendy , j. et al . 2000 , in preparation kotilainen , j. k. , ward , m. j. , & williger , g. m. 1993 , , 263 , 655 krolik , j. h. 1997 , emission lines in active galaxies : new methods and techniques , ed . b. m. peterson , f .- z . cheng , & a. s. wilson ( san francisco : asp ) , 459 magorrian , j. et al . 1998 , , 115 , 2285 nelson , c. h. , & whittle , m. 1995 , , 99 , 67 netzer , h. , & peterson , b.m . 1997 , in astronomical time series , ed . d. maoz , a. sternberg , & e. m. leibowitz ( dordrecht : kluwer ) , 85 peterson , b. m. , et al . 2000 , , in press peterson , b. m. , & wandel , a. 2000 , , in press richstone , d. o. , & tremaine , s. 1988 , , 327 , 82 richstone , d. et al . 2000 , in preparation sanders , d. b. 1999 , ap&ss , 266 , 331 schwarzschild , m. 1979 , , 232 , 236 simien , f. , & de vaucouleurs , g. 1986 , , 302 , 564 smith , e. , heckman , t. , & illingworth , g. 1990 , , 356 , 399 terlevich , e. , daz , a.i . , & terlevich , r. 1990 , , 242 , 271 van der marel , r. p. , rix , h .- w . , carter , d. , franx , m. , white , s.d.m . , & de zeeuw , p. 1994 , , 268 , 521 van der marel , r.p . , cretton , n. , de zeeuw , p.t . , & rix , h .- w . 1998 , , 493 , 613 wandel , a. 1999 , , 519 , l39 wandel , a. , peterson , b. m. , & malkan , m. a. 1999 , , 526 , 579	= 10000 = 10000 black hole ( bh ) masses that have been measured by reverberation mapping in active galaxies fall significantly below the correlation between bulge luminosity and bh mass determined from spatially resolved kinematics of nearby normal galaxies . this discrepancy has created concern that one or both techniques suffer from systematic errors . we show that bh masses from reverberation mapping are consistent with the recently discovered relationship between bh mass and galaxy velocity dispersion . therefore the bulge luminosities are the probable source of the disagreement , not problems with either method of mass measurement . this result underscores the utility of the bh mass velocity dispersion relationship . reverberation mapping can now be applied with increased confidence to galaxies whose active nuclei are too bright or whose distances are too large for bh searches based on spatially resolved kinematics .
the nasa deep space network ( dsn ) is a world - wide facility for communicating with deep space missions . transient observatory ( dto ) is a signal processing facility that can monitor up to four dsn telemetry bands for astronomically interesting signals . since telemetry signals occupy only a small fraction of the entire band , the rest of the band can be used for commensal scientific investigations . the signal processing requirements may be quite different for these projects . dto supports these by using different firmware on the same platform . currently , the primary scientific objective is monitoring mars for electrostatic discharges . such bursts , reported by @xcite , occurred during a convective dust storm . since the dsn is receiving data from spacecraft at mars almost continuously , a statistically significant database can reveal the conditions which give rise to such events . other planets can also be monitored for transient phenomena . additional investigations include searching for astronomical transients such as fast radio burts ( frbs ) , pulsars and signals resulting from extraterrestrial technologies . there is one 70 m diameter antenna in goldstone , california , with others near canberra , australia , and madrid , spain , roughly equidistant in longitude to enable continuous communication with robotic missions exploring the solar system . each site also has at least three 34-m antennas . table [ tab : bands ] @cccc@ band & frequencies & bandwidth & polarization + & ( ghz ) & ( mhz ) + s & 2.265 - 2.305 & 40 & lcp @xmath0 rcp + x & 8.25 - 8.65 & 400 & lcp & rcp + ka & 31.85 - 32.25 & 400 & lcp & rcp + the polarization for s - band is configured at the beginning of a tracking session . it is normally rcp . only some 34-m antennas have s - band . [ tab : bands ] shows the bands used for deep space to earth communications . the initial signal processing is done with two roach1 boards , each handling up to two 640mhz - wide baseband signals . each roach1 has a 10 gb ethernet ( 10 gbe ) interface with a gpu - equipped debian linux workstation for additional processing . the monitoring is done commensally with reception of deep space mission signals . since deep space missions move slowly across the sky , the time for acquiring data from a given direction can be quite long , though the direction is not under the investigator s control . at minimum elongation , venus moves about one beamwidth per hour at x - band for a 70 m antenna or at ka - band for a 34 m antenna . for mars the minimum dwell time is about 100 min . for jupiter and saturn it is about 5@xmath1 and 11 hr respectively . if , one the other hand , one wishes to survey more sky quickly , for a 34 m antenna that is not moving , the dwell time for a position on the sky is @xmath2s at s - band , @xmath3s at x - band and 3.8s at ka - band . a non - tracking 34-m antenna can cover 1% of the sky in 17 earth rotations at s - band , 59 at x and 227 at ka . dto can also be used during tracking sessions assigned to observations of radio astronomical sources . the facility will be available to the astronomical community through a peer - reviewed proposal process . figure [ fig : overview ] shows an overview of the signal processing facility . a commercial @xmath4 intermediate frequency ( if ) matrix switch made by jfw industries , inc . , directs up to four station ifs to the inputs of two roach1 signal acquisition and processing units . two high - speed , multi - core , gpu - equipped workstations accept processed data from each roach over a 10 gb ethernet ( 10gbe ) port . the powerpcs on the roach boards boot their kernel images from the master controller dto . dto also has the file systems of the powerpcs which control the operation of a xilinx xc5vsx95 t field programmable gate array ( fpga ) . the bit - files which configure the roach fpgas are loaded by the powerpcs . so , in effect , dto provides the firmware for the fpgas , allowing rapid switching of firmware under computer control . the sample clock generator ( samgen ) subsystem provides the clock signals for the analog - to - digital convertors ( adc ) , optional synchronization pulses and if amplification and low - pass filtering . figure [ fig : samgen ] has a schematic diagram . samgen has a valon dual synthesizer so the two roach board adcs can be independently clocked . it also distributes 1pps synchronization pulses to the kat adcs . in addition to the controller dto and the two ppcs , there are two high - performance computers . each post - processing ppc has two intel westmere 2.66ghz x5650 processors with six cores , allowing it to run twelve threads . it has 96 gb of ram , two gtx 580 gpu processors each with 512 cores , and 48 tb of fast disk . 32 t are available for data storage . all hosts have the debian linux operating system . on 2006 june 8 deep space station 13 ( dss-13 ) detected non - thermal bursty emissions from mars @xcite that showed resonances similar to schumann resonances on earth @xcite . transient non - thermal radio bursts can be distinguished from gaussian thermal noise by computing the kurtosis ( normalized fourth moment ) of the signal voltage @xcite . for such studies we have designed specifically for dto a 1024-ch firmware spectrometer that computes kurtosis as well as power . the design is shown in figure [ fig : kurtspec ] . schematic diagram of the kurtosis spectrometer firmware , including all the registers and data stores.,width=672 ] for mars , the integration time for these calculations , a parameter which can be adjusted , is 1 ms . this allows a signal kurtosis spectrogram to be examined for the presence of modulation by very low frequency waves propagating between the surface and ionosphere of mars . the dsn has telemetry with at least one spacecraft at mars for an average of 20 hrs every day . one of dto s main goals is to monitor mars diligently for further evidence for electrostatic discharges . it would be unlikely to detect radio spectral line emission from an arbitrary direction in the sky but planetary atmospheres can produce such emission from very low density upper atmospheres . one of the available firmwares is a 32k - channel spectrometer . it was developed for the tidbinbilla agn maser survey ( tams ) project led by harvard smithsonian astrophysical observatory @xcite and is part of the 17 - 27 ghz receiver system on dss-43 @xcite . with a 640 mhz bandwidth this gives a spectral resolution of 20 khz . the doppler velocity resolution is 2.5 km s@xmath5 at s - band , 0.7 at x - band and 0.2 km s@xmath5 at ka - band . this firmware is most likely to be used for antenna time assigned to astronomical research . it would also be useful for monitoring radio frequency interference ( rfi ) and spacecraft transmitters which have lost lock and drifted in frequency . extraterrestrial civilizations that are located close to the sun s ecliptic plane could be aware of earth and its life - supporting potential from earth s transits across the sun . such a civilization might direct a beacon towards earth to initiate possible contact . an interesting strategy is to conduct searches for extraterrestrial civilizations ( seti ) along the ecliptic plane @xcite . the dsn , in communication with spacecraft at or travelling towards other planets , has many antennas pointed close to the ecliptic . a commensal seti search may have enhanced chances of success compared to similar searches at radio observatories @xcite . there are 82 k and g stars within one kpc of earth within this zone @xcite , so that over a time comparable to the orbital periods of the planets , they would be within the beam of antennas communicating with deep space missions . amplitude modulated signals transmitted by putative extraterrestrial civilizations are expected to be most easily detected by the carrier tone , which is expected to be extremely narrow . an extremely high resolution spectrometer design , serendip vi , will be adapted from roach2 . the firmware performs 4k channelization . the samples from each channel of the firmware output are further channelized in software using gpus . a 128k pfb on each channel from the firmware spectrometer will yield a spectral resolution of 1.2 hz . serendip vi has been implemented in setiburst , a multi - purpose signal processor @xcite similar in concept to dto . the high resolution of the final spectra allows masking of spacecraft telemetry and known interference signals . since the amount of data is still huge , additional back - end processing will decimate the data . for example , the jpl `` setispec '' used in the gavrt seti project @xcite , reports for each of 4096 coarse channels , the number of and power detected in the strongest high - resolution sub - channel . the monitor and control ( m&c ) software has a server / client architecture . all control software is written in python . inter - process communication ( ipc ) is managed with pyro . overview of the dto software.,width=470 ] figure [ fig : sw_oview ] gives a schematic overview of the software . the * scheduler * processes the dsn 7-day schedule for the goldstone antennas and selects antennas and frequency bands according to a rule set determined by the science team . the * supervisor * is a client of the hardware servers and monitors and controls all aspects of dto the state of the if switches , the frequency and power of the sample clock generators , the gain of the katadc rf sections , the firmware loaded into the fpgas , etc . it also collects metadata from the station s monitor data server using the dsn s ipc protocol . the can be overseen by a human through the supervisor client . this has a set of classes and functions through which an expert user can communicate with the supervisor through a python command line . a graphical user interface ( gui ) can invoke the same classes and functions to provide a more convenient overview . the * roach power pcs * control most of the roach operation . the ppc filesystems are on dto . the supervisor loads the appropriate firmware for a session into the roachs . the user s client software works through a tunnel @xcite which is created by a user with the requisite authority ( username and two - factor passcode ) to log in to the firewall s gateway . only one log - in is needed for a session . data generated by the firmware streams through 10gbe connections to the signal processing hosts . because of the generally high data rate involved , this software is normally written multi - threaded c. python s libraries for gpu support are being investigated . the ouput is written to disk for further real - time analysis . in the final software , the data are stored temporarily in a circular buffer . these data are examined for `` events '' . this software is the most complex as it must separate real from false events , generate alerts for events requiring prompt attention and store data which events of interest . dto is awaiting shipment to the goldstone deep space communications complex ( gdscc ) . the hardware , firmware and m&c software for the initial science program has been verified in the laboratory at jpl . the location and access to power and signals at goldstone have been assigned . paperwork required for installation in the configuration - controlled dsn operations environment has been completed . dto will become operational at goldstone in the late summer of 2016 . dto will be operated as an open facility . the jpl interplanetary network directorate ( ind ) which manages the dsn will accept proposals from outside investigators who bring their own firmware to conduct research with different goals . because of the flexibility built into the casper development ecosystem , dto should not be considered limited to the applications described here , but rather a prototype for general purpose commensal science signal processing . for example , the high resolution spectrometer design was recently implemented as four spectrometers on a single roach2 . additional modifications , such as extending the kurtosis based designs for flexible integration times , and development of higher order ( 6th , 8th moment ) statistics are also ongoing . the instituto nacional de tcnica aeroespacial ( inta ) and ingeniera de sistemas para la defensa de espaa ( isdefe ) have made available four roach1 boards together with a controller host to the radio astronomy department of the madrid deep space communications complex ( mdscc ) . inta and jpl will collaborate in assembling a second dto for mdscc which will be operated by inta scientists . a fifth roach1 board will be used to provide a spectrometer for the partner educational antenna ( proyecto acadmico del radiotelescopio de nasa en robledo ) , a project also managed by inta . the deep space network is operated by the california institute of technology jet propulsion laboratory for the national aeronautics and space administration ( nasa ) . this work was funded by the science and technology investment program of the offices of the jpl chief scientist and chief technologist . we are endebted to jonathan kocz for completing the kurtosis firmware s 10 gbe output capability . we thank chris ruf and nilton renno for insightful discussions on kurtosis and the science of electrostatic discharges in the martian atmosphere . the advice and support of the jpl interplanetary network directorate chief scientist , joseph lazio , is gratefully acknowledged . chuck naudet leads the dsn science automation task which develops software that will enhance dto operation . dong shin assisted in assembling dto .	the dsn transient observatory ( dto ) is a signal processing facility that can monitor up to four dsn downlink bands for astronomically interesting signals . the monitoring is done commensally with reception of deep space mission telemetry . the initial signal processing is done with two casper roach1 boards , each handling one or two baseband signals . each roach1 has a 10 gbe interface with a gpu - equipped debian linux workstation for additional processing . the initial science programs include monitoring mars for electrostatic discharges , radio spectral lines , searches for fast radio bursts and pulsars and seti . the facility will be available to the scientific community through a peer review process . ; ; ;
the study of moduli spaces is an integral part of modern algebraic geometry and representation theory . it is thus very natural , if one is studying noncommutative surfaces , to wish to understand the various moduli spaces that can be associate to them . however , even in the commutative case , let alone the noncommutative one , very few examples have been explicitly computed which is what we aim to achieve in this paper . a rich class on noncommutative surfaces , that has been extensively studied , is that of orders on surfaces , which we now define . let @xmath5 be a normal integral surface . an * order * @xmath6 on @xmath5 is a coherent torsion free sheaf of @xmath7-algebras such that @xmath8 is a central simple @xmath9-algebra . @xmath5 is called the * centre * of @xmath6 . for example , if @xmath5 is as above , then any azumaya algebra on @xmath5 is an order on @xmath5 . furthermore , it is in fact a * maximal order * in the sense that it is not properly included in any other order . for a great reference on orders on surfaces , see @xcite and @xcite . since orders are finite over their centres they are in some sense only mildly noncommutative and many classical geometric techniques can be used to study them . in this paper we first fix an order @xmath6 on @xmath0 ramified on a union of two conics , and study two of its moduli spaces : * the moduli space of line bundles on @xmath6 ( see definition [ linebundle ] ) , with a fixed set of chern classes , denoted by @xmath10 , and * the moduli space of left quotients of @xmath6 , with a fixed set of chern classes , denoted by @xmath11 . the first moduli space should be thought of as the picard scheme of @xmath6 , but one should note that since @xmath6-line bundles are only one sided modules , this is not a group scheme . borrowing terminology from its commutative counterparts , the second moduli space will be referred to as the hilbert scheme of @xmath6 and should be thought of as the space parametrising noncommutative curves on @xmath6 . not surprisingly , these two moduli spaces are intrinsically linked ; in fact we will prove that @xmath11 is a ruled surface over @xmath10 and that @xmath12 is a genus two curve . furthermore , by analysing the universal family on @xmath11 we will show that @xmath11 maps to @xmath13 with branch locus being two conics and their four bitangents . the inspiration behind this paper comes from @xcite where the authors , chan and kulkarni , study the moduli space of line bundles on an order ramified on a smooth quartic . the reader is highly encouraged to read that paper in order to better understand our motivation . we will explain similarities and differences between our approaches as we go . to enable us to begin our project , we use the noncommutative cyclic covering trick , described in chapter [ nctrick ] , to construct our order on @xmath0 . the key ingredient to this construction , is a double cover @xmath14 , a line bundle @xmath15 and a morphism @xmath16 where @xmath17 is the covering involution . using this data one constructs a sheaf of algebras @xmath6 on @xmath18 which is an order on @xmath19 . the main tool we use for studying @xmath6-modules is the simple observation that any such module is also naturally an @xmath20-module . in particular , this allows us to talk about the chern classes and semistability of @xmath6-modules when viewed as @xmath20-modules . furthermore , we will see that any @xmath6-line bundle is a rank two vector bundle on @xmath18 , and so their study is rather different to the study of the picard scheme of @xmath18 and much closer related to the study of rank two vector bundles . the main points of difference are that , first of all , @xmath6-line bundles do not form a group for they are only left @xmath6-modules and so their moduli space is not naturally a group scheme . furthermore , the second chern class , which is zero when one looks at line bundles in the usual setting , plays a crucial role in their study , as do semistability considerations . more precisely , we are interested in studying those @xmath6-line bundles which have minimal second chern class . it is certainly not obvious that one can place a bound on the second chern class of @xmath6-line bundles and hence talk about those @xmath6-line bundles with `` minimal second chern class '' . for chan and kulkarni , this was achieved easily from the fact that for them , @xmath21 was an isomorphism which implied ( proposition 3.8 in @xcite ) that any @xmath6-line bundle was automatically @xmath22-semistable and so by invoking bogomolov s inequality , this aim was achieved . the authors used the @xmath22-semistability property further by noting by simply forgetting the extra @xmath6-module structure , one obtains the map @xmath23 it is the careful analysis of this map that allowed chan and kulkarni to prove that their moduli space was a genus two curve . in our case , @xmath21 will not be an isomorphism , and even though we will be able to deduce a lower bound for the second chern class ( proposition [ propdelta ] ) , the above map of moduli spaces will not be available for us , simply because @xmath6-modules will turn out to be not @xmath22-semistable in general . thus we will use a totally different approach . having bound the second chern class we will show that it suffices to consider only two possible first chern classes : @xmath24 with corresponding minimal @xmath25 and @xmath26 with corresponding minimal @xmath27 . the former case will be rather simple and we will prove that the moduli space in that case is just one point . the latter case will be far more interesting and will be the prime focus of this paper . we will prove , in theorem [ quotientscomputation ] , that for any @xmath6-line bundle @xmath28 with this set of chern classes we have the following exact sequence @xmath29 where @xmath30 is a quotient of @xmath6 . this establishes a connection between the moduli space of line bundles with minimal second chern class and the hilbert scheme of @xmath6 which parametrises quotients of @xmath6 with specified chern classes . we will explore this connection in depth and ultimately prove : [ mainthm ] let @xmath10 be the moduli space of @xmath6-line bundles with @xmath31 and @xmath27 and @xmath11 the hilbert scheme of @xmath6 , parameterising quotients of @xmath6 with @xmath32 and @xmath27 . then @xmath10 is a smooth genus @xmath33 curve . @xmath34 is a smooth ruled surface over @xmath10 . furthermore , @xmath11 exhibits an @xmath35 cover of @xmath0 , ramified on a union of @xmath33 conics and their @xmath36 bitangents . in their paper , chan and kulkarni had a remarkably similar result concerning the moduli of line bundles with minimal @xmath37 . they also reduced the study of their moduli space of line bundles with minimal second chern class to two possible first chern classes . in the first case , the moduli space was a point and in the second case , also a genus two curve . + acknowledgements : this paper is a summary of the author s phd thesis . consequently , the author would like to thank his phd supervisor daniel chan for all his help , patience and inspiration . the author is also very grateful to kenneth chan and hugo bowne - anderson for all the helpful discussions . we begin by briefly reviewing the relevant theory of orders on surfaces . after this , the rest of the paper is primarily devoted to making sense of , and proving theorem [ mainthm ] . in section [ mod ] we will define and study line bundles with minimal second chern classes on the order @xmath6 from construction [ mainconstruction ] . afterwards , we will introduce the hilbert scheme of @xmath6 , which parameterises left sided quotients of @xmath6 . we will compute its dimension and prove that it is smooth . it is here that we will also explore the bizarre covering of @xmath0 that it exhibits and study its ramification . in the last section , we will prove that the hilbert scheme is in fact a ruled surface over the moduli space . finally using the map to @xmath0 we will be able to compute the self intersection of the canonical divisor of the hilbert scheme which will allow us to compute the genus of the moduli space . we have already defined the notion of an order on a surface . we will now describe the aforementioned noncommutative cyclic covering trick which we will later use to construct the order whose moduli spaces we will be studying . this `` trick '' was introduced by chan in @xcite and the reader is advised to look there , in particular sections 2 and 3 for all the relevant details and proofs . the setup is as follows : let @xmath38 be a normal integral cohen - macaulay scheme and @xmath39 with @xmath40 for some minimal @xmath41 . further , assume that @xmath42 is a scheme . given any @xmath43 , we can form the @xmath44-bimodule @xmath45 such that @xmath46 and @xmath47 . suppose we have an effective cartier divisor @xmath48 and an @xmath49 such that there exists a non - zero map of @xmath44-bimodules @xmath50 satisfying the * overlap condition * ; namely that the two maps @xmath51 and @xmath52 are equal on @xmath53 . then @xmath54 is an order on @xmath5 with multiplication given by : @xmath55{ll } l_\sigma^{\otimes(i+j)},&i+j < e\\ l_\sigma^{\otimes(i+j)}\stackrel{1\otimes\phi\otimes 1}{\longrightarrow}l_\sigma^{\otimes\left ( i+j - e \right)},&i+j\geq e \end{array } \right.\ ] ] which is independent of any choice that needs to be made when applying the map @xmath56 due to the overlap condition . orders constructed in this manner are called * cyclic orders. we will almost always regard @xmath6 as an @xmath44-bimodule on @xmath38 , in which case we pay special consideration to the fact that it is not @xmath44-central . note that if we want to use this method to construct an order on a specific scheme @xmath5 we also need a way of finding a scheme @xmath38 and an automorphism @xmath39 such that @xmath57 . we can do so , using the classical cyclic covering construction . [ cons ] let @xmath5 be a normal integral scheme , let @xmath58 be an effective divisor and @xmath59 such that @xmath60 . then @xmath61 is a cyclic cover of @xmath5 . see chapter 1 , section 17 of @xcite for more details . note that if @xmath17 is the generator of @xmath62 then @xmath57 . to construct an order on @xmath5 using the noncommutative cyclic covering trick , let @xmath63 be another effective divisor on @xmath5 and let @xmath64 . find an @xmath43 and a non - zero morphism ( if one exists ) @xmath65 satisfying the overlap condition . then as described above , we can construct an order on @xmath5 which we will denote by @xmath66 . this order is ramified on @xmath67 , see @xcite theorem 3.6 for a proof of this . we suppress @xmath68 and @xmath48 from the notation . in this section we will use the noncommutative cyclic covering trick to construct a del pezzo order on @xmath0 ramified on a union of two conics . it is the moduli space and hilbert scheme of this order that we will be investigating for the remainder of this paper . [ mainconstruction ] let @xmath69 and @xmath70 be a double cover ramified on a smooth conic @xmath71 and let @xmath17 be the covering involution . it is well known that @xmath72 , @xmath73 and that @xmath74 . let @xmath75 be the inverse image of a general line in @xmath19 . it is a @xmath76-divisor and is ample . let @xmath77 be a second smooth conic , intersecting @xmath78 in @xmath36 distinct point , let @xmath79 which is a smooth @xmath80-divisor , let @xmath81 and fix once and for all a morphism @xmath82 . any such @xmath21 satisfies the overlap condition and so @xmath83 is a maximal order , in a division ring , on @xmath19 ramified on @xmath67 . see @xcite chapter 1 for full proofs . as mentioned previously , in @xcite the authors also consider a maximal order on @xmath0 ramified , in their case , on a smooth quartic . more importantly , the relation used in their construction was of the form @xmath84 . as we shall see this small difference makes their techniques for the study of @xmath10 , unusable in our case . to finish off the introduction we would like to explain in what sense our order @xmath6 is del pezzo . we begin with the definition of the canonical bimodule which is the analogue of the canonical sheaf on a scheme . [ delpezzodef ] let @xmath5 be a normal integral scheme and @xmath6 an order on @xmath5 . the canonical bimodule of @xmath6 is defined to be @xmath85 mimicking the commutative definition , we say that @xmath6 is del pezzo if @xmath86 is ample . for more details see @xcite section 3 . if @xmath5 is gorenstein , then @xmath87 . using the reduced trace map , we can identify @xmath88 as an @xmath6-subbimodule of @xmath89 and so @xmath90 can be identified as an @xmath6-subbimodule of @xmath91 . the next theorem allows us to determine , in the case where @xmath6 is constructed using construction [ cons ] , precisely what this subbimodule is . knowledge of @xmath92 will be very valuable to us in the future for various homological computations . [ canonical ] let @xmath5 be a normal integral gorenstein scheme . let @xmath93 be an order on @xmath5 as described in construction [ cons ] and let @xmath94 be the reduced pullback of @xmath78 to @xmath38 . then @xmath95 in @xmath96 . from lemma 17.1 of @xcite and the adjunction formula we know that @xmath97 thus , using the reduced trace map we have : @xmath98where @xmath99 for @xmath100 , and so : @xmath101thus : @xmath102 applying this theorem to our specific order @xmath6 we get : let @xmath83 be as in construction [ mainconstruction ] . then @xmath103 . in particular , @xmath6 is del pezzo . we simply apply theorem [ canonical ] and use the well known fact that @xmath104 from now on , unless explicitly stated otherwise , @xmath6 denotes @xmath105 the order constructed in construction [ mainconstruction ] . in this section we will study line bundles on @xmath6 . [ linebundle ] let @xmath5 be a normal integral scheme and @xmath106 an order in a division ring @xmath107 on @xmath5 . let @xmath28 be a sheaf of left @xmath106-modules . we say @xmath28 is a line bundle * on @xmath106 if @xmath28 is locally projective as a @xmath106-module and @xmath108 . the set ( not group ) of isomorphism classes of @xmath106-line bundles will be denoted by @xmath109 . the following proposition gives a very useful criterion for checking whether an @xmath6-module is in fact an @xmath6-line bundle . [ proplocfree ] @xmath110 if and only if @xmath28 is an @xmath6-module such that @xmath111 is a rank two locally free sheaf on @xmath18 . follows from the fact that , @xmath6 is locally of global dimension @xmath33 . see proposition 2.02 in @xcite for a full proof . [ atensorn ] suppose @xmath112 . then @xmath113 is an @xmath6-line bundle since it is clearly an @xmath6-module and is locally free of rank two over @xmath18 . in this section we study the possible chern classes of line bundles on @xmath6 . recall that whenever we speak of chern classes for any @xmath110 we imply that we are talking about the @xmath20-module @xmath111 . the first natural question to ask about any @xmath6-line bundle is what could be its first chern class . we answer this in the following proposition . as it turns out , the possibilities are fairly limited . let @xmath110 . then @xmath114 for some @xmath115 . conversely , given any such @xmath116 , @xmath117 with @xmath118 . first note that we have a chain of @xmath20-submodules @xmath119 which means @xmath120 is an exact sequence . let @xmath121 . the above then becomes : @xmath122 now @xmath123 is a locally free sheaf on @xmath48 of rank 2 , and so @xmath124 and hence @xmath30 must be line bundles on @xmath48 . consequently , @xmath125 and so @xmath126 . hence @xmath127 and the result follows . to see the converse , first note that by example [ atensorn ] we know that @xmath128 is indeed an @xmath6-line bundle . furthermore , @xmath129 . having classified all the possible first chern classes of @xmath6-line bundles , we move on to see what can be said about the second chern class . as we shall see , the second chern class has a strict lower bound analogous to bogomolov s inequality , which we now recall . let @xmath5 be a smooth projective surface and @xmath130 a torsion free coherent sheaf on @xmath5 with chern classes @xmath131 , @xmath37 and rank @xmath132 . fix an ample divisor @xmath75 on @xmath5 . the * gradient * of @xmath130 is defined to be @xmath133 @xmath130 is said to be @xmath22-semistable if for any subsheaf @xmath134 we have @xmath135 . bogomolov s inequality ( theorem 12.1.1 in @xcite ) states , that if @xmath130 is semistable then @xmath136 thus , if considering any class of semistable sheaves on @xmath5 with a fixed first chern class , the second chern class is bounded from below . in @xcite the authors were able to show to that for their cyclic order @xmath6 , any @xmath6-line bundle was automatically @xmath22-semistable as sheaf on @xmath18 and could thus bound the second chern class using bogomolov s inequality . we modify their proof and achieve a slightly weaker result for our order . [ mybogomolov ] let @xmath110 and let @xmath137 be an @xmath20-subsheaf . then @xmath138 note that @xmath28 is locally free of rank @xmath33 over @xmath18 . thus the result is clear if rank @xmath139= 2 and so we assume rank @xmath140 . observe that @xmath141 . now @xmath142 and so @xmath143 . it is easy to see that this inequality is tight . for example the @xmath6-line bundle @xmath6 has gradient @xmath144 and an @xmath20-submodule @xmath145 with @xmath146 . thus @xmath6-line bundles are in general not @xmath22-semistable and so we can not apply bogomolov s inequality to give a lower bound for the second chern class . furthermore , as mentioned earlier , this implies we can not use the map ( ) from page in order to study the moduli space of line bundles , simply because this map does not exist for us . luckily , due to a deep theorem by langer in @xcite the result of proposition [ mybogomolov ] is good enough to achieve a lower bound on @xmath37 . [ propdelta ] let @xmath110 with chern classes @xmath131 and @xmath37 . then @xmath147 follows immediately from theorem 5.1 of @xcite with @xmath148 and proposition [ mybogomolov ] . the above theorem can also be proven using rather elementary techniques , without needing the generality of @xcite . in section [ chernproof ] we will prove that @xmath149 is in fact a sufficient condition to guarantee that there exists am @xmath6-line bundle with these chern classes . having shown that for a fixed first chern class , the second chern class of any @xmath6-line bundle is bounded from below , we begin studying those line bundles , with minimal second second chern class . in particular , we would like to determine what the moduli space of such bundles is . the existence of a projective coarse moduli scheme parametrising @xmath6-line bundles with minimal @xmath37 follows easily from theorem 2.4 in @xcite and proposition [ propdelta ] . in fact this moduli space is smooth because @xmath6 is del pezzo . for a full explanation and proof , see @xcite chapter 2 . [ remarkonc ] since the functor @xmath150 is a category autoequivalence of @xmath6-mod , it induces an automorphism of the moduli space of @xmath6 . note that for any @xmath110 , @xmath151 . since by the previous proposition , @xmath152 for some @xmath153 we me may assume that @xmath154 or @xmath155 . before we begin our analysis of @xmath6-line bundles with minimal @xmath37 , we need to examine the inequality ( [ ahs ] ) we met in proposition [ mybogomolov ] a little further . let @xmath5 be a surface and @xmath156 a vector bundle on @xmath5 . we say @xmath156 is * almost semistable * if for any subbundle @xmath157 , @xmath158 . [ ahsprop ] let @xmath5 be a surface and @xmath156 a vector bundle on @xmath5 . 1 . @xmath156 is almost semistable if and only if @xmath159 is almost semistable for all @xmath160 . 2 . if @xmath156 is rank @xmath33 and almost semistable , then so is @xmath161 . @xmath162 1 . suppose @xmath156 is almost semistable and @xmath163 . then @xmath164 and so @xmath165 thus @xmath166 and so @xmath167 . to see the converse simply let @xmath168 . 2 . follows from ( 1 ) and the fact that @xmath169 . as we have seen in proposition [ mybogomolov ] , @xmath6-line bundles are almost semistable . we will use the above proposition later on for proving various properties regarding line bundles on @xmath6 . as mentioned in remark [ remarkonc ] the problem of studying the moduli space of @xmath6-line bundles with minimal @xmath37 naturally breaks up into two parts @xmath24 or @xmath170 . in this subsection we examine the former case . by proposition [ propdelta ] the minimal @xmath25 and this corresponds to @xmath171 , the smallest value possible . it is easy to see that the moduli space of @xmath6-line bundles with these chern classes is nt empty for clearly @xmath6 itself , regarded as a left @xmath6-module , has the desired chern classes . as it turns out , this is in fact the only such @xmath6-line bundle . [ thm0 ] let @xmath110 with @xmath24 and @xmath25 . then @xmath172 . in particular , the coarse moduli space of @xmath6-line bundles with these chern classes is a point . by the riemann - roch theorem @xmath173 . on the other hand @xmath174 and @xmath175 . as we saw in proposition [ mybogomolov ] , @xmath28 is almost semistable , and so by proposition [ ahsprop ] , @xmath176 is also almost semistable and so @xmath177 . thus @xmath178 and so @xmath179 which gives an injection of @xmath6-modules since their first chern classes equal , the map must be an isomorphism . finally @xmath181 where the first equality follows from proposition 2.6 of @xcite which asserts that there is a natural isomorphism of functors @xmath182 for any @xmath112 . see chapter 3 exercise 5.6 of @xcite for the cohomology of @xmath183 . thus the tangent space at the point corresponding to the @xmath6-line bundle @xmath6 is @xmath184-dimensional and so the moduli space is just a point . we now study the second case mentioned in remark [ remarkonc ] : the case where @xmath31 . by proposition [ propdelta ] the minimal @xmath27 which corresponds to @xmath185 which is its second smallest value for clearly @xmath186 must be even . note that @xmath187 is an @xmath6-line bundle by example [ atensorn ] and has the desired chern classes . thus the moduli space of such @xmath6-line bundles is not empty . from now on @xmath10 will denote the moduli space of @xmath6-line bundles with @xmath31 and @xmath27 . we first establish all the possible @xmath20-module structures that such @xmath6-line bundles can have . [ ystruc ] let @xmath188 with @xmath31 and @xmath27 . then either @xmath189 as and @xmath20-module or @xmath190 as @xmath6-modules where @xmath191 is either a @xmath192 or a @xmath193-divisor . the beginning of this proof is very similar to the proof of theorem [ thm0 ] so we skip some details which we have already explained there . let @xmath194 . then @xmath195 and @xmath196 . thus by the riemann - roch theorem @xmath197 . however , @xmath198 whilst @xmath199 is almost semistable with gradient @xmath200 and so @xmath201 and so @xmath202 . thus we know @xmath203 . now if there exists a bigger @xmath20-line bundle ( ordered by inclusion ) which embeds into @xmath28 then @xmath204 embeds into @xmath28 where @xmath191 is either a @xmath192 or a @xmath193-divisor . this extends to an embedding @xmath205 of @xmath6-line bundles and so comparison of the first chern classes guarantees that @xmath206 . suppose on the other hand that @xmath207 is the biggest line bundle which embeds into @xmath28 . let the quotient be @xmath30 and note that it is torsion free . by proposition 5 ( ii ) in @xcite @xmath208 for some @xmath209 and @xmath210 being the ideal sheaf of some @xmath184-dimensional subscheme . computing chern classes we see that @xmath211 and @xmath212 . finally , @xmath213 an so we see that as an @xmath20-module @xmath214 . this result is very different to what chan and kulkarni encountered in @xcite . in their example if an @xmath6-module was split as an @xmath20-module then they prove that the module must be of the form @xmath215 for some @xmath216 . furthermore , any rank two vector bundle on @xmath18 could be given at most two @xmath6-module structures . in our case , as the above theorem at least suggests , the @xmath20-vector bundle @xmath217 can be given an infinite number of non - isomorphic @xmath6-module structures . in the following proposition , we prove that this is indeed the case . [ propdim ] the tangent space to * pica at the point corresponding to @xmath218 and @xmath187 has dimension @xmath219 . the dimension of the tangent space is given by : + @xmath220 the other case is identical . thus at least one connected component of this moduli space is a smooth curve with all , except at most @xmath33 points , corresponding to @xmath6-modules with the underlying @xmath20-module structure being @xmath217 . we finish off the section with an algebraic description of the @xmath6-line bundles . [ propmina ] let @xmath188 with @xmath31 and @xmath27 . then @xmath221 further , if @xmath222 then @xmath223 is injective . we consider all the possibilities from theorem [ ystruc ] . if @xmath224 then @xmath225 if , on the other hand , @xmath226 as an @xmath20-module then : @xmath227 since @xmath28 and @xmath6 are torsion free , any non zero map @xmath228 must be injective . to understand better how @xmath28 sits inside @xmath6 we need to understand the all the possible cokernels . we do so , in the next theorem . [ quotientscomputation ] let @xmath188 with @xmath31 and @xmath27 . then for any @xmath229 there exists an exact sequence of @xmath6-modules @xmath230 where : 1 . if @xmath231 ( respectively @xmath232 ) then @xmath233 where @xmath191 is a @xmath192 ( respectively @xmath193 ) divisor ; 2 . if @xmath234 then @xmath235 as an @xmath20-module , where @xmath236 is a @xmath76-divisor . from the previous proposition , we know @xmath237 is injective . let us compute the cokernel . 1 . we prove only the case where @xmath231 because the other is similar . note that @xmath238 and so all @xmath6-module morphisms arise from an @xmath20-module morphism @xmath239 via @xmath240 . since any non zero morphism @xmath239 gives rise to the following exact sequence @xmath241 for some @xmath192-divisor @xmath191 and because @xmath6 is flat over @xmath18 , the result follows . 2 . note that with respect to the @xmath20-module decomposition @xmath242 we have @xmath243{ll } h^0(y,{\mathcal{o}}_y(-1,-1)^)&{\rm end}_y({\mathcal{o}}_y(-1,-1))\\ h^0(y,{\mathcal{o}}_y(-1,-1)^)&{\rm end}_y({\mathcal{o}}_y(-1,-1 ) ) \end{array } \right ) . \end{aligned}\ ] ] thus any @xmath20-module morphism @xmath237 is given by @xmath244{ll } \varphi_1 & \lambda_1\\ \varphi_2 & \lambda_2 \end{array } \right)$ ] where @xmath245 and @xmath246 which acts as right multiplication on the row vector @xmath217 . for this to be in fact an @xmath6-module morphism further conditions on @xmath5 need to be imposed . in particular @xmath223 needs to be injective and so @xmath247 are not both zero . + we claim that @xmath248 and that we have the following exact sequence @xmath249 with @xmath250 given by right multiplication by @xmath251{cl } \left(\begin{array}[]{c } \lambda_1+\lambda_2\\-(\varphi_1+\varphi_2 ) \end{array } \right)&\text{if $ \lambda_1+\lambda_2\neq 0$}\\ \left ( \begin{array}[]{c } \lambda_1\\-\varphi_1 \end{array } \right)&\text{if $ \lambda_1+\lambda_2=0.$ } \end{array } \right.\ ] ] since @xmath228 must be injective , @xmath252 and so , @xmath30 is isomorphic , as an @xmath20-module , to @xmath253 for some @xmath76-divisor @xmath236 . the proof of this claim is just a routine local computation and is done in lemma 2.4.5 in @xcite . the above theorem suggests that we should study quotients of @xmath6 . in particular , we should try to better understand the component(s ) of the hilbert scheme of @xmath6 containing the @xmath6-modules whose underlying @xmath20-module structure is @xmath253 where @xmath236 is a @xmath76-divisor . we do this in the following section . in this section we will study the hilbert scheme of @xmath6 the moduli space of left sided quotients of @xmath6 with a fixed set of chern classes . this is a closed subscheme of the classical quot scheme of @xmath6 , which is projective provided we fix a hilbert polynomial . see chapter 3 in @xcite for all the details . mimicking the commutative case , one should think of a quotient of @xmath6 , which is supported on a curve on @xmath18 , as a noncommutative curve lying on @xmath6 . as mentioned at the end of the last section , we are primarily interested in those quotients of @xmath6 which are supported on a @xmath76-divisor on @xmath18 . recall that in theorem [ quotientscomputation ] we saw a link between the moduli space of @xmath6-line bundles with with @xmath31 and @xmath27 and quotients of @xmath6 , or noncommutative curves on @xmath6 , with @xmath32 and @xmath27 . [ propker ] let @xmath255 be a scheme . let @xmath130 be a flat family of quotients of @xmath6 on @xmath255 with chern classes @xmath32 and @xmath27 . let @xmath256 . then @xmath257 is a flat family of @xmath6-line bundles on @xmath255 with chern classes @xmath31 and @xmath27 . @xmath257 is flat over @xmath255 because @xmath258 and @xmath130 are . restricting to the fibre above any @xmath259 we get @xmath260 of @xmath6-modules which is exact because @xmath130 is flat over @xmath255 and so @xmath261 . since @xmath262 we see that @xmath263 and @xmath264 . @xmath265 is torsion free and so @xmath266 because it is reflexive and hence locally free over @xmath18 . by proposition [ propdelta ] we have @xmath267 and so @xmath268 . having established a relationship between flat families of @xmath6-line bundles and flat families of quotients of @xmath6 , we now use theorem [ ystruc ] to classify all the possible @xmath20-module structures that quotients of @xmath6 may possess . as we shall see some ( and , as we shall later see , most ) must all also be quotients of @xmath20 . [ corquot ] let @xmath30 be a quotient of @xmath6 with @xmath32 and @xmath27 . then either : @xmath269 ( as an @xmath6-module ) where @xmath191 is either a @xmath192 or @xmath193-divisor ; or * @xmath235 ( as an @xmath20-module ) for some @xmath17-invariant @xmath76-divisor @xmath270 . the above proposition asserts that the kernel of @xmath271 is an @xmath6-line bundle with @xmath31 and @xmath27 . we have already classified all such line bundles and their respective cokernels in proposition [ ystruc ] and theorem [ quotientscomputation ] . the fact that @xmath236 must be @xmath17 invariant follows from the fact that in order to be an @xmath6-module there must be a non - zero map @xmath272 which is only possible if @xmath273 . [ smoothsupport ] let @xmath30 be a quotient of @xmath6 with @xmath32 and @xmath27 . if the support @xmath30 is smooth ( i.e. its the support is @xmath274 ) then @xmath30 is also quotient of @xmath20 . obvious from the previous corollary because the support of @xmath275 is not smooth . from now on @xmath254 will denote the hilbert scheme of @xmath6 corresponding to quotients of @xmath6 with @xmath32 and @xmath27 . we now proceed to study its properties . [ propdim ] the dimension of @xmath254 at the point corresponding to @xmath276 , where @xmath191 is a @xmath192 or @xmath193-divisor is , @xmath33 . we have @xmath277 let @xmath278 . the dimension of the tangent space is given by : @xmath279 unfortunately , we were unable to compute the dimension of the tangent space at any other points as directly as in the above proposition . we thus proceed by first showing that @xmath11 is smooth and later , after a considerable amount of work , that it is connected . this will of course prove that @xmath11 is a smooth projective surface . @xmath254 is smooth . let @xmath30 be a quotient of @xmath6 corresponding to some point @xmath280 . let @xmath28 the kernel of @xmath271 . we have an exact sequence @xmath281 where by proposition [ propker ] @xmath110 . obstruction to smoothness at @xmath282 is given by @xmath283 which we now compute . from corollary [ corquot ] there are only three cases to consider : * @xmath231 and @xmath233 where @xmath191 is a @xmath192 divisor . let @xmath284 which is a @xmath193-divisor . @xmath285 * @xmath232 and @xmath233 where @xmath191 is a @xmath193 divisor . the proof is the same as in the case above . * @xmath234 as an @xmath20-module and @xmath235 as an @xmath20-module for some @xmath76-divisor @xmath236 . using serre duality , we have : @xmath286 using the local - global spectral sequence we have @xmath287 @xmath288 since @xmath253 is a torsion sheaf . furthermore , ( ) is a locally projective @xmath6-module resolution of @xmath253 and so we get @xmath289 finally , since @xmath290 and @xmath291 we see that @xmath292 and so the result follows . thus , so far we know that at least one connected component of @xmath254 is a smooth projective surface . as mentioned earlier , in the next section we will see that in fact @xmath254 is connected , which will prove that this must be its only component . corollary [ corquot ] says that some quotients of @xmath6 are in fact also quotients of @xmath20 . in particular , they are isomorphic to @xmath253 where @xmath236 is a @xmath17-invariant @xmath76-divisor . furthermore , the support of @xmath275 is @xmath293 which is also a @xmath17-invariant @xmath76-divisor . since the tangent space at the points corresponding to @xmath275 is two , whilst dim @xmath294 it must be the case that every connected component of @xmath11 has a dense subset whose points correspond to quotients of @xmath6 that are also quotients of @xmath20 . we may thus expect that there is at least a rational map from the hilbert scheme of @xmath6 to the hilbert scheme of @xmath18 . we now explore this further . note first of all , that all @xmath17-invariant @xmath76-divisors are equal to @xmath295 where @xmath296 is a line on @xmath19 . furthermore , lines on @xmath19 are parameterised by @xmath13 . thus we can view @xmath297 as the parameter space of @xmath17-invariant @xmath76-divisors . [ thm : map]let @xmath130 be the universal family of quotients of @xmath6 on + @xmath298 . there exists a regular map @xmath299 from the surjective morphism @xmath300 we get a morphism @xmath301 . define @xmath302 and note that @xmath303 since at any on point on @xmath11 corresponding to quotient of the form @xmath275 the map @xmath304 has a non - trivial cokernel . let @xmath305 be the open subset @xmath306 where @xmath307 is the natural projection map . thus @xmath308 is a flat family of @xmath20-quotients over @xmath309 . since @xmath310 precisely for those @xmath311 which correspond to quotients of @xmath6 which are also quotients of @xmath20 , from the discussion preceding this theorem , we know that @xmath309 is dense in @xmath11 . thus we get a rational map @xmath312 . from corollary [ corquot ] we know that quotients of @xmath6 which are also quotients of @xmath20 are isomorphic as @xmath20-modules to @xmath253 where @xmath236 is a @xmath17-invariant @xmath76-divisor . thus @xmath313 . we will see in lemma [ lemmafinitepsi ] that @xmath314 is in fact finite to one and so each connected component of @xmath11 has at most dimension @xmath33 . thus from chapter 2 section 3 theorem 3 of @xcite @xmath314 is not regular at at most only a finite number of points . we claim that @xmath314 is in fact regular everywhere . to see this , let @xmath315 be the resolution of indeterminacy of @xmath314 . let @xmath316 and let @xmath106 be a smooth curve in @xmath11 such that @xmath317 ; i.e. the only point of @xmath106 not corresponding to a quotient of @xmath20 is @xmath282 . denote by @xmath318 its strict transform . since @xmath106 is smooth @xmath319 and so we may compare families over the two curves . we have a map @xmath320 and we denote the corresponding flat family over @xmath318 of @xmath20-quotients by @xmath321 . let @xmath322 and note that @xmath255 is a family of @xmath20-quotients on @xmath106 but we do nt know that it is flat over @xmath106 and so we proceed rather subtly . note that @xmath323 on @xmath324 . proposition 9.8 of chapter 3 in @xcite implies that once we have a flat family over @xmath325 then there is only one way to complete it to a flat family over @xmath106 and that is by taking the scheme theoretic closure in @xmath326 . however , @xmath255 is closed and so @xmath327 . since @xmath321 is a family of quotients of @xmath20 with @xmath32 and @xmath27 it follows that @xmath328 and so @xmath329 . thus , regardless of the choice of curve @xmath106 , the image of the point @xmath282 does nt change . hence @xmath330 and so @xmath314 is regular . in summary , the map @xmath314 does the following : every closed point on @xmath11 corresponds to some quotient of @xmath6 . there are two possibilities : either * it is also a quotient of @xmath20 , in which case it is isomorphic , and an @xmath20-module , to @xmath253 where @xmath236 is a @xmath17-invariant @xmath76-divisor , or * it is not a quotient of @xmath20 , then it is isomorphic , as an @xmath6-module , to @xmath275 where @xmath191 is either a @xmath193 or @xmath192-divisor . the crucial point is that the support of @xmath275 is also a @xmath17-invariant @xmath76-divisor . thus to every closed point on @xmath11 one can associate a @xmath17-invariant @xmath76-divisor . since @xmath17-invariant @xmath76-divisors are parameterised by @xmath297 , we get a natural set - theoretic map from ( closed points of @xmath11 ) @xmath331 ( closed points of @xmath297 ) . the above theorem proves that this map is in fact morphism of schemes . we want to study the map @xmath314 , in particular we want to understand its ramification for then we will be able to later compute @xmath333 . this amounts to computing the number of quotients of @xmath6 which have support a @xmath17-invariant @xmath76-divisor and @xmath27 . corollary [ corquot ] implies that this question will be answered provided we can understand the number of @xmath6-module structures that @xmath253 can be given , where @xmath236 is a @xmath17-invariant @xmath76-divisor . to give a coherent sheaf @xmath334 on @xmath18 an @xmath6-module structure amounts to giving a left @xmath20-module morphism @xmath335 satisfying the necessary associativity condition . two such morphisms @xmath336 give rise to isomorphic @xmath6-modules provided there exists @xmath337 such that@xmath338 ^ -\varphi\ar[d]_{{\rm id}\otimes\psi}\ar@<1.3ex>@{}[d]_(0.6){{\begin{rotate}{90 } { $ \sim$ } \end{rotate}}}&{\mathcal{g}}\ar[d]^\psi_(0.6){{\begin{rotate}{90 } { $ \sim$ } \end{rotate}}}\\ { a\otimes_y}{\mathcal{g}}\ar[r]_-{\varphi'}&{\mathcal{g}}}\ ] ] commutes . in general it may be rather difficult to determine whether such a @xmath339 exists , and consequently , whether two seemingly different @xmath6-module structures are actually isomorphic . the problem becomes increasingly difficult as the size of @xmath340 increases . luckily , in our case , this issue is easily manageable . we can illustrate of the above phenomenon with two ( related ) examples . recall from theorem [ ystruc ] that an @xmath6-line bundle had two possible @xmath20-module structures : either it was @xmath341 or @xmath342 . the former , as we later saw , had infinitely many non - isomorphic @xmath6-module structures whilst the latter , only had one . the fact that that @xmath343 has only one @xmath6-module structure is only clear , when it is written as @xmath344 which clearly only has one @xmath6-module structure . hence , if one does not realise that @xmath345 then determining the fact that all possible @xmath6-module structures are isomorphic may be very hard indeed . a similar phenomenon occurs for quotients of @xmath6 . let @xmath346 and forget the natural @xmath6-module structure , and ask : how many ( non - isomorphic ) @xmath6-module structures can @xmath30 have ? if one does not realise that at least as an @xmath20-module @xmath347 it will be difficult to prove that all the potentially different @xmath6-module structures are in fact isomorphic . furthermore , as we are about to see , for most @xmath17-invariant @xmath76-divisors @xmath236 , @xmath253 will have several , but finitely many , @xmath6-module structures . the reason for the difference in the number of @xmath6-module structures is partly due to the size of the endomorphism ring of the modules . in the first example , @xmath348 whilst @xmath349 . a larger automorphism group means it is `` easier '' for two @xmath6-modules structures to be isomorphic . we now study the number of @xmath6-module structures that @xmath253 may possess . for any @xmath350 we will denote by @xmath351 the corresponding line in @xmath0 and we let @xmath352 which is a @xmath17-invariant @xmath76-divisor . as we saw , for every @xmath353 , giving @xmath354 an @xmath6-module structure amounts to giving a left @xmath20-module map @xmath355 satisfying the necessary associativity condition . in order to better understand this we first introduce some notation : we let @xmath356 and @xmath357 then , since @xmath358 and because @xmath359 this condition is equivalent to giving a map @xmath360 such that @xmath361 is the identity . note that given such a map @xmath362 , the map @xmath363 gives a different , non isomorphic @xmath6-module structure to @xmath354 . this observation gives us the following : [ propinv ] there exist an involution @xmath364 sending an @xmath6-module structure given by @xmath362 to the one given by @xmath363 . the fixed points are those which corresponds to quotients of @xmath6 that are not quotients of @xmath20 . if @xmath365 sends the @xmath6-module structure given by @xmath362 to the one given by @xmath363 then if the module is also a quotient of @xmath20 then as we just saw , these two @xmath6-module structures are not isomorphic . if the module is not a quotient of @xmath20 then by corollary [ corquot ] it must be isomorphic to @xmath275 which can only be given one @xmath6-module structure . [ corfactor ] the map @xmath332 factors through @xmath366 . i.e. we have the following commutative diagram@xmath367\ar[dr]\\ & { { \bf hilb}\;a}/{\langle \tau \rangle}\ar[dl]\\ { ( { \mathbb{p}}^2)^\vee}}\ ] ] clear from the above proposition and theorem [ thm : map ] . we can view @xmath362 as an element of @xmath368 and , up to multiplication by @xmath369 , the associativity condition then simply says that we need @xmath370 , where each such @xmath362 gives rise to two @xmath6-module structures . since @xmath371 is a finite number of points we have proved the following lemma , which also finishes off the proof of the theorem [ thm : map ] : [ lemmafinitepsi ] the map @xmath314 is finite . this way of thinking , allows us to view the problem of giving @xmath354 an @xmath6-module structure geometrically . as we are about to see , the number of @xmath6-module structures that @xmath354 can be given depends primarily how many points @xmath351 intersects with @xmath78 and @xmath372 . note also that the dual of a smooth conic in @xmath0 is another smooth conic in @xmath373 . we denote the duals of @xmath78 and @xmath372 by @xmath374 and @xmath375 respectively . the picture one should keep in mind is this : ( 0,0)(12,12 ) ( 2.5,2.5)(0.8,1.5)(2.5,2.5)(1.5,0.8)(0,0)(5,5 ) ( 2,3.6)@xmath374 ( 1,2.5)@xmath375 ( 3.8,4.6)@xmath373 ( 3.7,1 ) ( 7,0)(12,5 ) ( 9.5,2.5)(0.8,1.5)(9.5,2.5)(1.5,0.8)(9,3.6)@xmath78 ( 8,2.5)@xmath372 ( 10.4,4.6)@xmath69 ( 11.5,3.3)(8.3,0.6 ) ( 5.1,2.5)(6.9,2.5 ) ( 7.0,7.0)(12.0,12.0 ) ( 9.5,6.9)(9.5,5.1 ) ( 9.5,6)@xmath376 ( 9.8,9.3 ) ( 0,0)(0.5,2.2 ) ( 8.5,0.4)@xmath351 ( 3.92,0.78)@xmath282 ( 10.9,9.8)(10.9,10.8)(10.1,11)(9.6,9.7)(9.6,8.4)(10.5,8.5 ) ( 8.2,7.4)@xmath377 ( 11,8.5)@xmath48 ( 10.7,11.6)@xmath378 ( 10.9,9.83)(10.9,10.8)(9.6,9.79)(9.6,8.43 ) ( 9.6,1.7)(10.87,2.8 ) ( 9.06,1.25 ) ( 10.27,2.28 ) we mark where @xmath351 intersects @xmath78 with a `` @xmath379(0,0.1)\endpspicture\;}$ ] '' and where @xmath351 intersects @xmath372 with a `` @xmath380 '' . the problem of giving @xmath354 an @xmath6-module structure breaks up into two cases : 1 . @xmath351 is not tangential to @xmath78 . in this case we get @xmath381 is a @xmath382 cover ramified at two points an hence @xmath383 , in particular it is smooth . we analyse this case first , in section [ sec1 ] . 2 . @xmath351 is tangential to @xmath78 . in this case @xmath381 is ramified at only one point and hence @xmath377 is the union of two @xmath274 s , in particular it is singular . we analyse this case second , in section [ sec2 ] . from now on , in any subsequent diagrams , any vertical conic on @xmath19 will be @xmath78 , any horizontal one will be @xmath372 and similarly with @xmath374 and @xmath375 on @xmath297 and hence will not longer be labelled . as mentioned earlier , we begin by studying the first of the two cases mentioned above . recall that @xmath377 is smooth , in fact @xmath383 , precisely when @xmath351 is not tangential to @xmath78 or , equivalently , when @xmath282 does nt lie on @xmath374 . in this case , from corollary [ smoothsupport ] we know that all quotients of @xmath6 with this support have their underlying @xmath20-module structure isomorphic to @xmath253 . this happens when @xmath351 is not a tangent to @xmath78 which is equivalent to @xmath282 not lying on @xmath374 . in this case , since @xmath384 we have @xmath385 and so to give @xmath354 an @xmath6-module structure corresponds to choosing two points @xmath386 such that @xmath387 . as mentioned earlier , any such choice gives rise to precisely two @xmath6-module structures . there are several cases that need to be considered depending on precisely where @xmath282 lies . 1 . @xmath282 does not lie on either @xmath374 or @xmath375 nor on any of the four bitangents to them and so we see that this is the generic case . in summary we have : + c\|c\|c position of @xmath353 & position of @xmath388 & @xmath381 + + ( 0,0)(5,5.4 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 2,0.3)(4.7,3.0 ) ( 2,4.7)(4.7,2.0 ) ( 0.3,2)(3.0,4.7 ) ( 0.3,3)(3,0.3 ) ( 3.7,1 ) ( 3.92,0.78)@xmath282 + & + ( 0,0)(5,5 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 4.5,3.3)(1.3,0.6 ) ( 2.06,1.25 ) ( 3.27,2.28 ) ( 1.7,0.4)@xmath351 ( 2.6,1.7)(3.87,2.8 ) + & + ( 0,0)(5,5 ) ( 2.5,3.5)(1.5,0.8 ) ( 0.5,1)(4.5,1 ) ( 2.5,2.3)(2.5,1.2 ) ( 1,1)(4,1 ) ( 2,1)(3,1)(2,2.77)(3,2.77)(2,4.22)(3,4.22 ) + thus there are @xmath36 choices for @xmath389 which results in @xmath390 different @xmath6-module structures on @xmath354 . in order for us to later study the ramification of @xmath314 we also include the column which shows which branch corresponds to which module structure . + p3.8cm\|p3.8cm\|p3.8 cm + 1 @xmath389 + & + 1 branches above @xmath282 corresponding to @xmath389 + & + 1 + no . of @xmath6-quotients with support @xmath377 + + + ( 0,0)(3.5,3.3 ) ( 2,1.5)(1.5,0.8 ) ( 1.5,2.22)(2.5,2.22 ) + & + ( 0,0)(3.5,2.6 ) ( 0.5,1)(3.5,1 ) ( 0.5,2)(3.5,2 ) ( 4,2)@xmath391 ( 4,1)@xmath392 + & + + ( 0,0)(3.5,2.6 ) ( 2,1.5)(1.5,0.8 ) ( 1.5,0.77)(2.5,0.77 ) + & + ( 0,0)(3.5,2.6 ) ( 0.5,1)(3.5,1 ) ( 0.5,2)(3.5,2 ) ( 4,2)@xmath393 ( 4,1)@xmath394 + & + + ( 0,0)(3.5,2.6 ) + ( 2,1.5)(1.5,0.8 ) ( 1.5,2.22)(2.5,0.77 ) + & + ( 0,0)(3.5,2.6 ) ( 0.5,1)(3.5,1 ) ( 0.5,2)(3.5,2 ) ( 4,2)@xmath395 ( 4,1)@xmath396 + + + ( 0,0)(3.5,2.6 ) + ( 2,1.5)(1.5,0.8 ) ( 1.5,0.77)(2.5,2.22 ) + & + ( 0,0)(3.5,2.6 ) ( 0.5,1)(3.5,1 ) ( 0.5,2)(3.5,2 ) ( 4,2)@xmath397 ( 4,1)@xmath398 + we may thus conclude that @xmath314 is an @xmath35 cover of @xmath297 . the other cases are used to study the ramification of this map . 2 . @xmath282 lies on @xmath375 but not on @xmath374 nor on any of the four bitangents . + c\|c\|c position of @xmath353 & position of @xmath388 & @xmath381 + + ( 0,0)(5,5.4 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 2,0.3)(4.7,3.0 ) ( 2,4.7)(4.7,2.0 ) ( 0.3,2)(3.0,4.7 ) ( 0.3,3)(3,0.3 ) ( 2.5,1.7 ) ( 2.5,2.1)@xmath282 + & + ( 0,0)(5,5 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 4.5,2.63)(1.3,0.2 ) ( 2.4,1.02 ) ( 3.09,1.55 ) ( 1.5,0.8)@xmath351 ( 3.62,2 . ) + & + ( 0,0)(5,5 ) ( 2.5,3.5)(1.5,0.8 ) ( 0.5,1)(4.5,1 ) ( 2.5,2.3)(2.5,1.2 ) ( 1,1)(4,1 ) ( 2.5,1)(2.5,2.64)(2.5,2.8)(2.5,4.2)(2.5,4.36 ) + there are now only @xmath399 choices for @xmath389 as we see in the table below . + p3.8cm\|p3.8cm\|p3.8 cm + 1 @xmath389 + & + 1 branches above @xmath282 corresponding to @xmath389 + & + 1 no . of @xmath6-quotients with support @xmath377 + + + ( 0,0)(3.5,3.3 ) ( 2,1.5)(1.5,0.8 ) ( 2,2.33)(2,2.18 ) + & + ( 0,0)(3.5,2.6 ) ( 0.5,1)(3.5,1 ) ( 0.5,2)(3.5,2 ) ( 4,2)@xmath391 ( 4,1)@xmath392 + + + ( 0,0)(3.5,2.6 ) ( 2,1.5)(1.5,0.8 ) ( 2,0.8)(2,0.67 ) + & + ( 0,0)(3.5,2.6 ) ( 0.5,1)(3.5,1 ) ( 0.5,2)(3.5,2 ) ( 4,2)@xmath393 ( 4,1)@xmath394 + & 6 + + ( 0,0)(3.5,2.6 ) ( 0,-1.4 ) ( 2,1.5)(1.5,0.8 ) ( 2,0.73)(2,2.28 ) + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,2)(3.5,2 ) ( 1,1.5)0.3590270 ( 4,2)@xmath395 ( 4,1)@xmath397 + + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,2)(3.5,2 ) ( 1,1.5)0.3590270 ( 4,2)@xmath396 ( 4,1)@xmath398 3 . @xmath282 lies on one exactly one of the four bitangents but not where they meet the conics . + c\|c\|c position of @xmath353 & position of @xmath388 & @xmath381 + + ( 0,0)(5,5.4 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 2,0.3)(4.7,3.0 ) ( 2,4.7)(4.7,2.0 ) ( 0.3,2)(3.0,4.7 ) ( 0.3,3)(3,0.3 ) ( 3.3,1.6 ) ( 3.6,1.3)@xmath282 + & + ( 0,0)(5,5 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 4,4.45)(1.3,0.2 ) ( 2,1.3 ) ( 3.2,3.2 ) ( 1.3,0.8)@xmath351 ( 2.26,1.75)(3.2,3.2 ) + & + ( 0,0)(5,5 ) ( 2.5,3.5)(1.5,0.8 ) ( 0.5,1)(4.5,1 ) ( 2.5,2.3)(2.5,1.2 ) ( 1,1)(4,1 ) ( 2.5,1)(2.5,2.73)(2.5,4.28)(1,1)(1,3.41)(1,3.59 ) + there are now only @xmath33 choices for @xmath389 as we explain in the table below . + p3.8cm\|p3.8cm\|p3.8 cm + 1 @xmath389 + & + 1 branches above @xmath282 corresponding to @xmath389 + & + 1 no . of @xmath6-quotients with support @xmath377 + + + ( 0,0)(3.5,3.3 ) ( 0,-1.4 ) ( 2,1.5)(1.5,0.8 ) ( 2,2.28)(0.54,1.5 ) + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,2)(3.5,2)(1,1.5)0.3590270 ( 4,2)@xmath391 ( 4,1)@xmath397 + + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,2)(3.5,2)(1,1.5)0.3590270 ( 4,2)@xmath392 ( 4,1)@xmath398 + & + 4 + + ( 0,0)(3.5,2.6 ) ( 0,-1.4 ) ( 2,1.5)(1.5,0.8 ) ( 2,0.74)(0.54,1.5 ) + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,2)(3.5,2 ) ( 1,1.5)0.3590270 ( 4,2)@xmath393 ( 4,1)@xmath395 + + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,2)(3.5,2 ) ( 1,1.5)0.3590270 ( 4,2)@xmath394 ( 4,1)@xmath396 4 . @xmath282 is chosen to be the point of intersection of two bitangents to @xmath374 and @xmath375 . + c\|c\|c position of @xmath353 & position of @xmath388 & @xmath381 + + ( 0,0)(5,5.4 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 2,0.3)(4.7,3.0 ) ( 2,4.7)(4.7,2.0 ) ( 0.3,2)(3.0,4.7 ) ( 0.3,3)(3,0.3 ) ( 4.2,2.5 ) ( 4.2,2)@xmath282 + & + ( 0,0)(5,5 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 4,4)(1,1 ) ( 1.8,1.8 ) ( 3.2,3.2 ) ( 1.3,0.8)@xmath351 ( 1.8,1.8)(3.2,3.2 ) + & + ( 0,0)(5,5 ) ( 2.5,3.5)(1.5,0.8 ) ( 0.5,1)(4.5,1 ) ( 2.5,2.3)(2.5,1.2 ) ( 1,1)(4,1 ) ( 4,1)(4,3.41)(4,3.59)(1,1)(1,3.41)(1,3.59 ) + there is now only @xmath219 choice for @xmath389 as we explain in the table below . + p3.8cm\|p3.8cm\|p3.8 cm + 1 @xmath389 + & + 1 branches above @xmath282 corresponding to @xmath389 + & + 1 no . of @xmath6-quotients with support @xmath377 + + + ( 0,0)(3.5,3.3 ) ( 0,-1.4)(2,1.5)(1.5,0.8 ) ( 3.46,1.5)(0.53,1.5 ) + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,1.5)(3.5,1.5 ) ( 1.15,2)(3.5,2 ) ( 1.15,0.5)(3.5,0.5 ) ( 1.29,1.25)0.53 - 180 - 100 ( 1.29,1.25)0.53100180 ( 1,1)0.3590150 ( 1,1.5)0.355 - 155 - 90 ( 4,2)@xmath391 ( 4,1.5)@xmath393 ( 4,1)@xmath395 ( 4,0.5)@xmath397 + & + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,1.5)(3.5,1.5 ) ( 1.15,2)(3.5,2 ) ( 1.15,0.5)(3.5,0.5 ) ( 1.29,1.25)0.53 - 180 - 100 ( 1.29,1.25)0.53100180 ( 1,1)0.3590150 ( 1,1.5)0.355 - 155 - 90 ( 4,2)@xmath392 ( 4,1.5)@xmath394 ( 4,1)@xmath396 ( 4,0.5)@xmath398 5 . @xmath282 lies on the intersection of one of the bitangents and @xmath375 . + c\|c\|c position of @xmath353 & position of @xmath388 & @xmath381 + + ( 0,0)(5,5.4 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 2,0.3)(4.7,3.0 ) ( 2,4.7)(4.7,2.0 ) ( 0.3,2)(3.0,4.7 ) ( 0.3,3)(3,0.3 ) ( 3.84,2.14 ) ( 4,1.8)@xmath282 + & + ( 0,0)(5,5 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 4.4,2.85)(1,3.85 ) ( 1.3,4.2)@xmath351 + & + ( 0,0)(5,5 ) ( 2.5,3.5)(1.5,0.8 ) ( 0.5,1)(4.5,1 ) ( 2.5,2.3)(2.5,1.2 ) ( 1,1)(4,1 ) ( 1,0.92)(1.08,3.41)(1.08,3.59)(1,1.08)(0.98,3.41)(0.98,3.59 ) + there is now only @xmath219 choice for @xmath389 as we explain in the table below . + p3.8cm\|p3.8cm\|p3.8 cm + 1 @xmath389 + & + 1 branches above @xmath282 corresponding to @xmath389 + & + 1 no . of @xmath6-quotients with support @xmath377 + + + ( 0,0)(3.5,3.3 ) ( 0,-1.4)(2,1.5)(1.5,0.8 ) ( 0.54,1.41)(.54,1.59 ) + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,1.5)(3.5,1.5 ) ( 1.15,2)(3.5,2 ) ( 1.15,0.5)(3.5,0.5 ) ( 1.29,1.25)0.53 - 180 - 100 ( 1.29,1.25)0.53100180 ( 1,1)0.3590150 ( 1,1.5)0.355 - 155 - 90 ( 4,2)@xmath391 ( 4,1.5)@xmath393 ( 4,1)@xmath395 ( 4,0.5)@xmath397 + & 2 + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,1.5)(3.5,1.5 ) ( 1.15,2)(3.5,2 ) ( 1.15,0.5)(3.5,0.5 ) ( 1.29,1.25)0.53 - 180 - 100 ( 1.29,1.25)0.53100180 ( 1,1)0.3590150 ( 1,1.5)0.355 - 155 - 90 ( 4,2)@xmath392 ( 4,1.5)@xmath394 ( 4,1)@xmath396 ( 4,0.5)@xmath398 we now analyse the second case mentioned on page . here @xmath377 is singular , in fact it is the union of two @xmath274 s crossing at one point . this occurs precisely when @xmath351 is tangential to @xmath78 or , equivalently , when @xmath282 lies on @xmath374 . let @xmath400 where @xmath401 is a @xmath192-divisor and @xmath402 which is a @xmath193-divisor . in this case @xmath403 . thus @xmath404 and so to give @xmath354 an @xmath6-module structure corresponds to choosing two points @xmath386 one lying on @xmath401 the other on @xmath405 such that @xmath406 . as before , any such choice gives rise to precisely two @xmath6-module structures . since we must choose one point from @xmath401 and the other from @xmath405 ( and can not choose both points to lie on @xmath401 nor on @xmath405 ) implies that we have `` lost '' some quotients of @xmath6 corresponding to @xmath282 . from a geometric view point , this means that @xmath407 and @xmath408 do not correspond to @xmath6-module structures on @xmath354 . however we are now in the case where corollary [ smoothsupport ] no longer applies , and so not all quotients of @xmath6 have their underlying @xmath20-module structure equal to @xmath253 for some @xmath76-divisor @xmath236 . in fact from corollary [ corquot ] we know that for every @xmath282 lying on @xmath375 there are two additional quotients of @xmath6 ( in the sense that they have no analogue in cases 1 - 5 because they are not quotients of @xmath20 ) with support @xmath377 and they are @xmath409 and @xmath410 . it is thus natural to think of the above two choices of @xmath389 as giving rise to these two quotients of @xmath6 and so we make this association in our future analysis of @xmath314 . 1 . @xmath282 lies on @xmath374 but not on @xmath375 nor on any of the four bitangents . + c\|c\|c position of @xmath353 & position of @xmath388 & @xmath381 + + ( 0,0)(5,5.4 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 2,0.3)(4.7,3.0 ) ( 2,4.7)(4.7,2.0 ) ( 0.3,2)(3.0,4.7 ) ( 0.3,3)(3,0.3 ) ( 1.7,2.5 ) ( 2.1,2.4)@xmath282 + & + ( 0,0)(5,5 ) ( 5,0)(0,0 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 4.5,2.63)(1.3,0.2 ) ( 2.4,1.02 ) ( 3.09,1.55 ) ( 3.62,2 ) ( 3.62,2 ) ( 4.5,1)@xmath351 + & + ( 0,0)(5,5 ) ( 0.5,2)(4.5,4 ) ( 0.5,4)(4.5,2 ) ( 0.5,1)(4.5,1 ) ( 2.5,2.5)(2.5,1.4 ) ( 2.5,0.91)(2.5,1.09 ) ( 3.2,1)(4,1)(3.2,2.64)(3.2,3.37)(4,2.25)(4,3.76 ) + there are now the full @xmath36 choices for @xmath389 , however they only gives rise to six quotients of @xmath6 as we explain below . + p3.8cm\|p3.8cm\|p3.8 cm + 1 @xmath389 + & + 1 branches above @xmath282 corresponding to @xmath389 + & + 1 no . of @xmath6-quotients with support @xmath377 + + + ( 0,0)(3.5,3.3 ) ( 0.5,2.3)(3.5,0.7 ) ( 0.5,0.7)(3.5,2.3 ) ( 2.5,1.78)(3,2.04 ) + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,2)(3.5,2)(1,1.5)0.3590270 ( 4,2)@xmath391 ( 4,1)@xmath392 + & + + + ( 0,0)(3.5,2.6 ) ( 0.5,2.3)(3.5,0.7 ) ( 0.5,0.7)(3.5,2.3 ) ( 2.5,1.24)(3,0.97 ) + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,2)(3.5,2)(1,1.5)0.3590270 ( 4,2)@xmath393 ( 4,1)@xmath394 + & 6 + + ( 0,0)(3.5,2.6 ) ( 0.5,2.3)(3.5,0.7 ) ( 0.5,0.7)(3.5,2.3 ) ( 2.5,1.78)(3,0.97 ) + & + ( 0,0)(3.5,2.6 ) ( 0.5,1)(3.5,1 ) ( 0.5,2)(3.5,2 ) ( 4,2)@xmath395 ( 4,1)@xmath396 + + + ( 0,0)(3.5,2.6 ) ( 0.5,2.3)(3.5,0.7 ) ( 0.5,0.7)(3.5,2.3 ) ( 2.5,1.24)(3,2.04 ) + & + ( 0,0)(3.5,2.6 ) ( 0.5,1)(3.5,1 ) ( 0.5,2)(3.5,2 ) ( 4,2)@xmath397 ( 4,1)@xmath398 + let us explain further why branches @xmath391 and @xmath392 come together here and why this case is different to case 1 . recall that to picking @xmath407 and @xmath411 we associate not a total of four @xmath6-module structure on @xmath354 but the two quotients of @xmath6 that are not quotients of @xmath20 with support @xmath377 , namely @xmath275 and @xmath412 . we also saw that the involution @xmath365 from proposition [ propinv ] fixes points of @xmath11 corresponding to @xmath275 and that by corollary [ corfactor ] the map @xmath314 factors through @xmath365 . hence the branches @xmath391 and @xmath392 must intersect at precisely points corresponding to @xmath275 . the same argument applies to explain why the branches @xmath393 and @xmath394 also merge . @xmath282 lies on the intersection of @xmath374 and @xmath375 . + c\|c\|c position of @xmath353 & position of @xmath388 & @xmath381 + + ( 0,0)(5,5.4)[case7 ] ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 2,0.3)(4.7,3.0 ) ( 2,4.7)(4.7,2.0 ) ( 0.3,2)(3.0,4.7 ) ( 0.3,3)(3,0.3 ) ( 3.18,3.18 ) ( 2.9,2.9)@xmath282 + & + ( 0,0)(5,5 ) + ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 2,4.7)(4.7,2.0 ) ( 4.5,1.8)@xmath351 + & + ( 0,0)(5,5 ) ( 0.5,2)(4.5,4 ) ( 0.5,4)(4.5,2 ) ( 0.5,1)(4.5,1 ) ( 2.5,2.5)(2.5,1.4 ) ( 2.5,0.91)(2.5,1.09 ) ( 4,0.92)(4,1.08)(4,2.18)(4,2.32)(4,3.68)(4,3.84 ) + there are now only @xmath36 choices for @xmath389 as we explain in the table below . + p3.8cm\|p3.8cm\|p3.8 cm + 1 @xmath389 + & + 1 branches above @xmath282 corresponding to @xmath389 + & + 1 no . of @xmath6-quotients with support @xmath377 + + + ( 0,0)(3.5,3.3 ) ( 0.5,2.3)(3.5,0.7 ) ( 0.5,0.7)(3.5,2.3 ) ( 2.8,2)(2.8,1.84 ) + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,2)(3.5,2)(1,1.5)0.3590270 ( 4,2)@xmath391 ( 4,1)@xmath392 + & + + ( 0,0)(3.5,2.6 ) ( 0.5,2.3)(3.5,0.7 ) ( 0.5,0.7)(3.5,2.3 ) ( 2.8,1.15)(2.8,0.99 ) + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,2)(3.5,2)(1,1.5)0.3590270 ( 4,2)@xmath393 ( 4,1)@xmath394 + & 4 + + ( 0,0)(3.5,2.6 ) ( 0,-1.4)(0.5,2.3)(3.5,0.7 ) ( 0.5,0.7)(3.5,2.3 ) ( 2.8,1.1)(2.8,1.94 ) + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,2)(3.5,2)(1,1.5)0.3590270 ( 4,2)@xmath395 ( 4,1)@xmath397 + + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,2)(3.5,2)(1,1.5)0.3590270 ( 4,2)@xmath396 ( 4,1)@xmath398 3 . @xmath282 is one of the points of intersection of the bitangents with @xmath374 . + c\|c\|c position of @xmath353 & position of @xmath388 & @xmath381 + + ( 0,0)(5,5.4 ) ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 2,0.3)(4.7,3.0 ) ( 2,4.7)(4.7,2.0 ) ( 0.3,2)(3.0,4.7 ) ( 0.3,3)(3,0.3 ) ( 2.85,3.85 ) ( 2.6,3.6)@xmath282 + & + ( 0,0)(5,5 ) + ( 2.5,2.5)(0.8,1.5 ) ( 2.5,2.5)(1.5,0.8 ) ( 0,0)(5,5 ) ( 2.8,4.7)(3.8,1 ) ( 4.2,1.1)@xmath351 + & + ( 0,0)(5,5 ) ( 0.5,2)(4.5,4 ) ( 0.5,4)(4.5,2 ) ( 0.5,1)(4.5,1 ) ( 2.5,2.5)(2.5,1.4 ) ( 2.5,0.91)(2.5,1.09 ) ( 4,1)(4,2.24)(4,3.76)(2.5,3)(2.5,1 ) + there are now only @xmath33 choices for @xmath389 as we explain in the table below . + p3.8cm\|p3.8cm\|p3.8 cm + 1 @xmath389 + & + 1 branches above @xmath282 corresponding to @xmath389 + & + 1 no . of @xmath6-quotients with support @xmath377 + + + ( 0,0)(3.5,3.3 ) ( 0.5,2.3)(3.5,0.7 ) ( 0.5,0.7)(3.5,2.3 ) ( 2,1.5)(2.8,1.94 ) + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,1.5)(3.5,1.5 ) ( 1.15,2)(3.5,2 ) ( 1.15,0.5)(3.5,0.5 ) ( 1.29,1.25)0.53 - 180 - 100 ( 1.29,1.25)0.53100180 ( 1,1)0.3590150 ( 1,1.5)0.355 - 155 - 90 ( 4,2)@xmath391 ( 4,1.5)@xmath392 ( 4,1)@xmath397 ( 4,0.5)@xmath398 + & + + ( 0,0)(3.5,3.3 ) ( 0.5,2.3)(3.5,0.7 ) ( 0.5,0.7)(3.5,2.3 ) ( 2,1.5)(2.8,1.1 ) + & + ( 0,0)(3.5,2.6 ) ( 1,1)(3.5,1 ) ( 1,1.5)(3.5,1.5 ) ( 1.15,2)(3.5,2 ) ( 1.15,0.5)(3.5,0.5 ) ( 1.29,1.25)0.53 - 180 - 100 ( 1.29,1.25)0.53100180 ( 1,1)0.3590150 ( 1,1.5)0.355 - 155 - 90 ( 4,2)@xmath393 ( 4,1.5)@xmath394 ( 4,1)@xmath395 ( 4,0.5)@xmath396 + note that the two @xmath6-module structures with support @xmath413 are @xmath275 and @xmath412 . by carefully following which branch connects to which branch we can see that @xmath11 is in fact connected and thus we may conclude that @xmath11 is in fact a smooth projective surface . in this section we tie up one loose end that we have left from section [ chernclasses ] and prove the existence of lines bundles with all possible combinations of chern classes , provided they satisfy our bogomolov - type inequality . we continue with the same notation as before . let @xmath414 and @xmath415 such that @xmath149 . then there exists an @xmath188 with these chern classes . before we begin the proof , we need the following lemma : [ na ] let @xmath236 be a smooth , @xmath17-invariant @xmath76-divisor on @xmath18 and @xmath416 . endow @xmath253 with an @xmath6-module structure , which we saw is always possible from cases 1 - 5 previously . then @xmath139 inherits an @xmath6-module structure from @xmath253 . we need give an @xmath20-module morphism @xmath417 satisfying the required associativity condition . suppose @xmath418 is the morphism which gives @xmath253 its @xmath6-module structure . then @xmath419 is the required morphism . the discriminant of any rank two vector bundle @xmath28 , defined to be the integer @xmath420 , is unchanged by tensoring with a line bundle ( see chapter 12.1 of in @xcite ) and so as we saw before we can thus assume @xmath24 or @xmath421 . we deal with these two cases separately although the proofs will be very similar . fix for the remainder of the proof a smooth @xmath17-invariant @xmath76-divisor @xmath236 and an @xmath6-module structure on @xmath253 . we will now construct an @xmath6-line bundle with @xmath421 and @xmath422 for an arbitrary @xmath423 . using lemma [ na ] endow @xmath424 with an @xmath6-module structure . note that @xmath425 we claim that there is at least one morphism @xmath426 which is surjective . from the above computation , we see that any @xmath6-module morphism @xmath427 arises from an @xmath20-module morphism @xmath428 . choose @xmath21 in such a away that @xmath429 , where @xmath430 is the skyscraper sheaf at @xmath431 , with the @xmath431 lying in the azumaya locus of @xmath6 . then , since @xmath432 , when we extend @xmath21 to a morphism @xmath433 we must have @xmath434 for the simple representations of @xmath435 are all two dimensional . letting @xmath436 we have @xmath437 it is easy to check that @xmath188 with @xmath438 and @xmath439 . constructing an @xmath6-line bundle with @xmath24 and @xmath422 for an arbitrary @xmath423 is an almost identical process where one finds a surjective morphism @xmath440 in the same manner as before and then proves that the kernel must be a line bundle . a simple computation shows that this kernel has the desired chern classes . in this section we establish a link between the moduli space of @xmath6-line bundles with @xmath31 and @xmath27 , which as before we denote by @xmath10 , and the hilbert scheme of @xmath6 , which parameterises quotients of @xmath6 with @xmath32 and @xmath27 , which as before is denoted by @xmath11 . in particular we will show that @xmath11 is a ruled surface over @xmath10 . thus by using the map @xmath314 from the previous section , we will calculate @xmath333 , which will allow us to determine the genus of @xmath10 . * @xmath235 as an @xmath20-module , which occurs precisely when @xmath110 with @xmath234 as an @xmath20-module , or * @xmath347 , where @xmath191 is a @xmath192 ( respectively @xmath193 ) divisor , which occurs precisely when @xmath441 ( respectively @xmath442 ) . let @xmath130 be the universal family on @xmath11 . from proposition [ propker ] @xmath446 is a flat family of @xmath6-line bundles on @xmath11 and so we get a map @xmath447 . @xmath448 being smooth and together with proposition [ propdim ] implies one of its components is a curve . however , from the previous section we know that @xmath11 is a smooth projective surface and thus @xmath10 must in fact be connected and hence must be a smooth projective curve . it thus suffice to show that every fibre of @xmath449 is isomorphic to @xmath274 which is clear from proposition [ propmina ] since @xmath11 is a ruled surface over @xmath10 we can determine the genus of @xmath10 using corollary 2.11 in chapter 5 of @xcite which states that @xmath450 furthermore , we can determine @xmath333 using the map @xmath314 . as discussed above , all that we need to do is compute @xmath333 . recall from before that we have an @xmath35 map @xmath451 . thus using formula 19 of section 16 in chapter 1 of @xcite we have : @xmath452 where @xmath453 is the ramification divisor on @xmath11 . let us describe @xmath453 . looking at case 2 of section [ sec1 ] we define @xmath454 and @xmath455 to be the divisors such that @xmath456 . similarly looking at case 6 in section [ sec2 ] we define @xmath457 and @xmath458 to be such that @xmath459 . denote by @xmath460 the four bitangents to @xmath461 . looking at case 3 of section [ sec1 ] we see that @xmath462 is two divisible and we let @xmath463 be such that @xmath464 . thus @xmath465 . * @xmath469 * @xmath470 . similarly , * @xmath471 . * @xmath472 for @xmath473 . * @xmath474 from case 7 on page . * @xmath475 for @xmath473 . similarly , * @xmath476 for @xmath473 . * @xmath477 for @xmath478 . we can see from case 2 , 5 and 7 that @xmath481 is an tale double cover of @xmath372 . thus @xmath482 where both @xmath483 and @xmath484 have genus zero . we now use the adjunction formula ( proposition 1.5 in chapter v of @xcite ) to compute @xmath485 . we have @xmath486thus @xmath487 and an identical computation shows @xmath488 . thus @xmath489 . note that at no stage did we use the fact that @xmath11 is ruled in order to calculate @xmath333 . in particular , we did nt use the fact that we knew in advance that @xmath333 is a multiple of eight . we could have simplified the computation above if we had done so , but it seemed nice to spend the extra work and get an independent confirmation that fact . as we saw in the above proof @xmath457 is the union of two @xmath274 s . these @xmath274 s are fibres of @xmath494 above the two very special points on @xmath10 corresponding to the @xmath6-line bundles @xmath344 and @xmath442 . since @xmath454 is also a union of two @xmath274 s it would have been nice to find the two @xmath6-line bundles which they are fibres of , but unfortunately , we were unable to do so . + + author : boris lerner + email : [email protected] + address : school of mathematics and statistics + university of new south wales + sydney , 2052 , nsw + australia .	orders on surfaces provided a rich source of examples of noncommutative surfaces . in @xcite the authors prove the existence of the analogue of the picard scheme for orders and in @xcite the picard scheme is explicitly computed for an order on @xmath0 ramified on a smooth quartic . in this paper , we continue this line of work , by studying the picard and hilbert schemes for an order on @xmath0 ramified on a union of two conics . our main result is that , upon carefully selecting the right chern classes , the hilbert scheme is a ruled surface over a genus two curve . furthermore , this genus two curve is , in itself , the picard scheme of the order . throughout this paper we assume all objects and maps are defined over an algebraically closed field @xmath1 of characteristic zero . we denote the dimension of any cohomology group over @xmath1 by the name of the group written with a non - capital letter for e.g. @xmath2 and similarly for @xmath3 and @xmath4 .
it has been known for more than thirty years that the ` quiet ' photosphere contains magnetic fields @xcite . most prominent are compact , mixed - polarity flux elements of the order of 10@xmath1 mx and arcsec sizes that vary over tens of minutes @xcite . the quiet - sun magnetic field is known by various names with the term internetwork ( in ) field in wide use . small fluxes and sizes with rapid time changes make the compact in fields difficult to observe and characterize ( e.g. keller et al . @xcite detected an additional component consisting of a close association between granulation and a fluctuating @xmath01 g vertical component of the quiet sun magnetic field that they call a granular magnetic field . most previous in field observations have been made at or near disk center using line - of - sight ( los ) magnetograms that reveal properties of the vertical component of the in field ( e.g. socas - navarro , martnez pillet , & lites 2004 ) . there is scant information about the center - to - limb variation of the in or its possible horizontal components . @xcite presented in field observations showing little , if any , center - to - limb variation of the los component implying that the in fields are more isotropically oriented than the network fields . @xcite used vector magnetograms near disk center to discover sporadic short - lived ( 5 min ) , arcsec - scale horizontal in field elements that they named hifs . they associated hifs with the eruption of small bipolar elements of magnetic flux from the solar interior . @xcite made sensitive 1-d scans across the disk and concluded that the in field consisted of relatively strong , mainly vertically - oriented features and weaker , mainly horizontal components . @xcite , observing at a heliocentric angle of 38 found several los magnetic features , longer lived than hifs , which were interpreted as being mainly horizontally oriented and closely related to granular flow dynamics . in this letter we present results from time sequences of los component magnetograms of the quiet sun made with the solis vector spectromagnetograph ( vsm ) @xcite and the gong network instruments @xcite . using different methods , the instruments provide the difference between the wavelengths of zeeman sensitive lines in right and left circularly polarized light . these differences are expressed as the homogeneous los field strength in gauss that would produce the measured splitting . since the field is generally inhomogeneous , we obtain only lower limits of true los field strengths . our observations are unique in combining full - disk coverage and comparatively high time cadence , with good sensitivity and moderate spatial resolution . these properties have revealed that there is a ubiquitous , spatially - structured , nearly horizontal field component that varies strikingly over a wide range of time periods . we made time sequences of los magnetograms with the vsm ( 90 s cadence , 3.2 h duration on 2006 december 15 , 11 pixels ) and with gong ( 10 min cadence of 10 min averages coinciding with the vsm data and also a 7 h duration on 2006 december 6 , 25 pixels ) . each time sequence was registered to a fixed solar image centering and then disk features were rotated to a selected time using an assumed representation of solar rotation . these steps led to movies that emphasized real solar changes . near disk center the movies show relatively slow variations consisting of evolution of the network fields and the mixed - polarity in fields . increasingly obvious away from disk center , another mixed - polarity , more dynamic component is present everywhere in otherwise quiet areas . it is patchy with sizes ranging from our resolution limit of a few arcsec to @xmath015 that remain visible for a few minutes to more than 15 min at our noise levels of @xmath01 and @xmath00.2 g per pixel for gong and vsm data respectively . 15 g.,width=312 ] the heretofore unrecognized patchy field is visible in single observations such as figure 1 as an increasingly mottled background structure as one looks from disk center to the limb . in contrast , the visibility of network fields decreases toward the limb . figure 2 better emphasizes these dynamic background structures by subtracting a long time average from a single magnetogram . in addition , we prepared figure 3 from 7 h of magnetograms to study the center - to - limb behavior of the changing fields . it shows the temporal rms of the time - varying fields over the solar disk . to emphasize the patchy field variations , we excluded any measurements with absolute field strengths @xmath25.5 g and set such areas to white in the figure . remaining is the temporal rms variation of the dynamic background field and other sources of changes . these other sources include seeing and registration variations , proper motion of magnetic features , appearance , disappearance , and shape changes of magnetic features , barely resolved in fields , and instrumental noise . the latter is a combination of camera and photon noise being relatively low near the bright disk center and greater near the darker limb . we modeled this noise and find it to vary with radius only slowly except close to the limb . the majority of the signal variation in figure 3 is caused by the dynamic background field . 7.5 g.,width=312 ] 5.5 g were not included in the calculation of the temporal rms and are set to white . note the increase of the rms toward the limb and the absence of any large - scale departure from radial symmetry.,width=312 ] the obvious increase of the magnetic field fluctuations toward the limb suggests that the dynamic features are mainly horizontally oriented . figure 4 ( upper curve ) is a radial average of the los data in figure 3 ( excluding strong fields ) and supports this idea . horizontally - oriented structures should strengthen with the sine of heliocentric angle , i.e. , a linear increase with distance from disk center . the upper curve nicely shows a linear trend but also contains instrumental noise ( and a seeing - noise spike at the limb ) . the dashed curve is a model of camera and photon noise . the lower curve is the quadratic difference of the observed rms minus the instrumental noise model . the corrected rms is not linear , suggesting that the fluctuating field is not strictly horizontal . the dotted curve is a model for which the field is inclined to the vertical by 74 and fits the data . the rms near the limb could also be reduced by loss of resolution due to foreshortening and a possible height variation of the dynamic field , factors that would make the inferred field direction more nearly horizontal . 5.5 g. ( _ dashed curve _ ) model of noise due to the camera and photon statistics . ( _ lower curve _ ) observed rms corrected for the noise model . ( _ dotted curve _ ) model of the corrected rms.,width=288 ] vsm data confirm the gong results with lower noise and higher spatial resolution and cadence . figure 5_a _ is a 1880 by 83cut from one frame of a vsm time series of los magnetograms ( offset @xmath02 from disk center ) . the time variation over 3.2 h along a trace through this area is shown in figure 5_b_. the varying horizontal magnetic field is seen near the edges as mottling . in comparison , the mottling is nearly absent in quiet areas near disk center . the temporal rms of network - free regions near disk center measures 0.8 g while near the limb it is 1.9 g. quadratically subtracting these values gives 1.7 g as the temporal rms of the horizontal field near the limb . this can be compared with the lower resolution gong value of @xmath01.4 g near the limb ( bottom curve in figure 4 ) . higher spatial resolution data would no doubt give a larger value for the temporal rms of the horizontal field . figure 5_c _ is data from the gong instrument located at cerro tololo processed to try to match the vsm data . the gong data is noisier and the registration is imperfect , especially just left of center , but similarity of the background solar signals is evident here and in movies . figure 5_d _ is a series of power spectra of the data in figure 5_b _ covering the frequency range from 0.09 to 5.2 mhz . figure 5_e _ is a spline - smoothed fit to the lower envelope of the power spectra to show the enhanced background power toward the limbs . going beyond a simple rms analysis , figure 6 shows average power spectra for data near the limb ( full line ) and near disk center ( dotted line ) . the near - limb spectrum is dominated by a @xmath3 slope at low frequency up to about 1.5 mhz . at higher frequencies , after a short transition , the spectrum is essentially flat due to instrumental and registration noise . there is no obvious indication of excess power around 3 mhz . near the disk center , where the horizontal field component is small , the average power spectrum is weaker and more complicated . at low frequencies it becomes steeper than -1.4 . from 0.3 to @xmath02.0 mhz the power varies as @xmath4 . at higher frequencies the spectrum is flattened by noise with no sign of extra power at 3 mhz . we discovered a ubiquitous , nearly horizontal component of the solar magnetic field in quiet regions of the photosphere . its reality is confirmed using observations with different instruments , spectrum lines and measurement techniques . this component exhibits wide ranges of spatial and temporal scales : from our resolution limit of a few arcsec up to @xmath015and from times of several minutes to hours . in movies of the los field , this component looks like a seething pattern of mottling . at a spatial resolution of 5 the average temporal rms of the horizontal field variation near the limb is 1.4 g. doubling the spatial resolution to 25 increases the temporal rms value to 1.7 g. based on its temporal and spatial scales , we speculate that the seething horizontal field is driven by granular and supergranular convection , and by field line reconfigurations in response to evolving flux distributions in the nearby network and in . @xcite observed eruption and subsequent shredding of a few in magnetic flux elements consistent with this notion . if connections with existing network flux elements are important , we might expect a dependence of the strength of the horizontal component upon the amount of neighboring large - scale magnetic flux . the lower right part of figure 3 contains such a location and there is no evidence to support this expectation . this finding , and the absence of any evidence of latitudinal or longitudinal dependence , favors the idea that the horizontal field is mainly created and driven by local processes . recent numerical simulations by @xcite show that it is impossible to separate temporal and spatial components of a solar convective turbulence spectrum . so we can not make a simple interpretation of the observed @xmath3 power variation over more than a decade of frequency . the absence of any but feeble hints of excess power at 3 mhz suggests that p - mode oscillations play little , if any , role in the dynamics of the horizontal field . the sporadic hifs observed by @xcite are probably small bipolar flux elements erupting from the interior . they are too infrequent to explain the ubiquitous , nearly horizontal field fluctuations that we found . numerical magnetoconvection simulations can provide insight into the physics of the horizontal field . for example , @xcite find a weak , predominantly horizontal field in the photospheric layers of granules associated with strong horizontal flows . it would be interesting to see what more advanced models , such as those of @xcite , would predict on a scale larger than granulation . more extensive and detailed observations are certainly required to clarify the physical nature of the nearly horizontal field . many questions remain unanswered : is a similar field observed higher in the solar atmosphere ? is the field a miniature version of canopy fields found around strong flux concentrations . what is the detailed relationship with the photospheric convective and oscillatory velocity fields ? what is the association with the granular magnetic field of @xcite ? how does the field fit with hanle - effect observations of a microturbulent magnetic field ( e.g. stenflo , keller , & gandorfer 1998 ) ? although most attention has been directed to the vertical component of the quiet sun magnetic field , the ubiquitous presence of a nearly horizontal component suggests that additional studies may prove it to be at least as significant in improving our understanding of solar magnetism . a practical consequence of the field involves extrapolations of photospheric field measurements to the corona . it is usually assumed that the observed los fields are radially oriented in order to estimate the surface distribution of magnetic flux . this assumption is acknowledged as wrong in active regions . now we know that it is also incorrect for quiet regions observed near the limb . however , time and spatial averaging of observations may mitigate this effect . finally , we note that it is overly simplistic to consider the in field as being composed of independent vertical and horizontal components . it is most likely that these are just observational manifestations of a dynamically interacting field of wonderful complexity . we gratefully acknowledge the solis and gong team members for their devoted and skilled work that provides high - quality data to the research community . solis / vsm data used here were produced with support from nsf and nasa . this work utilizes data obtained by the global oscillation network group ( gong ) program , managed by the national solar observatory . the real - time gong data were acquired by instruments operated by the big bear solar observatory and cerro tololo interamerican observatory and obtained from publically available archives . the national solar observatory is operated by the association of universities for research in astronomy , inc . ( aura ) , under cooperative agreement with the national science foundation . de pontieu , b. 2002 , , 569 , 474 gadun , a. s. , solanki , s. k. , sheminova , v. a. , & ploner , s. r. o. 2001 , , 203 , 1 georgobiani , d. , stein , r. f. , and nordlund , . 2007 , in asp conf . 354 , solar mhd theory and observations , eds . h. uitenbroek , j. leibacher , & r. f. stein ( san francisco : asp ) , 109 harvey , j. & gong instrument development team 1988 , in esa sp-286 , seismology of the sun & sun - like stars , ed . e. j. rolfe ( noordwijk : esa ) , 203 keller , c. u. , deubner , f .- l . , egger , u. , fleck , b. , & povel , h. p. 1994 , , 286 , 626 keller , c. u. , harvey , j. w. , & giampapa , m. s. 2003 , proc . spie , 4853 , 194 khomenko , e. v. , shelyag , s. , solanki , s. k. , & vgler , a. 2005 , , 442 , 1059 lin , h. & rimmele , t. 1999 , , 514 , 448 lites , b. w. , leka , k. d. , skumanich , a. , martnez - pillet , v. , & shimizu , t. 1996 , , 460 , 1019 livingston , w. & harvey , j. 1971 , in iau symp . 43 , solar magnetic fields , ed . r. howard ( dordrecht : reidel ) , 51 livingston , w. & harvey , j. 1975 , , 7 , 346 martin , s. f. 1988 , , 117 , 243 meunier , n. , solanki , s. k. , & livingston , w. c. 1998 , , 331 , 771 smithson , r. 1975 , , 7 , 346 socas - navarro , h. , martnez pillet , v. , & lites , b. w. 2004 , , 611 , 1139 stenflo , j. o. , keller , c. u. , & gandorfer , a. 1998 , , 329 , 319 22 g. ( _ b _ ) vsm data time variation for 3.2 h along a trace through the area shown in ( _ a _ ) . the display saturates at @xmath511 g and the spatial glitch in the middle is a data reduction artifact . ( _ c _ ) same except using gong data . registration with ( _ b _ ) is best near the limbs . coarser spatial and time gong samples were interpolated to match the vsm data . note higher noise level compared to vsm data . ( _ d _ ) log of 3 decades of power spectra of the columns in ( _ b _ ) displayed on a log frequency scale from 0.09 at the bottom to 5.2 mhz at the top . note absence of any obvious periodic signal . ( _ e _ ) spline - smoothed fit of the background of ( _ d _ ) . note the larger background power levels toward the limbs.,width=672 ]	the photospheric magnetic field outside of active regions and the network has a ubiquitous and dynamic line - of - sight component that strengthens from disk center to limb as expected for a nearly horizontal orientation . this component shows a striking time variation with an average temporal rms near the limb of 1.7 g at @xmath03 resolution . in our moderate resolution observations the nearly horizontal component has a frequency variation power law exponent of -1.4 below 1.5 mhz and is spatially patchy on scales up to @xmath015 arcsec . the field may be a manifestation of changing magnetic connections between eruptions and evolution of small magnetic flux elements in response to convective motions . it shows no detectable latitude or longitude variations .
despite recent successes in developing new methods such as gate - set tomography ( gst)@xcite to fully and accurately characterize a given quantum process , as well as simplified methods@xcite to avoid scalability limitations , quantum process tomography ( qpt)@xcite remains a benchmark standard to which the results of the new evolving methods must be compared . in this paper , we review single qubit process tomography and present some new findings on the properties of the process matrix in the familiar @xmath0 representation and demonstrate their utility via application to nontrace - preserving processes such as qubit leakage errors . in particular , we examine the general form of constraints for numerical fitting of experimental data , and extract simplified forms , which indicate explicit relationships among the various elements of the process matrix , one of which is the familiar one , @xmath1 ( in the pauli basis ) for a trace - preserving process . the other three derived relationships for a trace - preserving process , exclusively involve the off - diagonal elements and thus provide further insight into the structure of the process matrix . knowledge of these can thus serve as useful tools for an experimentalist interested in measuring quantum gates to determine error models . we illustrate their utility with several example process matrices , including some models of leakage errors . [ fig : one ] shows a device under test ( dut ) upon which qubits impinge in a quantum state described by the density matrix @xmath2 . the output qubits density matrix is denoted by @xmath3 . ordinarily , quantum state tomography produces normalized states ; however , the measurement rates contain additional information on the loss to characterize a non - trace - preserving process . to use the loss information , the density matrix of the output state @xmath3 includes a scaling factor ( @xmath4 ) to account for any loss of qubits as they traverse the dut . following@xcite , the output state in figure 1 , @xmath3 , can then be written as @xmath5 where @xmath6 is an operator representing the effect of the dut on the input state . this can be further expanded as @xcite @xmath7 where @xmath8 s comprise a set of at most four operators describing the effect of the dut . now these operational elements can be expressed in terms of a _ fixed set of basis operators _ , @xmath9 , i.e. , we can write @xmath10 as a result , @xmath11 where @xmath12 . since indices @xmath13 and @xmath14 each run from 1 through 4 , @xmath15 is a 4 x 4 matrix , called the _ process matrix_. this matrix is hermitian . therefore , it has at most @xmath16 independent parameters . additionally , it is nonnegative definite , i.e. , its eigenvalues are zero or greater . now , invoking the fact that for a trace - preserving process , @xmath17 , one obtains from eq . [ eq:5 ] @xmath18 these are , in effect , four constraints on the elements , @xmath15 . these constraints then reduce the number of independent parameters of the @xmath0 matrix from 16 to 12 . in general , including nontrace - preserving processes@xcite , @xmath19 where @xmath20 note that the matrix @xmath21 is nonegative - definite hermitian . in what follows , we choose the pauli basis , i.e. , we set @xmath22 , where @xmath23 , @xmath24 , @xmath25 , and @xmath26 . it can be shown that for this _ fixed _ set of basis operators , @xmath27 , which then equals one for a trace - preserving process because in that case , @xmath28 . [ eq:8 ] further implies that the eigenvalues of the @xmath21 matrix ( defined in eq . [ eq:9 ] ) are each greater than or equal to zero and also less than or equal to one . for the choice @xmath22 , we find @xmath29 @xmath30 where @xmath31 @xmath32 are the two @xmath21-matrix eigenvalues appearing in the above inequalities , eqs . [ eq:10 ] and [ eq:11 ] , which the @xmath0 matrix must , in general , satisfy ( we assume a positive sign for the radical sign in eq . [ eq:12 ] ) . adding eqs . [ eq:10 ] and [ eq:11 ] yields @xmath33 , which is normally quoted in literature ; however , eq . [ eq:10 ] indicates a much tighter constraint , involving both the diagonal elements and the off - diagonal elements . when the process is trace preserving , the equality holds , which then requires that all three terms under the radical sign in eq . [ eq:12 ] be individually equal to zero because @xmath1 . in other words , not just @xmath1 , but the entire set @xmath34 @xmath35 @xmath36 @xmath37 must hold in any numerical fit to the experimental data to yield a physical @xmath0 matrix . to our knowledge , this explicit form of constraints has not been cited or discussed in the past , although sets of equations of the form , eq . [ eq:7 ] , typically have been employed directly as constraints in numerical optimization procedures to obtain a fitted physical ( trace - preserving ) process matrix from experimental data ( see , e.g.,@xcite ) . note that this set of constraints can also be derived directly by solving the linear equations embodied in eq . [ eq:7 ] . from eq . [ eq:11 ] , it further follows that @xmath38 . using the fact that both @xmath39 ) and @xmath40 are nonnegative , eqs . [ eq:10 ] and [ eq:11 ] can now be rewritten as @xmath41 @xmath42 these two inequalities serve as general constraints that must be satisfied in a quantum process . below we give some some examples to corroborate the above results : the process matrix for the hadamard gate is given by @xmath43.\label{eq:19}\ ] ] eq . [ eq:13 ] is satisfied as @xmath44 . further there are no complex coefficients , and the first - row elements are all zero , so eqs . [ eq:14]-[eq:16 ] are all true and identically zero , as is @xmath40 . therefore , eqs . [ eq:17 ] and [ eq:18 ] are satisfied as well . this is a nontrace - preserving process . the @xmath0 matrix is given by @xmath46 . \label{eq:20}\ ] ] @xmath47 , which is less than 1 , as expected . additionally , the value of @xmath40 , using eq.[eq:12 ] , is also equal to 1/2 . the constraints , eqs . [ eq:17 ] and [ eq:18 ] , are satisfied . violations occur in eqs . [ eq:14]-[eq:16 ] . + + in addition to @xmath48 for a nontrace - preserving process , what specific violations occur in eqs . [ eq:14]-[eq:16 ] can also be an indication of the type of nontrace - preserving process . we illustrate this with respect to a leakage error model for quantum computing . qubit leakage is of two types : 1 ) coherent leakage , where the qubit represented by a two - level subsystem of a multi - level system like the trapped ion , leaks out of its hilbert space and then transitions back to it ; 2 ) loss , where the qubit permanently transitions out of its hilbert space , i.e. , never returns to it and is thus considered lost . in this paper , we focus on the latter , where , for example , the qubit in the first excited state ( @xmath49 ) of the multi - level system , may be further excited outside of the qubit s computational hilbert space , and never returns to it ( or returns to it after a very long time , so for practical purposes it is considered lost ) . the process is therefore nontrace preserving . following@xcite , @xmath50 where @xmath51 represents the error operation , @xmath2 is the input state , @xmath3 is the output state and @xmath52 is the leakage error probability . it follows from above that @xmath53 indicating that the qubit is lost with a probability @xmath52 when it is in the excited state and remains stable when it is in the ground state ( @xmath54 ) . using eq . [ eq:5 ] and eq . [ eq:21 ] , the process matrix is @xmath55.\label{eq:23}\ ] ] @xmath56 for @xmath57 . in this case , when @xmath58 , this is no longer a trace - preserving process , so eq . [ eq:16 ] is violated in proportion to the leakage probability @xmath52 . in fact , all the nonzero , non - identity elements deviate from the corresponding elements of the ideal identity gate by an amount identical in magnitude ( @xmath59 ) , which is proportional to the leakage probability @xmath52 . consider now the case where in eq . [ eq:21 ] , the pauli operator , @xmath60 is replaced by @xmath61 . this is a nontrace - preserving process with @xmath62 given by eq . [ eq:22 ] , but with @xmath60 replaced with @xmath61 . this corresponds to a noisy environment where the state @xmath63 stays stable , and the state @xmath64 leaks out with probability @xmath52 . on the other hand , @xmath54 and @xmath49 , which comprise the @xmath65 and the @xmath66 states , leak out with the same probability , @xmath67 the corresponding process matrix is given by @xmath68 . \label{eq:24}\ ] ] here the violation , indicative of a nontrace - preserving process , occurs in eq . [ eq:14 ] , instead of eq . [ eq:16 ] , signifying a different nontrace - preserving process , even though @xmath69 remains unchanged . the positioning of the nonzero elements , except the first diagonal element here , has shifted within the @xmath0 matrix , suggestive of the change in the nature of the nontrace - preserving process . this manner of shift is predictable if one is specifically working with a general leakage error model in which @xmath60 in eq . [ eq:21 ] is replaced with @xmath70 . we further extend the model of eq . [ eq:21 ] to qubits , where the ground state ( @xmath71 may also leak out , although with a low probability compared to the excited state ( @xmath49 ) as , for example , in superconducting phase qubits@xcite . the leakage process here can be represented by the following equation : @xmath72 where an extra term has been added to eq . [ eq:21 ] to account for the leakage of the ground state as seen below : @xmath73 the ground state leakage probability , from eq . [ eq:26 ] , is @xmath74 . while the excited state leakage probability is @xmath75 . the process matrix is the same as the one given in eq . [ eq:23 ] , except that the nonzero off - diagonal elements are now changed to @xmath76 , an indication of the change of the nature of the nontrace - preserving process , namely , the presence of leakage from the ground state as well . we also note here that the left hand side of eq . [ eq:17 ] , @xmath77 evaluates to @xmath78 for this model in contrast to the value of 1 obtained for eqs . [ eq:23 ] and [ eq:24 ] , which can be another distinguishing feature . thus , we see that simplification of the constraints , eq . [ eq:9 ] into the set , eqs . [ eq:13]-[eq:16 ] , can provide insight into the structure of the trace - preserving process matrix ; the three newly derived explicit forms , eqs . [ eq:14]-[eq:16 ] , express clear relationships among the off - diagonal elements ; we have not seen these relationships mentioned or discussed in the literature before . violations of these constraints is an indication of a nontrace - preserving process , and the nature of the violations , as we have illustrated above , can help discriminate one type of a nontrace - preserving process from another . furthermore , it must be emphasized that for a quantum process known to be nontrace - preserving like the polarizer ( where @xmath79 , ideally ) , or for a process suspected to be not strictly trace - preserving like a quantum gate with leakage errors , or simply for a dut whose behavior is not known a priori ( a true black box ) , the general constraints , eqs . [ eq:17 ] and [ eq:18 ] , must be invoked in the fitting of data . in summary , we have revisited the theoretical aspects of single qubit quantum process tomography to determine the behavior of a quantum device . more specifically , we have reexamined the well - known constraints for the process matrix ( in the @xmath0 representation ) , and recast them into more insightful forms . in the case of a trace - preserving process , specific relationships among the various elements of the process matrix emerge that then shed light on its basic generic structure . knowledge of these new constraint relationships permit an enhanced understanding of the interpretation and analysis of the experimental data . we have illustrated their validity and utility with several examples , with specific attention to leakage errors , which are of significant importance in quantum computing . we tested the efficacy of constraints , eqs . [ eq:17 ] and [ eq:18 ] , in fitting data by adding noise to the above ideal @xmath0 matrices for the hadamard gate , the polarizer , and the leakage error models considered in this paper . we simulated gaussian hermitian complex noise using the matlab r2015b function randn which returns a number from a normal distribution with zero mean and a standard deviation equal to 1 . this noise is then scaled by a variable scaler ranging from @xmath80 to @xmath81 and added to the process matrix , after which the process matrix is optimized ; one fixed value of the scaler is used at one time . we use toolboxes yalmip version 19-sep-2015 @xcite with sedumi 1.32 @xcite for optimization within matlab . in the numerical simulations , we frequently observed the noisy @xmath0 matrices to have negative eigenvalues , eigenvalues exceeding unity , and/or trace exceeding unity . imposing the requirements of nonnegative definiteness , hermiticity and the constraints , eqs . [ eq:17 ] and [ eq:18 ] to fit these noisy @xmath0 matrices always restored physicality ; the eigenvalues were then nonnegative and less than or equal to 1 . improperly constraining the system , e.g. , imposing only @xmath82 , without eqs . [ eq:17 ] and [ eq:18 ] , led to unphysical output states computed from @xmath0 , even though the requirements of nonnegative definiteness and hermiticity for the @xmath0 matrix were still in place . further it is worth noting that in many examples examined , the fidelity between the target process matrix and each of the two types of optimizations is similar , especially when it is high , and in this case does not aid one in detecting optimization errors . next we give two specific examples showing an initial noisy process matrix and the results after applying the complete constraints . first we consider the hadamard gate as given by eq . [ eq:19 ] . after adding noise scaled by @xmath83 , we obtain , as an example , the following : @xmath84 . \label{eq : m1}\ ] ] this initial matrix has one eigenvalue greater than one and two negative eigenvalues and is therefore unphysical . we also note that the set of eqs . [ eq:13]-[eq:16 ] is violated here . here and in the following examples , we show rounded results , while full precision is used to compute reported derived quantities . under the assumption of a trace - preserving process , we perform numerical fitting using eqs . [ eq:13]-[eq:16 ] , as constraints . the result is @xmath85 . \label{eq : m2}\ ] ] eqs . [ eq:13]-[eq:16 ] are now satisfied . if , on the other hand , the quantum process is suspected to be not strictly trace - preserving ( due to the possibility of leakage errors ) , one must replace the constraints , eqs . [ eq:13]-[eq:16 ] , with the the general constraints , eqs.[eq:17 ] and [ eq:18 ] . the result , after fitting with these constraints , is @xmath86 . \label{eq : m3}\ ] ] eq . , [ eq : m2 ] and [ eq : m3 ] are very similar , however , the latter s trace is 0.9999 , so it is not trace preserving , but it is a valid physical process . as a second example , we consider the leakage error model described by eq . [ eq:25 ] ( a nontrace - preserving process ) with @xmath87 , @xmath88 , and the gaussian noise scaler equal to @xmath83 . an instance of the noisy process matrix is @xmath89 . \label{eq : m4}\ ] ] it has two negative eigenvalues , and is therefore unphysical . in addition , eq . [ eq:17 ] is violated as the left - hand side evaluates to a value of 1.0034 . after optimization with constraints , eqs.[eq:17 ] and [ eq:18 ] , the process matrix is @xmath90 . \label{eq : m5}\ ] ] the optimized result is nonnegative definite and satisfies the required constraints , eqs . [ eq:17 ] and [ eq:18 ] . one of us ( nap ) acknowledges research sponsored by the laboratory directed research and development program of oak ridge national laboratory , managed by ut - battelle , llc , for the u. s. department of energy . rb acknowledges useful communications with joel wallman , joseph emerson , andrzej veitia , and robin blume - kahout . rb and nap contributed to the conception , implementation , and analysis of these results . all authors have reviewed the manuscript . * competing financial interests * the authors declare no competing financial interests . & in _ _ ch . 8 , 389 - 393 ( , ) . , , , & . _ _ * * , ( ) . _ et al . _ . _ _ * * , ( ) . , & preprint at http://arxiv.org/abs/1412.4126v2 ( ) . . _ _ * * , ( ) . . in _ _ ( , ) . & , _ _ . available at : . ( accessed : 4th november 2015 )	we briefly review single - qubit quantum process tomography for trace - preserving and nontrace - preserving processes , and derive explicit forms of the general constraints for fitting experimental data . these new forms provide additional insight into the structure of the process matrix . we illustrate their utility with several examples , including a discussion of qubit leakage error models and the intuition which can be gained from their process matrices .
uninterrupted series of successes of quantum mechanics support a belief that quantum formalism applies to all of physical reality . thus , in particular , the objective classical world of everyday experience should emerge naturally from the formalism . this has been a long - standing problem , in fact already present from the very dawn of quantum mechanics ( see e.g. the writings of bohr @xcite and heisenberg @xcite for some of the earlier discussions and e.g. @xcite for some of the modern approaches , relevant to the present work ) . perhaps the most promising approach is decoherence theory ( see e.g. @xcite ) , based on a system - environment paradigm : a quantum system is considered not in an isolation , but rather interacting with its environment . it recovers , under certain conditions , a classical - like behavior of the system alone in some preferred frame , singled out by the interaction and called a _ pointer basis _ , and explains it through information leakage from the system to the environment ( the system is monitored by its environment ) . however , as zurek noticed recently @xcite , decoherence theory is silent on how comes that in the classical realm information is _ redundant_same record can exist in a large number of copies and can be independently accessed by many observers and many times . to overcome the problem , he has introduced a more realistic model of environment , composed of a number of independent fractions , and argued using several models ( see e.g. refs . @xcite ) that after the decoherence has taken place , each of these fractions carries a nearly complete classical information about the system . then zurek argues that this huge information redundancy implies objective existence @xcite . this model , called quantum darwinism , although very attractive ( see ref . @xcite for some experimental evidence ) , has a certain gap which make its foundations not very clear . postponing the details to section [ qdcond ] , the criterion used in quantum darwinism to show the information redundancy is motivated by _ entirely classical _ reasoning and a priori may not work as intended in the quantum world . there is however another basic question : is there a fundamental physical process , consistent with the laws of quantum mechanics , which leads to the appearance in the environment of multiple copies of a state of the system ? in other words , how does nature create a bridge from fragile quantum states , which can not be cloned @xcite , to robust classical objectivity ? zurek is aware of the difficulty when he writes @xcite : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ quantum darwinism leads to appearance in the environment of multiple copies of the state of the system . however the no - cloning theorem prohibits copying of unknown quantum states . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ however , he does not provide a clear answer to the question @xcite : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ quick answer is that cloning refers to ( unknown ) quantum states . so , copying of observables evades the theorem . nevertheless , the tension between the prohibition on cloning and the need for copying is revealing : it leads to breaking of unitary symmetry implied by the superposition principle , [ ... ] _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ but the no - cloning theorem prohibits only _ uncorrelated _ copies of the state of the system , whereas it leaves open a possibility of producing _ correlated _ ones . this is the essence of _ state broadcasting_a process aimed at proliferating a given state through correlated copies @xcite . in this work we identify a weaker form of state broadcasting_spectrum broadcasting _ , introduced in ref . @xcite , as the fundamental physical process , consistent with quantum mechanical laws , which leads to the perceived objectivity of classical information , and as a result recover quantum darwinism ( as a limiting point ) . we do it first in full generality , using a definition of objective existence due to zurek @xcite and bohr s notion of non - disturbance @xcite . then , in one of the emblematic examples of decoherence theory and quantum darwinism : a small dielectric sphere illuminated by photons ( see e.g. refs . the recognition of the underlying spectrum broadcasting mechanism has been possible due to a paradigmatic shift in the core object of the analysis . from a partial state of the system ( decoherence theory ) or information - theoretical quantities like mutual information ( quantum darwinism ) to a full quantum state of the system and the observed environment . this also opens a possibility for direct experimental tests using e.g. quantum state tomography @xcite . what does it mean that something _ objectively exists _ ? what does it mean for information ? for the purpose of this study we employ the definition from ref . @xcite : [ obj ] a state of the system @xmath0 exists objectively if [ ... ] many observers can find out the state of @xmath0 independently , and without perturbing it . in what follows we will try to make this definition as precise as possible and investigate its consequences . the natural setting for this is quantum darwinism @xcite : the quantum system of interest @xmath0 interacts with multiple environments @xmath1 ( denoted collectively as @xmath2 ) , also modeled as quantum systems . the environments ( or their collections ) are monitored by independent observers ( _ environmental observers _ ) and here we do not assume symmetric environments they can be all different . the system - environment interaction is such that it leads to a full decoherence : there exists a time scale @xmath3 , called _ decoherence time _ , such that asymptotically for interaction times @xmath4 : i ) there emerges a unique , stable in time preferred basis @xmath5 , so called _ pointer basis _ , in the system s hilbert space ; ii ) the reduced state of the system @xmath6 becomes stable and diagonal in the preferred basis : [ decoh ] _ s_e_s : e_i p_i \| i i , where @xmath7 s are some probabilities and by @xmath8 we will always denote asymptotic equality in the deep decoherence limit @xmath9 . we emphasize that we assume here the _ full decoherence _ , so that the system decoheres in a basis rather than in higher - dimensional pointer superselection sectors ( decoherence - free subspaces ) . coming back to the definition [ obj ] , we first add an important _ stability requirement _ : the observers can find out the state of @xmath0 without perturbing it _ repeatedly and arbitrary many times_. in our view , this captures well the intuitive feeling of objectivity as something stable in time rather than fluctuating . thus , if definition [ obj ] is to be non - empty , it should be understood in the time - asymptotic and hence decoherence regime , which in turn implies that the state of @xmath0 which can possibly exist objectively , is determined by the decohered state ( [ decoh ] ) . we will show it on a concrete example we study later . next , we specify the observers . apart from the environmental ones , we also allow for a , possibly only hypothetical , _ direct observer _ , who can measure @xmath0 directly . we feel such a observer is needed as a reference , to verify that the findings of the environmental observers are the same as if one had a direct access to the system . it is clear that what the observers can determine are the eigenvalues @xmath7 of the decohered state ( [ decoh])they otherwise know the pointer basis @xmath5 , as if not , they would not know what the information they get is about . hence , the state in definition [ obj ] , which gains the objective existence , is the classical part of the decohered state ( [ decoh ] ) , i.e. its spectrum @xmath10 . the word find out we interpret as the observers performing von neumann ( as more informative than generalized ) measurements on their subsystems . by the independence condition , they act independently , i.e. there can be no correlations between the measurements and the corresponding projectors must be fully product : [ prod ] ^m_s_i^m_1_j_1^m_n_j_n , where all @xmath11 s are mutually orthogonal hermitian projectors , @xmath12 for @xmath13 . now the crucial word perturbation needs to be made precise . the debate about its meaning has been actually at the very heart of quantum mechanics from its beginnings , starting from the famous work of einstein , podolsky and rosen ( epr ) @xcite and the response of bohr @xcite . it is quite intriguing that this debate appears in the context of objectivity . the exact definitions of the epr and bohr notions of non - disturbance are still a subject of some debate and we adopt here their formalizations from ref . @xcite : the sufficient condition for the epr non - disturbance is the _ no - signaling principle _ , stating that the partial state of one subsystem is insensitive to measurements performed on the other subsystem ( after forgetting the results ) @xcite . quantum mechanics obeys the no - signaling principle , but bohr argued that the epr s notion is too permissive , as it only prohibits mechanical disturbance , and proposed a stricter one , which can be formally stated @xcite that the whole _ state must stay invariant under local measurements on one subsystem ( after forgetting the results ) . for the purpose of this study we adopt bohr s point of view , adapted to our particular situation we assume that _ neither of the observers bohr - disturbs the rest _ ( in the @xmath14 direction it is our formalization of the definition [ obj ] , while in the @xmath15 it follows from the repetitivity requirement ) . together with the product structure ( [ prod ] ) , this implies that on each @xmath16 there exists a _ non - disturbing measurement _ , which leaves the whole asymptotic state @xmath17 of the system and the observed environment invariant ( we will specify the size of the observed environment later ) . for the system @xmath0 it is obviously defined by the projectors on the pointer basis @xmath5 , as by assumption this is the only basis preserved by the dynamics . for the environments we allow for a general higher - rank projectors @xmath18 , @xmath19 , and not necessarily spanning the whole space , as the environments can : i ) have inner degrees of freedom not correlating to @xmath0 and ii ) correlate to @xmath0 only through some subspaces of their hilbert spaces ( we will later encounter such a situation in the concrete example ) . when more than one observer preform the non - disturbing measurements , a further specification of bohr - nondisturbance is needed . allowing for general correlations @xmath20 $ ] may lead to a disagreement : if one of the observers measures first , the ones measuring afterwards may find outcomes depending on the result of the first measurement ( if the observers do not discard their results an meet to compare them later ) . this can hardly be called objectivity and we thus add to the definition [ obj ] an obvious _ agreement requirement _ : ... observers can find out the _ same _ state of @xmath0 independently , ... , leading to a natural conclusion @xcite : [ agree ] ( p_ij_1 j_n0 i = j_1= ... =j_n ) p_ii i=1 , i.e. the environmental bohr - nondisturbing measurements must be _ perfectly correlated _ with the pointer basis . hence , after forgetting the results , the asymptotic post - measurement state @xmath21 reads ( by @xmath22 we denote @xmath23 asymptotic ) : & & ^m_s : e()=_i , j_1 , , j_np_ij_1 j_n_ij_1 j_n^s : e()= + & & _ i ii_i_s : e()ii_i , where @xmath24 . now we are ready for _ the crucial step _ : we impose the relevant form of the bohr - nondisturbance condition : [ maxcorr ] _ i ii_i_s : e()ii_i=_s : e ( ) , whose only solution @xcite are the , so called , classical - quantum ( cq ) states @xcite : [ wism ] _ s : e()=_i p_i ii^e_i , where @xmath7 are the probabilities from eq . ( [ decoh ] ) and @xmath25 are some residual states in the space of all the environments with mutually orthogonal supports : @xmath26 for @xmath27 . hence , @xmath28 are perfectly distinguishable @xcite through the assumed non - disturbing measurements @xmath29 , projecting on their supports . the derived form ( [ wism ] ) sheds some light on the word many in the definition [ obj ] : the compatible states ( [ wism ] ) are necessarily @xmath30 separable , while we argue that _ generically _ , for large systems , the unitary system - environment evolution @xmath31 leads to @xmath30 entanglement ( see e.g. ref . @xcite for the definition of the latter ) . we first recall that the initial states , weather pure or mixed , are always assumed to be @xmath30 product the system and the environment did not interact in the remote past and there is _ no prior information _ about the system in the environment . the entanglement generation is then clear for pure initial states , as entanglement is the only form of correlation for such states and without a @xmath30 correlation there can be no decoherence ( [ decoh ] ) . for mixed initial states the situation is more subtle as in finite - dimensional state - spaces there exist non - zero volume separable balls around the identity operator @xcite . if the @xmath30 state is initially in this ball , the unitary evolution will not lead it out of it , while building enough correlations for the decoherence ( [ decoh ] ) to happen . however , for large dimensions , the radius of the largest separable ball decreases as @xmath32 @xcite and for infinite - dimensional spaces becomes strictly zero ( see e.g. ref . this is the case here : the environment must be of a large dimension if it is to have a large informational capacity , needed to carry a large number of copies of a state of @xmath0 . thus , the @xmath30 entanglement is generically produced during the evolution , as hitting the separable ball becomes highly unprobable due to its vanishing measure . the only way then to eventually obtain a separable state from an entangled one is by forgetting subsystems some portions of the environment pass unobserved , as it is actually always the case in reality . thus , slightly abusing the language and identifying observers with the fractions of the environment they observe , we can interpret many as sufficiently many but not all some loss of information is necessary . in what follows the total observed fraction of the environment will be denoted by @xmath33 or @xmath34 ( depending on the context ) and all the states above should be understood as @xmath35 . finally , let us look at the residual states @xmath25 in eq . ( [ wism ] ) . we comeback to the demand of independent ability to determine the state of @xmath0 , already used in eq . ( [ prod ] ) , and we further interpret it as a _ strong independence _ : _ the only correlation between the environments should be the common information about the system_. in other words , conditioned by the information about the system , there should be no correlations between the environments . thus , once one of the observers finds a particular result @xmath36 , the conditional state should be fully product . since the direct observer is already uncorrelated by eq . ( [ wism ] ) , this implies that : _ i^fe=^e_1_i^e_fn_i . and the states @xmath37 must be perfectly distinguishable for each environment @xmath38 independently : [ distng ] ^e_k_i^e_k_i=0 ii , since by the bohr - nondisturbance ( [ maxcorr ] ) for any @xmath39 it holds @xmath40 and @xmath41 for @xmath27 . gathering all the above facts together , we finally obtain : if there is a decoherence mechanism that asymptotically leads to an objectively existing state of @xmath0 in the sense of definition [ obj ] , then the asymptotic joint state of the system and the observed environment fraction ( after the necessary tracing out of some of the environment ) must be of a special classical - classical @xcite form : @xmath42 where all @xmath37 satisfy ( [ distng ] ) . from the quantum information point of view , state ( [ br2 ] ) is a final state of a process similar to _ quantum state broadcasting _ @xcite . the latter is a task ( described by a linear map ) , which aims at producing from a given state @xmath43 a multipartite state @xmath44 , called an n - party broadcast state for @xmath43 , such that for every reduction @xmath45 , thus proliferating @xmath43 , but in a more subtle manner then by cloning . remarkably there is a weaker form of broadcasting , _ spectrum broadcasting _ @xcite a task aiming at proliferating merely a spectrum of a quantum state , or equivalently a classical probability distribution . we define it as follows : @xmath46 is a spectrum broadcast state for @xmath43 , with @xmath47 , if for every reduction @xmath38 there exist encoding states @xmath37 such that : _ e_1 e_k e_n^s - br_e_1 ip_i^e_k_i ^e_k_i^e_k_ii=0 ( comparing to ref . @xcite we allow for arbitrary encoding states @xmath37 , as long as they are perfectly distinguishable ) . for a given @xmath43 , a spectrum broadcast state @xmath46 allows then one to locally recover perfect copies of the spectrum @xmath48 ( through the projective measurements of the supports of @xmath37)the spectrum is _ redundantly proliferated_. this is clearly the case of the state ( [ br2 ] ) due to the distinguishability ( [ distng ] ) : ( [ br2 ] ) is a spectrum broadcast state for the decohered state ( [ decoh ] ) . condition ( [ distng ] ) forces the correlations in ( [ br2 ] ) to be entirely classical and thus the detailed structures of @xmath37 ( e.g. their ranks ) become irrelevant for the correlations . one can even pass to the purifications @xmath49 @xcite of @xmath37 , which by ( [ distng ] ) will be mutually orthogonal for @xmath27 . in the equivalent language of quantum channels @xcite , the redundant classical information transfer from the system to the observed environment is asymptotically described by a @xmath50-type channel defined by ( [ br2 ] ) @xcite . the result ( [ br2 ] ) can be then re - stated as : in the presence of decoherence , spectrum broadcasting is a necessary condition for objective existence , in the sense of definition [ obj ] , of the classical state of @xmath0 ( = the spectrum of ( [ decoh ] ) ) . in other words , if a decoherence mechanism leads to a redundant production of classical information records about the system , and hence to objectively existing classical state of @xmath0 , it is necessarily achieved ( in the asymptotic limit ) through spectrum broadcasting . conversely , a spectrum broadcasting process resulting in a state ( [ br2 ] ) ( with the crucial property ( [ distng ] ) ) leads to the objective existence in the sense of definition [ obj ] of the classical state @xmath10 . indeed , projections on the pointer basis @xmath5 and on the disjoint supports of @xmath37 constitute the preferred , non - disturbing measurements . performing them independently , the observers will all detect the same probability distribution @xmath10 without bohr - disturbing the quantum state of the system ( [ decoh ] ) and the measurements can be repeated arbitrary many times . summarizing , under the assumptions elaborated above , we have proven the following implications , identifying spectrum broadcasting as the physical process responsible for the appearance of the classical objectivity : @xmath51 we also note that the form ( [ br2 ] ) resolves the apparent puzzle appearing within quantum darwinism @xcite and mentioned in the introduction : how can multiple information records be produced during a quantum evolution when state cloning is forbidden in quantum mechanics @xcite ? the answer from ( [ br2 ] ) is that : i ) only state s spectrum is proliferated and ii ) instead of clones rather _ classically correlated copies _ are produced . it may seem that by the time - stability requirement of objectivity , our reasoning may exclude time evolving classical objective states and lead to the classical zeno paradox . this is however not so . moving outside the strict decoherence framework , within which our results have been derived , one can allow for a changing in time pointer basis @xmath52 , and hence probabilities @xmath53 ( cf . ( [ decoh ] ) ) , but evolving on a much slower time - scale than that of the decoherence . this is the case in most of the realistic situations , as the decoherence time - scales are usually very short , and it opens the possibility for objectively existing , time - evolving classical states @xmath54 iff the spectrum broadcast state ( [ br2 ] ) is formed fast enough for every @xmath55 . as a final touch , we quote the results of refs . @xcite on the epistemological versus ontological interpretation of a quantum state itself : under suitable assumptions , a state of a quantum system is a _ property of the system _ rather than a state of knowledge about it . this somewhat strengthens our result and justifies the use of quantum states for studying objective existence : the latter gains a certain ontological status , as it intuitively should . in the studies of quantum darwinism the objective existence has been so far argued based on a single functional condition , which we will call _ quantum darwinism condition _ ( see e.g. refs . @xcite and references therein ) : [ zurek ] i(_s : fe)=h_s , where @xmath56 is the quantum mutual information , @xmath57 stands for the von neumann entropy , and @xmath58=h(\{p_i\})$ ] is the entropy of the decohered state ( [ decoh ] ) . condition ( [ zurek ] ) has been shown to hold in several models , including environments comprised of photons @xcite and spins ( see e.g. ref . @xcite ) . for finite times @xmath55 , the equality ( [ zurek ] ) is not strict and holds within some error @xmath59 , which defines the _ redundancy _ @xmath60 as the inverse of the smallest fraction of the environment @xmath61 , for which @xmath62=[1-\delta(t)]h_s$ ] . when satisfied , ( [ zurek ] ) implies that the mutual information between the system and the environment fraction is a constant function of the fraction size @xmath33 ( up to an error @xmath63 for finite times ) and the plot of @xmath64 against @xmath33 exhibits a characteristic plateau , called the _ classical plateau _ ( see e.g. ref . the appearance of this plateau has been heuristically explained in the quantum darwinism literature as a consequence of the _ redundancy _ : classical information about the system exists in many copies in the environment fractions and can be accessed independently and without perturbing the system by many observers , thus leading to objective existence of a state of @xmath0 @xcite . those far reaching statements has been based only on the condition ( [ zurek ] ) . but the motivation behind using ( [ zurek ] ) to prove the objective existence is somewhat doubtful as it comes solely from the classical world @xcite : in the classical information science condition ( [ zurek ] ) is equivalent to a perfect correlation of both systems @xcite . that is one system has a full information about the other and indeed in a multipartite setting this information thus exists objectively , in accord with the definition [ obj ] . but in the quantum world the situation is very different @xcite : surprisingly , quantum darwinism condition ( [ zurek ] ) alone _ is not sufficient _ to guarantee objectivity in the sense of definition [ obj ] ( see also ref . @xcite in this context ) . it is clear that the spectrum broadcast states ( [ br2 ] ) satisfy ( [ zurek ] ) , but there are also _ entangled _ states satisfying it , thus violating the form ( [ br2 ] ) , derived from the definition [ obj ] as a necessary condition for objectivity . as a simple example consider the following state of two qubits : _ abp p_(a\| 00 + b\| 11 ) + ( 1-p)p_(a\| 01 + b\| 10 ) , where @xmath65 , @xmath66 , @xmath67 and @xmath68 . then the partial state @xmath69 , @xmath70 is diagonal in the basis @xmath71 and moreover @xmath72 ( the binary shannon entropy @xcite ) , so that the quantum darwinism condition holds : @xmath73 , @xmath74 , but the systems are nevertheless entangled , which one verifies directly through the ppt criterion @xcite . thus , by the results of the previous section , we argue that the functional criterion ( [ zurek ] ) is not enough and the objective existence , as defined by definition [ obj ] , should be proven at the _ structural _ level of quantum sates . in particular , if the spectrum broadcasting form ( [ br2 ] ) can be asymptotically derived in a given model , this will guarantee the objective existence . the paradigmatic shift with respect to the earlier works on decoherence theory and quantum darwinism we propose here , is that the core object of the analysis should be the structure of the full quantum state of the system @xmath0 and the observed environment @xmath34 , rather than the partial state of the system only ( decoherence theory ) or information - theoretical functions ( quantum darwinism ) . below we present such a state - based analysis and explicitly derive spectrum broadcasting states in the emblematic example for decoherence theory and quantum darwinism : a small dielectric sphere illuminated by photons ( see e.g. refs . of radius @xmath75 and relative permittivity @xmath76 is illuminated by a constant flux of photons ( represented by green spots ) . the photons constitute the environments @xmath2 of the sphere . the sphere can be at two possible locations @xmath77 and @xmath78 , separated by a distance @xmath79 , much larger than the effective photon wavelengths @xmath80 . photons scatter elastically and slightly differently depending on where the sphere is , but this difference is vanishingly small for each individual scattering the information about the sphere s position is diluted in the photonic environment . however , when grouped into macroscopic fractions , the photons become collectively almost perfectly resolving and the classical information about the sphere becomes available in the environment in multiple copies . we calculate the full post - scattering state of the sphere and a macroscopic fraction of the photons in the dipole approximation @xmath81 and show that this redundant proliferation of information is described by spectrum broadcasting ( [ br2 ] ) . for technical reasons , we use box normalization : the sphere and the photons are enclosed in a large cubic box of edge @xmath82 and the photon momentum eigenstates @xmath83 obey periodic boundary conditions . [ general ] ] we first introduce the model , following the usual treatment ( see e.g. refs . the system @xmath0 is a sphere of radius @xmath75 and relative permittivity @xmath76 , bombarded by a constant flux of photons , which constitute the multiple environments ( see fig . [ general ] ) and decohere the sphere . the sphere can be located only at two positions : @xmath77 or @xmath78 , so that effectively its state - space is that of a qubit @xmath84 with a preferred orthonormal ( due to the mutual exclusiveness ) basis @xmath85 , @xmath86 , which will become the pointer basis . this greatly simplifies the analysis , yet allows the essence of the effect to be observed . the sphere is sufficiently massive , compared to the energy of the incoming radiation , so that the recoil due to the scattering photons can be totally neglected and photons energy is conserved , i.e. the scattering is _ elastic_. the environmental photons are assumed not energetic enough to individually resolve the sphere s displacement @xmath87 : [ soft ] kx1 , where @xmath88 is some characteristic photon momentum ( the exact sens of it will be clear in what follows ) . otherwise , each individual photon would be able to resolve the position of the sphere and studying multiple environments would not bring anything new . on the technical side , following the traditional approach @xcite , we describe the photons in a simplified way using box normalization : we assume that the sphere and the photons are enclosed in a large box of edge @xmath82 and volume @xmath89 ( see fig . [ general ] ) and photon momentum eigenstates @xmath83 obey periodic boundary conditions . although a more rigorous treatment was developed in ref . @xcite with well localized photon states , we choose this traditional heuristic approach as , at the expense of a mathematical rigor , it allows to expose the physical situation more clearly , without unnecessary mathematical details ( we remark that the findings of ref . @xcite agree , up to an insignificant numerical factor , with the previous works using box normalization ) . after dealing with formally divergent terms , we remove the box through the thermodynamic limit ( signified by @xmath90 ) @xcite : [ thermod ] v , n , = , that is we expand the box and add more photons , keeping the photon density constant , as the relevant physical quantity is the radiative power , proportional to @xmath91 . the thermodynamic limit is crucial in the sense that it defines micro- and macroscopic regimes , which will turn to be qualitatively very distinct . the detailed dynamics of each _ individual _ scattering is irrelevant the individual scatterings are treated asymptotically in time . the interaction time @xmath55 enters the model differently , thought the number of scattered photons . it may be called a macroscopic time . assuming photons come from the area of @xmath92 ( see fig . [ general ] ) at a constant rate @xmath93 photons per volume @xmath94 per unit time , the amount of scattered photons from @xmath95 to @xmath55 is : [ nt ] n_tl^2ct , where @xmath96 is the speed of light . throughout the calculations we work with a fixed time @xmath55 and pass to the asymptotic limit @xmath97 ( signified by @xmath8 or @xmath22 ) at the very end . since multiphoton scatterings can be neglected and all the photons are treated equally ( _ symmetric environments _ ) , the effective sphere - photons interaction up to time @xmath55 is of a controlled - unitary form : [ u ] u_s : e(t)_i=1,2\| x_i x_i \|_n_t , where ( assuming translational invariance of the photon scattering ) @xmath98 is the scattering matrix ( see e.g. ref . @xcite ) when the sphere is at @xmath99 , @xmath100 is the scattering matrix when the sphere is at the origin , and @xmath101 is the photon momentum operator . due to the elastic scattering , @xmath102 s have non - zero matrix elements only between the states @xmath83 of the same energy @xmath103 . in the sector ( [ soft ] ) the interaction ( [ u ] ) is vanishingly small at the level of each _ individual _ photon @xcite : in the thermodynamic limit @xmath104 ( in a suitable sense we clarify later ) , and hence @xmath105 . surprisingly , this will not be true for macroscopic groups of photons . we also note that unlike in the previous treatments @xcite , already at this moment we explicitly include in the description _ all _ the photons scattered up to the fixed time @xmath55 . finally , the preferred role of the basis @xmath106 is already singled out now by the form of the interaction ( [ u ] ) @xcite . following our critique of the quantum darwinism condition ( [ zurek ] ) , we analyze the model at the level of states . we need several ingredients . first , the initial , pre - scattering in state , is as usually assumed a _ full product _ : [ init ] _ s : e(0)^s_0(^ph_0)^n_t , with @xmath107 having coherences in the preferred basis @xmath106 and @xmath108 some initial states of the photons ( the environments are by assumption symmetric ) . next , we introduce a crucial _ environment coarse - graining _ @xcite : the full environment ( i.e. all the @xmath109 photons ) is divided into a number of _ macroscopic fractions _ , each containing @xmath110 photons , @xmath111 ( fig . [ division ] ) . by _ macroscopic _ we will always understand scaling with the total number of photons @xmath109 . by definition , these are the environment fractions accessible to the independent observers from section [ tw ] . such a division may seem artificial and arbitrary , as e.g. the choice of @xmath112 is unspecified . however , observe that in typical situations detectors used to monitor fractions of the environment , e.g. eyes , have some minimum detection thresholds some minimum amount of radiative energy delivered in a given time interval is needed to trigger the detection . each macroscopic fraction @xmath110 is meant to reflect that detection threshold . its concrete value ( the fraction size @xmath112 ) is for our analysis irrelevant it is enough that it scales with @xmath109 . this coarse - graining procedure is analogous to the one used e.g. in the description of liquids @xcite : each point of a liquid ( a macro - fraction @xmath112 here ) is in reality composed of a suitable large number of microparticles ( individual photons ) . it is also employed in mathematical approach to von neumann measurements using , so called , macroscopic observables ( see e.g. ref . @xcite and the references therein ) . thus , we divide the detailed initial state of the environment @xmath113 into @xmath114 macroscopic fractions : _ n_t&=&_mn_t _mn_t + & & _ m,[init_mac ] where @xmath115 is the initial state of each macroscopic fraction ( _ macro - state _ for brevity ) . ( [ nt ] ) ) , into @xmath116 equal macroscopic fractions @xmath110 . only one fraction ( bounded by the red cubic cage ) is shown for clarity . the macro - fractions represent sensitivity of the detectors used to observe the scattered photons , e.g. an eye . the exact size of the fraction given by the number @xmath117 $ ] is irrelevant for our analysis , it is enough that it scales with the total photon number @xmath109 . [ division ] ] after all the @xmath109 photons have scattered , the asymptotic ( in the sense of the scattering theory ) out-state @xmath118 , is given from eqs . ( [ u],[init],[init_mac ] ) by & & _ s : e(t)= + & & _ i=1,2x_i \|^s_0 x_i\| x_i x_i \|_m[se1 ] + & & + _ ijx_i \|^s_0 x_j\| x_i x_j \|_m[se2 ] where [ rho_i ] _ i^mac(t)(s_i^ph_0 * s_i^)^mn_t , i=1,2 . by the argument of section [ tw ] , in order to have a chance to observe the broadcasting state ( [ br2 ] ) , we trace out some of the environment . in the current model it is important that the forgotten fraction must be _ macroscopic _ : we assume that @xmath119 , @xmath120 out of all @xmath116 macro - fractions of eq . ( [ init_mac ] ) are observed , while the rest , @xmath121 , is traced out . the resulting partial state reads ( cf . ( [ se1],[se2 ] ) ) : [ sfe ] & & _ s : fe(t)=_i=1,2x_i \|^s_0 x_i\| x_i x_i & & + _ ijx_i \|^s_0 x_j(_i^ph_0 s_j^)^(1-f)n_t\| x_i x_j \| + & & ( s_i^ph_0 s_j^)^fn_t . [ i ne j ] we finally demonstrate that in the soft scattering sector ( [ soft ] ) , the above state is asymptotically of the broadcast form ( [ br2 ] ) by showing that in the deep decoherence regime @xmath4 two effects take place : 1 . the coherent part @xmath122 given by eq . ( [ i ne j ] ) vanishes in the trace norm : [ znikaogon ] 2 . the post - scattering macroscopic states @xmath123 ( cf . ( [ rho_i ] ) ) become perfectly distinguishable : [ nonoverlap ] _ 1^mac(t)_2^mac(t)0 , or equivalently using the generalized overlap @xcite : & & b + & & 0,[nonoverlap_norm ] despite of the individual ( microsopic ) states becoming equal in the thermodynamic limit . the first mechanism above is the usual decoherence of @xmath0 by @xmath34the suppression of coherences in the preferred basis @xmath106 . some form of quantum correlations may still survive it , since the resulting state ( [ i = j ] ) is generally of a classical - quantum ( cq ) form @xcite . those relict forms of quantum correlations are damped by the second mechanism : the asymptotic perfect distinguishability ( [ nonoverlap ] ) of the post - scattering macro - states @xmath123 . thus , the state @xmath35 becomes of the spectrum broadcast form ( [ br2 ] ) for the distribution : [ pi ] p_i = x_i \|^s_0 x_i , which by implications ( [ impl ] ) gains objective existence in the sense of definition [ obj ] . for greater transparency , we first demonstrate the mechanisms ( [ znikaogon],[nonoverlap ] ) , and hence a formation of the broadcast state ( [ br2 ] ) , in a case of pure initial environments : [ pure ] _ ph^0\| k_0 k_0 same momenta @xmath124 , @xmath125 , satisfying ( [ soft ] ) . to show ( [ znikaogon ] ) , observe that @xmath122 , defined by eq . ( [ i ne j ] ) , is of a simple form in the basis @xmath106 : [ matrix ] _ s : fe^ij(t)= , where @xmath126 and @xmath127 . since @xmath102 s are unitary and @xmath128 , @xmath129 , we obtain : & & \|\|_s : fe^ij(t)\|\|_= + & & \|\|(s_1^ph_0 s_1^)^fn_t+\|\|(s_2^ph_0 s_2^)^fn_t + & & = 2\|x_1 \|^s_0 x_2\|\|_1^ph_0 s_2^\|^(1-f)n_t[\|i ne j\| ] the decoherence factor @xmath130 for the pure case ( [ pure ] ) has been extensively studied before ( see . e.g. refs . let us briefly recall the main results . under the condition ( [ soft ] ) and using the classical cross section of a dielectric sphere in the dipole approximation @xmath131 , one obtains in the box normalization : [ psipsi ] & & k_0\|s_2^_1k_0=1+i + & & - ( 3 + 11 ^ 2)+o , where @xmath132 is the angle between the incoming direction @xmath133 and the displacement vector @xmath134 and @xmath135^{1/3}$ ] . this implies : & & \|_1^ph_0 s_2^\|^(1-f)n_t=\|k_0\|s_2^_1k_0\|^(1-f)n_t + & & ^l^2(1-f)ct[linia2 ] + & & e^-t.[decay_ogon ] in the second line above we used eq . ( [ psipsi ] ) up to the leading order in @xmath136 ; in the last line we removed the box normalization through the thermodynamical limit ( [ thermod ] ) and thus obtained the decoherence time @xcite : ^-1x^2 c k_0 ^ 6 a^6 ( 3 + 11 ^ 2 ) . ( [ \|i ne j\|],[decay_ogon ] ) imply that @xmath137 , since the sequence @xmath138 is monotonically increasing . as a result , whenever we forget a macroscopic fraction of the environment ( @xmath139 ) , the resulting coherent part @xmath122 decays in the trace norm exponentially , with the characteristic time @xmath140 . this completes the first step ( [ znikaogon ] ) . the asymptotic orthogonalization ( [ nonoverlap ] ) is also straightforward to show in the case of pure environments . the post - scattering states of the environment macro - fractions , eq . ( [ rho_i ] ) , are all pure : _ i^mac(t)=(s_i\| k_0 k_0 \|s_i^)^mn_t\| _ i^mac(t ) _ i^mac(t ) \| , so it is enough to consider their overlap : & & \|_2^mac(t)\|_1^mac(t)\|=\|k_0\|s_2^_1k_0\|^l^2mct + & & e^-t.[ortogonalizacja ] thus , for @xmath23 the states of the macro - fractions @xmath141 _ asymptotically orthogonalize _ and moreover on the same timescale @xmath3 as the decay of the coherent part described by eq . ( [ ortogonalizacja ] ) ( note that @xmath142 so the timescales from eqs . ( [ decay_ogon],[ortogonalizacja ] ) do not differ considerably ) . this shows the asymptotic formation of the broadcast state ( [ br2 ] ) with pure encoding states @xmath143 : & & _ s : fe(0)=^s_0_fm _ s : fe()= + & & _ i=1,2p_i\| x_i x_i \| _ fm , + [ b - state ] where @xmath7 is given by eq . ( [ pi ] ) and @xmath144 emerges as the non - disturbing environmental basis in the space of each macro - fraction , spanning a two - dimensional subspace , which carries the correlation between the macro - fraction and the sphere ( this basis depends on the initial state @xmath145 ) . thus , the correlations become effectively among the qubits . the full process ( [ b - state ] ) is a combination of the measurement of the system in the pointer basis @xmath106 and spectrum broadcasting of the result , described by a cc - type channel @xcite : ^sfe_(_0^s)_ix_i \|^s_0 x_i\| i^mac i^mac \|^fm . [ l ] quantum darwinism condition ( [ zurek ] ) and the classical plateau follow now form the eq . ( [ b - state ] ) : [ qd ] i[_s : fe(t)]h_s , because of the conditions ( [ znikaogon],[nonoverlap_norm ] ) ( see appendix [ ihs ] for the details ) . thus the mutual information becomes asymptotically independent of the fraction @xmath33 ( as long as it is macroscopic ) . we stress that in our analysis eq . ( [ qd ] ) is derived as a consequence of the spectrum broadcasting . in quantum darwinism simulations for finite , fixed times @xmath55 ( see e.g. refs . @xcite ) , one can observe that the formation of the plateau is stronger driven by increasing the time rather than the macro - fraction @xmath33 ( keeping all other parameters equal ) . this can be straightforwardly explained by looking at the eqs . ( [ decay_ogon],[ortogonalizacja ] ) : the fractions @xmath146 are by definition at most @xmath147 , and hence have little effect on the decay of the exponential factors , while @xmath55 can be arbitrarily greater than @xmath3 , thus accelerating the formation of the broadcast state ( [ b - state ] ) . , corresponding to the sphere being at @xmath148 ( represented by the small solid slabs on the left ) become identical in the thermodynamic limit ( cf . ( [ micro ] ) ) and hence completely indistinguishable . they carry vanishingly small amount of information about the sphere s localization , which is due to the assumed weak coupling between the sphere and each individual environmental photon ( [ soft ] ) . on the other hand , the collective states of _ macroscopic _ fractions @xmath149 ( represented by the big solid slabs on the right ) become by eq . ( [ ortogonalizacja ] ) more and more distinguishable in the thermodynamic ( [ thermod ] ) and the deep decoherence @xmath23 limits . together with the decoherence mechanism ( [ znikaogon ] ) this leads to a formation of the spectrum broadcast state ( [ br2 ] ) with pure environmental states , and hence to the objective existence of the ( classical ) state of the sphere in the sense of definition [ obj ] . [ fig_ort ] ] there is a very distinct difference in the macro- and microscopic behavior of the environment , already alluded to in refs . @xcite . from eq.([psipsi ] ) it follows that within the sector ( [ soft ] ) the post - scattering states of _ individual _ photons ( _ micro - states _ ) @xmath150 , become identical in the thermodynamic limit and hence encode no information about the sphere s localization : [ micro ] ^mic_2\|^mic_1k_0\|s_2^_1k_01 . this is not surprising due to the condition ( [ soft ] ) . on the other hand , and despite of it , by eq . ( [ ortogonalizacja ] ) macroscopic groups of photons are able to resolve the sphere s position and in the asymptotic limit resolve it perfectly ( fig . [ fig_ort ] ) . it leads to an appearance of different information - theoretical phases in the model , which we now describe . we stress that the macro - fraction @xmath112 can be arbitrarily small ( which only prolongs the orthogonalization time , cf . ( [ ortogonalizacja ] ) ) , but must scale with the total number of photons @xmath109 . indeed , for a microscopic , i.e. not scaling with @xmath109 , fraction @xmath151 the limit ( [ micro ] ) still holds : @xmath152^\mu\xrightarrow{\text{therm.}}1 $ ] . thus , if the observed portion of the environment is _ microscopic _ , the asymptotic post - scattering state is in fact a product one : & & _ s : e(0)=^s_0(_0^mac)^ _ s : e()= + & & _ i=1,2p_i\| x_i x_i \| ( s_i\| k_0 k_0 \|s_i^)^= + & & ( _ i=1,2p_i\| x_i x_i \| ) \| ^mic ^mic \|^,[product ] where @xmath153 because of eq . ( [ micro ] ) ( and @xmath90 denotes equality in the thermodynamic limit ( [ thermod ] ) ) . we call it a product phase , in which @xmath154=0 $ ] . conversely , if we have access to the full environment , ignoring perhaps only a microscopic fraction @xmath151 , the arguments leading to eqs . ( [ decay_ogon],[ortogonalizacja ] ) do not work anymore , since from eq . ( [ micro ] ) : \|_1^ph_0 s_2^\|^ 1 , and thus there is no decoherence nor orthogonalization . the post - scattering state contains then the full _ quantum _ information about the system due to the unsuppressed system - environment entanglement produced by the controlled - unitary interaction ( [ u ] ) . as a result , the mutual information attains in the thermodynamical limit its maximum value @xmath155 ( equal to @xmath156 for a pure @xmath157 ) and we call this regime a full information phase . we note that the rise of @xmath158 above @xmath159 certifies the presence of entanglement @xcite . the intermediate phase described by eq . ( [ b - state ] ) , we propose to call a broadcasting phase . the resulting schematic phase diagram is presented in fig . [ phase ] . ) and in the deep decoherence regime @xmath4 . the horizontal axis is the macroscopic fraction @xmath33 of the environment @xmath2 under the observation . vertical axis represents the asymptotic mutual information between the system @xmath0 and the macroscopic fraction @xmath34 , @xmath160 $ ] . the plot shows two phase transitions : the first one occurs at @xmath161 from the product phase of eq . ( [ product ] ) to the broadcasting phase @xmath162 of eq . ( [ b - state ] ) . the second one is from the broadcasting phase to the full information phase at @xmath163 , when the observed environment is quantumly correlated with the system . due to the thermodynamic limit each value of the fraction @xmath33 should be understood modulo a microscopic fraction , i.e. a fraction not scaling with the total photon number @xmath109 ( cf . ( [ nt ] ) ) . [ phase ] ] the quantity experiencing discontinuous jumps is the mutual information between the system @xmath0 and the observed environment @xmath34 , and the parameter which drives the phase transitions is the fraction size @xmath33 . as discussed above , each value of @xmath33 has to be understood modulo a micro - fraction . the appearance of the phase diagram is a reflection of both the thermodynamic and the deep decoherence limits and its form is in agreement with the previously obtained results ( see e.g. refs . @xcite ) . we now move to a more general case when the environmental photons are initially in a mixed state . unlike in the previous studies ( see e.g. refs . @xcite ) , we will not assume the thermal blackbody distribution of the photon energies , but consider a general state , diagonal in the momentum basis @xmath83 and concentrated around the energy sector ( [ soft ] ) : [ mu ] ^ph_0=_k p(k ) \| k k eigenstates @xmath83 are discrete box states and the summation is over the box modes . the partial post - scattering state @xmath164 is given by the same eqs . ( [ rho_i]-[i ne j ] ) with the above @xmath108 . the first step ( [ znikaogon ] ) , i.e. the decay of the coherent part , is the same as before , as nowhere in eqs . ( [ matrix]-[\|i ne j\| ] ) the purity was used , but the decoherence factor is now modified . in the leading order in @xmath136 it reads @xcite : & & \|_1^ph_0 s_2^\|^(1-f)n_t + & & ^(1-f)n_t[bareta ] + & & , [ decay_ogon_mix ] where the modified decoherence time @xmath165 is given by @xcite : [ tau_d_mix ] ^-1x^2 c a^6 k^6 ( 3 + 11 ^ 2_k ) , and @xmath166 denotes the averaging with respect to @xmath167 . completing the second step ( [ nonoverlap_norm ] ) is more involved ( our calculation is partially similar to that of ref . we first calculate the bhattacharyya coefficient @xmath168 for the individual states @xmath169 . let : ^mic_2_1(_k , k \|)s_1^,[m0 ] where : m_k k _ kp(k ) k\| s_1^_2 kk\| s_2^_1 k . + [ m ] by eq . ( [ mu ] ) it is supported in the sector ( [ soft ] ) , and we diagonalize it in the leading order in @xmath136 . for that , we first decompose matrix elements @xmath170 in @xmath136 and keep the leading terms only . let us write : _ 1^_2=1 * -(1 -s_1^_2)-b . matrix elements of @xmath171 between vectors satisfying ( [ soft ] ) are of the order of @xmath136 at most . indeed , by eq . ( [ psipsi ] ) the diagonal elements @xmath172 . the off - diagonal elements are , in turn , determined by the unitarity of @xmath173 and the order of the diagonal ones : @xmath174 for any fixed @xmath175 satisfying ( [ soft ] ) ( there is a single sum here ) , where we again used eq . ( [ psipsi ] ) . hence : [ offdiag ] _ k _ kk\|b_kk\|^2=_kk\|k\|s_1^_2k\|^2 = o ( ) . as a byproduct , by the above estimates in the energy sector ( [ soft ] ) , @xmath176 in the strong operator topology : @xmath177 for any @xmath178 from the subspace defined by ( [ soft ] ) . coming back to @xmath170 , from eqs . ( [ psipsi],[offdiag ] ) in the leading order : m_k k&=&p(k)^2_kk-p(k)^3/2b^_kk + & -&p(k)^3/2b_kk+ o().[m ] the first term is non - negative and is of the order of unity , while the rest is of the order @xmath136 and forms a hermitian matrix . we can thus calculate the desired eigenvalues @xmath179 of @xmath170 using standard , stationary perturbation theory of quantum mechanics ( see e.g. ref . @xcite ) , treating the terms with the matrix @xmath171 as a small perturbation . assuming a generic non - degenerate situation ( the measure @xmath167 in eq . ( [ mu ] ) is injective ) , we obtain : [ m ] m(k)=p(k)^2(1-b^_kk - b_kk)+o ( ) , and : & & = m + & & _ kp(k)_kp(k)(1-b_kk ) + & & = 1+_k(-p(k))= + _ k\|k\|s_1^_2k\|^2 + & & + _ k_kk\|k\|s_1^_2k\|^2 1-,[rhorho ] where we have used eqs . ( [ m0],[m],[psipsi],[m ] ) in the respective order , and introduced : & & \|(1-_kp(k)\|k\|s_1^_2k\|^2 ) ( c)^-1[eta ] + & & \|_k_kkp(k)\|k\|s_1^_2k\|^2 [ eta ] ( in eq . ( [ eta ] ) we have used eqs . ( [ bareta],[tau_d_mix ] ) ) . this implies for the micro - states : b(^mic_1,^mic_2 ) = 1 - 1 , [ ort_micmix ] since @xmath180 are of the order of unity in @xmath136 by eqs . ( [ offdiag],[eta ] ) . thus , under ( [ soft ] ) , the states @xmath181 become equal . this is the mixed stated analog of eq . ( [ micro ] ) , employing the generalized overlap @xmath168 . passing to the macro - states @xmath182 ( cf . ( [ rho_i ] ) ) , we in turn obtain : & & b= ( ) ^mn_t + & & ( 1-)^mn_t , [ ort_mix ] where @xcite : [ alpha ] and we have used eq . ( [ eta ] ) . thus , whenever @xmath183 , the macroscopic states satisfy @xmath184\approx 0 $ ] for @xmath185 , despite eq . ( [ ort_micmix ] ) . that is , they become supported on orthogonal subspaces and hence perfectly distinguishable through orthogonal projectors on their supports @xcite . the latter are within the subspaces @xmath186 ( cf . ( [ mu ] ) ) , rotated by @xmath187 and @xmath188 respectively . this shows the asymptotic formation of the spectrum broadcasting state ( [ br2 ] ) : [ b - state_mix ] _ s : fe()=_i=1,2p_i\| x_i x_i \| ^fm with @xmath189 , and hence the objective existence in the sense of definition [ obj ] of the classical state ( [ pi ] ) of the sphere for the mixed environments ( [ mu ] ) . thus , all our previous pure - case findings apply equally well here too : for @xmath190 we asymptotically observe the broadcasting phase ( [ b - state_mix ] ) and recover the quantum darwinism condition ( [ zurek ] ) by the same eq . ( [ qd ] ) ( see appendix [ ihs ] for the details ) . moreover , from eqs . ( [ ort_micmix],[ort_mix ] ) , all the pure - case considerations regarding micro- and macro - regimes ( cf . ( [ micro ] ) and the following paragraphs ) hold true and the same phase diagram of fig . [ phase ] emerges . this is a deep feature of the model . however , there is one remarkable difference with respect to the pure case . comparing eqs . ( [ decay_ogon_mix ] ) and ( [ ort_mix ] ) one sees that in the mixed case the timescales of decoherence ( [ znikaogon ] ) and distinguishability ( [ nonoverlap ] ) are a priori different : @xmath165 and @xmath191 respectively . since @xmath192 the latter time is in general larger and the broadcast state is fully formed for @xmath193 . mixedness of the environment thus slows down the process of formation of the broadcast state . if the difference @xmath194 is sufficiently large , then for the intermediate times @xmath195 the state @xmath164 is approximately a cq state , whose mutual information is given by the holevo quantity @xcite : @xmath196=s_{\text{vn}}\left[\sum_i p_i \varrho_i^{mac}(t)^{\otimes fm}\right ] -(fm)\sum_i p_i s_{\text{vn}}[\varrho_i^{mac}(t)]$ ] . those different time scales were already discovered and discussed in ref . @xcite , where @xmath197 was called the environment receptivity and @xmath198 the redundancy rate . however , the presented physical interpretations of those quantities were rather heuristic , based loosely on the quantum darwinism condition ( [ zurek ] ) and not grounded in the full state analysis , as we have presented above . moreover , the measure @xmath167 studied in ref . @xcite was of a special , product form : @xmath199 , where @xmath200 is the thermal distribution of the energies and the photons were assumed to come from a portion of the celestial sphere of an angular measure @xmath201 . above , we have shown the effect for a general , diagonal in the momentum eigenbasis state ( [ mu ] ) . let us recall after refs . @xcite that for an isotropic illumination when @xmath202 ( all the directions are equally probable ) , @xmath203 @xcite and there is no broadcasting of the classical information : perfectly mixed directional states of the photons can not store any localization information of the sphere , neither on the micro- nor at the macro - level ( cf . ( [ ort_micmix],[ort_mix ] ) ) . by eqs . ( [ decay_ogon],[ortogonalizacja ] ) and eqs . ( [ decay_ogon_mix],[ort_mix ] ) , the asymptotic formation of the spectrum broadcast states relies , among the other things , on the full product form of the initial state ( [ init ] ) and the interaction ( [ u ] ) in each block @xmath36 . however , from the same equations it is clear that one can allow for correlated / entangled fractions of photons , as long as they stay microscopic , i.e. do not scale with @xmath109 . the corresponding terms then factor out in front of the exponentials in eqs . ( [ decay_ogon],[ortogonalizacja],[decay_ogon_mix],[ort_mix ] ) and the formation of the spectrum broadcast states is not affected . we finish with a surprising application of the classical perron - frobenius theorem @xcite , leading to `` singular points '' of decoherence . let the initial state of the sphere be @xmath204 . then , in the spectrum broadcast states ( [ b - state],[b - state_mix ] ) there appears a ( unitary-)stochastic matrix @xmath205 ( cf . ( [ pi ] ) ) . by the perron - frobenius theorem it possesses at least one stable probability distribution @xmath206 : @xmath207 and such a distribution exists for _ any _ initial eigenbasis @xmath208 of @xmath0 . let us now choose it as the spectrum of the initial state @xmath157 : @xmath209 . then , the scattering process ( [ u ] ) not only leaves this distribution unchanged , but broadcasts it into the environment : & & ( _ 0^mac)^fm _ s : fe()= + & & = _ i(_jp_ij()_j())\| x_i x_i \| ( _ i^mac)^fm + & & = _ i_i()\| x_i x_i \| ( _ i^mac)^fm . the initial spectrum does not decoherethat is why we have called it a singular point of decoherence . this perron - frobenius broadcasting process , first introduced in ref . @xcite , can thus be used to faithfully ( in the asymptotic limit above ) broadcast the classical message @xmath210 through the environment macro - fractions . in this work we have identified spectrum broadcasting of ref . @xcite , a significantly weaker form of quantum state broadcasting , as the fundamental quantum process , which leads to objectively existing classical information . more specifically , adopting the multiple environments paradigm , the suitable definition of objectivity ( definition [ obj ] ) , and bohr s notion of non - disturbance , we have proven that the only possible process which makes transition from quantum state information to the classical objectivity is spectrum broadcasting . this process constitutes a formal framework and a physical foundation for the quantum darwinism model , which , as we have pointed out , in its information - theoretical form does not produce a sufficient condition for objectivity , since it allows for entanglement . we have shown that in the presence of decoherence , spectrum broadcasting is a necessary and sufficient condition for the objective existence of a classical state of the system . it filters a quantum state and then broadcasts its spectrum i.e. a classical probability distribution , in multiple copies into the environment , making it accessible to the observers . in the picture of quantum channels , this redundant classical information transfer from the system to the environments is described by a cc - type channel . we have illustrated spectrum broadcasting process on the emblematic example for decoherence theory : a small dielectric sphere embedded in a photonic environment . in particular , we have explicitly shown the asymptotic formation of a spectrum broadcasting state for both pure and general ( not necessarily thermal ) mixed photon environments . then , we have derived in the asymptotic limit of deep decoherence the information - theoretical phase diagram of the model . depending on the observed macroscopic fraction @xmath33 of the environment , it shows three phases : the product , broadcasting and full information phase , and is a complete agreement ( up to some error @xmath63 for finite times ) with the classical plateau of the original quantum darwinism studies . there are two phase transitions taking place : i ) from the product phase to the broadcasting phase ( at @xmath211 ; ii ) from the broadcasting phase ( @xmath162 ) to the full information phase ( at @xmath163 ) , when the observed environment becomes quantumly correlated with the system . in addition , we have pointed out that a special form spectrum broadcasting the perron - frobenius broadcasting , can be used to faithfully ( in the asymptotic limit ) broadcast certain classical message through the noisy environment fractions . from an experimental point of view , our work opens a possibility to develop an experimentally friendly framework for testing quantum darwinism . our central object , the broadcast state ( [ br2 ] ) , is in principle directly observable through e.g. quantum state tomography a well developed , successful , and widely used technique . in contrast , the original quantum darwinism condition ( [ zurek ] ) relies on the quantum mutual information and it is not clear how to measure it . we finish with a series of general remarks and questions . first , there is a straightforward generalization of the illuminated sphere model to a situation where classical correlations are spectrum broadcasted @xcite . consider several spheres , each with its own photonic environment , and separated by distances @xmath212 much larger than the photon wavelengths , @xmath213 ( cf . ( [ soft ] ) ) . the effective interaction is then a product of the unitaries ( [ u ] ) , e.g. : u_s_1s_2:e_1e_2(t)_i , j=1,2\| x_i x_i \|\| y_j y_j \| _ i^n_t_j^n_t , for two spheres , where @xmath214 are the spheres positions and @xmath215 are the corresponding scattering matrices , and the asymptotic spectrum broadcast state carries now the joint probability , e.g. @xmath216 ( cf . eq ( [ pi ] ) ) . it is measurable by observers , who have an access to photon macro - fractions , originating from all the spheres . second , in the example we have studied , and in the majority of decoherence models @xcite , the system - environment interaction hamiltonian is of a product form : [ hint ] h_int = g a_s _ k=1^n x_e_k , where @xmath217 is a coupling constant and @xmath218 are some observables on the system and the environments respectively . the pointer basis appears then trivially as the eigenbasis of @xmath219it is arguably put by hand by the choice of @xmath220 . it is then an interesting question if there are more general interaction hamiltonians , without a priori chosen pointer basis , which nevertheless lead to an asymptotic formation of a spectrum broadcast state : [ br3 ] _ s : fe(t)_i i i \|_k _ i^e_k , ^e_k_i^e_k_ii=0 . are there _ truly dynamical _ mechanisms leading to stable pointer bases and objective classical states ? viewing eq . ( [ br3 ] ) form a different angle , we note that spectrum broadcasting defines a split of information contained in the quantum state @xmath221 into classical and quantum parts . as it is well known , every quantum state can be convexly decomposed in many ways into mixtures of pure states , so a priori such a split does not exist . some additional process is needed . spectrum broadcasting is an example of it : by correlating to the preferred basis @xmath5 , it endows the corresponding probabilities @xmath222 with objective existence , in the sense of definition [ obj ] , and defines them as a `` classical part '' of @xmath6 , leaving the states @xmath223 as a `` quantum part '' ( cf . no - local - broadcasting theorem of ref . @xcite ) . third , there appears to be a deep connection between the non - signaling principle and objective existence in the sense of definition [ obj ] : the core fact that it is at all possible for observers to determine _ independently _ the classical state of the system is guaranteed by the non - signaling principle : @xmath224 . there is no contradiction with the bohr - nondisturbance , as the latter is a strictly _ stronger _ condition than the non - signaling @xcite(this is the core of bohr s reply @xcite to epr ) . in fact , the above connection reaches deeper than quantum mechanics . in a general theory , where it is possible to speak of probabilities @xmath225 of obtaining results @xmath226 when performing measurements @xmath227 ( however defined ) , whatever the definition of objective existence may be , the requirement of the _ independent _ ability to locally determine probabilities by each party seem indispensable . this is guaranteed in the non - signaling theories , where all @xmath225 s have well defined marginals . in this sense non - signaling seems a _ prerequisite of cognition_. this connection will be the subject of a further research . finally , one may speculate on a relevance of our results for life processes . already in 1961 , wigner tried to argue that the standard quantum formalism does not allow for the self - replication of biological systems @xcite . it seemed to be confirmed by the famous no cloning theorem @xcite . however , now we see that cloning is not the only possibility . as we have shown , spectrum broadcasting implies a redundant replication of classical information in the environment . this is indispensable for the existence of life : one of the most fundamental processes of life is watson - crick alkali encoding of genetic information into the dna molecule and self - replication of the dna information . it can not be thus a priori excluded that spectrum broadcasting may indeed open a classical window for life processes within quantum mechanics . this research is supported by erc advanced grant qolaps and national science centre project maestro dec-2011/02/a / st2/00305 . we thank m. piani for discussions on strong independence . p.h . and r.h . acknowledge discussions with k. horodecki , m. horodecki , and k. yczkowski . here we present an independent derivation of the quantum darwinism condition ( [ zurek ] ) for the illuminated sphere model from section [ sphere ] ( cf . ( [ qd ] ) ) . although illustrated on a concrete model , our derivation is indeed more general : instead of a direct , asymptotic calculation of the mutual information @xmath196 $ ] in the model ( cf . refs . @xcite ) , we will show that eq . ( [ zurek ] ) follows from the mechanisms of i ) decoherence , eq . ( [ znikaogon ] ) , and ii ) distinguishability , eq . ( [ nonoverlap_norm ] ) , once they are proven . let the post - interaction @xmath228 state for a fixed , finite box @xmath82 and time @xmath55 be @xmath229 . it is given by eqs . ( [ i = j],[i ne j ] ) and now we explicitly indicate the dependence on @xmath82 in the notation . then : & & \|h_s - i\| + & & \|i - i\|[coh ] + & & + \|h_s - i\|,[ort ] where @xmath230 is the decohered part of @xmath229 , given by eq . ( [ i = j ] ) . we first bound the difference ( [ coh ] ) , decomposing the mutual information using conditional information @xmath231 : i(_s : fe)=s_(_s)-s_(_s : fe\|_fe ) , so that : & & \|i - i\| + & & \|s_-s_\|+[ss ] + & & \|s _ + & & -s_\| . [ scond ] from eq . ( [ soft ] ) , the total @xmath228 hilbert space is finite - dimensional for a finite @xmath232 : there are @xmath233@xmath234 photons ( cf . ( [ nt ] ) ) and the number of modes of each photon is approximately @xmath235 . hence , the total dimension is @xmath236 and we can use the fannes - audenaert @xcite and the alicki - fannes @xcite inequalities to bound ( [ ss ] ) and ( [ scond ] ) respectively . for ( [ ss ] ) we obtain : & & \|s_-s_\| + & & _ e(l , t)(d_s-1)+h , where @xmath237 is the binary shannon entropy and : & & _ e(l , t)\|\|_s(l , t)-^i = j_s(l , t)\|\|_tr + & & = \|\|^ij_s(l , t)\|\|_tr2\|c_12\|^l^2ct [ elt ] with @xmath238 , where we have used the reasoning ( [ matrix]-[decay_ogon ] ) , or ( [ decay_ogon_mix]-[tau_d_mix ] ) for the mixed environments , but with @xmath161 . for ( [ scond ] ) the same reasoning and the alicki - fannes inequality give : & & \|s_-s_\| + & & 4_fe(l , t)d_s+2 h , with : _ fe(l , t ) & & \|\|_s : fe(l , t)-^i = j_s : fe(l , t)\|\|_tr + & = & \|\|^ij_s : fe(l , t)\|\|_tr + & & 2\|c_12\|^l^2(1-f)ct.[eflt ] above @xmath232 are big enough so that @xmath239 . ( [ ss]-[eflt ] ) give an upper bound on the difference ( [ coh ] ) in terms of the decoherence speed ( [ znikaogon ] ) . to bound the `` orthogonalization '' part ( [ ort ] ) , we note that since @xmath230 is a cq - state ( cf . ( [ i = j ] ) ) , its mutual information is given by the holevo quantity @xcite : i=\{p_i,_i^mac(t)^fm } , where @xmath7 is given by eq . ( [ pi ] ) . from the holevo theorem it is bounded by @xcite : [ hol ] i_max(t)\{p_i,_i^mac(t)^fm}h(\{p_i})h_s , where @xmath240 $ ] is the fixed time maximal mutual information , extractable through generalized measurements @xmath241 on the ensemble @xmath242 , and the conditional probabilities read : [ pie ] ^e_j\|i(t)(here and below @xmath36 labels the states , while @xmath243 the measurement outcomes ) . we now relate @xmath244 to the generalized overlap @xmath245 $ ] ( cf . ( [ nonoverlap_norm ] ) ) , which we have calculated in eq . ( [ ort_mix ] ) . using the method of ref . @xcite , slightly modified to unequal a priori probabilities @xmath7 , we obtain for an arbitarry measurement @xmath246 : & & i(^e_j\|ip_i)=i(^e_i\|j^e_j ) = h(\{p_i})-_j=1,2^e_jh(^e_1\|j ) + & & + & & h(\{p_i})-2_j=1,2^e_j + & & = h(\{p_i})-2_j=1,2 , where we have first used bayes theorem @xmath247 , @xmath248 , then the fact that we have only two states : @xmath249 , so that @xmath250 , and finally @xmath251 . on the other hand , @xmath252 @xcite . denoting the optimal measurement by @xmath253 and recognizing that @xmath254 , we obtain : & & i_max(t)ih_s- + & & -2b + & & = h_s-2b^fm inserting the above into the bounds ( [ hol ] ) gives the desired upper bound on the difference ( [ ort ] ) : & & \|h_s - i\|2b^fm + & & where the generalized overlap is given by eq . ( [ ort_mix ] ) : & & b + & & ^l^2mct . [ bh ] gathering all the above facts together finally leads to a bound on @xmath255\right\|$ ] in terms of the speed of i ) decoherence ( [ znikaogon ] ) and ii ) distinguishability ( [ nonoverlap_norm ] ) : & & \|h_s - i\|h+ 2h+ + & & [ gen1 ] + & & 4_fe(l , t)2 + 2b^fm,[gen2 ] where @xmath256 , @xmath257 , @xmath258 $ ] are given by eqs . ( [ elt ] ) , ( [ eflt ] ) , and ( [ bh ] ) respectively . choosing @xmath232 big enough so that @xmath259 ( when the binary entropy @xmath260 is monotonically increasing ) , we remove the unphysical box and obtain an estimate on the speed of convergence of @xmath261 $ ] to @xmath159 : & & _ l\|h_s - i\| h(\|c_12\|e^- ) + & & + 2h(2\|c_12\|e^-t ) + 8\|c_12\|e^-t2 + & & + 2e^-t . this finishes the derivation of the quantum darwinism condition ( [ qd ] ) . we note that the result ( [ gen1],[gen2 ] ) is in fact a general statement , valid in any model where : i ) the system @xmath0 is effectively a qubit ; ii ) the system - environment interaction is of a environment - symmetric controlled - unitary type : let a two - dimensional quantum system @xmath0 interact with @xmath93 identical environments , each described by a finite - dimensional hilbert space , through a controlled - unitary interaction : u(t)_i=1,2iiu_i(t)^n . let the initial state be @xmath262 and @xmath263 . then for any @xmath162 and @xmath55 big enough : & & \|h(\{p_i})-i\|h+ 2h+ + & & [ gen1 ] + & & 4_fe(t)2 + 2b^fn,[gen3 ] where : & & p_ii\|_0^s\|i,_i(t)u_i(t)_0^eu_i(t)^ , + & & _ e(t)\|\|_s(t)-^i = j_s\|\|_tr , + & & _ fe(t)\|\|_s : fe(t)-^i = j_s : fe(t)\|\|_tr . n. bohr , discussions with einstein on epistemological problems in atomic physics in p. a. schilpp ( ed . ) , _ albert einstein : philosopher - scientist _ , library of living philosophers , evanston , illinois ( 1949 ) ; n. bohr , collected works in j. kalckar ( ed . ) , _ foundations of quantum mechanics i ( 1926 - 1932 ) _ vol . 6 , north - holland , amsterdam ( 1985 ) . e. joos , h. d. zeh , c. kiefer , d. giulini , j. kupsch , and i .- o . stamatescu , _ decoherence and the appearancs of a classical world in quantum theory _ , springer , berlin ( 2003 ) ; w. h. zurek , rev . phys . * 75 * , 715 ( 2003 ) ; m. schlosshauer , rev . phys . * 76 * , 1267 ( 2004 ) ; m. schlosshauer , _ decoherence and the quantum - to - classical transition _ , springer , berlin ( 2007 ) . h. d. zeh , found . * 1 * , 69 ( 1970 ) ; h. d. zeh , found . phys . * 3 * , 109 ( 1973 ) ; w. h. zurek , phys . d * 24 * , 1516 ( 1981 ) ; zurek , phys . d * 26 * , 1862 ( 1982 ) ; w. h. zurek , phys . today * 44 * , 36 ( 1991 ) ; h. d. zeh , roots and fruits of decoherence , in b. duplantier , j .- raimond , v. rivasseau ( eds . ) , _ quantum decoherence _ , birkhuser , basel ( 2006 ) . r. brunner , r. akis , d. k. ferry , f. kuchar , and r. meisels , phys . * 101 * , 024102 ( 2008 ) ; a. m. burke , r. akis , t. e. day , g. speyer , d. k. ferry , and b. r. bennett , phys . lett . * 104 * , 176801 ( 2010 ) . let us show eq . ( [ agree ] ) more formally , considering for simplicity only two observers . if one of them measures first and gets a result @xmath36 , then the joint conditional state becomes @xmath264 , @xmath265 and the subsequent measurement by the second observer will yield results @xmath243 with conditional probabilities @xmath266 . if for some @xmath36 , @xmath267 for @xmath13 , then comparing their results after a series of measurements at some later moment , the observers will be confused as to what exactly the state the system @xmath0 was : with the probability @xmath268 the second observer will obtain different states @xmath13 , while the first observer measured the same state @xmath36 . one would not the observers findings objective , unless for every @xmath36 there exists only one @xmath269 such that @xmath270 ( actually @xmath271 , which follows from the normalization @xmath272 , so that the distributions @xmath273 are all deterministic ) . reversing the measurement order and applying the same reasoning , we obtain that for every @xmath243 there can exist only one @xmath274 such that @xmath275 , where by the bayes theorem @xmath276 , @xmath277 . these two conditions imply that the joint probability @xmath278 ( after an eventual renumbering ) . applying the above argument to all the pairs of indices , one obtains eq . ( [ agree ] ) . the fact that cq and qc states carry some form of non - classical correlations has been shown e.g. through the no - local - broadcasting theorem in ref . @xcite or through entanglement activation in m. piani , s. gharibian , g. adesso , j. calsamiglia , p. horodecki , and a. winter , phys . 106 * , 220403 ( 2011 ) . to prove it , we calculate @xmath284 for @xmath202 . or more precisely , since we are working in the box normalization , the measure is @xmath285 , where @xmath286 is the number of the discrete box states @xmath83 with the fixed length @xmath287 . in the continuous limit @xmath286 approaches @xmath288 . as the scattering is by assumption elastic , matrix elements @xmath289 are non - zero only for the equal lengths @xmath290 and hence : [ sectors ] * s_1^_2_k u_k , u_k^u_k=1_k . decomposing the summations over @xmath291 into the sums over the lengths @xmath292 and the directions @xmath293 and using ( [ sectors ] ) , we obtain : & & _ k , kp(k)\|k\|s_1^_2k\|^2= _ k _ n(k),n(k)\|k\|s_1^_2k\|^2= + & & _ k ( p_k s_1^_2 p_k s_2s*_1^ ) = _ k p(k)=1,[1 ] where @xmath294 is a projector onto the subs - space of a fixed length @xmath39 , and hence @xmath295 . comparing with eq . ( [ rhorho ] ) , eq . ( [ 1 ] ) leads to @xmath296 , and hence by definition ( [ alpha ] ) to @xmath203 . e. p. wigner , the probability of the existence of a self - reproducing unit , in _ the logic of personal knowledge : essays presented to michael polany on his seventieth birthday _ , routledge & kegan paul , london ( 1961 ) .	quantum mechanics is one of the most successful theories , correctly predicting huge class of physical phenomena . ironically , in spite of all its successes , there is a notorious problem : how does nature create a bridge from fragile quanta to the robust , objective world of everyday experience ? it is now commonly accepted that the most promising approach is the decoherence theory , based on the system - environment paradigm . to explain the observed redundancy and objectivity of information in the classical realm , zurek proposed to divide the environment into independent fractions and argued that each of them carries a nearly complete classical information about the system . this quantum darwinism model has nevertheless some serious drawbacks : i ) the entropic information redundancy is motivated by a priori purely classical reasoning ; ii ) there is no answer to the basic question : what physical process makes the transition from quantum description to classical objectivity possible ? here we prove that the necessary and sufficient condition for objective existence of a state is the spectrum broadcasting process , which , in particular , implies quantum darwinism . we first show it in general , using multiple environments paradigm , a suitable definition of objectivity , and bohr s notion of non - disturbance , and then on the emblematic example for decoherence theory : a dielectric sphere illuminated by photons . we also apply perron - frobenius theorem to show a faithful , decoherence - free form of broadcasting . we suggest that the spectrum broadcasting might be one of the foundational properties of nature , which opens a window for life processes .
classical lattice statistical - mechanical models with local constraints have been of great interest for decades . by `` local constraint '' , we mean a local rule which restricts the allowed configurations . a famous example is that of the hard - core close - packed dimer model @xcite . the degrees of freedom are dimers stretching between adjacent sites of a lattice , while the hard - core and close - packing constraints mean that each site of the lattice is touched by exactly one dimer . another famous example is baxter s three - color model , where each link is covered by one of three `` colors '' of dimers , with the constraint that each site is touched by all three colors @xcite . another oft - studied constraint is to require that the degrees of freedom be `` loops '' , i.e. one - dimensional objects without ends . for example , both close - packed hard - core dimers and the three - color model can be viewed as loop models . in the latter case , the links colored by two of the colors ( say @xmath1 and @xmath2 ) form closed loops of alternating @xmath1 and @xmath2 colors . since every vertex has one @xmath1 and one @xmath2 touching , the three - color model is therefore equivalent to a fully - packed loop model ( every site has one loop going through it ) . each loop receives a weight @xmath3 , since there are two possible ways of ordering @xmath1 and @xmath2 around each loop . one interesting limit of constrained models is at infinite temperature , where each allowed configuration has the same boltzmann weight . the partition function in this limit is a purely combinatorial quantity : it simply counts the number of configurations . because of the constraints , the physics of such models is still very rich . for example , the three - color model is critical , as is the hard - core close - packed dimer model on the square lattice @xcite . obviously , not all constrained models are critical : dimers on the triangular lattice ( or any non - bipartite lattice ) have exponentially decaying correlators @xcite . the purpose of this paper is to present a constrained lattice model that has several rather interesting properties . there are three equivalent ways of defining the model . one is as an ising model on the honeycomb lattice with a constraint around each hexagon ; one is with a constraint on the three - color model , and a third is as three coupled ising models . the latter form is most naturally given in terms of loops representing ising domain walls , and is also the representation where its properties are most transparent . this model is of interest for several reasons . it is defined in terms of simple local degrees of freedom and constraints , yet exhibits fascinating conservation laws . as we will detail , the transfer matrix decomposes into sectors which are labeled by non - abelian ( and non - local ) charges . the number of distinct sectors _ exponentially _ increases as the size of the system increases . this symmetry enables us to do exact diagonalization of the transfer matrix for systems of sizes up to 36 sites across , i.e. a hilbert space initially of size @xmath4 . we know of no non - trivial system with such a property . despite the fact that the conserved charges are non - local , the configuration space of the model has the striking property that it is connected under simple local moves . even though the three - color model is closely related to ours , to relate all its different configurations requires changing degrees of freedom arbitrarily far apart @xcite . since we give a precise relation between the three - color model and ours , we thus have located the obstruction to connectivity ( the `` 11th '' vertex discussed below ) in the three - color model . not only does this mean that our model is amenable to monte carlo situations , but it should prove interesting to study its classical dynamics @xcite . a new reason to be interested in two - dimensional classical lattice models with constraints comes from _ quantum _ physics . the motivation is to find phases with topological order , where there is no non - vanishing local order parameter , but only non - local ones . the idea for building such a quantum model by starting with a classical magnet with local constraints came long ago @xcite , and a theoretical triumph in proving they exist came from a quantum eight - vertex model @xcite and a quantum dimer model on the triangular lattice @xcite . the two - dimensional quantum models are defined by using each configuration in a two - dimensional classical lattice model as a basis element of the hilbert space . one characteristic of topological order is that the number of ground states depends on the genus of two - dimensional space . constrained lattice models give natural ways of defining the different sectors which , with appropriate choice of hamiltonian @xcite , correspond to different ground states in the quantum theory . when writing the eight - vertex or dimer models as loop models , the different ground states are labeled by the number ( mod 2 ) of loops which wrap around cycles of the torus . in section [ sec : models ] we introduce these models and show that they are equivalent . we relate our model to several others in appendix [ sec : relations ] , enabling us to put upper and lower bounds on the entropy . in section [ sec : flip ] , we discuss the dynamics under local moves , showing that the configurations on the plane or sphere are all connected by simple local moves . on surfaces with non - contractible cycles , we classify the infinitely many separate dynamical sectors . in section [ sec : ft ] , we give arguments which suggest our model is not critical . we also develop in section [ sec : transfer ] and solve in section [ sec : gauge ] a closely related model which has a gauge symmetry . we exploit this symmetry in section [ sec : numerics ] to show how to reduce dramatically the size of the original transfer matrix . this enables us to exactly diagonalize the transfer matrix for quite large lattices , and the results again suggest that the model is not critical . in section 9 , we present our conclusions , and discuss applying our results to build a quantum model with topological order . the model we are introducing can be described in three equivalent ways . here we present them , and then demonstrate their equivalence . [ [ model-1 ] ] model 1 : + + + + + + + + the degrees of freedom of the ising model are `` spins '' @xmath5 taking values of @xmath6 at each site @xmath7 of some lattice . the energy in general from nearest - neighbor interactions is given by @xmath8 the ising model on the honeycomb lattice has a critical point when @xmath9 @xcite . our model 1 is the ising model on the honeycomb lattice , with the constraint that there must be three up spins and three down spins around each hexagon , i.e. @xmath10 this is quite a strong constraint , retaining only 20 of the original 64 possibilities for the spins around each hexagon . we will mostly discuss the infinite temperature limit @xmath11 or @xmath12 , in which each allowed configuration has equal weight . [ [ model-2 ] ] model 2 : + + + + + + + + the degrees of freedom in the three - color model are three colors , say @xmath1 , @xmath2 , and @xmath13 , which are placed on the links on the honeycomb lattice . the usual constraint in the three - color model is to require that at each site of the lattice , all three colors appear . in other words , links of the same color can never touch . when the partition function is simply the sum over all allowed configurations ( i.e. in the infinite - temperature limit ) , the model is critical and integrable @xcite . our model 2 is the three - color model with an additional constraint forbidding configurations which have the same two colors alternating around any given hexagon . in a picture , @xmath14 where @xmath15 can be any of @xmath1 , @xmath2 , or @xmath13 this forbids 6 of the 66 allowed configurations around a hexagon in the three - color model . imposing the constraint ( [ constraint2 ] ) in the fully - packed loop formulation of the three - color model forbids the shortest loops , of length 6 . the constraint is symmetric under permutations of @xmath1 , @xmath2 , and @xmath13 , so it forbids all `` short '' loops , no matter which two colors are chosen to form the loops . [ [ model-3 ] ] model 3 : + + + + + + + + consider now ising spins @xmath16 on the triangular lattice . instead of studying a single ising model on this lattice , we instead consider _ three _ identical ising models , on each of the three identical triangular sublattices of the triangular lattice . this has an ( ising@xmath17 critical point when @xmath18 @xcite . the domain walls for an ising model separate unlike spins ; they live on the links of the dual lattice . for each of our three ising models on triangular lattice , its dual lattice is the honeycomb lattice made up of the sites of the other two ising models . it is not possible for the domain walls of a given ising model to cross or even touch , but the walls of the different decoupled models can cross and touch . our constraint couples the three ising models by not allowing the walls of different models to cross ( although they can touch ) . a configuration in this model is displayed in figure [ fig:3ising ] . in terms of the spins , consider a hexagon on the triangular lattice , comprised of six sites surrounding a given site . label the six spins around this hexagon by @xmath19 , so that @xmath20 , @xmath21 and @xmath22 are in one of the three ising models , while @xmath23 , @xmath24 and @xmath25 are in another . denote @xmath26 , with the subscripts interpreted mod @xmath27 . a domain wall occurs when @xmath28 . the constraint that domain walls not cross in terms of these spins is then @xmath29 where @xmath30 is the site on the triangular lattice at the center of the hexagon . ) requires that the domain walls do not cross . ] the allowed domain walls inside this hexagon are of the types illustrated in figure [ fig:10vertex2 ] below . model 3 can equivalently be described in terms of closed mutually - avoiding loops on the triangular lattice , with the added restriction that loops must turn by @xmath31 degrees at every site . these three models are equivalent to each other under local reformulations of the degrees of freedom . first let us recall the mapping of the three - color model without constraint ( [ constraint2 ] ) to an ising model on the honeycomb lattice @xcite . the ising variables represent chiralities in the three - color model ; this chirality representation occurs in the superconducting - array realization of the three - color model @xcite . consider given configuration in the three - color model . there are six possible configurations of the three - color model around each site of the honeycomb lattice : put an ising spin @xmath32 on the site if the colors on the three links touching it are @xmath33 clockwise , and @xmath34 if the three are @xmath33 counter - clockwise . going around each hexagon , it is easy to check that there are either @xmath35 , @xmath36 or @xmath27 up spins . it is also easy to check for any configuration with @xmath35 , @xmath36 or @xmath27 up spins , one can reverse the map and find a configuration in the three - color model . ignoring boundary conditions , there are three configurations in the three - color model for each in the ising model , so the map can be made one - to - one by specifying the color on one link . the three - color model at infinite temperature therefore maps onto the ising model on the honeycomb lattice with @xmath12 and the requirement the sum of the @xmath5 around hexagon obeys @xmath37 . one can generalize the three - color model to include interactions equivalent to a non - zero @xmath38 if desired ; this is easily done in the domain - wall formulation given below . the equivalence of model 1 to model 2 is now obvious . hexagons in the three - color model with alternating colors as in ( [ constraint2 ] ) correspond to having @xmath39 in the ising model . these are forbidden in model 1 by ( [ constraint1 ] ) , and in model 2 by ( [ constraint2 ] ) . to show the equivalence of model 1 with model 3 , we reexpress the degrees of freedom in model 1 in terms of _ anti_domain walls . every time adjacent spins are different , we draw an antidomain wall on the link of the dual lattice separating them . these antidomain walls therefore form loops on the dual triangular lattice . each of the configurations on each hexagon obeying the constraint ( [ constraint1 ] ) correspond to one of the types of antidomain - wall configurations illustrated in figure [ fig:10vertex ] . there are 10 different configurations of three different types : the empty one , six ( related by 60 degree rotations ) with two antidomain walls , and three ( related by 60 degree rotations ) with four antidomain walls . ising domain walls have a weight @xmath40 per link . since antidomain walls are simply the complement of the domain walls , they can be taken to have weight @xmath41 per link . and @xmath42 to distinguish them from the ising spins in model 3 , which are denoted by @xmath43 . ] these antidomain walls in model 1 correspond to _ walls in model 3 . the triangular lattice for model 3 is simply the dual lattice of the honeycomb lattice for model 1 . the domain walls in a triangular - lattice ising model make a @xmath31 degree turn at every site , just like the antidomain walls in figure [ fig:10vertex ] . if the three ising models in model 3 were decoupled , there would be 16 different domain - wall configurations going through each site of the triangular lattice , because there are four possibilities for each of the two ising models whose domain walls go through this point . there are only 10 possibilities in figure [ fig:10vertex ] . model 1 and model 3 are therefore equivalent if we restrict to these 10 , which are redrawn in figure [ fig:10vertex2 ] . as is obvious from the figures , the ones disallowed are those where the domain walls cross . disallowing crossings leaves exactly the 10 , so the non - crossing constraint is the only one . in figure [ fig:3ising ] , we drew the domain walls for the three different ising models with solid , dashed , and dot - dashed lines , to emphasize the fact that they do not cross , with each forming closed loops . in figure [ fig:10vertex2 ] , we drew these with the dotted and dashed lines , but the same 10 vertices occur at any point on the triangular lattice with the appropriate types of lines . we have therefore shown that model 3 is the same as model 1 , up to unimportant constants in front of the partition functions . our model is therefore a `` 10-vertex model '' on the triangular lattice . these vertices are a subset of those in the 32-vertex model discussed in @xcite . these proofs of course mean that model 2 is equivalent to model 3 as well , so the three - color model with and without ( [ constraint2 ] ) can also be written in terms of a vertex model on the triangular lattice . the usual three - color model also allows the 11th vertex pictured in figure [ fig:11thvertex ] . ) ] this vertex is a source / sink of domain walls , and so the three - color model without constraint ( [ constraint2 ] ) can not be mapped onto three ising models . in model 1 , this 11th vertex corresponds to a hexagon with all up or all down spins , i.e. @xmath39 . in the appendix , we relate our model to two others : hard hexagons on the triangular lattice , and a generalized ising antiferromagnet . our model is found from these by relaxing constraints , so these models give lower bounds on the entropy of ours . an important question in many physical applications of two - dimensional geometrically constrained models is whether the space of states is connected under local moves . it is essential if one is to study either classical or quantum dynamics , and is also useful for doing monte carlo simulations @xcite . for example , to build the quantum models discussed in the introduction , without connectivity under local moves , the hamiltonian is non - local . the three - color model is not connected : any closed loop of bonds of the honeycomb lattice that contains only two colors will give a different configuration if those two colors are permuted . even though there exist short loops on the lattice ( the shortest loop is a single hexagon ) , the space of states is not connected unless the dynamics is able to permute arbitrarily large loops @xcite . in this section we discuss the properties of our model under local dynamics . we show that , unlike the three - color model , the connected sectors can be enumerated simply and correspond to topological classes of sets of nonintersecting loops in the plane . since we have shown that the constraint ( [ constraint2 ] ) turns the three - color model into our model , this result illuminates the reason why the three - color model is not connected by local moves . the most - local dynamics of the ising variables of model 1 that conserves the constraint ( [ constraint1 ] ) is to act on hexagons where the spins alternate up and down around the hexagon . flipping each up spin to down and each down to up around such a hexagon preserves the constraint not only on the original hexagon , but also on each of its six neighbors as well . we display this flip in figure [ fig : flippable ] . in model 3 , this corresponds simply to flipping the ising spin at the center of this hexagon , i.e. sending @xmath44 . this is the only local move necessary to connect configurations . this is easiest to see in the loop representation . since in the model 3 , the flip changes the spin at the center of this hexagon , it simply flips the model-3 loop variables on the hexagon surrounding this hexagon of model 1 . an example is illustrated in figure [ fig : moveloop ] . . ] for example , if all six of the links on the surrounding hexagon are empty , the flip creates a loop of minimal length . if they are all full , this is a minimal - length loop surrounding the hexagon , and the flip removes the loop . in other cases , it shrinks or expands the loop without creating any loose ends . it is now easy to see how the flip connects configurations . a loop of minimal length has a flippable hexagon inside it , so these can be removed by one flip . longer loops can be shrunk and then removed by repeatedly flipping . if there are loops inside other loops , then the ones inside need to be removed first . when space is topologically a sphere , all configurations are therefore connected to the empty one . since all processes can be reversed , this means all configurations on the sphere are connected . when space has non - contractible cycles , however , not all loops can be removed . in order to use the formulation of model 3 , the periodic boundary conditions around a cycle must identify sites of the same triangular sublattice . when this is done , the loops are of three distinct types , as seen in figure [ fig:3ising ] . since flips can not move two loops of different types through each other , loops which wrap around a non - contractible cycle can only be removed if they adjacent to another of the same type . the flip illustrated in figure [ fig : moveloop2 ] turns two adjacent non - contractible loops of the same type into two contractible ones . when space is a cylinder , the different sectors can be enumerated simply : a sector is given by a sequence of loop colors ( those encountered reading from left to right along the cylinder , for example ) , with an even number of adjacent occurrences of the same colors being equivalent to the identity . mathematically , this set is isomorphic to the free group on three elements @xmath45 , with the relations @xmath46 . putting two cylinders next to each other defines a group action on the set of topological sectors , and this group action is non - abelian : for example , @xmath47 is not the same as @xmath48 . [ fig : toposector ] shows an example of how the non - intersection constraint can prevent annihilation of two loops of the same color . each sector corresponds to a conserved charge in the transfer matrix ; we will define these in section [ sec : transfer ] . when building a quantum model based on this classical model , each of these sectors will correspond to a ground state of an appropriately defined hamiltonian . we will discuss the quantum model further in section [ sec : topological ] . finally , when the model is defined on a torus , all noncontractible loops must go around the same cycle . this cycle can be labeled as @xmath49 , where @xmath50 and @xmath51 are integers , and @xmath52 and @xmath53 define the torus . most topological sectors on the torus can be labeled by @xmath54 , where @xmath55 is a non - trivial element of the free group defined above with the additional requirement that products must be interpreted cyclically , and @xmath56 is an element of @xmath57 , the group of modular transformations of the torus . @xmath57 is generated by exchanging @xmath58 , and shifting @xmath59 . topological sectors not of this form are the trivial sector @xmath60 ( i.e. no @xmath56 ) , and sector with a single loop ( i.e. @xmath61 or @xmath62 ) , where @xmath63 . a basic question about our model is if it is critical . we can gain insight into this question by studying the field theories valid near two critical points which occur when by relaxing or increasing the constraints . three decoupled ising models are critical when @xmath38 is appropriately tuned . in the continuum limit , the critical point can be described by using conformal field theory @xcite . one important thing conformal field theory allows one to do is classify all the operators of the theory . the ising model has only two relevant rotationally - invariant ones , the spin field , and the energy operator @xmath64 . perturbing the critical point by the latter corresponds to changing the temperature , i.e. taking @xmath38 away from @xmath65 , so in the lattice model we can identify @xmath66 so that @xmath67 corresponds to a domain wall between @xmath7 and @xmath30 . a useful symmetry of the ising model is kramers - wannier duality , which shows the equivalence of high- and low - temperature partition functions . in terms of the spins / fields , it takes @xmath68 and @xmath69 . to reach our model 3 , one must perturb the ( ising@xmath17 critical point to enforce the constraint @xmath70 from ( [ constraint3 ] ) . by construction , @xmath70 when the domain walls through this hexagon do not cross , and @xmath71 when they cross . thus to reduce the weight of configurations where domain walls cross , we add @xmath72 to the energy with positive coefficient @xmath73 , i.e.@xmath74 where @xmath75 is the energy of three decoupled ising models . the constraint ( [ constraint3 ] ) is enforced in the @xmath76 limit . this perturbation is clearly relevant , since it includes the energy operators @xmath77 in the three individual models , and marginal terms which couple the two models . a key fact to notice is that @xmath72 is not invariant under any of the dualities of the three ising models . an important result in two - dimensional statistical mechanics is the existence of a `` @xmath62-theorem '' @xcite . the @xmath62-theorem says that there is a function @xmath62 of the parameters of the theory satisfying a very important property : it can not increase under renormalization group flows . moreover , at a critical point its value is known from conformal field theory it is a quantity called the central charge . thus if one starts at a known critical point and perturbs by a relevant operator , the fact that @xmath62 must decrease means that either the flow must end up at a critical point with a smaller value of @xmath62 , or at no critical point at all . the ising critical point has @xmath78 . since our model is a relevant perturbation of three ising models , this implies that either our model is not critical , or if it is critical , it should have @xmath79 . the three - color model ( at infinite temperature ) is critical , and has @xmath80 @xcite . thus imposing constraint [ constraint2 ] on the three - color model should move the model away from the three - color critical point . this is in accord with the numerics discussed below . an obvious question is if our model is critical at some value of @xmath38 . while it is conceivable , it does not seem likely . one can cancel the relevant piece @xmath77 of @xmath72 by changing the temperature of the three ising models . this leaves the marginal terms quadratic in @xmath77 . these marginal terms can change the dimensions of operators , so a fine - tuned model could be critical . however , since the constraint @xmath70 violates dualities , this critical point is not likely to be the ( ising@xmath17 one . this argument does not preclude a flow to a critical point with a lesser central charge . one candidate for a flow is the hard - hexagon model . the critical point in this model has @xmath81 ; it is in the same universality class as the three - state potts model @xcite . however , it occurs very far from the hard - hexagon model of interest , for several reasons . first , to get our model , one must allow configurations not present in the hard - hexagon model . second , the latter s critical point occurs when the weight @xmath82 per hard hexagon is @xmath83 . to get model 3 at infinite temperature , the configurations must all be of equal weight , i.e.@xmath84 . perturbing @xmath82 away from @xmath85 is relevant . it is not clear whether allowing the additional configurations is relevant or not . it is conceivable that the two perturbations could effectively cancel , leaving one at the hard - hexagon critical point , but we have no evidence for this . there are no unitary critical points with @xmath86 symmetry and @xmath87 , so our model can not be critical with these central charges . there are several with @xmath88 , so it is conceivable that it could be critical with these central charges , but we have found no evidence for this . in this section we define the transfer matrix , and show that it possesses some remarkable and unusual properties at infinite temperature . we exploit these properties in the next section to solve our model on a different lattice , and then in section [ sec : numerics ] to do numerics on very large systems . consider the model formulated in terms of non - crossing domain walls on the links of the triangular lattice . take the transfer matrix to act perpendicular to one of the three axes . the transfer matrix @xmath89 acts on the space of states on a zig - zag line ; each link is labeled by an index @xmath90 , where @xmath91 is the number of hexagons across the original lattice . we take the convention that links @xmath7 and @xmath92 meet at a vertex when @xmath7 is odd . the degrees of freedom are the domain walls on the links . we denote @xmath93 if there is a domain wall on link @xmath7 , and @xmath94 if there is not . the space of states is then of dimension @xmath95 . the interactions are at the vertices of the lattice : the fact that there are only 10 vertices must be enforced . it is most convenient to write @xmath89 in the form @xmath96 where @xmath97 moves you to the next zig - zag line , which has the property that links @xmath7 and @xmath92 meet at a vertex when @xmath7 is even . @xmath97 therefore imposes the weights at @xmath91 vertices . the operator @xmath98 is the translation operator , which shifts all the spins by one site . the transfer matrix with periodic boundary conditions in both directions is therefore @xmath99 for a lattice of @xmath100 hexagons . since @xmath89 and @xmath101 have the same eigenvectors , the eigenvectors of @xmath89 are the same as those of @xmath102 , and the eigenvalues are simply related . some interesting conservation laws follow immediately from the fact that domain walls do not cross . the total number of domain walls must be conserved mod 2 , so @xmath103 is conserved mod 2 . when @xmath104 is a multiple of three , this conservation law is much more powerful : the transfer matrix locally conserves the number of domain walls mod 2 on _ each _ of the three sublattices . namely , just as the sites can be divided into three sublattices , the links can as well ; these result in the three types of the domain walls illustrated in figure [ fig:3ising ] . the power of the non - crossing constraint is that adjacent domain walls of different types can not change places or annihilate as the transfer matrix evolves the system across the lattice . the distinct sectors on the cylinder described at the end of section [ sec : flip ] are a consequence of this symmetry . this symmetry is already quite powerful . by studying @xmath97 , we find even more remarkable properties . @xmath97 commutes with @xmath105 local symmetry generators , so the model with transfer matrix @xmath97 instead of @xmath89 has a _ gauge symmetry_. there are two different types of local conservation laws . the first one is easy to see . say two consecutive links meeting at a vertex are both occupied , i.e. @xmath106 . then examine the 10 vertices in figure [ fig:10vertex2 ] , and take the transfer matrix to act in the vertical direction . there is only one possible vertex where both incoming links are covered , the last one drawn . this vertex has both outgoing links covered as well . thus acting with @xmath97 keeps @xmath106 , while all other vertices have @xmath107 before and after @xmath97 acts . thus @xmath108 is conserved by @xmath97 for any integer @xmath30 . the second local conservation law is not as obvious . it involves two adjacent vertices connected by a horizontal link . this horizontal link is of the same type as the links @xmath109 and @xmath110 , so an incoming domain wall on these links can turn by 120 degrees onto the horizontal link . this conservation law arises from the facts that there are no allowed vertices which have just one or three walls touching them , and that the number of walls of a given type is conserved _ locally _ mod 2 . to illustrate this , first consider the case where links @xmath109 and @xmath110 are either both occupied , or both unoccupied . computing @xmath97 requires summing over the two possibilities for the horizontal link . when the horizontal link is unoccupied , the only allowed contribution to @xmath97 is to leave the configuration unchanged . when the horizontal link is occupied , the only allowed configuration is that both - occupied annihilates into both - unoccupied , or vice versa . therefore @xmath97 here does not conserve @xmath111 and @xmath112 individually , but it does preserve the number of incoming lines mod 2 . defining @xmath113 , here we have @xmath114 before and after @xmath97 acts . when one of the two links @xmath109 and @xmath110 is occupied and the other unoccupied , @xmath115 . in this case , when the horizontal link is unoccupied , @xmath97 leaves the configuration unchanged , and when the horizontal link is occupied , the two configurations change place . thus @xmath116 remains @xmath117 before and after @xmath97 acts . thus @xmath116 is a local conserved quantity . note that @xmath118 , where @xmath119 is the total number of walls ( which is indeed conserved mod @xmath3 ) . the quantities @xmath120 and @xmath116 are not conserved in the full model , because they do not commute with translation operator @xmath98 . however , in the infinite - temperature case @xmath12 , they do allow the non - zero eigenvalues of the full transfer matrix @xmath89 to be found from much smaller matrices . for example , we show in section [ sec : numerics ] how this gives the largest eigenvalue of @xmath89 for @xmath121 from a 5-by-5 matrix , considerably smaller than the @xmath122 transfer matrix obtained without exploiting any symmetries ! the key simplification in the @xmath12 limit is that @xmath97 becomes a sum of projection operators . precisely , for each set of values of the @xmath120 and @xmath116 , define a matrix @xmath123 acting on the @xmath124 states on the zig - zag line . the matrix elements @xmath125 are defined to be 1 if both states @xmath126 and @xmath127 have the charges @xmath128 , and 0 if either or both do not . then the result is that @xmath129 where the sum is over all possible values of @xmath130 and @xmath131 . note that not all values are possible : for example , if @xmath132 and @xmath133 , then @xmath116 must be @xmath134 as well . the decomposition ( [ up ] ) follows from an extension of the arguments which led to @xmath135=[u , r_j]=0 $ ] . there we saw that each initial state leads to an outgoing state with the same charges at most once . since @xmath12 , all allowed configurations have the same weight 1 , so every entry of @xmath97 must be @xmath35 or @xmath134 . moreover , by explicitly examining all the possibilities for each set of four successive sites @xmath136 and the horizontal links touching the two vertices , it is easy to see that @xmath97 takes any initial state with a given values of @xmath137 and @xmath138 , to any final state with the same values . thus @xmath97 indeed is block diagonal , with each block given by the operator @xmath123 . let us give an explicit example with @xmath139 and periodic boundary conditions . we denote a state with domain walls at @xmath140 by @xmath141 , and the empty state as @xmath142 . consider the sector which has have the same conserved charges as the empty state , which are @xmath143 , and @xmath144 . the other states which have these charges are @xmath145 . it is then easy to check that on these three states @xmath146 there are two states in each of the other sectors with @xmath143 . when @xmath147 , the sector is comprised of @xmath148 and @xmath149 , when @xmath150 it consists of @xmath151 and @xmath152 , and when @xmath153 , it consists of @xmath154 and @xmath155 . within any of these sectors , @xmath156 @xmath97 for each of the 7 states with @xmath157 and/or @xmath158 is diagonal : there is only one state in each sector . the crucial property of @xmath97 at @xmath12 is that it is proportional to a projection operator . namely , the product of two different projection operators is zero , and each @xmath159 @xmath160 , where @xmath51 is the number of states in this sector . each @xmath159 has only a single non - zero eigenvalue @xmath51 , and the corresponding eigenstate is the equal - amplitude sum over all states in the sector . thus most states in the hilbert space are annihilated by @xmath89 . the eigenstates of @xmath89 with non - zero eigenvalues have an important property , following from the fact that all states in the same sector end up with the same coefficient after acting with @xmath97 . since @xmath97 is the last part of of @xmath89 , the final state after acting with @xmath89 must have the same property : all states with the same values of @xmath161 and @xmath162 have the same coefficient in the end . this means that at @xmath12 , all eigenstates of @xmath89 with non - zero eigenvalue must have the same property as well ! we can therefore work in a space of states vastly reduced in size , by keeping just one state in each sector . how to work out the explicit transfer matrix in this reduced basis is explained in section [ sec : numerics ] . we emphasize that the @xmath120 and @xmath116 are not conserved charges for the full transfer matrix @xmath89 , like they are for @xmath97 . the eigenstates of @xmath89 do not have definite values of the @xmath120 and @xmath116 , but are a sum over states with different values . our result here says that for eigenstates of @xmath89 with non - zero eigenvalues , all states in a given sector must have the same coefficient . this is not a symmetry , because the coefficients are not the same for eigenstates with zero eigenvalue . since the matrix @xmath97 commutes with all the local symmetry generators , using it as a transfer matrix results in a model with a gauge symmetry . because of the gauge symmetry , the resulting `` model @xmath97 '' can be reduced to a one - dimensional model and solved exactly . in this respect it is quite similar to the two - dimensional ising gauge theory . however , the solution of model @xmath97 has some very striking properties of its own : the eigenvalues of the transfer matrix are given in terms of fibonacci numbers . we derive this here . model @xmath97 is the ising model with constraint ( [ constraint1 ] ) around each plaquette of the lattice pictured in figure [ fig : modelu ] . it is the square lattice , with an extra site added to all the horizontal links . it is therefore not rotationally invariant . . ] we find explicit expressions for the eigenvalues of @xmath97 in the limit @xmath12 , where we can exploit the fact that its transfer matrix @xmath97 can be written as the sum ( [ up ] ) . this means that the eigenstates are the sum over all states in a given sector , and the corresponding eigenvalue is the number of states in that sector . this turns out to be an amusing combinatorial problem . let us consider the sector including the empty state , which has all @xmath163 and all @xmath114 . having @xmath164 means that the links @xmath109 and @xmath165 are not both occupied by walls . having @xmath114 means that either both of the links @xmath109 and @xmath110 are occupied , are neither one is . the eigenvalue for @xmath166 is then the number of states @xmath167 satisfying these constraints . to count these , note that if both links @xmath109 and @xmath110 are occupied , then links @xmath165 and @xmath168 must be unoccupied , in order to preserve @xmath169 . but if these latter two links are unoccupied , then links @xmath170 and @xmath171 must be unoccupied as well , to keep @xmath172 . this rule gives a way of counting the configurations in this sector using one - dimensional transfer matrix @xmath173 , which propagates the system by two sites . start at one end . if @xmath174 , then @xmath175 can be either @xmath35 or @xmath134 . however , if @xmath176 , then @xmath177 . iterating this procedure along the whole line gives @xmath178 the first row and column of @xmath173 correspond to unoccupied links , while the second correspond to occupied ones . it is simple to show by induction that @xmath179 where @xmath180 is the @xmath126th fibonacci number ( @xmath181 , @xmath182 , and @xmath183 for the rest ) . thus @xmath184 which for large @xmath91 grows as @xmath185 , where @xmath186 is the golden mean . using a transfer matrix in one dimension makes it possible to write an expression for all the eigenvalues . first consider the case with all @xmath114 except @xmath187 , and all @xmath163 . if @xmath188 , then @xmath189 , but if @xmath190 , then @xmath191 can be either @xmath35 or @xmath134 . thus the 1d transfer matrix for @xmath192 is @xmath193 , where @xmath194 the eigenvalue @xmath195 for the case where one of the @xmath116 is flipped to @xmath117 is therefore @xmath196 in general , when a given @xmath197 , one simply inserts @xmath198 at the @xmath199th site . thus if @xmath200 with all others remaining @xmath134 , we have eigenvalue @xmath201 by using various identities for fibonacci numbers , one finds @xmath202 letting some of the @xmath120 be @xmath134 can be handled in a similar fashion . as noted above , having @xmath203 means that the walls on links @xmath204 and @xmath205 automatically follow from knowing @xmath206 and @xmath116 . this is handled in the transfer - matrix formalism by inserting the matrix @xmath207 at the site of every @xmath203 , where @xmath208 thus when @xmath203 for some @xmath199 while all other @xmath163 and all @xmath209 , we have eigenvalue @xmath210 this eigenvalue is smaller than @xmath195 and @xmath211 ; the eigenvalue @xmath167 is the largest , with @xmath195 the next highest . continuing in this fashion , one obtains the general formula @xmath212 the conservation laws still hold when @xmath213 , so @xmath97 remains block diagonal . however , the blocks are no longer projection operators , so there is generically more than one non - zero eigenvalue per block . we suspect however that the gauge symmetry makes it possible the eigenvalues here in terms of a one - dimensional transfer matrix like ( [ general1],[general2 ] ) . to use exact diagonalization on the transfer matrix of the full model at @xmath12 , we utilize the trick described in section [ sec : transfer ] to reduce its size . this enables us to find its largest eigenvalue for cylinders of up to @xmath214 hexagons ( 36 ising sites ) . 5.5 in@ c c c c c c c width ( @xmath215 ) & & & + & model 3&three - color&model 3 & three - color&model 3 & three - color + + 3&0.4621&0.4621 + 6&0.3911&0.4028&1.880&1.569 & 0.3674&0.3830 + 9&0.3771&0.3900&2.000&1.829&0.3659&0.3798 + 12&0.3722&0.3853&1.990&1.914&0.3660&0.3793 + 15&0.3700&&1.965&&0.3660 + 18&0.3688&&1.942&&0.3661 + @xmath216&@xmath216&@xmath216 + @xmath217&0.3661&0.3791&&1.99&0.3661&0.3791 + theory & & 0.379114&&2&&0.379114 + each state in this new space is labeled by the values @xmath161 and @xmath218 , which for short we call @xmath13 . after @xmath102 acts , giving every element in the same block the same coefficient , we label the blocks @xmath219 . to work out the transfer matrix in this new basis , first one needs to list all the states in a given block @xmath13 . pick one and act with @xmath98 , i.e. just shift the whole thing over by one site . compute the new values of @xmath161 and @xmath218 after the shift , or equivalently , compute @xmath220 and @xmath221 , which we collectively label @xmath222 . the block @xmath219 reached from acting with @xmath102 on this element @xmath13 is then labeled by @xmath161 and @xmath218 , where @xmath223 and @xmath224 for all @xmath30 . one does this for each element in the block @xmath13 : work out @xmath225 and then @xmath219 for each , and then increase the element @xmath226 by one . going through all the blocks gives the reduced transfer matrix @xmath227 . since the eigenvectors of @xmath102 are the same as those of @xmath89 , and the eigenvalues are simply related , we focus on this . to give an example , for @xmath139 , we have @xmath228 which has eigenvalues @xmath229 . note that it is not symmetric . there are three blocks here . the block @xmath230 has three states @xmath231 , the block @xmath232 has just @xmath233 , and the block @xmath234 has just @xmath235 . upon acting with @xmath98 , @xmath142 goes to the block @xmath142 , which means @xmath102 takes it to all the members of this block . thus we increase @xmath236 by 1 . acting with @xmath237 on @xmath238 takes it to @xmath233 , so we increase @xmath239 by 1 . acting with @xmath98 on @xmath240 gives @xmath235 , so @xmath241 . doing this for the other two blocks gives @xmath242 this has eigenvalues @xmath243 as we want , and is symmetric it just lost some zero eigenvalues . the ground - state sector for @xmath244 is even easier . there are 4 states in the block @xmath134 : @xmath245 . these all go to the same block under @xmath102 , so the reduced transfer matrix is simply a number : @xmath246 . this is indeed the largest eigenvalue here . for higher @xmath91 , the size of @xmath227 still increases exponentially , but not as quickly . to give an example of how much this reduces the size of the matrix , exploiting translation invariance and parity as well means that the largest eigenvalue of @xmath102 for @xmath121 is the same as that of the 5-by-5 matrix @xmath247 in this and all the examples we have examined , @xmath227 is upper _ left _ triangular . we have used this reduced transfer matrix for the domain - wall loop representation in numerical simulations of transfer matrices with width up to 18 hexagons in model 1 ( 36 ising variables ) . the resulting largest eigenvalues for widths that are multiples of 3 are shown in table 1 . the entropy per hexagon converges to a number right in the middle of the upper ( from the three - color model ) and lower ( from the hard - hexagon model ) bounds given in ( [ bounds ] ) . expanding the largest eigenvalue in a power series in @xmath248 gives additional valuable information . when the system is at a conformally invariant critical point , the subleading piece is universal and proportional to the central charge @xcite , which must obey @xmath249 in any system with positive boltzmann weights . if the system is not critical , this piece should fall off to zero as @xmath250 . the precise formula for our case is @xmath251 here @xmath91 is the width in hexagons , @xmath252 is the largest eigenvalue of @xmath102 , and the geometrical factor @xmath253 results from the ratio between the width and length of the transfer matrix step . the resulting estimates of central charge for our model do not converge even at the largest system sizes , while for the three - color model , extrapolation from smaller sizes gives a central charge consistent with the expected value @xmath80 @xcite . the conclusion of this transfer - matrix study is that our model is most likely not described by the @xmath80 critical theory of the three - color model , even though large system sizes are required to see the difference . since the central charge does not seem to be converging to anything , the numerical results are in harmony with the field - theory arguments of section [ sec : ft ] in suggesting that our model is non - critical ( i.e. , has a finite correlation length ) . we can not categorically rule out that it is critical with @xmath254 , but have no evidence for this scenario . in this paper we discussed constrained classical lattice models . by imposing some simple constraints on ising spins , we found a variety of intriguing properties . in particular , we showed that the space of states on the sphere is connected under local moves , and that on surfaces with non - contractible cycles , different sectors can be labeled by loop configurations . we also presented substantial ( if not conclusive ) evidence that the model is not critical . in the introduction , we mentioned a quantum motivation for studying classical lattice models with constraints . the results of this paper imply that our model has the right characteristics to yield a quantum model with a topological phase , with the added intriguing possibility that the excitations have non - abelian statistics . we therefore will conclude this paper with a discussion of the quantum model in more detail . the connection between quantum and classical models comes from a trick due to rokhsar and kivelson @xcite . let the basis elements for the hilbert space for the quantum model consist of configurations in the two - dimensional classical lattice model . then one can construct a quantum hamiltonian acting on these states with a ground state consisting of a _ superposition _ of these states , with each term having an amplitude corresponding to its weight in the classical model . correlators in the ground state of the quantum model are then related to the correlators in the classical model . in the quantum model , one need not impose the constraints directly on the hilbert space , but rather one can add an energy penalty for configurations which violate the constraint . the ground state then contains only configurations satisfying the constraint . violating the constraint locally then corresponds to a quasiparticle excitation . thus for a given classical model , one can obtain very different quantum models depending on which defects are allowed and which are not . let us make this explicit in terms of model 1 . here we can take the hilbert space to be comprised of two - state ising variables on the sites of the honeycomb lattice . one simply imposes an energy penalty on configurations violating constraint ( [ constraint1 ] ) , i.e. for each hexagon @xmath255 one includes @xmath256 in the hamiltonian . the off - diagonal terms in hamiltonian are given by flip we defined in section [ sec : flip ] , as displayed in figure [ fig : flippable ] . the trick of rokhsar and kivelson is to add a potential which , combined with the flip , is a projector . namely , we add a potential term which counts the number of flippable hexagons . then the hamiltonian is @xmath257 where @xmath258 is the flip and @xmath259 is the number of flippable plaquettes . it is simple to write @xmath260 explicitly in terms of ising spins , but the expression is quite unwieldy . the lowest eigenvalue of @xmath260 is zero , since it is the sum of projectors and a positive diagonal term . the ground state on the sphere is unique , and consists of the equal - amplitude sum over all configurations satisfying constraint ( [ constraint1 ] ) . on surfaces with contractible cycles , the local flip can not change the sectors described above . there will be a ground state for each of these sectors , consisting of the equal - amplitude sum of all the configurations in the sector satisfying the constraint . the reason for the interest in quantum models of this type is that they often have topological order . topological order means that there is no local order parameter with a non - vanishing expectation value , but only non - local ones . by using the rokhsar - kivelson trick , it was demonstrated that a quantum eight - vertex model @xcite and a quantum dimer model on the triangular lattice @xcite indeed have topological order . the quantum model with hamiltonian @xmath260 indeed should have topological order , since the number of ground states depends on the genus of the surface , a telltale sign . one reason why models with topological order are interesting is that they can lead to excitations with fractional statistics . non - topological solid and superfluid quantum phases near the rokhsar - kivelson point corresponding to our model , and possible unconventional phase transitions , are discussed using a two - component quantum height model in ref . . the excitations in the quantum model with hamiltonian @xmath260 correspond to hexagons with different numbers of up and down spins . the specific @xmath261 we chose means the lowest - energy defects have @xmath262 . these are illustrated in figure [ fig : defects ] . . ] in the loop language , they correspond to joining loops of different types . with the hamiltonian @xmath260 , these defects have no dynamics , but one can of course add terms allowing them to move . changing the potential to favor other kinds of defects gives different theories . allowing just @xmath263 defects gives a variation on the `` odd '' ising gauge theory described in depth in @xcite ; these defects correspond to allowing loops ( the domain walls of model 3 ) to cross . allowing just @xmath39 defects gives a quantum version of the three - color model , i.e. the defects are the 11th vertex shown in figure [ fig:11thvertex ] . note however that even though the ground states of all three models we have introduced are identical , their local defects can be quite different . the defects just described are nonlocal in the color representation of model 2 , because the three colors become rotated upon circling , i.e. , each bond no longer has a uniquely defined color@xcite . likewise , another kind of defect we could introduce would be to treat the _ loops _ in model 3 as the degrees of freedom for the quantum model . then we can allow defects to correspond to loops with ends ( like the end of a flux tube in gauge theory ) . the defects we have discussed have an important property : although the energy @xmath256 associated with them is local , they are attached to zero - energy defect lines , which can only end in another defect . for example , a @xmath262 defect has two types of domain walls attached , which must eventually end in another defect . this property makes it likely that the corresponding quasiparticles have fractional statistics , because when particles are exchanged , they must pass through these defect lines . the @xmath262 defects illustrated in figure [ fig : defects ] are particularly intriguing . since the model has an @xmath264 symmetry under exchange of the ising models , these defects can be classified in representations of this non - abelian symmetry . ( note also that the larger symmetry generated by the global conserved quantities of the classical transfer matrix is non - abelian as well . ) this makes it possible that a suitable choice of hamiltonian will result in non - abelian braiding of the excitations , a topic of great current interest because of potential application to topological quantum computation @xcite . these arguments make it likely that one can realize a phase with topological order using our model as a starting point . to prove this , more work needs to be done . one needs to prove the quasiparticles are deconfined , i.e. that lines connecting the defects have no energy per unit length in the quantum theory . this does seem very plausible , given that @xmath261 is non - zero only at the location of the defect . in the three - color model , these defects have binding free energy that scales as a power - law @xcite , which is critical between confinement and deconfinement . a related question is proving that the ground state of the hamiltonian contains macroscopically long loops even in the continuum limit ; many examples are known of lattice loop models where the average loop length ( in terms of the lattice spacing ) remains finite . also , the hamiltonian @xmath260 does not allow the defects and defect lines to cross through each other , making it impossible to understand the fractional statistics precisely . we leave these very interesting open questions for future study . we thank ashvin vishwanath for useful conversations . this work was supported by the nsf under grant dmr-0412956 ( p.f . ) and dmr-0238760 ( j.e.m . and c.x ) . valuable intuition and information can be gained by relating our model to two well - studied models , the triangular - lattice ising antiferromagnet and the hard hexagon model . our model can be found by _ relaxing _ constraints on these two . since adding constraints reduces the entropy , the maps described in this section give lower bounds on the entropy of our model . moreover , both have critical points different from that of the three - color model . the ising antiferromagnet on the triangular lattice is one of the classic examples of geometrically frustrated magnetism @xcite . at zero temperature in the classical model , each fundamental triangle contains either two up spins and one down , or two down spins and one up . to avoid confusing these ising spins with the earlier ones , we label them as @xmath265 , so the zero - temperature constraint is that the sum of spins around every fundamental triangle is @xmath266 . by drawing each frustrated bond as a dimer on the dual lattice , this model is identical to the close - packed hard - core dimer model on the honeycomb lattice , which is known to be critical @xcite . we now consider a model with the same constraint around each triangle , but where the degrees of freedom can take _ any _ half - integer value @xmath267 , not just @xmath268 as in the ising antiferromagnet . we call this a `` height '' , and prove here that this model is equivalent to ours . the ising spin @xmath269 of our model 1 is defined on the sites of dual honeycomb lattice by @xmath270 where even and odd @xmath7 are the sites on the two equivalent sublattices of the honeycomb lattice . the constraint ( [ constraint1 ] ) follows automatically from this definition . any height configuration obeying ( [ sigmah ] ) therefore defines a configuration in model 1 . to finish the proof of equivalence , we now show that each ising spin configuration satisfying constraint ( [ constraint1 ] ) generates , up to two arbitrarily specified half - integers , a unique configuration of heights . fix a configuration of ising spins on the honeycomb lattice . pick two adjacent sites on the dual triangular lattice , and assign them arbitrary half - integer heights @xmath271 , and @xmath272 . if one knows two of the heights around a triangle , and the value of the ising spin at the center of the triangle , then ( [ sigmah ] ) determines the third height uniquely . consider the sites illustrated in figure [ fig : dualhex ] . applying ( [ sigmah ] ) to the two triangles containing both @xmath271 and @xmath272 gives the heights @xmath273 and @xmath274 . applying ( [ sigmah ] ) again to the triangles involving @xmath275 and @xmath276 gives two more heights @xmath277 and @xmath278 . there is now another height @xmath279 which belongs to a triangle involving @xmath280 as well as to a triangle with @xmath281 . again applying ( [ sigmah ] ) to either of these triangles gives @xmath279 ; because of the constraint ( [ constraint1 ] ) it is determined uniquely . this therefore determines all the heights on the six triangles involving @xmath271 . repeating this process for the triangles around the heights @xmath282 then determines the heights on another concentric ring . in this fashion all the heights follow from the ising - spin configuration . the constraint ( [ constraint1 ] ) ensures that these are unique , up to the two original choices of @xmath271 and @xmath272 . the indeterminacy of these two half - integers can be understood simply by noting that , if three integers @xmath45 are added to the height variables globally on the three sub - lattices of the triangular lattice , then as long as @xmath283 , the resulting ising spin configuration on the honeycomb lattice is unchanged . our model is obtained by relaxing a constraint on the zero - temperature ising antiferromagnet , so it provides a lower bound on the entropy of our model . the honeycomb - dimer model equivalent to the former has an entropy of @xmath284 per hexagon @xcite . a slightly better lower bound can be obtained by relating our model to another interesting model , the hard hexagon model . the hard hexagon model is defined by placing particles on the sites of the triangular lattice , so that no two particles are adjacent or on the same site . each particle can equivalently be viewed as a `` hard '' hexagon with the length of a link : the restriction that particles can not be placed on adjacent sites means that the hexagons may not overlap @xcite . a typical configuration is drawn in figure [ fig : hardhex ] . the relation between model 3 and hard hexagons comes by drawing lines surrounding any clusters of hexagons , as shown in figure [ fig : hardhex ] . each of these loops corresponds to a domain wall in one of the three ising models on the three triangular sublattices . by construction , these loops do not cross , although they can touch . thus each configuration in the hard - hexagon model corresponds to one in model 3 . the converse is not true : there are configurations in model 3 not in the hard hexagon model . in model 3 , one can have domain - wall loops inside of other loops , as long as they do not cross . if there is a domain - wall loop of one ising model inside that of another , this corresponds in the hard hexagon model to placing a hexagon on top of others . this is forbidden there . both the hard hexagon model and the three - color model without constraint ( [ constraint2 ] ) are integrable . in both cases , one can compute the asymptotic behavior of the number of configurations as the number of sites gets large @xcite . since our model has more configurations than the hard hexagon model and less than the three - color model , this gives lower and upper bounds on the entropy @xmath285 in this limit : @xmath286 where @xmath104 is the number of sites on the triangular lattice in model 3 ( the number of hexagons in the honeycomb lattice in model 1 ) . our numerics discussed in section [ sec : numerics ] give @xmath287 , consistent with these bounds .	we study a constrained statistical - mechanical model in two dimensions that has three useful descriptions . they are 1 ) the ising model on the honeycomb lattice , constrained to have three up spins and three down spins on every hexagon , 2 ) the three - color / fully - packed - loop model on the links of the honeycomb lattice , with loops around a single hexagon forbidden , and 3 ) three ising models on interleaved triangular lattices , with domain walls of the different ising models not allowed to cross . unlike the three - color model , the configuration space on the sphere or plane is connected under local moves . on higher - genus surfaces there are infinitely many dynamical sectors , labeled by a noncontractible set of nonintersecting loops . we demonstrate that at infinite temperature the transfer matrix admits an unusual structure related to a gauge symmetry for the same model on an anisotropic lattice . this enables us to diagonalize the original transfer matrix for up to 36 sites , finding an entropy per plaquette @xmath0 and substantial evidence that the model is not critical . we also find the striking property that the eigenvalues of the transfer matrix on an anisotropic lattice are given in terms of fibonacci numbers . we comment on the possibility of a topological phase , with infinite topological degeneracy , in an associated two - dimensional quantum model .
low - mass protostars have been grouped into an evolutionary sequence , class 0 through class iii , based on their spectral energy distributions ( lada & wilking 1984 ; lada 1987 ; myers et al . 1987 ; strom et al . 1989 ; adams , lada , & shu 1987 ; andr , ward - thompson , & barsony 1993 ) . class 0 sources are in the early protostellar collapse stage , with spectral energy distributions ( seds ) that resemble blackbodies with t @xmath1 30k ( andr et al . the majority of the source mass resides in the infalling envelope . even at this early stage of collapse ( of order a few @xmath2 10@xmath3 yr ) , these sources exhibit powerful , bipolar molecular flows . the class i stage is a later stage of protostellar collapse which lasts a few @xmath2 10@xmath4 yrs . these sources display very broad seds that peak near 100 @xmath0 m . their envelope masses are similar to the mass of the central pre - main - sequence core . they have well - developed accretion disks and their envelopes have bipolar cavities excavated by outflows ( terebey , chandler , & andr 1993 , kenyon et al . 1993b , tamura et al . the class ii stage is characterized by the presence of excess infrared emission above that expected for a stellar photosphere , with the sed emission peak occuring in the near - infrared . this is the signature of a classical t tauri star surrounded by an accretion disk . infall from the cloud has ceased due to dispersal of the remnant infall envelope by the combined effects of infall and outflow . class ii sources have ages of about @xmath5 yrs ( strom et al . the class iii sources , also known as `` weak - lined '' t tauri stars , have sed s that resemble a stellar photosphere . mid - ir radiation from remant disk material , if it exists , should be detectable by the upcoming space infrared telescope facility ( sirtf ) . it is not known for certain whether class iii sources are more evolved than the class ii sources , or whether they have simply lost most of their circumstellar material on a faster timescale . the class iii stage can last for about @xmath6 years , ending in a zero - age main - sequence star ( palla 1999 ) . interpretations of the seds of pre - main sequence stars have traditionally relied on 1-d or 1.5-d radiative transfer models of spherical envelopes for class 0-i sources ( adams et al . 1987 , kenyon , calvet , & hartmann 1993a ; jayawardhana , hartmann , & calvet 2001 ) and flat or flared - disk models for class ii sources ( kenyon & hartmann 1987 ; chiang & goldreich 1997 ; dalessio et al . 1998 , 1999 ; dullemond , dominik , & natta 2001 ; dullemond 2002 ; dullemond & natta 2003 ) . while the class ii models do a good job matching observations of pure disk sources , the models for younger sources generally do not take into account bipolar cavities created by outflows , vertically extended disks in class 0-i sources , inclination effects , or intermediate stages between class i ( envelopes ) and ii ( disks ) sources . these models appeared sufficient to explain the previously sparsely sampled seds , though it was clear early on that they failed to account for the large optical / near - ir flux from most class i sources ( adams et al . kenyon et al . ( 1993b ) showed that bipolar cavities allow a sufficent amount of near - ir flux to scatter out to match the observations . now with more complete sed information on several sources showing more complicated seds ( liu et al . 1996 , chandler , barsony , & moore 1998 , rebull et al . 2003 , wolf , padgett , & stapelfeldt 2003 ) , and with the upcoming sirtf mission which will sample more frequencies in the mid - ir , it is clear that 2-d and 3-d models are required to interpret these sources . in a previous paper ( whitney et al . 2003 , hereafter paper i ) , we showed that including a flared disk and bipolar cavity in models of class i sources leads to more complicated seds than those from 1-d models or 2-d models with rotationally- flattened envelopes only ( efstathiou & rowan - robinson 1991 ) . we showed that the near - ir and mid - ir colors of 2-d models are substantially bluer than those from 1-d models for a given envelope mass . thus , using 1-d models to interpret sources that are not spherical will lead to an underestimate of the mass of the envelope and an overestimate of the evolutionary state . this paper follows up paper i with models comprising an evolutionary sequence from class 0 - iii . our goal is to provide more accurate interpretations of observations . we find substantial overlap between colors of sources at different evolutionary stages in the mid - ir wavelengths that sirtf will observe . the information provided by seds alone may not be sufficient to interpret a source s evolutionary state , since , for example , a more pole - on class i source can resemble an edge - on class ii source . combining more information from images and polarization , which help determine inclination and size , can resolve this degeneracy . 2 of this paper describes the models , 3 describes the resulting seds , colors , images and polarization spectra , and 4 summarizes the results . for the infalling envelopes , we use the density structure of a rotating envelope in free - fall collapse ( ulrich 1976 ; equation 1 of paper i ) . the disk density is a flared disk in hydrostatic equilibrium ( shakura & sunyaev 1973 , pringle 1981 , equation 3 of paper i ) . the envelopes have curved bipolar cavities filled with constant density gas and dust . the cavity shape follows @xmath7 , where @xmath8 is the cylindrical radius . we computed six models comprising an evolutionary sequence from class 0 to iii . this evolutionary sequence is characterized by a decreasing envelope infall rate , increasing disk radius , increasing bipolar cavity opening angle , and decreasing cavity density . the density structures for each model are shown in the right - hand panels of figure 3 . for more detailed plots of the disk / envelope density in the inner regions , see paper i , figure 2 . table 1 shows the parameters common to all models . the stellar parameters are typical of a low - mass t tauri star . the disk parameters are based on models of t tauri disks in hydrostatic equilibrium ( dalessio et al . 1998 ) and models of observed images ( cotera et al . 2001 , wood et al . the disk density is proportional to @xmath9 . the disk scale height increases with radius as @xmath10 scaled to the value of @xmath11 at the stellar radius , given in table 1 . the disk viscosity parameter , @xmath12 , is taken from models of average passive t tauri accretion disks ( dalessio et al . 1999 ) , based on the @xmath13-disk prescription ( shakura & sunyaev 1973 , kenyon & hartmann 1987 , bjorkman 1997 ; hartmann 1998 ; see equations 5 - 6 of paper i ) . this is used in our calculation of disk accretion rate and luminosity ( table 2 ) . table 2 shows the parameters that vary between the six models . the envelope infall rates are based on previous models of class 0 and i sources ( adams et al . 1987 , kenyon et al . 1993a , b , jayawardhana et al . 2001 , stark et al . 2003 ) . the envelope mass is determined by a combination of several parameters : envelope infall rate , centrifugal radius (= disk outer radius ) , outer envelope radius , cavity size and shape and density . in the models with infalling envelopes ( class 0-i ) , the envelope mass grows with radius as @xmath14 . thus , the envelope mass tabulated is only that within 5000 au , our chosen outer envelope radius . the envelope inner radius and disk inner radius are chosen to be at or slightly larger than the dust destruction radius ( except for the class iii source ) ; this is larger in the more dense disks . the class iii disk inner radius is chosen to be 50 stellar radii since evolved disks can have large inner holes ( e.g. , hr 4796 , koerner et al . 1998 ) the disk mass is chosen to be 0.01 @xmath15 for all evolutionary states except class iii . this mass is typical for class 0-ii sources ( beckwith et al . 1990 , terebey et al . 1993 , dalessio , calvet , & hartmann 2001 , looney , mundy , & welch 2003 ) . the disk size is expected to grow with time for envelopes which initially rotated as a solid - body . our choices of disk size are based on previous modeling and observations ( beckwith et al . 1990 , kenyon et al . 1993a , b , stark et al . 2003 , padgett et al . the envelope centrifugal radius is chosen to be the same as the disk outer radius . the disk accretion rate and contribution to luminosity varies between the models due to different disk densities . we note that its effect on the seds and images is small . the class iii disk mass is chosen to be @xmath16 . for more complete treatments of class iii disks and debris disks exploring a wider range of parameter space in mass and dust properties , see wood et al . ( 2002b ) and wolf & hillenbrand ( 2003 ) . the bipolar cavity densities are based on observations of molecular outflow densities ( moriarty - schieven et al . 1992 , 1995a , b ) and our assumption that these would decrease with evolution . the cavity opening angles are also expected to increase with age and our choices are based on images of molecular outflows and envelopes ( chandler et al . 1996 , tamura et al . 1996 , hogerheijde et al . 1998 , padgett et al . 1999 ) and models of images ( stark et al . 2003 ) . the 90 degree opening angle for the class ii and iii sources is used to give a constant density envelope at typical molecular cloud ambient densities . the density of the class iii ambient cloud is lower than the other sources since the disk density is low and it would not be surprising to see class iii sources in less dense regions of the molecular cloud . the radiative transfer method is the same as described in paper i. it uses the radiative equilibrium solution method of bjorkman & wood ( 2001 ) incorporated into a 3-d spherical - polar grid . complicated 3-d density distributions can be solved with the same ease as 1-d spherical densities . the models properly account for scattering and inclination effects . most of the source luminosity comes from the central star , with a small amount of accretion luminosity from the disk . the input stellar spectrum is a kurucz model atmosphere ( kurucz 1994 ) for the stellar parameters given in table 1 . both the stellar and disk accretion luminosity are reprocessed by the dusty disk and envelope . an addition to the models in this paper is the capability to include different grains in different regions . a grain model is assigned to each grid cell . several variables used for the temperature calculation in a cell , e.g. , the rosseland mean opacity , are computed for each grain model used in the calculation . previous modeling of seds of disks has shown that to fit images and seds of class ii sources ( disks ) requires large grains with maximum sizes up to 1 mm ( dalessio et al . 2001 , wood et al . 2002a , wolf et al . 2003 ) . on the other hand , to match the colors , images and polarization maps of class i sources ( mainly envelopes ) in the taurus molecular cloud , grain models similar to the diffuse ism work reasonably well ( kenyon et al . 1993b , whitney et al . 1997 , lucas & roche 1997 , 1998 , stark et al . 2003 , wolf et al . for this reason we allow the grain properties to vary in different regions , as shown in figure 1 . for the dense regions of the disk ( n@xmath17 @xmath18 ) , we use the large grain model that wood et al . ( 2002a ) used to fit the hh30 disk sed . for the less dense regions of the disk , we use the same grain model cotera et al . ( 2001 ) used to model the hh30 near - ir scattered light images . these grains are larger than ism grains but not as large as the disk midplane grains . for the envelope we use the grain model from paper i which has a size distribution that fits an extinction curve typical of the more dense regions of the taurus molecular cloud , with @xmath19 , the ratio of total - to - selective extinction , equal to 4.3 ( whittet et al . 2001 ; cardelli , clayton , & mathis 1989 ) . these grains also include a water ice mantle covering 5% of the radius . for the outflow , since small grains can condense in outflows , we use ism grains which are on average slightly smaller than the envelope grains ( kim , martin , & hendry 1994 ) . the grain properties are shown in figure 2 , and briefly summarized in table 3 . figure 2 shows that the larger grains ( e.g. , the disk midplane grains shown by the dashed line ) have have a flatter opacity , larger albedos ( at longer wavelengths ) , larger @xmath20 ( average cosine scattering angle ) , and lower polarization ( @xmath21 ) . this section displays spectra , colors , polarization spectra , and images for the six evolutionary stages . all of the sources have luminosity l=1 @xmath22 , and are placed at a distance of d=140 pc , for comparison to nearby low - mass star formation regions such as the taurus molecular cloud . figure 3 shows densities and seds for the 6 evolutionary states . the smallest densities displayed ( @xmath23 gm @xmath18 ) are near the ambient density of @xmath24 gm @xmath18 or @xmath25 @xmath18 . in the inner regions , not shown on the scales plotted here , the densities increase to about @xmath26 gm @xmath18 in the most dense regions of the disk and about @xmath27 gm @xmath18 , depending on evolutionary state , in the inner envelope . the seds in figure 3 show the emission emergent from the entire envelope . in nearby star formation regions , the flux is often measured in apertures that are smaller than the envelope size . this affects the optical / near - ir flux the most , with smaller apertures giving less flux , as we will show later . for each model , ten inclinations are plotted , the top curve corresponding to pole - on ( colored pink ) and the bottom to edge - on ( green ) . the largest variations with inclination appear in the mid - ir , with edge - on sources showing a broad dip near 10 @xmath0 m in the class i - ii sources . the 9.8 @xmath0 m amorphous silicate feature is much narrower than this . the broad dip is due to two effects : the large extinction in the disk midplane that blocks thermal radiation from the inner disk+envelope ; and the low albedo that prevents radiation from scattering out the polar regions . the optical extinction , a@xmath28 , through the disk midplane is @xmath29 in the class 0 sources and @xmath30 in the class i - ii sources ( see paper i , figure 3 for a plot of a@xmath28 vs. inclination angle ) . shortward of 10 @xmath0 m , the class i and ii envelopes allow optical / near - ir radiation to scatter out through the outflow cavities ( or upper layers of the disk in the case of the class ii source ) . thus the edge - on class i and ii sources are double - peaked , with the short - wavelength peak due to scattering , and the long - wavelenth peak due to thermal emission . the class ii model seds are similar to previous models ( kenyon & hartmann 1987 ; chiang & goldreich 1997 ; dalessio et al . 1998 , 1999 ; dullemond et al . 2001 ; dullemond 2002 ; dullemond & natta 2003 ) except at nearly edge - on inclinations ( @xmath31 ) , where extinction and scattering dominate at near- and mid - ir wavelengths . dalessio et al . ( 1999 ) include these effects and their models show similar behavior with inclination as ours . dullemond ( 2002 ) calculated the 2-d structure of disks and herbig ae / be stars and found that the inner wall puffs up and shadows outer regions in these sources . since we do not compute the 2-d hydrostatic equilibrium solution , we do not have this effect in our models . a forthcoming paper will consider this effect in t tauri disks ( wood et al . 2003 , private communication ) . the more embedded class 0 sources have less scattered flux at optical / near - ir wavelengths due to higher extinction in the envelope and cavity , so the `` dip '' at 10 @xmath0 m is not as striking in these sources . the class 0 source shows little variation with inclination except for the pole - on source and at near - ir wavelengths . the class iii source also shows little variation with inclination because it is optically thin at all wavelengths . examining the variation of seds with evolutionary state , we see a tendency for increased shortwave ( 0 - 10 @xmath0 m ) flux and decreased longwave ( 100 - 1000 @xmath0 m ) flux with age . as stated previously , the seds shown in figure 3 include the flux emitted by the entire envelope . normally , observed fluxes are integrated within a given aperture size . figure 4 shows seds of the class 0 and i sources computed in different aperture sizes , with 1000 au radius apertures depicted by the solid lines and 5000 au apertures by the dashed lines . the class ii and iii sources are not shown since they are smaller than either of these apertures and thus do not vary in these apertures . three inclinations are shown for each model ( @xmath32 ) in three colors ( pink , blue , green respectively ) . the large aperture results show much more scattered flux at wavelengths less than 10 @xmath0 m . the class 0 source has no near - ir flux in the small aperture . this suggests that care should be taken in comparing different sets of observations with either different aperture sizes or different source distances . figure 5 shows that a low - luminosity class i and a high - luminosity class ii source at different inclinations can resemble each other . our class ii model was scaled up by a factor of 5 to get the fluxes to agree with the class i model . this gives higher far - ir flux than the class i source but better agreement in the near - ir where the class i source scatters more light from its large envelope . the black lines correspond to a class i source , and the grey lines are a class ii source . the dashed lines show both sources at an inclination of @xmath33 , or nearly edge - on . the solid lines show the class i source at @xmath34and a class ii source at @xmath35 . in both cases , the class ii source is viewed fairly close to edge - on so the central source is obscured at visible wavelengths . in the left panel , the class i fluxes are summed in the 1000 au radius aperture , and at right , the fluxes are summed in the 5000 au aperture . the smaller - aperture results do not show as much agreement . thus in nearby star formation regions , it is possible to distinguish class i and ii sources based on images and aperture photometry . however , in far - away star formation regions where the class i source may not be spatially resolved , it may not be so easy to distinguish without additional information such as polarization ( section 3.3 ) . figure 6 shows the effects of varying grain properties in the different regions of the disk / envelope . the solid lines show our standard models used throughout this paper , with the different grain models in the four different regions ( figure 1 ) . the grey line plots models which have the envelope grains throughout the disk / envelope / cavity . in models where the envelope emission ( and scattering ) dominates , we should see little difference between the seds . this is the case for the class 0 source , which shows a difference only at optical / near - ir wavelengths for the pole - on source , due to the slightly higher extinction in the cavity of the envelope grains compared to the outflow grains . the late class 0 source shows less of this effect due to lower cavity density , but shows a difference at wavelengths longer than 100 @xmath0 m for more pole - on viewing angles . this is because some disk emission can emerge into pole - on inclinations at long wavelengths , and differences in the grain properties between the two models are apparent . the class i through ii sources show more differences at these wavelengths , especially at pole - on inclinations . in more evolved or pole - on sources , more disk radiation can penetrate through the envelope . this difference is greatest in the class ii source where emission comes only from the disk , and therefore different grains in the disk have a noticeable effect on the longwave sed . edge - on , we can see differences near 10 @xmath0 m in the late class i through iii sources due to the different extinction opacities in the disk . the class iii models show little difference because the disk is optically thin and the multi - grain model ( in black ) uses the disk atmosphere grains ( table 3 and figure 2 ) which are not substantially different from the envelope grains . note that we have not attempted to created a realistic grain model for a class iii source that would compute size distributions based on shattering models . more detailed models can be found in kenyon et al . ( 1999 ) , augereau et al . ( 1999 ) , wolf & hillenbrand ( 2003 ) , and aigen & lunine ( 2003 ) . to turn the results of figure 6 around , grain growth in the disk is not detectable in the seds of class 0 sources , due to the dominance of envelope emission . grain growth is detectable in class i sources by the slope of the longwave emission ( @xmath36 m ) which is flatter for large grains in a similar manner to class ii disks ( beckwith , henning , & nakagawa 2000 ) . these results agree with those of wolf et al . ( 2003 ) who fit near - ir images of the iras 04302 + 2247 with ism - like grains in the envelope , but required large grains in the disk to fit the submm images . in anticipation of sirtf observations , we show in figures 7 and 8 colors and magnitudes for the near - ir j , h , and k bands ; the mid - ir sirtf irac bands at 3.6 , 4.5 , 5.8 , and 8 @xmath0 m ; and the far - ir sirtf mips bands at 24 , 70 , and 160 @xmath0 m . we also show n - band ( 10 @xmath0 m ) magnitudes for comparison to current ground - based observations . the colored points are our models with colors corresponding to evolutionary state as shown in the legend at top left . the size of the circles correspond to inclination from pole - on ( large ) to edge - on ( small ) . in figures 7a and 8a , the fluxes have been computed in an aperture size of 1000 au radius , or 7.1at a distance of 140 pc . figures 7b and 8b show results for a 5000 au radius aperture . these plots make it clear that in comparing results of star formation regions at different distances , the aperture size should be taken into account . all of the models have fluxes brighter than 1 @xmath0jy at the wavelengths shown except for some of the class 0 points , which are fainter at wavelengths smaller than 5 @xmath0 m . for these sources , we show fluxes as faint as 0.01 @xmath0jy to account for the possibility of a high luminosity class 0 source in a nearby star forming region . most of the shortwave class 0 fluxes are too faint to show in the small aperture plots ( fig . 7a ) except at long wavelengths or pole - on inclinations . the small black symbols in figure 7 are computed colors of a variety of different kinds of objects based on complete 1 - 35 @xmath0 m infrared spectra using the sky program ( wainscoat et al . 1992 , cohen 1993 . ) . the crosses correspond to main sequence , red giant and supergiant stars ; the diamonds are agb stars , squares are planetary nebula , the asterisks are reflection nebulae , the small x is an hii region and the big x is a t tauri star ( cohen & witteborn 1985 ) . since the sky spectral library does not exist at wavelengths longward of 35 @xmath0 m , the bottom right panel of figure 7 does not show sky results . this figure shows a lot of overlap between our predicted star formation colors and those of agb stars and planetary nebulae . however , the bottom left panel , [ 3.6]-[5.8 ] vs [ 8.0]-[24 ] , shows significant separation . for the most part , the main sequence , red giant and supergiants are well separated from the star formation models . the arrows show reddening vectors for the standard diffuse ism law ( in grey ) and for our envelope dust model , more representative of dark clouds . in the near - ir , the star formation points will overlap with reddened main sequence and other sources ( top two panels ) . however , several of the mid - ir panels show reasonable separation from reddened galactic point sources , especially the bottom left panel again . the overlap with agb stars , planetary nebulae , and reflection nebulae is only a problem for distant star formation regions in the galactic plane , where we may expect to see a total of no more than 300 of these sources per square degree . how well do the colors distinguish evolutionary states in the star formation models ? in general , the younger sources appear to be more red , though there is overlap due to inclination effects and aperture size ( fig . 7b ) . as figure 4 showed , the class 0 source has little measurable flux below @xmath37 m for all but the pole - on source . thus for many of the panels , only the pole - on inclination is shown in figure 7a , the smaller aperture results . in the near - ir sequences ( top two panels ) , the pole - on class 0 point falls in the middle of the other models . at longer wavelengths , the class 0 points that are observable ( in dark blue ) tend to lie in a separate region from the other sources , in the small aperture results ( figure 7a ) . however , in the large aperture results of figure 7b , the class 0 sources are in many cases _ bluer _ than the more evolved sources , even some of the class iii sources , especially at k - n , [ 5.8]-[8.0 ] , [ 8.0]-[24 ] in figure 7b . this shows again the importance of considering aperture size because of the contribution of scattered light on large scales in the envelope especially in the youngest sources . the class 0 points are redder in k-[3.6 ] , [ 3.6]-[4.5 ] but only by about 0.5 - 1 magnitude . they are much redder at [ 24]-[70 ] , as shown in the bottom right of fig . 7b . in several of the near - ir and mid - ir sequences ( top four panels ) notice that the bluest sources are edge - on class i sources . this is true for both the small and large aperture results . the reason for this is that the edge - on disk blocks all of the reddened stellar flux , leaving only the scattered flux from the envelope which is relatively blue ( due to decreasing scattering opacity with increasing wavelength ) . the j - h vs k-[3.6 ] and k-[3.6 ] vs k - n can be compared to ground based observations of j - h vs k - l and k - l vs k - n , for example the taurus - aurigae cluster at 140 pc ( wood et al . 2002b ) and the ngc 2024 cluster at a distance of 415 pc ( haisch et al . 2001 ) . the taurus cloud is sufficiently nearby for our small aperture results to apply , and indeed our results in figure 7a agree with those shown in wood et al . ( 2002b , their figure 7 ) . it is interesting to note that our large - aperture k-[3.6 ] vs k - n plot includes class 0 sources in the same region the haisch et al estimate class ii sources to lie . in the j - h vs k-[3.6 ] panel , the class 0 sources also fall in the same region that haisch et al . estimate class i sources to lie ( their figure 7 ) . and the class i sources overlap with the class ii source region . thus , in more distant star formation regions , it will be more difficult to separate evolutionary states by their colors alone . we emphasize that scattered light , especially when measured in large apertures or distant sources makes class 0 and i sources appear much bluer than is perhaps generally thought . the sequences in figure 8a , the 1000 au radius aperture results , show that in general the edge - on sources are fainter , and the younger sources are redder . note the large range in magnitude ( y - axis ) for all of these sources , despite the fact that _ all of these models have the same luminosity_. even more striking is the variation in magnitude from a given model due to inclination . not surprisingly , this effect is larger for the younger sources due to their large variation of extinction with inclination . as expected , the class ii and iii sources have fairly constant flux with inclination until they become edge on . because the edge - on sources are fainter , they could be mistaken for much lower luminosity sources such as brown dwarfs ( walker et al . the variation of magnitude with inclination is much smaller at 70 @xmath0 m because of the lower envelope / disk extinction and more isotropic thermal emission . figure 8b shows again the overlap in colors of all the evolutionary states when viewed through large apertures ( 5000 au radius ) . at j - k , the class 0 source has the same colors as a class ii and iii source . at [ 5.8]-[8 ] , the class 0 source is bluer than all except the edge - on class i and ii sources . the [ 3.6]-[5.8 ] sequence shows more spread , but notice that the entire evolutionary sequence ranges within 3 magnitudes . since these models all have the same luminosity , the class 0 sources are faintest in the near to mid - ir due to large envelope extinction . the class iii source is well - separated at [ 24]-[70 ] and [ 70]-[160 ] . since the lifetimes of class ii sources are 10 - 100 times longer than class 0 and i sources , we expect observed color - color and color - magnitude diagrams to be dominated by class ii and iii sources . since 90% of these sources are viewed at unobscured inclinations , their fluxes can be used to estimate luminosity . the caution is that there is perhaps a 15% contamination in these diagrams due to younger sources ( class 0 and i ) that appear blue because of scattering . figure 9 shows polarization plots for four of the evolutionary sequences . the class 0 models did not have enough signal - to - noise to show the spectral results ( current observing technology would probably suffer similar difficulties . ) the polarization of the class iii model is less than 0.2% so is not shown . in these models , the polarization arises from scattering from spherical grains in an asymmetric envelope geometry . there is no component due to aligned grains ( see whitney & wolff 2002 for models of scattering from aligned grains , and aitken et al . 2003 for models that include emission from aligned grains ) . in the class 0 and i sources , more radiation escapes the polar regions than the equatorial , giving rise to polarization with a position angle oriented perpendicular to the rotation axis ( the @xmath38-axis ) . in the class ii sources , most of the scattering takes place in the disk plane . viewed at most inclinations the polarization is oriented perpendicular to the disk plane , or parallel to the rotation axis . we define @xmath39 , where @xmath40 is measured from the rotation axis . because @xmath40 is either 0or 90 in these axisymmetric models , we can display it as @xmath41 which is either negative or positive . for class 0 and i sources , the @xmath41 polarization is mostly negative and for class ii sources , it is mostly positive as shown in figure 9 . the polarization is highest towards edge - on inclinations where the unpolarized central stellar flux becomes more extincted and the envelope scattering geometry is most asymmetric . the wavelength dependence of the polarization is due to several grain properties , especially the opacity , scattering albedo and polarization efficiency . the polarization spectrum of the late class 0 source is inversely correlated with the envelope extinction ( or dust opacity , see figure 2 , top left ) . the more the central unpolarized flux is extincted , the higher is the polarization of the emergent radiation . thus the polarization rises across the 3.1 @xmath0 m water ice feature and the 9.8 @xmath0 m silicate feature . between 3 and 8 @xmath0 m it decreases due to decreasing envelope extinction as well as decreasing scattering albedo . a competing effect is the polarization efficiency ( figure 2 , bottom right ) which increases from 1 to 10 @xmath0 m . in the late class 0 source , the envelope is still opaque at 10 @xmath0 m , so even though the scattering albedo is very low , there is essentially no unpolarized stellar flux to dilute the faint scattered flux . thus the polarization is higher at 10 @xmath0 m than at any other wavelength due to the increased polarization efficiency . in practice , measurement of polarization at this wavelength may be difficult due to the correspondingly low fluxes . in the class i source , the polarization increases at 10 @xmath0 m for the edge - on case where the disk extincts all stellar flux , but it is not as high as at 1 @xmath0 m . this is because the envelope scattering geometry is more asymmetric at 1 @xmath0 m than at 10 @xmath0 m . at 10 @xmath0 m the emerging flux comes from a wider angular range due to lower overall extinctions . in the late class i and class ii edge - on sources , the polarization is large and positive ( @xmath42 ) over a large range in wavelength , peaking near 9 @xmath0 m . at these wavelengths , most of the scattered ( polarized ) light emerges from the disk plane . the polarization is negative ( @xmath43 ) at visible wavelengths in the edge - on class ii source due to a higher amount of scattering from the ambient cloud in the polar regions . since our disk grains do not include the water ice features , there is no feature in the polarization spectrum in these models . the large polarization values in figure 9 suggest that near - ir polarization measurements could provide a way to disentangle the overlap in the colors of sources of different evolutionary states , especially in the large - aperture results . figure 10 shows the same results as in figure 7b except that the size of the symbols correspond to the k - band polarization of that source computed in the same aperture size as the flux . it is clear from this figure that the class 0 sources can be separated from the other evolutionary states on the basis of their k - band polarization . the edge - on class i and ii sources also show measurable ( 5 - 10% ) polarization . thus , we can see that if the very blue sources also have significant polarization , they are likely edge - on class i and ii sources . this provides a way to distinguish these sources from brown dwarfs . figure 11 shows 3-color images of our evolutionary sequence at selected wavelength ranges . the images have been computed through passbands corresponding to the hubble space telescope ( hst ) nicmos near - ir filters f110w , f160w , and f205w with approximate central wavelengths of 1.0 , 1.6 , and 2.0 @xmath0 m ; and the sirtf mid - ir irac and mips filters at approximately 3.6 , 4.5 , 5.8 , 8 , and 24 @xmath0 m . the images are displayed as `` true - color '' images with the shortest wavelength image in blue , the middle in green , and the longest in red . the near - ir images are combined in the left panels of figure 11 . in the middle , the sirtf irac bands are combined with 3.6 @xmath0 m image in blue , 4.5 and 5.8 @xmath0 m images averaged together in green , and the 8.0 @xmath0 m image in red . at right , the 8.0 , 24 , and 70 @xmath0 m images make up the three color planes . the displayed spatial resolution of the images in class 0-i sources is 0.4fwhm , which is finer than that expected from the sirtf irac camera by about a factor of 4 . the displayed resolution of the class ii and iii sources is 0.1and 0.04respectively . the images are viewed at an inclination angle of 80showing the inner 2000 au for the envelope sources ( class 0-i ) , 500 au of the class ii source , and 200 au for the class iii sources . the intensities are scaled to a source luminosity of 1 @xmath22 and a distance of 140 pc . in figure 11a , the minimum intensities displayed are 0.06 , 0.03 , and 15 mjy / srjy arcsec@xmath44 from left to right for the class 0-i sources . the peak intensities displayed are as high as @xmath45 times higher than the minimum intensities , though in many of the edge - on sources , the peak intensity is only 100 - 1000 times higher than the minimum intensity . the class 0 source shows no detectable near - ir emission at the displayed intensities . the near - ir images are similar to those presented by stark et al . the younger sources are redder . the class 0 source is not detected at near - ir wavelengths at the sensitivities shown here . as pointed out by stark et al . , the disk in the late class i source casts a shadow much larger than the disk size itself . hodapp et al . ( 2003 ) see a similar feature in the object asr 41 . the sirtf irac diffuse source sensitivity is expected to be approximately 0.026 mjy / sr at 5.8 @xmath0 m and lower in the smaller wavelength bands ( sirtf observing manual ) . thus , the diffuse emission from edge - on protostars in nearby star forming regions should be detectable with sirtf . the mid - ir images ( middle panels ) tend to be blue and the far - ir images ( right ) are red . this is most striking in the class i - ii sequence . given that the seds of these sources show a large dip centered at about 10 @xmath0 m , this result is understandable . the diffuse emission on large scales in the mid - ir images ( middle panels ) is due to scattering . the envelope grains ( figure 2 , table 3 ) have enough mid - ir albedo to give large detectable scattering nebulae in the envelope sources . as discussed in paper i , models of mid - ir images should provide useful information on the albedo and therefore grain sizes of protostars . the mid - ir images of class ii sources will also provide a useful test for the grain properties . as figure 11b shows , the disk midplane thickness varies with wavelength due to the decrease in dust opacity with increasing wavelength . comparison of near - ir and mid - ir images will provide direct information of the opacity variation with wavelength . modeling the brightness of the scattering nebulae over this wavelength range will provide information on the dust grain sizes . for example , mccabe , duchene & ghez ( 2003 ) have detected scattered light from the hk tau b disk at 11.8 @xmath0 m and concluded that grain sizes of 1.5 - 3 @xmath0 m are required to produce the necessary scattered light . the large - scale red emission in the far - ir images ( right panels ) is due to thermal emission at 70 @xmath0 m . in the far - ir images , the 8 @xmath0 m scattered light is very faint , and the 24 @xmath0 m emission is confined to a smaller region , mainly the disk , as indicated by the yellow color in the center and in the class ii source ( fig 11b top right ) . we note that the spatial resolution at 70 @xmath0 m shown here is much higher than will be obtained by sirtf ( @xmath46 ) . in figure 11b , the minimum intensities are 1 , 1 , and 80 mjy / sr from left to right for the class ii source ; and 0.5 , 0.3 , and 40 mjy / sr for the class iii source . for the class iii source , the peak intensity from the star is about 1000 times brighter than the displayed peak in the near - ir ( left ) and mid - ir ( middle ) panels . that is , the dynamic range between the faintest intensity shown and the peak intensity is about 10 million for the near- and mid - ir images . the corresponding dynamic range for the far - ir image ( right ) is @xmath47 . the stellar size shown in these images has a fwhm of 0.1 . thus detection of the diffuse flux would require some combination of high dynamic range , masking of the central source , excellent stellar point - spread - function subtraction , or adaptive optics techniques . also of interest is the fact that the ambient cloud material is nearly as bright as the disk flux at the near - ir wavelengths ( left ) . this ambient material is also seen as faint emission in the class ii source at near - ir wavelengths at the displayed intensities . figure 12 shows images for more a more pole - on inclination of 30 . the minimum intensities plotted are the same as in figure 12 . the near - ir and mid - ir images ( left and middle panels ) of the class i - iii sources are `` saturated '' ; that is , the central pixels have fluxes that are 1 - 2.5 orders of magnitude higher than the peak displayed fluxes . for typical observed dynamic ranges of @xmath45 and integration times suitable for measuring the stellar fluxes , the diffuse emission of the class i source may be not detected . this is demonstrated also in stark et al . thus pole - on class i sources could be mistaken for class ii sources . several `` flat - spectrum '' sources may be simply low - inclination class i sources , as suggested by calvet et al . however , the 24 @xmath0 m and 70@xmath0 m images do not require as much dynamic range to image the diffuse emission at pole - on inclinations due to the less centrally peaked emission . in the class 0 sources , the central source is still blocked even at 30 inclination . in fact , the class 0 sources are brighter at these lower inclinations than edge - on . this is in contrast to the class i sources which will reveal its faint diffuse emission best when viewed edge - on where the bright central source is blocked . we have attempted to present our models of an evolutionary sequence in a way that can be easily compared to observations . from examination of seds , colors , polarization , and images we conclude the following : \1 . edge - on class i and class ii sources show `` double - peaked '' seds , with a short - wavelength hump due to scattered light , and a long - wavelength hump due to thermal emission ( figure 3 , 6a ) . the dip in the middle is caused , on the short - wavelength side , by the lowering scattering albedo as wavelength increases from 1 to 10 @xmath0 m allowing less scattered light ; and on the long - wavelength side , by increasing extinction ( opacity ) as wavelength _ decreases _ from 100 to 10 @xmath0 m , allowing less thermal emssion to escape . the long - wavelength emission in class 0 sources arises from the opaque envelope . in class i sources , much of the longwave emission emerges from the disk ( figure 6 ) . thus , the longwave class i emission is sensitive to grain properties in the disk , as shown by wolf et al . ( 2003 ) . variations due to inclination cause overlaps in sed shapes between different evolutionary states . class i and ii sources of varying inclination can resemble each other ( figure 5 ) . this overlap in sed behavior is most apparent at mid - ir wavelengths . thus mid - ir color - color plots show large overlap between different evolutionary states ( figure 7 ) . ` flat - spectrum' sources could be intermediate - inclination ( @xmath48 ) class i sources ( fig 6c ) , as proposed by by calvet et al . ( 1994 ) . the aperture size in which fluxes and colors are computed has a large effect in sources surrounded by large envelopes ( class 0 and i sources ) . the colors can change by 1 - 3 magnitudes depending on aperture size . this is important to take into account when comparing different sets of observations or clouds at different distances . in particular , the large aperture results give very blue colors even for the class 0 sources , which in some cases can be bluer than the class iii source ( figure 7b ) . several of the model mid - ir color sequences are well - separated from reddened main sequence , red giant and supergiant stars . however , in most sequences , there is overlap with agb stars , planetary nebulae , and reflection nebulae . polarization can aid in separating evolutionary states in color - color diagrams . the class 0 sources have dramatically higher k - band polarization ( @xmath49% ) than the class i and ii sources . the edge - on class i and ii sources , which tend to be very blue and faint , have between 5 - 15% polarization and should be distinguishable from brown dwarfs and other faint sources . \7 . the emergent flux from class 0-i sources vary with inclination by several magnitudes ( figure 8) . the flux also varies between different evolutionary states with the same luminosity . thus , caution is required in computing luminosity functions . however , this would not be a problem for the unobscured class ii and iii sources , which may make up @xmath50% of the sources in a cloud . large scale diffuse emission in nearby edge - on late class 0 and class i sources should be detectable by sirtf in nearby star formation regions . the much smaller class ii sources will easily be detected by sirtf though at low spatial resolution . these images will provide useful tests of the dust grain properties because the image brightness will be very sensitive to grain albedo ( and thus grain size ) . we thank mike wolff for supplying the information in table 3 . this work was supported by the nasa astrophysics theory program ( nag5 - 8587 ) and the national science foundation ( ast-9909966 , ast-9819928 ) . k. wood acknowledges support from the uk pparc advanced fellowship . m. cohen thanks nasa for supporting his participation in the glimpse legacy science team through jpl , under award # 1242593 with uc - berkeley . lll @xmath51 & stellar radius & 2.09 @xmath52 + @xmath53 & stellar temperature & 4000 k + @xmath54 & stellar mass & 0.5 @xmath15 + @xmath55 & source luminosity & 1 @xmath22 + @xmath13 & disk radial density exponent & 2.25 + @xmath56 & disk scale height exponent & 1.25 + @xmath11 & disk scale height at @xmath51 & 0.01 + @xmath57 & disk viscosity parameter & 0.01 + envelope infall rate ( @xmath58yr ) & @xmath59 & @xmath60 & @xmath61 & @xmath62 & 0 & 0 + envelope mass ( @xmath15 ) & 3.73 & 0.37 & 0.19 & 0.037 & @xmath59 & @xmath63 + envelope inner radius ( @xmath51 ) & 7.5 & 7.5 & 7 & 7 & 7 & 50 + envelope outer radius ( au ) & 5000 & 5000 & 5000 & 5000 & 500 & 500 + disk mass ( @xmath15 ) & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & @xmath64 + disk inner radius ( @xmath51 ) & 7.5 & 7.5 & 7 & 7 & 7 & 50 + disk outer radius ( au ) & 10 & 50 & 200 & 300 & 300 & 300 + disk accretion rate ( @xmath58yr ) & @xmath65 & @xmath66 & @xmath67 & @xmath68 & @xmath68 & 0 + disk acc . lum . ( @xmath69 ) & 0.036 & 0.0069 & 0.0018 & 0.0012 & 0.0012 & 0 + cavity density ( @xmath70 @xmath18 ) & 10@xmath4 & @xmath71 & @xmath72 & 10@xmath3 & @xmath73 & @xmath74 + cavity opening angle ( ) & 5 & 10 & 20 & 30 & 90 & 90 + llll disk midplane & inner disk ( @xmath75 @xmath76 ) & 0.69 & 4.9 + disk atmosphere & upper layers of disk & 0.042 & 4.1 + envelope & infalling envelope & 0.048 & 4.3 + outflow & bipolar cavity & 0.026 & 3.6 +	we present model spectral energy distributions , colors , polarization , and images for an evolutionary sequence of a low - mass protostar from the early collapse stage ( class 0 ) to the remnant disk stage ( class iii ) . we find a substantial overlap in colors and seds between protostars embedded in envelopes ( class 0-i ) and t tauri disks ( class ii ) , especially at mid - ir wavelengths . edge - on class i - ii sources show double - peaked spectral energy distributions , with a short - wavelength hump due to scattered light and the long - wavelength hump due to thermal emission . these are the bluest sources in mid - ir color - color diagrams . since class 0 and i sources are diffuse , the size of the aperture over which fluxes are integrated has a substantial effect on the computed colors , with larger aperture results showing significantly bluer colors . viewed through large apertures , the class 0 colors fall in the same regions of mid - ir color - color diagrams as class i sources , and are even bluer than class ii - iii sources in some colors . it is important to take this into account when comparing color - color diagrams of star formation regions at different distances , or different sets of observations of the same region . however the near - ir polarization of the class 0 sources is much higher than the class i - ii sources , providing a means to separate these evolutionary states . we varied the grain properties in the circumstellar envelope , allowing for larger grains in the disk midplane and smaller in the envelope . in comparing to models with the same grain properties throughout we find that the sed of the class 0 source is sensitive to the grain properties of the envelope only that is , grain growth in the disk in class 0 sources can not be detected from the sed . grain growth in disks of class i sources can be detected at wavelengths greater than 100 @xmath0 m . our image calculations predict that the diffuse emission from edge - on class i and ii sources should be detectable in the mid - ir with the space infrared telescope facility ( sirtf ) in nearby star forming regions ( out to several hundred parsecs ) .
the discovery of carbon nanotubes ( cnts ) by iijima @xcite and their subsequent large - scale production @xcite was followed by the synthesis of cnts filled with atoms and/or molecules . these novel hybrid materials often exhibit one - dimensional characteristics and are presently the subject of fundamental studies as well as research aiming at their application in nanotechnology . for a review on cnts and their filling we refer to refs . and , respectively . self - assembled chains of c@xmath0 fullerene molecules inside single - walled carbon nanotubes ( swcnts ) , the so - called peapods @xcite , provide a unique example of such nanoscopic compound materials , and feature unusual electronic @xcite and structural properties . high - resolution transmission electron microscopy observations on cnts filled sparsely with c@xmath0 molecules @xcite demonstrate the motion of the fullerene molecules along the tube axis and imply that the interaction between c@xmath0 molecules and the surrounding nanotube wall is due to weak van der waals forces and not to chemical bonds . recently , the way the c@xmath0 molecules of a ( c@xmath0)@xmath1@swcnt peapod @xcite @xmath2 c@xmath0 molecules inside in a swcnt are packed in the encapsulating tube has been investigated both experimentally and theoretically @xcite . obviously , the structure of a peapod is governed by the interactions between the c@xmath0 molecules , and by the way a c@xmath0 molecule interacts with the surrounding tube wall . already when considering the stacking of cylindrically confined hard spheres , a possible rudimentary description of a ( c@xmath0)@xmath1@swcnt peapod , various chiral structures of the spheres stacking for varying tube radius are obtained @xcite . in ref . , hodak and girifalco calculated lowest - energy ( c@xmath0)@xmath1@swcnt peapod configurations by means of a continuum approach for the c@xmath0-tube interaction : both a swcnt and a c@xmath0 molecule are approximated as a homogeneous surface cylindrical and spherical , respectively . although in doing so any effect of tube chirality and/or molecular orientation can not be accounted for , such a model provides useful information about the spatial arrangement of the spherical molecules in the tube . ten different stacking arrangements were obtained for the tube radius @xmath3 ranging from @xmath4 to @xmath5 . the simplest configuration ( c@xmath0 spheres " aligned linearly along the tube axis ) occurs for the smallest tubes ( @xmath4 @xmath6 ) . other phases consist of zig - zag patterns or c@xmath0 balls forming helices . some of the predicted phases have been observed experimentally @xcite . interestingly , experimental observations of similar structures formed by c@xmath0 molecules inside bn nanotubes have been reported as well @xcite . an atomistic molecular dynamics study on the arranging of c@xmath0 molecules inside swcnts was carried out by troche et al . @xcite ; the c@xmath0-tube interaction was modelled by adding carbon - carbon lennard - jones 6 - 12 potentials . troche et al . @xcite concluded that the chirality of the encapsulating swcnt has only a minor effect on the lowest - energy configuration of the c@xmath0 molecules and their obtained arrangements , thus depending on the tube radius only , are in full agreement with those of hodak and girifalco @xcite . conclusions on the individual orientations of c@xmath0 molecules inside a swcnt were not given by troche et al . @xcite their goal was to study the packing of several molecules . molecular orientation effects are expected to come into play at sufficiently low temperatures when orientational motion is frozen , and indeed do so as was shown in refs . and , where the potential energy of a single c@xmath0 molecule confined to the tube axis of a swcnt , called nanotube field " , was calculated by treating the tube as a homogeneous cylindrical carbonic surface density but retaining the icosahedral features of a c@xmath0 molecule . a specific dependence on the tube radius was found ; three distinct molecular orientations were observed within the range @xmath7 . it is our opinion that , for calculating tube - c@xmath0 interactions , taking the detailed molecular structure of a c@xmath0 molecule into account has priority over the chiral structure of a nanotube . replacing a swcnt by a continuous cylindrical distribution of carbon atoms is intuitively justifiable , but treating a c@xmath0 molecule as a sphere ( as in ref . @xcite ) with no further structure is a more questionable approximation . indeed , whereas the carbon - carbon bonds in a cnt are of one type , a c@xmath0 molecule features longer ( single " ) and shorter ( double " ) bonds , arranged in pentagons electron - poor regions and hexagons electron - rich regions . the importance of taking the detailed molecular structure properly into account follows from refs . and ; but the neglect of the discrete atomistic structure of the tube when considering c@xmath0-tube interactions , although intuitively plausible , requires solid grounds . the goal of this paper is to answer the question how good a smooth - tube approximation really is , and to confirm the relevance of the precise structure of a c@xmath0 molecule , i.e. the importance of allowing for molecular orientational degrees of freedom . the content of the paper is as follows . in sec . [ nanotubefield ] , we discuss formulas for the calculation of the nanotube field of an encapsulated c@xmath0 molecule for both a continuous " and a discrete " tube . then ( sec . [ comparison ] ) , we plot nanotube fields for a selection of representative nanotubes and make preliminary visual comparisons between the two approaches . in sec . [ casestudy ] , we present an all - variable treatment and apply it for tubes with intermediate and small tube radii . finally , general conclusions are given ( sec . [ discussion ] ) . we consider a c@xmath0 molecule in a swcnt , the molecule assuming a centered position in the tube , and set up a cartesian system of axes @xmath8 so that the @xmath9-axis coincides with the tube s long axis and contains the molecule s center of mass ( fig . [ figswcnt ] ) . the potential energy @xmath10 of the c@xmath0 molecule then depends on the orientation of the molecule , which can be characterized by three euler angles @xmath11 , on the position of the molecule along the tube , i.e. the @xmath9-coordinate of the molecular center of mass for which we write @xmath12 , and on the tube indices @xcite @xmath13 : @xmath14 for the euler angles we use the convention of ref . @xcite : a coordinate function @xmath15 is transformed as @xmath16 , where @xmath17 stands for the succession of a rotation over @xmath18 about the @xmath9-axis , a rotation over @xmath19 about the @xmath20-axis , and a rotation over @xmath21 about the @xmath9-axis again . the @xmath22- , @xmath20- and @xmath9-axes are kept fixed . note that the * coordinate * transform associated with the euler angles reads @xmath23 and that the rotation of the c@xmath0 molecule over @xmath24 about the @xmath9-axis is performed last . as the starting orientation [ @xmath25 we take the so - called standard orientation [ fig . [ orientations](a ) ] : twofold molecular symmetry axes then coincide with the cartesian axes and every cartesian axis intersects two opposing double bonds . ( we recall that the carbon - carbon bonds of a c@xmath0 molecule can be divided into two categories : 60 single bonds , fusing pentagons and hexagons , and 30 double bonds , fusing hexagons . the latter are somewhat longer than the former @xcite . ) bearing in mind the results of refs . and and anticipating the results obtained in the present work , we point out two more molecular orientations of importance . the first is the pentagonal " orientation class , obtained by the euler transformation @xmath26 , resulting in two opposing pentagons of the c@xmath0 molecule being perpendicular to the @xmath9-axis [ fig.[orientations](b ) ] . the second is the category of hexagonal " orientations , a result of the euler transformation @xmath27 , making two opposing hexagons lie perpendicular to the @xmath9-axis [ fig . [ orientations](c ) ] . the angle @xmath28 is related to the dihedral angle @xmath29 ( the inner angle between adjacent faces ) of a regular icosahedron : @xmath30 . other @xmath31 pairs yield pentagonal " , hexagonal " and double - bond " orientations as well : 12 pairs correspond to a pentagonal " , 20 pairs to a hexagonal " , and 30 pairs to a double - bond " orientation since a c@xmath0 molecule has 12 pentagons , 20 hexagons and 30 double bonds . for the description of the interaction between the c@xmath0 molecule and the nanotube we follow earlier work @xcite and treat the c@xmath0 molecule as a rigid cluster of interaction centers ( ics ) . not only c atoms ( ` a ' ) act as ics , but also double bonds ( ` db ' ) and single bonds ( ` sb ' ) . we label the 60 atoms by the index @xmath32 . in the center of every of the 60 single bonds an ic is put , labelled by the index @xmath33 . on each of the 30 double bonds , 3 ics dividing the bond in four equal parts are put , totalling to 90 db ics , labelled @xmath34 . such a construction was originally introduced for modelling intermolecular interactions in solid c@xmath0 ( c@xmath0 fullerite ) ; having three ics per double bond reflects the electronic density being smeared out along a double bond @xcite . every ic of the c@xmath0 molecule interacts with every atom of the nanotube via a pair interaction potential @xmath35 , depending on the type of ic ( ` t ' = ` a ' , ` db ' , ` sb ' ) . the total potential energy is then obtained by summing over all pair interactions : where @xmath37 indexes the atoms of the tube and @xmath38 stands for their respective coordinates . as in refs . and , we use born mayer van der waals pair interaction potentials : @xmath39 again , the use of such pair potentials was originally introduced for studying c@xmath0 - c@xmath0 interactions in c@xmath0 fullerite @xcite ; it lead to a crystal field potential and a structural phase transition temperature @xcite in good agreement with experiments . the potential constants @xmath40 , @xmath41 and @xmath42 used are those of ref . . in eq.([discreteeq ] ) , the sum over tube atoms , labelled by the index @xmath37 and having coordinates @xmath43 , can be restricted to atoms in a certain vicinity of the c@xmath0 molecule , realized by imposing the criterion @xmath44 in refs . @xcite and @xcite , a smooth - tube approximation to eq . ( [ discreteeq ] ) was presented . the actual network of carbon atoms making up the swcnt is replaced by a homogeneous , cylindrical carbonic " surface density with value @xmath47 ( units @xmath48 ) . the c@xmath0 molecule - nanotube interaction energy is then rewritten as @xmath49 where @xmath50 is the cylindrical coordinate of a point on the tube ( @xmath51 , @xmath52 , @xmath53 ) and @xmath3 is the tube radius . the motivation for introducing approximation ( [ smootheq ] ) is twofold . one reason is the dependence of @xmath54 on the tube radius @xmath3 rather than on the tube indices @xmath13 . indeed , @xmath3 remains the only relevant tube - characteristic parameter and as such simplifies a systematic investigation of carbon nanotubes . a further consequence of the tube s cylindrical symmetry is the irrelevance of the euler angle @xmath55 ( a final rotation of the c@xmath0 molecule over @xmath24 about the tube axis does nt matter ) and of the @xmath9-coordinate @xmath12 ( for infinite or long - enough tubes ) . a second advantage of the smooth - tube ansatz is the possibility of performing an expansion of @xmath54 into symmetry - adapted rotator functions , a point we will return to in sec . [ discussion ] . we stress the limited dependence of @xmath54 by writing @xmath56 to distinguish the smooth - tube approximation from the discrete case , we add the subscript ` discrete ' : @xmath57 where the actual expression is given by eqs . ( [ discreteeq ] ) ( [ cutoff ] ) . in this paper we test the validity of smooth - tube approximation ( [ smootheq ] ) by comparing @xmath58 and @xmath54 for a selection of tubes . bearing in mind the three qualitatively different radii ranges ( @xmath59 , @xmath60 @xmath61 and @xmath62 @xmath63 ) obtained in ref . @xcite , we have selected zig - zag , armchair and chiral tubes with radii around @xmath64 , @xmath65 and @xmath66 . we have generated @xmath13 tubes starting from a graphene sheet with basis vectors @xmath67 and @xmath68 , where @xmath69 and @xmath70 are planar cartesian basis vectors , and performing the roll - up along the vector @xmath71 @xcite . the tube is then positioned so that the c atom originally ( before rolling up ) at @xmath72 lies in the @xmath73 plane with @xmath22-coordinate @xmath74 and @xmath20-coordinate @xmath3 and that the cylinder containing the c atoms has its long axis coinciding with the @xmath9-axis . the c@xmath0 molecule is initially positioned so that its center of mass lies at the origin ( @xmath75 ) ; a translation along the @xmath9-axis away from the initial position is measured via the center of mass @xmath9-coordinate @xmath12 . the radius of the tube with indices @xmath13 reads @xmath76 , with @xmath77 @xcite ; the corresponding surface density has the value @xmath78 a further tube parameter is its translational periodicity @xmath79 , relevant when considering the @xmath12-dependence of v@xmath80 . while @xmath79 is small for non - chiral i.e. zig - zag , @xmath81 , and armchair , @xmath82 tubes , the translational period can get very large for chiral tubes @xcite . a tube may also have an @xmath83-fold symmetry axis ( coinciding with the @xmath9-axis ) and therefore a rotational period @xmath84 . when considering a tube with @xmath83-fold rotational symmetry it suffices to examine the interval @xmath85 . the periodicities and other tube characteristics of our selected tubes are listed in table [ tubes ] . j. cambedouzou , v. pichot , s. rols , p. launois , p. petit , r. klement , h. kataura , and r. almairac , eur . j. b * 42 * , 31 ( 2004 ) ; j. cambedouzou , s. rols , r. almairac , j .- l . sauvajol , h. kataura , and h. schober , phys . b * 71 * , 041403(r ) ( 2005 ) .	we calculate the van der waals energy of a c@xmath0 molecule when it is encapsulated in a single - walled carbon nanotube with discrete atomistic structure . orientational degrees of freedom and longitudinal displacements of the molecule are taken into account , and several achiral and chiral carbon nanotubes are considered . a comparison with earlier work where the tube was approximated by a continuous cylindrical distribution of carbon atoms is made . we find that such an approximation is valid for high and intermediate tube radii ; for low tube radii , minor chirality effects come into play . three molecular orientational regimes are found when varying the nanotube radius .
the observation of the signal of the @xmath0 exotic baryon @xcite , broadly known as the pentaquark for its minimal five - quark fock space assignment in quantum chromodynamics , was stimulated by the prediction @xcite of a @xmath5 baryon antidecuplet . many experimental and theoretical studies have been devoted to this resonance @xcite . the original states proposed to form this antidecuplet @xcite were : @xmath6 where the first and last , explicitly exotic states , had not been observed at that time . isospin su(2 ) is expected to hold to very good accuracy and we do not list the @xmath7 quantum number . this assignment is now , however , challenged for several reasons . first , the na49 collaboration @xcite reported evidence for an exotic cascade @xmath8 , probably in the same antidecuplet , with a much lighter mass , 1860 mev . this is somewhat problematic as doubts have arisen @xcite , and it has not been seen in other experiments @xcite ( see , however , k. kadija s presentation at the pentaquark04 workshop @xcite with new reanalyses still supporting the findings of ref . . however , should the state be reconfirmed at 1860 mev , using the standard gell - mann okubo rule ( gmo ) of equal mass splittings for the su(3 ) antidecuplet , the mass of @xmath9 would have to be near 1647 mev , which is about 60 mev below the nominal one @xmath1 . furthermore , the mass of @xmath10 would have to be about @xmath11 mev . since a @xmath12 resonance is listed at @xmath13 mev with the same spin and parity , we will refer to the @xmath12 member of the antidecuplet as @xmath14 . the association of this state to the @xmath15 would not fit in that scenario @xcite . second , quark model calculations that have appeared after the report of the evidence of @xmath0 tend to predict an @xmath9 at around @xmath16 mev @xcite . these predicted the @xmath17 at @xmath18 mev @xcite , which is more in line with the experimental outcome than the original calculation of the chiral soliton model , although the latter can be readjusted ( then underestimating the @xmath19 ) . another difficulty arises because of the potential mixing of the nonexotic members of the multiplet , the @xmath9 and @xmath10 with members of pentaquark or ordinary three - quark octets @xcite . this would make the mass splitting between the physical states dependent on two mixing angles . a current conjecture is a mixing with the roper resonance @xcite that would , by level repulsion , push the @xmath1 further above the @xmath0 than predicted by the gmo rule . also with ideal mixing , the hidden strangeness @xmath20 wave function dominates the @xmath1 , thus raising its mass . however , such a strong mixing is not preferred by other authors @xcite . in summary , a new @xmath19 state would have to be searched for at a smaller mass if we were to impose perfect gmo rule . the models we work with in this paper are rather phenomenological . however , our method , based on symmetry principles , is suited to at least estimating meson cloud effects , which are important for the understanding of pentaquark properties . the main conclusion of this work is that the virtual `` two - meson cloud '' yields an attractive self - energy that provides about 20% of the pentaquark mass splittings . we believe that our study here will become useful when more data are available . the study presented here is complementary and looks for another source of mass splitting not contemplated by the gmo rule . it would come from the two - meson cloud . the possibility of constructing the @xmath0 as a @xmath21 bound state @xcite has been examined in some detail @xcite employing meson - meson and meson - baryon interactions from chiral lagrangians , where attraction was found but not strong enough to bind the system . yet , this result leaves one wondering as to what role the two - meson cloud could play in the stability of the state . coupling to multi - meson components is also implicit in the chiral soliton picture , which leads to small masses of the @xmath0 @xcite . in the present paper , we do not face the possible contribution of the one - meson cloud to the antidecuplet binding , which can be easily addressed as a minor correction to our results . the small width of the @xmath0 to @xmath22 , in spite of the appreciable phase space available , qualitatively demands that this contribution should be reasonably small ; in fact , it has been checked quantitatively in refs . the self - energy of @xmath0 with a two - meson cloud has been studied in parallel @xcite in the context of the medium modification of @xmath0 and possible formation of @xmath0 hypernuclei @xcite . we here report in full on vacuum results for not only the @xmath0 but also other members of the antidecuplet . an important experimental input relevant to the present study is the relatively large branching ratio of @xmath1 into @xmath2 , about @xmath23 @xcite . the branching ratio into @xmath2 with the two pions in an @xmath3 wave is @xmath24 and into @xmath25 , @xmath26 . this @xmath1 resonance and its baryon - meson - meson decay mode has been used in ref . @xcite to produce a good shape of the @xmath27 distribution in the @xmath28 reaction leading to the @xmath29 . in the present work , we assume that the @xmath1 has a large antidecuplet component @xcite , and we will perform a study of the @xmath2 @xmath3 wave and @xmath25 decay channels of this resonance and their influence on the masses of various members of the antidecuplet . certainly one has to accept a mixing with an octet component for realistic resonances in order , for instance , to explain the @xmath1 decay into @xmath30 , which is forbidden for its antidecuplet component @xcite . but we do not expect the mixing angle to be close to ideal , as this would imply a stronger @xmath31 branching ratio than @xmath26 , as observed experimentally . the decay pattern of @xmath1 and @xmath32 also supports the small mixing angle @xcite . the present study also provides information on the antidecuplet baryon - baryon - meson - meson @xmath33 contact interaction , which could be applied to the study of @xmath0 production with the @xmath34 and @xmath35 reactions . these reactions are studied in refs . @xcite and experimental information is becoming available @xcite . the paper is organized as follows . in sec . [ sec : lag ] , we construct various @xmath36 interactions with the two octet mesons and one baryon belonging to octets and with the other baryon to an antidecuplet . in sec . [ sec : self ] , we compute the contributions of two - meson and one - baryon loops to the mass splittings among the members of antidecuplet baryons . in sec . [ sec : results ] , we present numerical results and discuss the importance of two - meson contributions to the mass splittings and partial decay widths . as we will see , the contributions from the two - meson loops provide sizable contributions to supplement the mass splittings naively expected from strange quark counting . we will then discuss the range of interaction strengths of various coupling terms . section [ sec : summary ] is devoted to a summary . we also add appendices , where complete tables for the @xmath36 interactions are presented . following a common convention @xcite , we write the physical meson and baryon fields as follows @xmath37 the antidecuplet containing the exotic pentaquark states is a tensor @xmath38 totally symmetric in its three su(3 ) indices . the components of @xmath38 are related to the physical fields by @xmath39 where we have adopted the normalization in ref . @xcite , which is different from those used in refs . @xcite by a sign and/or a factor . now we consider the possible interaction lagrangians , constrained to be su(3 ) symmetric . we intend to address the process @xmath40 where an octet baryon @xmath41 and two octet mesons @xmath42 couple to an antidecuplet baryon @xmath43 . to have an su(3 ) invariant lagrangian , we couple first the two @xmath44 and then combine the resulting irreducible representations with the baryon @xmath45 to produce a @xmath46 representation . the group theoretical irreducible decomposition gives @xmath47 here @xmath48 and @xmath49 denote symmetric and antisymmetric combinations of the two - meson fields . hence we obtain four @xmath46 representations after recoupling @xmath50 , @xmath51 , @xmath52 and @xmath53 with @xmath41 . in constructing effective lagrangians , we follow the principle of using the minimum numbers of derivatives in the fields . this will be released later when we discuss possible structures involving derivatives . to construct @xmath48 from two @xmath44 , we have in tensor notation @xmath54 = & \phi_{i}{}^{a}\phi_{a}{}^{j}+\phi_{i}{}^{a}\phi_{a}{}^{j } -\frac{2}{3}\delta_{i}{}^{j}\phi_{a}{}^{b}\phi_{b}{}^{a } \nonumber \\ = & 2\phi_{i}{}^{a}\phi_{a}{}^{j } -\frac{2}{3}\delta_{i}{}^{j}\phi_{a}{}^{b}\phi_{b}{}^{a } . \label{eq : mm8s}\end{aligned}\ ] ] we combine this now with an @xmath45 to give an antidecuplet @xmath55 = & 2\phi_{l}{}^{a}\phi_{a}{}^{i } b_{m}{}^{j}\epsilon^{lmk } \nonumber \\ & + ( i , j , k\text { symmetrized } ) . \label{eq : mm8sb}\end{aligned}\ ] ] hence , the interaction lagrangian becomes @xmath56 where h.c . denotes the hermitian conjugate terms , in order to take into account the processes in which the antidecuplet is in the initial state . note also that two @xmath57 fields have appeared , and we have included a factor @xmath58 in order to make @xmath59 dimensionless ( @xmath60 is the pion decay constant @xmath61 mev ) . next we take the antisymmetric combination of the @xmath44 and @xmath44 , which for identical meson octets leads to @xmath62 = \phi_{i}{}^{a } \phi_{a}{}^{j } -\phi_{i}{}^{a } \phi_{a}{}^{j}=0 . \label{eq : mm8a}\ ] ] so given the identity of the meson octets , this combination is zero . the simplest way to construct the lagrangian of this structure is to introduce a derivative in one of the fields , which leads automatically to the vector current consisting of two meson fields . proceeding as before , we combine this structure with the @xmath45 to give @xmath63 then finally @xmath64 where @xmath65 is dimensionless . this interaction lagrangian contains the coupling of the @xmath1 with @xmath2 , the two pions in a @xmath66-meson type correlation . from the experimental branching ratio , we can determine the coupling constant @xmath65 . to construct the @xmath63 combination from two mesons , we have now @xmath67 = & \epsilon^{lmk}\phi_l{}^{i}\phi_m{}^{j } + ( i , j , k\text { symmetrized } ) \nonumber\\ = & \epsilon^{lmk}\phi_l{}^{i}\phi_m{}^{j } + \epsilon^{lmk}\phi_l{}^{j}\phi_m{}^{i } + \epsilon^{lmi}\phi_l{}^{j}\phi_m{}^{k } \nonumber \\ & + \epsilon^{lmi}\phi_l{}^{k}\phi_m{}^{j } + \epsilon^{lmj}\phi_l{}^{k}\phi_m{}^{i } + \epsilon^{lmj}\phi_l{}^{i}\phi_m{}^{k } \nonumber\\ = & 0 , \label{eq : mmbar10}\end{aligned}\ ] ] which is identically zero for equal meson octets . the expansion for the @xmath68 representation leads to @xmath69 = & \phi_{i}{}^{j } \phi_{k}{}^{l } + \phi_{i}{}^{l } \phi_{k}{}^{j } + \phi_{k}{}^{j } \phi_{i}{}^{l } + \phi_{k}{}^{l } \phi_{i}{}^{j } \nonumber\\ & -\frac{1}{5 } \left(\delta_{i}{}^{j}d_{k}{}^{l } + \delta_{i}{}^{l } d_{k}{}^{j } + \delta_{k}{}^{j } d_{i}{}^{l } + \delta_{k}{}^{l } d_{i}{}^{j}\right)\nonumber \\ & -\frac{1}{6 } \left(\delta_{i}{}^{j}\delta_{k}{}^{l } \phi_{a}{}^{b } \phi_{b}{}^{a } + \delta_{i}{}^{l } \delta_{k}{}^{j } \phi_{a}{}^{b } \phi_{b}{}^{a}\right ) , \label{eq : mm27}\end{aligned}\ ] ] where @xmath70 is defined in eq . . now the combination of @xmath53 to @xmath45 to give the @xmath63 representation leads to @xmath71 + \text{h.c . } , \label{eq:27lag}\end{aligned}\ ] ] where the first term gives us a new su(3 ) structure , but the second one is equal to @xmath72 given in eq . . to summarize briefly , for the possible su(3 ) symmetric couplings of @xmath36 , there are two independent terms with no derivatives , namely eqs . and . with one derivative , there are four more terms available , but we will consider only eq . , which has the structure for the decay of @xmath73-wave ) as observed experimentally . in the perturbative chiral lagrangian approach , one would like to implement chiral symmetry as a derivative expansion . in addition , one of the advantages of chiral lagrangians is that they relate coupling constants of different processes and , in particular , with increasing number of mesons . however , in the present case we can not take advantage of any of these relations , since the couplings for the present lagrangians are _ a priori _ completely arbitrary , and we are only interested in the two - meson problem . still , in this section we build the lowest - order chiral lagrangian , with two derivatives . let us remark that the chiral expansion with baryons is known to converge much more slowly than chiral perturbation theory with mesons , and this lowest - order lagrangian can only be expected to give a mere qualitative description of the physics . for that reason , to build the lagrangians of the previous secs . [ subsec:8slag ] and [ subsec:27lag ] we just relied on flavor su(3 ) . still , we will check here that the lack of chiral symmetry in those lagrangians does not have much relevance to the mass splittings and decays we are interested in , since already with the leading - order lagrangian we get qualitatively the same results . in other words , the relevant symmetry here is su(3 ) , not chiral symmetry . to show this , we write a chiral invariant lagrangian by making the substitution @xmath74 in eq . such that @xmath75 where @xmath76 is the axial current written in terms of the chiral field @xmath77 : @xmath78 with @xmath79 . to the leading order in meson field , @xmath80 , we find the interaction lagrangian induced from eq . by making the replacement @xmath81 obviously , the su(3 ) structure is not affected by this procedure , although the use of lagrangians involving derivatives will introduce some degree of su(3 ) breaking due to the momenta of mesons . hence , it is useful to verify that this chiral invariant lagrangian will lead eventually to the same results as those obtained from the lagrangians without derivatives in the fields . we also perform self - energy calculations using this @xmath48 chirally symmetric lagrangian , eq . ( [ eq : chirallag ] ) . in this section , we consider the su(3 ) breaking interaction term within the context of chiral lagrangians . without using derivatives in the fields , the only possible term is a mass term that violates both su(3 ) and chiral symmetry , but in the way demanded by the underlying qcd lagrangian @xcite . the mass term appears through the combination @xmath82 with the mass matrix , written in terms of the meson masses , @xmath83 then it leads to the lagrangian @xmath84 in the expansion of @xmath85 , we have two meson fields with the structure @xmath86 substituting @xmath87 for @xmath85 in eq . , we obtain the desired mass lagrangian . the antidecuplet self - energies deduced from one of the interaction lagrangians can be obtained by @xmath88 where the index @xmath89 stands for @xmath90 , @xmath91 , @xmath92 , @xmath93 and @xmath94 for corresponding lagrangians , , , and ; @xmath95 denotes the antidecuplet states @xmath96 , @xmath9 , @xmath10 and @xmath17 ; the argument @xmath97 is the energy of the antidecuplet baryon ; and the factors @xmath98 are @xmath99 in eq . , @xmath100 are su(3 ) coefficients that come directly from the lagrangians when evaluating the different matrix elements . we compile the results in appendix [ sec : coef ] . the function @xmath101 of argument @xmath97 ( the energy of the assumed state of the antidecuplet at rest ) is the two - loop integral with two mesons and one baryon as shown in fig . [ fig : loop2meson ] . @xmath102 where @xmath103 in these expressions , @xmath94 and @xmath104 are the masses of a baryon and mesons . the more complicated integrand in @xmath105 arises because of the @xmath106 factor when one derivative is included as in eq . . we neglect the negative - energy intermediate baryon propagator as this is suppressed by a further power of @xmath107 , leading only to a small relativistic correction . the @xmath108 and @xmath109 integrations of eq . are easily carried out , and we obtain @xmath110 the real part of this integral is divergent . we regularize it with a cutoff @xmath111 in the three momentum on @xmath112 and @xmath113 , which is a parameter of the calculation and its value must be somewhat larger than the scale of the typical pion momenta . on the other hand , we use low - energy lagrangians with one or two derivatives at most , and thus the cutoff should not be too large ; otherwise , terms with more derivatives could become relevant . in this work we will take @xmath111 in the range 700 800 mev , roughly the order of magnitude of the cutoff used to regularize the meson - baryon loops in the study of the @xmath114 interaction @xcite . with the @xmath115 of sec . [ subsec : chirallag ] , the cutoff is smaller in order to reproduce analogous results to those with the @xmath90 lagrangian . the imaginary part of the diagram represents the decay width , in accordance with the optical theorem . the total decay width of a member of the antidecuplet to any @xmath116 states is given by @xmath117 while the partial decay width to a particular channel is given by @xmath118 as an example , let us give in detail the contribution from @xmath119 to the @xmath120 self - energy @xmath121 , \label{eq : example1}\end{aligned}\ ] ] and the contribution from @xmath72 to the @xmath17 self - energy @xmath122 . \label{eq : example2}\end{aligned}\ ] ] the expression for all cases can be derived from tables ix xii in appendix [ sec : formulae ] . in eq . ( [ eq : selfenergy ] ) , we gave a contribution to the self - energy from one interaction lagrangian @xmath123 . for the total self - energy , the sum should be taken over the five interactions ( @xmath124 and @xmath94 ) at each vertex . this means that at each vertex function , we should make the replacement as @xmath125 . we shall , however , not take into account interference between the @xmath91 term and the others because of the @xmath4-wave nature of the term . it is known that @xmath126 wave ) occurs through the @xmath127 decay . in order to keep close to the experimental information , we shall also assume that the pair of mesons in the @xmath91 case reconstruct a vector meson . hence , we replace the contact interaction of the @xmath119 to account for the vector meson propagator ( fig . [ fig : loopvec ] ) and include the factor @xmath128 in each @xmath129 vertex . the consideration of these contributions needs extra work on the loop integrals since we introduce new poles . the imaginary part of the integrals ( associated to placing on - shell the @xmath116 intermediate states ) can be easily accounted for by multiplying the integrand of eq . by @xmath130 where @xmath131 accounts for the width of the vector meson ( @xmath66 or @xmath132 depending on the @xmath133 ) incorporating the energy dependence through the factor @xmath134 multiplied to the nominal width , with @xmath135 the relative three momentum of the mesons in the decay of the vector meson in the rest frame and @xmath136 . for the real part , one must sort out the poles of the vector meson and the intermediate @xmath116 state , which is technically implemented by means of the integral @xmath137 where p.v . stands for the principal value . here , we neglected the width of the vector meson , which does not play much of a role in the off - shell regions of integrations . the @xmath108 and @xmath109 integrations can be performed analytically , and one obtains the simple expression @xmath138 where @xmath139 is the on - shell energy of the vector meson . next we present some numerical results that illustrate the antidecuplet mass shifts and decay widths to three - body channels . one of the most exciting aspects in the antidecuplet is that the @xmath0 is located about @xmath140 mev below the @xmath141 threshold . hence , it can not decay into this or any other @xmath116 channels to which it couples . for the interaction lagrangians , we obtain the @xmath142 coefficients from the experimentally allowed decay amplitudes of the @xmath1 . we give several examples that illustrate the general behavior of the two - meson cloud , common to the lagrangians described in previous sections . before studying each of the lagrangians , let us recall that the mass splitting of the antidecuplet has a contribution which follows the gmo rule , and it would be originated by the difference of the masses of the constituent quarks and their correlations . to this , we add the splitting coming from the real part of the self - energy due to the meson cloud that we are studying . thus , the masses of the antidecuplet are approximately given by @xmath143 where @xmath144 is the bare mass of the antidecuplet and @xmath145 is the gmo mass splitting , part of which simply comes from the difference of the constituent quark masses . in the constituent quark model , @xmath145 is related to the difference between the constituent masses of @xmath146 , @xmath147 and @xmath3 quarks , @xmath148 . certainly , quark correlations can also contribute to the experimental value of @xmath145 . the difference between the light and strange quark masses has been obtained , for example from hyperfine splittings , in ref . @xcite , @xmath149 whereas for baryons @xmath150 but other differences like @xmath151 , @xmath152 or @xmath153 suggest a wider range , from 122 to 190 and 250 mev , respectively . as we will see , the values of @xmath154 needed within this work are of this order of magnitude but somewhat larger , leaving room for extra quark correlations effects . [ [ antidecuplet - mass - shift - with - mathcal - l8s - and - mathcal - l8a ] ] antidecuplet mass shift with @xmath155 and @xmath156 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ to fix the couplings of the lagrangians , we start by taking @xmath72 and @xmath119 defined above and adjusting the coupling constants to obtain the partial decay widths of the @xmath1 to @xmath157 wave , isoscalar ) and @xmath158 wave , isovector ) respectively . these are controlled by the imaginary part of the self - energies , which are finite and independent of the cutoff . the central values in the particle data group ( pdg ) @xcite are @xmath159 and the uncertainties ( counting those of the branching ratio and the total width ) can be a large fraction of these numbers . a fit to these central values gives us @xmath160 with these couplings , we calculate the real part of the self - energies for all the antidecuplet . for the bare antidecuplet mass @xmath97 as input , we take an average value of @xmath161 mev . we also performed a calculation with different values of @xmath97 and found that the results have the same qualitative trend , but the depth of the binding varies . to estimate the binding , we show the mass shift from the @xmath72 with respect to @xmath97 in fig . [ fig : pzerodep ] . we see that , independently of the values of @xmath97 , all the self - energies are attractive , and that the interaction is more attractive the larger the strangeness ; hence , the @xmath120 is always more bound . in fig . [ fig : totresult ] we show the results for the contributions from @xmath72 and total contributions of @xmath119 and @xmath72 , with @xmath161 mev and cutoffs @xmath163 and @xmath164 mev . the numerical values of the mass shifts are displayed in table [ tbl : totresult ] . we see that @xmath72 provides more binding than @xmath119 for the same cutoff . the total binding for @xmath120 ranges from 90 to about 130 mev , depending on the cutoff . the splitting between the @xmath120 and @xmath17 states is about 45 mev for a cutoff of 700 mev and 60 mev for a cutoff of 800 mev . since the experimental splitting is 320 mev for the @xmath165 and @xmath166 , the splitting provided by the two - meson cloud is on the order of 20% of the experimental one . t. nakano _ et al . _ , ( leps collaboration ) phys lett . * 91 * , 012002 ( 2003 ) . d. diakonov , v. petrov , and m. v. polyakov , z. phys . a * 359 * , 305 ( 1997 ) . m. praszalowicz , in _ workshop on skyrmions and anomalies _ , edited by m. jezabek and m. praszalowicz , ( world scientific , singapore , 1987 ) p. 112 . c. alt _ et al . _ , ( na49 collaboration ) phys * 92 * , 042003 ( 2004 ) . h. g. fischer and s. wenig , eur . phys . j. c * 37 * , 133 ( 2004 ) . k. t. knopfle ( hera - b collaboration ) , m. zavertyaev , and t. zivko , j. phys . g * 30 * , s1363 ( 2004 ) . 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our aim here is to present the holomorphic classification of complex rational surfaces in a simple way . we base ourselves on the fact that the topological classification of real surfaces is very intuitive as one can see by following our pictures . one should keep in mind that a real surface is a two dimensional object while a rational surface is a four dimensional object . hence , we can follow the classification of real surfaces by drawing pictures and use the intuition built in this case to understand the classification of rational surfaces , for which we are no longer able to draw pictures . when classifying real surfaces it is natural to take the following steps . 1 . basic surfaces : where we give the first elementary examples of real surfaces . 2 . two types of surfaces : where we show that real surfaces are naturally divided into two categories , namely orientable and non - orientable . 3 . constructing new surfaces out of basic ones : where we define the operation of connected sum and show how to build new examples of real surfaces . the classification theorem : that says that all real surfaces can be obtained out of the basic ones by means of the connected sum operation . we give an outline of the proof of the classification theorem for real surfaces . then we present the classification of rational surfaces following exactly the same four steps as above with the appropriate modifications . 1 . basic surfaces : where we give the first elementary examples of complex surfaces . 2 . two types of surfaces : where we show that complex surfaces are naturally divided into two categories , namely rational and irrational . 3 . constructing new surfaces out of basic ones : where we define the operation of blowing up and show how to build new examples of complex surfaces . the classification theorem : that says that all rational surfaces can be obtained out of the basic ones by means of the blowing up operation . we give an informal presentation and indicate references for detailed proofs . we only assume some elementary knowledge of smooth manifolds and some linear algebra . by a real surface we mean a smooth manifold of two real dimensions , which is connected and compact . by a complex surface we mean a complex smooth manifold of two complex dimensions ( hence four real dimensions ) which is connected and compact . * 1 . basic surfaces . * we begin with examples of real surfaces . * the sphere @xmath0 * the torus @xmath1 ( which looks like a doughnut ) * the n - torus @xmath2 ( which looks like a doughnut with @xmath3 holes ) + * the orientable surfaces * * the real projective plane @xmath4 for @xmath5 * the klein bottle @xmath6 which can be viewed as a twisted torus , constructed by gluing the two ends of a cylinder with a twist . + * a non - orientable surface * * 2 . two types of surfaces . * there is a natural division in the classification of real surfaces , given by orientability . orientable surfaces are intuitively the ones that have two sides , that is inside and outside , like @xmath7 @xmath8 and @xmath9 orientable real surfaces are the so called riemann surfaces . the non - orientable surfaces are the ones containing a mbius band , which therefore do nt have outside and inside , like @xmath10 and @xmath11 + * 3 . constructing new surfaces out of the basic ones . * when one wants to study the topology of real surfaces , there is a basic operation , which allows us to construct new surfaces out of old ones . this operation is called connected sum ( represented by # ) , and works as follows . given two surfaces @xmath12 and @xmath13 cut out open discs @xmath14 in @xmath12 and @xmath15 in @xmath16 and then glue the two surfaces @xmath17 and @xmath18 by identifying the boundaries of @xmath14 and @xmath19 here are some examples . * @xmath20 # @xmath0 = @xmath20 for any real surface @xmath21 that is , the sphere works as an identity for the # operation * @xmath1 # @xmath1 = @xmath22 that is , the 2-torus is obtained by gluing together two tori ( intuitively , just put together two doughnuts one after the other and you get a 2-torus . ) * @xmath2 # @xmath23 = @xmath24 which is a generalization of the previous example . * @xmath10 # @xmath10 = @xmath11 this case is not intuitively obvious , but is very easy to show by cuting and pasting . we give a proof of this in section iii . * 4 . the classification theorem . * once we have the basic surfaces and the operation for constructing new surfaces out of old ones , we can build all real surfaces . the classical structure theorem is the following ( see [ 2 ] ) . every compact connected real surface is obtained from either the sphere @xmath0 or the torus @xmath1 or the real projective plane @xmath10 by connected sums . in this section we outline the proof of the classification theorem for real surfaces . for a detailed proof see [ 2 ] . by the very definition of the connected sum operation , we know that performing connected sums on real surfaces , we always obtain a real surface . however , it is not obvious that all real surfaces are obtained by connected sums starting only with the sphere , the torus and the real projective plane . so in this section , we will give an idea of how to show this . to prove this theorem one very useful tool is to construct surfaces by identifying sides of polygons . here one should keep in mind that we want a topological classification , so that an object is considered the same as a continuous deformation of it which can be reversed . for example , from a topological point of view a disc and a square are considered the same topological object , since each can be continuously deformed to the other . here are some examples of representing real surfaces by polygons with sides identified . we will name our polygons according to their sides read counterclockwise . * the polygon representations * now we also need to represent connected sums using this method of polygons . this is easily seen by an example . we can represent any of our real surfaces by a polygon with sides identified . however this representation is not unique . to understand this look at the following sequences which shows that @xmath25 equals the klein bottle . let us simply continue from the previous sequence of pictures . we cut our polygon with sides @xmath26 through a diagonal side @xmath27 . then we glue the two triangles back together by their @xmath28 sides to obtain the sequence of sides @xmath29 which is the klein bottle . now we are ready to understand the proof of the structure theorem of real surfaces . the proof goes as follows . every real surface is obtained by identifying sides of a polygon . this is intuitively believable . the rigorous mathematical reason is that every surface has a triangulation . then to prove the theorem one wants to see that one can choose a representation which shows the correspondence with a connected sum of tori , spheres or real projective planes . first of all it is clear that each letter appears in two sides of the polygon since we want to obtain a closed surface . hence for every surface we get a sequence of letters where each letter appears twice . we put the exponent @xmath30 when a letter appears in a side pointing in the clockwise direction . then we use the operations of cutting and pasting as we did in the above example to get the polygon to a nice form . one shows that cutting and pastings can be done to get the sequence of letters to be always of one of the types @xmath31 , @xmath32 or @xmath33 following each other any number of times . for example we can have @xmath34 which is a connected sum of n tori . mathematically one gets a group generated by expressions of the form @xmath31 , @xmath32 and @xmath33 . each geometric property of a surface gets translated into a property of the group . for example , the fact that the sphere is an identity element for connected sums gets translated into the fact that @xmath35 in the group language . the best way to understand this result is to draw polygons on pieces of papers and perform cutting and pasting until one gets to the form of connected sums of the basic elements . summing up , given a surface , one represents it by a polygon with sides to be identified . then by cutting and pasting one can always get to a simple representation of the surface as a polygon containing only sequences of sides of the form @xmath36 and @xmath33 . these are exactly the expressions that correspond to @xmath37 and @xmath1 . here we imitate step by step the classification of real surfaces presented in section ii . we begin with some examples . + * 1 . basic surfaces . * here we give examples of complex surfaces . for the first two examples , recall that @xmath41 for @xmath42 to define the hirzebruch surfaces , we need the concept of a vector bundle . intuitively , a rank n vector bundle over a manifold @xmath43 is given by attaching to each point of the manifold a rank n vector space @xmath44 which is called the fiber . in other words a vector bundle over @xmath43 is a family of vector spaces parametrized by the points of @xmath45 one very basic property of vector bundles that one should keep in mind is called local triviality . this means that when we look at a small open set @xmath46 in the manifold @xmath47 then the vector bundle over @xmath46 is isomorphic to a trivial product @xmath48 here we only need vector bundles over @xmath49 and we will only define these . for more on vector bundles see [ 4 ] . first of all we choose charts for the complex projective line as follows : @xmath50 where @xmath51 with intersection @xmath52 and transition function @xmath53 topologically @xmath54 is just the sphere @xmath0 and one can look at the chart @xmath46 as covering the sphere minus the north pole , while @xmath55 covers the sphere minus the south pole . now we define the rank one complex vector bundle @xmath56 over @xmath54 ( rank one complex means that the fibers will be copies of @xmath57 ) . to define this bundle we start with two trivial products @xmath58 and @xmath59 then we need to give a transition function that tells us how the fibers get put together in the overlap of the charts . for each n we give the function @xmath60 which means that we identify these fibers by the vector space isomorphism taking @xmath61 to @xmath62 formally we define @xmath56 by charts @xmath63 @xmath64 with intersection @xmath65 and transition function @xmath66 now we construct rank two vector bundles over @xmath54 using the operations we know for vector spaces coming from basic linear algebra . recall that we have an operation of direct sum for vector spaces , which for vector spaces @xmath67 of rank @xmath68 and @xmath69 of rank @xmath70 associates the vector space @xmath71 of rank @xmath72 now starting with two vector bundles @xmath73 and @xmath74 over a manifold @xmath75 we construct a new bundle over @xmath43 called the whitney sum @xmath76 whose fibers are just the vector space sum of the fibers of @xmath73 and @xmath77 that is if @xmath73 has a fiber @xmath67 at the point @xmath78 and @xmath74 has a fiber @xmath69 at the point @xmath78 then @xmath79 has fiber @xmath71 at the point @xmath80 for example , taking the whitney sum of the bundles @xmath81 and @xmath82 will give us a rank two bundle over @xmath54 denoted by @xmath83 now , just as we constructed @xmath10 by projectivizing @xmath84 and @xmath38 by projectivizing @xmath85 it is possible to projectivize any vector bundle @xmath86 thus obtaining a bundle @xmath87 whose fibers are projective spaces . also here the idea is to perform the projectivization at each of the fibers . if we start with the rank two bundles @xmath88 then projectivizing we transform the fibers into projective lines @xmath89 recall that these are all bundles over @xmath89 hence what we obtain is actually a family of @xmath54 s parametrized by points of another copy of @xmath89 the reader should intuitively think of the number @xmath3 as a particular way of twisting such a family @xmath3 times when we go around the base . each hirzebruch surface @xmath91 is a complex surface . hence now we have an infinite set of complex surfaces , one for each @xmath92 we remark also that the first hirzebruch surface @xmath93 is simply the product @xmath94 intuitively in this case we have a family of @xmath54 s parameterized by a @xmath54 with no twisting hence a product . + * 2 . two types of surfaces . * just as real surfaces were naturally divided into two categories , also complex surfaces will be divided into two categories . however orientability does not make a division , because all complex surfaces are orientable . there is a natural division in the classification of complex surfaces is given by rationality . let us observe that all complex surfaces we mention here can be thought of as lying inside some complex projective space . in a projective space it always make sense to write down quotients of homogeneous polynomials of the same degree . one then defines a rational map to be a map that is locally given ( in each coordinate ) by quotients of polynomials . we remark a very special property of rational maps that is the fact that a rational map need not be defined over every point of its domain . clearly when we define a map by quotients of polynomials , then the map is not defined where the denominator vanishes . in fact a rational map needs only to be defined over a dense open subset of its domain and two rational maps are considered equal if they coincide on a dense open set . rational surfaces are intuitively the ones that are somehow similar to @xmath95 by definition , a complex surface @xmath96 is rational if there is a rational map @xmath97 whose inverse is also rational . where two rational maps are called inverse to each other if their composition is the identity defined over a dense open set . obviously @xmath38 itself is rational . also the hirzebruch surfaces are rational surfaces and these are in fact the simplest examples of rational surfaces . + * 3 . constructing new surfaces out of basic ones . * when one wants to study all rational surfaces , there is a basic operation , which allows us to construct new surfaces from old ones . this operation is called blow - up . when we start with a complex surface @xmath96 and blow up a point in @xmath96 , the result is another complex surface @xmath98 which is different from @xmath99 in fact topologically , the new surface @xmath98 is obtained from @xmath96 by performing a connected sum of @xmath96 together with a copy of @xmath38 with the reversed orientation . the orientation problem is a technical detail that we will explain after we define the blow - up . but right now it is important to see that when we do a blow - up , then from a topological point of view we are doing exactly the same kind of operation that we did for real surfaces in section ii . hence some of the geometric intuition we got from the pictures can be carried over to this section . however , we remark also that the blow - up operation is an analytic procedure in the sense that it takes us from a complex analytic manifold into another complex analytic manifold . that means that the blow - up operation is much more powerful then simply performing connected sums in a topological way as we did for real surfaces . we could phrase this as _ the blow - up is a complex analytic way of performing connected sums_. the blow - up operation is defined locally , in the sense that it is done in a coordinate chart . that is , to blow - up a point @xmath78 in a surface @xmath100 we take a coordinate chart around @xmath101 taking @xmath78 to @xmath102 then we blow up the origin in @xmath103 and construct the surface @xmath104 which is isomorphic to @xmath96 everywhere except at the point @xmath101 which gets replaced by a @xmath89 intuitively one picks a point and blows it up ( in the usual sense of the word , i.e. to explode ) to a projective line . the result is that the lines passing through p become disjoint . the next following picture represents the blow - up of @xmath105 at the origin . note that after the blowup we get a mobius band as the top and bottom edges of the strip are identified after giving a twist . a good way to understand this picture is to think of the blow - up as placing a line @xmath106 through origin at the height @xmath107 where @xmath107 is the angle of @xmath108 then the line corresponding to @xmath109 is to be identified to the line corresponding to zero , since they coincide in @xmath110 notice that if @xmath113 is nonzero , then @xmath114 determines a unique line @xmath115 namely the line through the origin in @xmath103 passing by @xmath116 hence , over @xmath117 @xmath118 is isomorphic to @xmath119 which is what we should expect since points outside of the origin are not blown - up . however , for @xmath120 we have that @xmath121 for every @xmath122 this gives a copy of @xmath54 which lies over the origin , called the exceptional divisor . this exceptional divisor `` separates '' the lines through the origin turning them into disjoint lines . one very nice exercise for the reader would be to show that @xmath123 is the line bundle @xmath124 over @xmath89 but for a first approach the reader may just take this as a result . a copy of @xmath54 inside a surface is called a line . the exceptional divisor is a special kind of line . for instance it is different from any line contained in @xmath95 we now assign numbers to lines according to the form of their neighborhoods , this number is know as the self - intersection number of the line . a line @xmath54 inside @xmath38 will be our standard line . to it we assign the number 1 . to the exceptional divisor we will assign the number -1 ( the technical reason for this is the exercise we just left to the reader ) . for a line which has a neighborhood isomorphic to @xmath56 we assign the number @xmath92 these numbers are the analogous to the directions for the arrows we had when dealing with real surfaces . only in that case only two direction were possible , so an arrow was enough . now we have `` twists '' and we label the lines according to the twists in their neighborhoods . a negative sign can be thought of as meaning twist in negative direction . it is this negative sign that accounts for the reversed orientation when we talked about the topology of the blow - up viewed as a connected sum . an interesting example is to start with @xmath38 and blow up two points @xmath78 and @xmath125 and then we blow down the line passing by @xmath78 and @xmath126 then , the resulting surface is @xmath127 this is a lot more technical , we refer the reader to [ 1 ] for a proof . not only the blow - up is topologically what we would expect for a generalization of the operation we had for real surfaces and even more , an analogous structure theorem holds . + * 4 . the classification theorem . * + once we have the basic surfaces and the operation for constructing new surfaces out of old ones , we can build all rational surfaces . the classical structure theorem is the following ( see [ 1 ] ) . a complete proof of this theorem would require that we develop quite a bit of theory . hence we give only some informal ideas . we start with a rational surface and look for special lines inside the surface . then there are two possibilities either we have lines with associated number -1 or we do not have any such lines . suppose @xmath96 is a surface containing a -1 type line @xmath122 then we can eliminate this line @xmath115 by performing what is called a blow - down , that is exactly the inverse operation of a blow - up . the blow - down of @xmath115 contracts the line into a point @xmath78 giving a new surface @xmath128 ( so that the blow - up of @xmath128 at @xmath78 equals @xmath129 . then if @xmath128 has no more -1 type lines we are done . otherwise we perform another blow - down on @xmath130 it is true that after finitely many blowing downs we arrive at a surface containing no more -1 type lines . then there are two possibilities , either the new surface has only type 1 lines and in this case it is @xmath38 or it has some line of type @xmath131 in which case it is a hirzebruch surface @xmath40 griffiths , p. and harris , j. _ principles of algebraic geometry . john wiley and sons , inc.(1978 ) _ harris , j. _ algebraic geometry a first course , graduate texts in mathematics 133 , springer verlag ( 1992 ) _ massey , w. s. _ algebraic topology , an introduction . graduate texts in mathematics 56 , springer verlag ( 1977 ) _ steenrod , n. _ the topology of fiber bundles , princeton university press ( 1951 ) _	this is an expository paper which presents the holomorphic classification of rational complex surfaces from a simple and intuitive point of view , which is not found in the literature . our approach is to compare this classification with the topological classification of real surfaces . # 1#2 = 0.10753 plus 2pt minus 1pt # 1 # 2 [ section ] [ guess]theorem [ guess]lemma [ guess]corollary
it is well known that the geometrical and topological effects play a central role in a large number of physical problems . they have important implications on all scales , from subnuclear to cosmological . in particular , in quantum field theory the properties of the vacuum crucially depend on the both geometry and topology of the background spacetime . in the present paper we consider combined effects of the geometry and topology on the vacuum current densities induced by magnetic flux tubes . as a background geometry we consider de sitter ( ds ) spacetime and the topological effects are induced by two types of sources . the first one will correspond to a planar angle deficit due to the presence of a cosmic string and the second one comes from the compactification of the spatial dimension along the cosmic string . the cosmic strings are among the most important types of topological defects that may have been formed by the phase transitions in the early universe @xcite . though the recent observations of the cosmic microwave background radiation have ruled out them as the primary source for primordial density perturbations , the cosmic strings give rise to a number of interesting physical effects such as the doubling images of distant objects or even gravitational lensing , the emission of gravitational waves and the generation of high - energy cosmic rays ( see , for instance , damo00 ) . recent developments on the formation of topological defects in superstring theories have led to a renewed interest in cosmic ( super)strings . in particular , a variant of their formation mechanism has been proposed in the framework of brane inflation @xcite . string - like defects also appear in a number of condensed matter systems , including liquid crystals and graphene - made structures . in the simplest theoretical model , the cosmic string is described by a planar angle deficit with the background geometry being locally flat except on the top of the string where it has a delta shaped curvature tensor . the corresponding non - trivial topology induces nonzero vacuum expectation values ( vevs ) for physical observables . specifically , the vev of the energy - momentum tensor associated with various fields has been developed by many authors @xcite-@xcite . moreover , considering a magnetic flux running along the strings , there appear additional contributions to the corresponding vacuum polarization effects for charged fields @xcite,@xcite-@xcite . the presence of a magnetic flux induces also vacuum current densities . this phenomenon was analyzed for massless sira and massive @xcite scalar fields . it has been shown that an azimuthal vacuum current appears if the ratio of the magnetic flux by the quantum one has a nonzero fractional part . the analysis of the induced fermionic currents in higher - dimensional cosmic string spacetime in the presence of a magnetic flux have been developed in @xcite . the fermionic current induced by a magnetic flux in ( 2 + 1)-dimensional conical spacetime and in the presence of a circular boundary has also been analyzed @xcite . in general , the analysis of quantum effects for matter fields in a cosmic string spacetime , consider this defect in a flat background geometry . for a cosmic string in a curved background , quantum effects associated with a scalar field have been discussed in @xcite for special values of the planar angle deficit . the vacuum polarization in schwarzschild space - time threaded by an infinite straight cosmic string is investigated in otte10 . in recent publications we have investigated the vacuum polarization effects for massive scalar @xcite and fermionic beze10 fields , induced by a cosmic string in ds spacetime . it has been shown that for massive quantum fields the background gravitational field essentially changes the behavior of the vacuum densities at distances from the string larger than the ds curvature radius , when compared with the case of the string in minkowski spacetime . depending on the specific value of the mass , at large distances two regimes are realized with monotonic and oscillatory behavior of the vevs . similar analysis for vacuum polarization effects , induced by a cosmic string in anti - de sitter spacetime , have been developed in @xcite and @xcite for massive scalar and fermionic fields , respectively . the choice of ds spacetime as the background geometry in the present paper is motivated by several reasons . first of all , this spacetime is a maximally symmetric solution of the einstein equation in the presence of a positive cosmological constant and , as a consequence of high degree of symmetry , a large number of physical problems are exactly solvable on its background . as it will be shown below , this is the case for the problem under consideration . the importance of ds spacetime as a gravitational background has essentially increased after the appearance of the inflationary scenario for the expansion of the universe at early stages . most versions of this scenario assume a period of quasiexponential expansion in which the geometry of the universe is approximated by a portion of ds spacetime . this gives a natural solution to a number of problems in standard cosmology . in addition , the quantum fluctuations in the inflaton field during the inflationary epoch generate inhomogeneities that are seeds for the formation of the large scale cosmic structures . more recently , astronomical observations of high - redshift supernovae , galaxy clusters , and the cosmic microwave background have indicated that at present the universe is accelerating and can be well approximated by the friedmann - robertson - walker cosmological model with the energy dominated by a positive cosmological constant - type source . if the universe is going to accelerate forever , this model will lead asymptotically to a ds spacetime as a future attractor for the dynamics of the universe . the second type of the topological effects we shall consider here is induced by the compactification of the spatial dimension along the cosmic string axis . the compact spatial dimensions are an inherent feature of most high - energy theories of fundamental physics , including supergravity and superstring theories . an interesting application of the field theoretical models with compact dimensions recently appeared in nanophysics . the long - wavelength description of the electronic states in graphene can be formulated in terms of the dirac - like theory in three - dimensional spacetime with the fermi velocity playing the role of speed of light ( see , e.g. , cast09 ) . in graphene - made structures , like cylindrical and toroidal carbon nanotubes , the background geometry for the corresponding field theory contains one or two compact dimensions . in quantum field theory , the periodicity conditions imposed on the field operator along compact dimensions modify the spectrum for the normal modes and as a result of this the vevs of physical observables are changed . recently the analysis of the induced fermionic current and the vev of the energy - momentum tensor in a compactified cosmic string spacetime in the presence of magnetic flux running along the string , have been developed in @xcite . the vev of the fermionic current in spacetimes with an arbitrary number of toroidally compactified spatial dimensions and in the presence of a constant gauge has been investigated in @xcite . furthermore , the combined effects of topology and the gravitational field on the vevs of the current density for charged scalar and fermionic fields in the background of ds spacetime with an arbitrary number of toroidally compactified spatial dimensions is considered in @xcite . the finite temperature effects on the current densities for scalar and fermionic fields in topologically nontrivial spaces have been discussed in @xcite . the present paper is organized as follows . in section [ sec2 ] we describe the background geometry and construct the complete set of normalized positive- and negative - energy fermionic mode functions obeying a quasiperiodic boundary condition with an arbitrary phase along the string axis . in addition , we assume the presence of a constant gauge field . in section [ sec3 ] , by using the mode - summation method , we first show that the vevs for the charge density and the radial current vanish . then we evaluate the renormalized vev of the azimuthal current density induced by a magnetic flux running along the string axis . it is decomposed into two parts : the first one corresponds to the geometry of a cosmic string in ds spacetime without compactification and the second one is induced by the compactification of the spatial dimension parallel to the string . the vev of the axial current density is investigated in section [ sec4 ] . this vev is a purely topological effect induced by the compactification and vanishes in the geometry of a straight cosmic string . the most relevant conclusions of the paper are summarized in section [ conc ] . throughout the paper we use the units with @xmath0 . the main objective of this section is to present the geometry of the spacetime , where we develop our analysis and also to obtain the complete set of solutions of dirac equation in this background . so we first write the line element , in cylindrical coordinates , corresponding to a cosmic string along the @xmath1-axis in ds spacetime @xmath2where @xmath3 , @xmath4 and @xmath5 , being @xmath6 . the parameter @xmath7 , bigger than unity , codifies the presence of the cosmic string . additionally we shall assume that the direction along the @xmath1-axis is compactified to a circle with the length @xmath8 : @xmath9 . the parameter @xmath10 in ( [ ds21 ] ) is related to the cosmological constant @xmath11 and the scalar curvature @xmath12 by the expressions @xmath13 and @xmath14 . in addition to the synchronous time coordinate @xmath15 , we introduce the conformal time @xmath16 according to @xmath17 in terms of this coordinate , the line element ( [ ds21 ] ) is confromally related to the geometry of a cosmic string in minkowski bulk , with the conformal factor @xmath18:@xmath19 by the coordinate transformation@xmath20and @xmath21 , with the function @xmath22 , the line element ( [ ds21 ] ) is presented in the static form @xmath23this line element has been previously discussed in @xcite . it is shown that , to leading order in the gravitational coupling , the effect of the vortex on de sitter spacetime is described by ( [ ds2st ] ) . the dynamics of a massive spinor field in curved spacetime in the presence of a four - vector potential , @xmath24 , is governed by the dirac equation @xmath25here , @xmath26 represents the dirac matrix in curved spacetime and @xmath27 the spin connection . both are expressed in terms of the flat space dirac matrices , @xmath28 , by the relations @xmath29where the semicolon stands for the standard covariant derivative for vector fields . in ( [ gammamu ] ) , @xmath30 is the tetrad basis satisfying the relation @xmath31 , with @xmath32 being the minkowski spacetime metric tensor . we assume that along the compact @xmath1-dimension the fermionic field obeys the quasiperiodicity condition as shown below : @xmath33 in the above equation , @xmath34 is a constant phase defined in the interval @xmath35 $ ] . the special cases @xmath36 and @xmath37 correspond to the periodic and antiperiodic boundary conditions ( untwisted and twisted fields , respectively ) . for the rotation around the @xmath1-axis we shall use the periodic boundary condition @xmath38 for a constant vector potential , @xmath24 , the latter may be excluded from the field equation ( [ direq ] ) by the gauge transformation@xmath39with @xmath40 . the new wave function obeys the equation @xmath41and the periodicity conditions@xmath42with the notations@xmath43note that the physical components @xmath44 and @xmath45 of the vector potential are related to the covariant components @xmath46 and @xmath47 by @xmath48 and @xmath49 . the parameters in the phases of the periodicity conditions can be expressed in terms of the magnetic flux along the string axis , @xmath50 , and flux enclosed by the @xmath1-axis , @xmath51 , by the formulas@xmath52with @xmath53 being the flux quantum . in what follows we will work in terms of the gauge transformed field @xmath54 omitting the prime . the current density is invariant under the gauge transformation ( gauge ) . our main interest in this paper is the evaluation of the vev of the fermionic current density , @xmath55 . this vev is expressed in terms of the two - point function @xmath56\|0\rangle $ ] , where @xmath57 and @xmath58 are spinor indices and @xmath59 is the vacuum state . for the vev one has@xmath60 in quantum field theory on curved backgrounds the choice of the vacuum is not unique ( see , for example , @xcite ) . in ds spacetime there exists a one - parameter family of maximally symmetric quantum states . in what follows we will assume that the field is prepared in the ds - invariant bunch - davies vacuum state @xcite . in the class of ds - invariant quantum states , the bunch - davies vacuum is the only one for which the ultraviolet behavior of the two - point functions is the same as in minkowski spacetime . let @xmath61 be a complete set of normalized solutions to the dirac equation specified by the set of quantum numbers @xmath62 . note that the background geometry under consideration is time - dependent and the energy is not conserved . however , we will refer to the solutions @xmath63 and @xmath64 as the positive- and negative - energy modes in the sense that in the limit @xmath65 they reproduce the positive- and negative - energy fermionic modes in minkowski spacetime . expanding the field operator in terms of the complete set of fermionic modes , the following mode - sum formula is obtained for the current density : @xmath66 . \label{current}\]]consequently , in this evaluation we need the fermionic modes for the geometry at hand . in order to find the mode functions , we will take the flat space dirac matrices according to @xcite @xmath67where @xmath68 , and @xmath69 are the @xmath70 pauli matrices . the basis of tetrads corresponding to the line element ( [ ds21 ] ) may have the form@xmath71for the curved space gamma matrices , in the coordinate system corresponding to ( [ ds21 ] ) , this choice leads to the representation@xmath72with the @xmath73 matrices@xmath74and @xmath75 . for the spin connection components one gets @xmath76 and @xmath77for @xmath78 . this leads to the following expression for the combination appearing in the dirac equation ( [ direq ] ) : @xmath79 the positive- and negative - energy mode functions obeying the periodicity conditions ( [ percond2 ] ) can be found in a way similar to that we have used in @xcite for the geometry of a straight cosmic string in ds spacetime in the absence of the magnetic flux . for the bunch - davies vacuum state these functions are given by @xmath80where @xmath81 , @xmath82 , @xmath83 , @xmath84 , @xmath85 . moreover , @xmath86 and @xmath87 are the bessel and hankel functions , respectively , and @xmath88 in ( [ psisigma+ ] ) , we have defined @xmath89with @xmath90 for @xmath91 and @xmath92 for @xmath93 . the quantum number @xmath94 determines the eigenvalues of the projection of the total momentum along the cosmic string and the quantum number @xmath58 corresponds to the eigenvalue of@xmath95 where @xmath96 with @xmath97 . the mode functions above are specified by the complete set of quantum numbers @xmath98 . in addition , the functions ( [ psisigma+ ] ) obey the periodicity condition ( [ percond1 ] ) . from the condition ( percond2 ) we find the eigenvalues for the quantum number @xmath99 : @xmath100with @xmath101 . the coefficients @xmath102 are determined by the orthonormalization condition@xmath103where @xmath104 is the determinant of the spatial metric tensor corresponding to the line element ( [ ds21 ] ) . the delta symbol in the rhs of ( [ norm ] ) is understood as the kronecker delta for the discrete indices @xmath105 and the dirac delta function for the continuous one @xmath106 . by using the wronskian for the hankel functions , we find @xmath107note that , if we write the parameter @xmath108 , defined in ( [ abet ] ) , in the form @xmath109where @xmath110 is an integer number , then , by shifting @xmath111 , we can see that the vevs of physical observables depend on @xmath112 only . as it is well known , in minkowski spacetime , the theory of von neumann deficiency indices leads to a one - parameter ( usually denoted by @xmath113 ) family of allowed boundary conditions in the background of an aharonov - bohm gauge field @xcite . additionally to the regular modes , these boundary conditions , in general , allow normalizable irregular modes . a special case of boundary conditions has been discussed in @xcite , where the atiyah - patodi - singer type nonlocal boundary condition is imposed at a finite radius , which is then taken to zero . similar approach , with the mit bag boundary condition , has been used in @xcite for a two - dimensional conical space with a circular boundary . in the geometry under consideration there are no normalizable irregular modes for @xmath114 in the case @xmath115 , the irregular mode corresponds to @xmath116 . for the mode functions ( [ psisigma+ ] ) with this value of the momentum , the boundary condition on the string axis is a special case of one - parameter family of conditions with the parameter @xmath117 . note that with this value and for a massless field both parity and chiral symmetry are conserved @xcite . the evaluation of the vev of the fermionic current for other boundary conditions on the string axis is similar to that described below . the contribution of the regular modes to the vev is the same for all boundary conditions and the results will differ by the parts related to the irregular modes . having the complete set of mode functions ( [ psisigma+ ] ) , we can evaluate the vev for the current density by making use of the mode - sum formula ( current ) where now the summation is specified by@xmath118with@xmath119of course , the expression in the rhs of ( [ current ] ) is divergent and a regularization with the subsequent renormalization is necessary . here we shall use a cutoff function to regularize without writing it explicitly . the special form of this function will not be important for the further discussion . an alternative way would be the point - splitting regularization procedure which corresponds to the evaluation of the expression under the sign of the limit in ( [ vevj ] ) for @xmath120 . however , in this case the calculations are more complicated . first let us consider the charge density : @xmath121substituting the mode functions ( [ psisigma+ ] ) and using the relation @xcite @xmath122with @xmath123 being the macdonald function , we obtain@xmath124 \notag \\ & & \times \sum_{\chi = -,+}\chi \left [ \|k_{1/2+\chi im\alpha } ( i\gamma \eta ) \|^{2}+\|k_{-1/2+\chi im\alpha } ( i\gamma \eta ) \|^{2}\right ] . \label{density3}\end{aligned}\]]by taking into account that @xmath125 , we conclude that the charge density vanishes . for the vev of the radial current density one has @xmath126substituting the corresponding gamma matrices and the fermionic mode functions from ( [ psisigma+ ] ) , it can be shown that all terms cancel and the resulting radial current is also zero . now we turn to the azimuthal current which is given by the expression ( current ) with @xmath127 . substituting ( [ psisigma+ ] ) , we can see that the positive- and negative - energy modes give the same contribution . by using ( [ rel ] ) , and after the summation over @xmath58 , the vev of the azimuthal current is presented in the form @xmath128 . \label{fc2n}\]]in order to separate explicitly the topological part , for the summation over @xmath129 , we apply the abel - plana formula in the form @xcite @xmath130 f(u ) \notag \\ & & \qquad + i\int_{0}^{\infty } du\left [ f(iu)-f(-iu)\right ] \sum_{\chi = \pm 1}% \frac{g(i\chi u)}{e^{lu+2\pi i\chi \tilde{\beta}}-1}\ , \label{fc3}\end{aligned}\]]choosing @xmath131 and @xmath132 . \label{fc4}\ ] ] the first term in the rhs of ( [ fc3 ] ) is responsible for the azimuthal current in the cosmic string background without compactification , that will be denoted below by @xmath133 . the second one corresponds to the contribution due to the compactification of the string along the @xmath1-axis , denoted by @xmath134 . therefore , the application of the summation formula ( [ fc3 ] ) allows us to decompose the azimuthal current as @xmath135as we shall see , the compactified part goes to zero in the limit @xmath136 . we start the evaluation with the part corresponding to the geometry of a straight cosmic string , @xmath133 . using the first term in the abel - plana formula , for this part we get @xmath137 , \label{fc6}\end{aligned}\]]where @xmath138 is given by the expression ( [ gamma ] ) . replacing the product of the macdonald functions by the integral representation wats44 @xmath139 , \label{fc7}\]]the integral over @xmath99 is evaluated directly . performing the integral over @xmath140 with the help of the formula below @xcite @xmath141 , \label{fc8}\]]we obtain@xmath142 } \mathcal{j}(q , a_{0},z ) , \label{fc9}\end{aligned}\]]where we have introduced the notation@xmath143 . \label{jcal}\]]the integration over @xmath144 in ( [ fc9 ] ) can be done explicitly and one finds@xmath145 , \label{fc13}\]]where we have introduced a new integration variable @xmath146 . the current density given by this formula is a periodic odd function of the flux along the string axis with the period equal to the flux quantum . the azimuthal current density , given by ( [ fc13 ] ) , depends on the radial and time coordinates through the ratio @xmath147 . this property is a consequence of the maximal symmetry of ds spacetime and the bunch - davies vacuum . by taking into account that the combination @xmath148 is the proper distance from the string , we see that @xmath147 is the proper distance measured in units of the ds curvature scale @xmath149 . for the further transformation of the expression in the rhs of ( [ fc13 ] ) , we follow the procedure used in @xcite for the summation over @xmath94 in ( [ jcal ] ) . introducing the notation @xmath150we can see that @xmath151 . for the series ( [ seriesi0 ] ) one has the integral representation @xmath152e^{z\cos ( 2\pi k / q ) } , \label{seriesi3}\end{aligned}\]]in which @xmath140 is an integer defined by @xmath153 and @xmath154 \cosh \left [ \left ( qa_{0}+\chi q/2 - 1/2\right ) x\right ] \ . \label{fqualf1}\]]in the specific case @xmath155 , the term @xmath156should be added to the rhs of ( [ seriesi3 ] ) . by taking into account ( seriesi3 ) , for the function ( [ jcal ] ) one gets@xmath157where@xmath158 \cosh \left [ \left ( 1/2-\chi a_{0}\right ) qx\right ] . \label{gqa}\]]in the case @xmath155 , the term @xmath159 must be added to the rhs of ( [ jcal2 ] ) . as is seen , in the absence of a magnetic flux along the string , @xmath160 , one has @xmath161 and the azimuthal current density @xmath133 vanishes which is the same result as for the flat space in the presence of the cosmic string @xcite . the azimuthal current density in ds spacetime induced by the magnetic flux is obtained as a special case with @xmath162 . in this case , for the function in the integrand of ( [ jcal2 ] ) one has@xmath163and the sum in the rhs of ( [ jcal2 ] ) is absent . for the function @xmath164 one gets @xmath165as a result , in the absence of the planar angle deficit the expression for the azimuthal current density is simplified to @xmath166}\,% \mathrm{re\,}\left [ k_{1/2+im\alpha } ( x)\right ] . \label{j2q1}\end{aligned}\ ] ] an equivalent expression for the azimuthal current in general case of @xmath7 is obtained substituting ( [ jcal2 ] ) into ( [ fc9 ] ) and integrating over @xmath1:@xmath167 ^{5/2}}\right ] , \label{fc9b}\end{aligned}\]]where we have made the change of variables @xmath168 . in the absence of the conical defect one has @xmath162 and from ( [ fc9b ] ) we find a simpler formula @xmath169^{5/2}}. \label{j2bq1}\]]for a massless field , integrating over @xmath170 , from the general formula ( fc9b ) we find@xmath171 . \label{fc16}\end{aligned}\]]the above result shows that @xmath133 is conformally related to the corresponding induced current in pure cosmic string spacetime @xcite , with the conformal factor @xmath172 , as expected . in the region near the string , @xmath173 , the dominant contribution to ( [ fc13 ] ) comes from large @xmath174 and we can use the asymptotic expression of the macdonald function for large arguments . the leading term is independent of the mass and it reduces to ( [ fc16 ] ) which diverges on the string with the inverse fourth power of the proper distance . at large distances from the string , @xmath175 , in ( [ fc13 ] ) we use the asymptotic form of the macdonald function for small values of the argument . the leading term in the asymptotic expansion of the azimuthal current behaves as @xmath176 } { ( \alpha r/\eta ) ^{4 } } , \label{j2slarge}\]]with a phase @xmath113 depending on the parameters @xmath177 , @xmath7 and @xmath178 . as a result , at large distances the azimuthal current in the geometry of a straight cosmic string damps oscillatory with the amplitude decaying as the inverse fourth power of the proper distance from the string . the oscillation frequency increases with increasing mass . in the region under consideration the influence of the gravitational field on the current density is essential . the behavior of the current density in ds spacetime , described by ( [ j2slarge ] ) , is crucially different from that for the string in minkowski bulk . in the latter case , at large distances from the string the current density for a massive field is suppressed exponentially , by the factor @xmath179 for @xmath180 and by the factor @xmath181 for @xmath182 . in the left panel of figure [ fig1 ] we have plotted the quantity @xmath183 , with @xmath148 being the proper distance from the string , as a function of the ratio @xmath184 ( proper distance measured in units of @xmath149 ) for @xmath185 and for separate values of the parameter @xmath7 ( the numbers near the curves ) . the full curves correspond to a massive field with @xmath186 and the dashed lines are for a massless field . in the latter case the combination @xmath187 does not depend on @xmath147 . from the asymptotic analysis given above it follows that @xmath188 which is also seen from the graphs . for large values of @xmath147 and for a massive field we see the characteristic oscillations described by ( [ j2slarge ] ) . in the right panel of figure fig1 the quantity @xmath189 is displayed as a function of the parameter @xmath112 for fixed values @xmath190 ( full curves ) and @xmath191 ( dashed curves ) . again , the numbers near the full curves correspond to the values of @xmath7 and we have taken @xmath186 . for the dashed curves the same values of @xmath7 are used and @xmath192 increases with increasing @xmath7 . [ cols="^,^ " , ] in this paper we have investigated the combined effects of the background gravitational field and topology on the vev of the current density for a massive fermionic field . this quantity is an important local characteristic of the quantum vacuum . in addition to describing the physical structure of a charged quantum field at a given point , the vev of the current density acts as the source in the semiclassical maxwell equations and plays an important role in modeling a self - consistent dynamics involving the electromagnetic field . in order to have an exactly solvable problem , we have taken ds spacetime as the background geometry . the latter is among the most popular gravitational backgrounds and plays an important role in cosmology . the topological effects are induced by a cosmic string and by its compactification along the axis . additionally , we have assumed the presence of a constant gauge field with nonzero axial and azimuthal components . by a gauge transformation , the problem is reduced to the one in the absence of a gauge field with the quasiperiodicity conditions ( [ percond1 ] ) and ( percond2 ) on the field operator . in the new representation , the information about the gauge field is encoded in the phases of these conditions . for the evaluation of the vev of the current density we have employed the direct summation over the complete set of fermionic modes . for the bunch - davies vacuum state , the corresponding mode functions are given by ( [ psisigma+ ] ) . for the model under consideration , in general , there is a one parameter family of boundary conditions imposed on the field operator at the location of the string . the fermionic modes used in the present paper correspond to the boundary condition when the mit bag boundary condition is imposed at a finite radius , which is then taken to zero . the formal expression for the vev of the induced fermionic current density is presented in the the mode - sum form ( [ current ] ) . because of the compactification of the cosmic string along its axis , the quantum number corresponding to the @xmath1-direction becomes discrete and for the corresponding summation we use the abel - plana - type formula ( [ fc3 ] ) . as a consequence , the current density is decomposed in two parts : the first part corresponds to the geometry of a straight cosmic string in ds spacetime and the second one is induced by the compactification . for a massless fermionic field , the problem under consideration is conformally related to the problem with a cosmic string in minkowski spacetime and the corresponding expressions for the vev of the current density are related by the standard conformal transformation . in the problem under consideration the vevs of the charge density and the radial current vanish . for the vev of the azimuthal current density we have considered first the part corresponding to the geometry without compactification , denoted by @xmath193 . two different integral representations for this part are provided , the expressions ( fc13 ) and ( [ fc9b ] ) . the azimuthal current is an odd periodic function of the magnetic flux along the string axis with the period equal to the flux quantum . it depends on the radial and conformal time coordinates through the ratio @xmath147 , which is the proper distance from the string measured in units of the ds curvature scale @xmath149 . near the string , the leading term in the asymptotic expansion of @xmath193 is independent of the mass and it behaves as the inverse forth power of the proper distance . in this limit the dominant contribution comes from the modes with small wavelengths and the effects of the curvature are small . for a massive field , the influence of the gravitational field on the vacuum current density is crucial at distances from the string larger than the curvature radius of the background spacetime . in this limit , corresponding to @xmath175 , the azimuthal current density exhibits a damping oscillatory behavior with the amplitude inversely proportional to the fourth power of the distance . note that for the string in minkowski spacetime and for a massive field , at large distances the current is exponentially suppressed . the contribution to the azimuthal current density coming from the compactification of the string along its axis is given by ( [ fc27 ] ) and the total current is given by ( [ j2tot ] ) . in the compactified geometry the azimuthal current is an odd periodic function of the magnetic flux along the string and an even periodic function of the flux enclosed by the @xmath1-axis . in both cases the period is equal to the flux quantum . near the string the total azimuthal current is dominated by the part @xmath194 . in this region the leading term in the expansion of topological part is given by ( [ j2cnear ] ) . this part vanishes on the string for @xmath195 , is finite for @xmath196 and diverges for @xmath115 . in the opposite limit of large distances from the string , the behavior of the azimuthal current density for a compactified cosmic string depends crucially on whether the parameter @xmath197 , defined by ( [ abet2 ] ) , is zero or not . for @xmath198 and @xmath199 , the leading term in the asymptotic expansion is given by ( [ j2far ] ) and the azimuthal current is suppressed by the factor @xmath200 with with @xmath201 . for @xmath202 the suppression is stronger , by the factor @xmath203 . for @xmath204 , at large distances the azimuthal current exhibits a damping oscillatory behavior described by ( [ j2far2 ] ) . the amplitude of the oscillations decay as @xmath205 and in this case the topological part @xmath206 dominates in the total current density . the vev of the axial current density is given by the expression ( axial3d ) . the appearance of the nonzero axial current is a purely topological effect induced by the compactification of the string along its axis . the axial current density is an even periodic function of the magnetic flux along the string axis and an odd periodic function of the flux enclosed by the @xmath1-axis with the periods equal to the flux quantum . the modulus of the axial current , @xmath207 , increases with increasing @xmath7 for the values of the parameter @xmath112 close to @xmath208 and decreases for @xmath112 close to @xmath209 . in the absence of the planar angle deficit one has @xmath210 and the general formula is reduced to ( [ j3q1 ] ) . for general case of the parameter @xmath7 , the corresponding asymptotic near the cosmic string is given by ( [ j3near ] ) and the axial current density vanishes on the string for @xmath195 and diverges for @xmath115 . at large distances from the string the effects of the planar angle deficit and of the magnetic flux on the axial current are small and to the leading order we recover the result for ds spacetime with a single compact dimension , described by ( [ j30 ] ) . for small values of the proper length of the compact dimension , the axial current is dominated by the part corresponding to the current density in the geometry without the cosmic string and magnetic flux along the @xmath1-axis . in this limit , the leding term in the asymptotic expansion is given by ( [ j3smalll ] ) and the string - induced corrections to this leading term are exponentially small . for large values of the ratio @xmath211 , the behavior of the axial current is described by ( [ j3largel ] ) . in this range the current density exhibits damping oscillations with power - law decaying amplitude as a function of the length of the compact dimension . this behavior is essentially different from the case of the string in minkowski bulk with the exponentially suppressed axial current for large values of @xmath8 . the results described above can be used , in particular , for the investigation of the effects induced by cosmic strings in the inflationary epoch . though the strings produced before or during early stages of inflation are diluted by the expansion , cosmic strings can be continuously formed during inflation by coupling the symmetry breaking and inflaton fields @xcite or by quantum - mechanical tunneling @xcite . another field of application corresponds to string - driven inflationary models with the cosmological expansion driven by the string energy turo88 . the authors thank conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq ) for the financial support . a. a. s. was supported by the state committee of science of the ministry of education and science ra , within the frame of grant no . scs 13 - 1c040 . s. sarangi and s.h.h . tye , phys . b * 536 * , 185 ( 2002 ) ; e.j . copeland , r.c . myers , and j. polchinski , jhep * 06 * , 013 ( 2004 ) ; g. dvali and a. vilenkin , jcap * 0403 * , 010 ( 2004 ) ; j. polchinski , arxiv : hep - th/0410082 . a.g . smith , in _ the formation and evolution of cosmic strings _ , proceedings of the cambridge workshop , cambridge , england , 1989 , edited by g.w . gibbons , s.w . hawking , and t. vachaspati ( cambridge university press , cambridge , england , 1990 ) . bezerra de mello , v.b . bezerra , a.a . saharian , and a.s . tarloyan , phys . d * 74 * , 025017 ( 2006 ) ; e.r . bezerra de mello , v.b . bezerraa , and a.a . saharian , phys . b * 645 * , 245 ( 2007 ) ; e.r . bezerra de mello , v.b . bezerra , a.a . saharian , and a.s . tarloyan , phys . d * 78 * , 105007 ( 2008 ) . saharian , _ the generalized abel - plana formula with applications to bessel functions and casimir effect _ ( yerevan state university publishing house , yerevan , 2008 ) ; preprint ictp/2007/082 ; arxiv:0708.1187 .	we investigate the vacuum fermionic currents in the geometry of a compactified cosmic string on background of de sitter spacetime . the currents are induced by magnetic fluxes running along the cosmic string and enclosed by the compact dimension . we show that the vacuum charge and the radial component of the current density vanish . by using the abel - plana summation formula , the azimuthal and axial currents are explicitly decomposed into two parts : the first one corresponds to the geometry of a straight cosmic string and the second one is induced by the compactification of the string along its axis . for the axial current the first part vanishes and the corresponding topological part is an even periodic function of the magnetic flux along the string axis and an odd periodic function of the flux enclosed by the compact dimension with the periods equal to the flux quantum . the azimuthal current density is an odd periodic function of the flux along the string axis and an even periodic function of the flux enclosed by the compact dimension with the same period . depending on the magnetic fluxes , the planar angle deficit can either enhance or reduce the azimuthal and axial currents . the influence of the background gravitational field on the vacuum currents is crucial at distances from the string larger than the de sitter curvature radius . in particular , for the geometry of a straight cosmic string and for a massive fermionic field , we show that the decay of the azimuthal current density is damping oscillatory with the amplitude inversely proportional to the fourth power of the distance from the string . this behavior is in clear contrast with the case of the string in minkowski bulk where the current density is exponentially suppressed at large distances . pacs numbers : 04.62.+v , 03.70.+k , 98.80.cq , 11.27.+d
with the recent explosion of interest in graphene , there are numerous experimental motivations for understanding the influence of impurities on its electronic and transport properties . for non - interacting electrons , the influence of a dilute concentration of impurities on transport properties has been investigated in some depth @xcite . here we shall instead study in some detail the physics associated with a _ single _ impurity carrying electrical charge @xmath1 . nanoscale studies of the electronic properties of a single graphene sheet have recently become possible @xcite , and so it should eventually be possible to observe the variation in the charge density and the local density of states as a function of distance from the impurity . we shall show here that this spatial structure is a sensitive probe of the strong correlations between the electrons in graphene , and of the unusual nature of screening in a two - dimensional semi - metal with a dirac dispersion spectrum . for non - interacting electrons , the influence of a coulomb impurity exerting a potential @xmath2 ( where @xmath3 is the distance from the impurity ) was studied some time ago @xcite . this case is equivalent to the familiar `` friedel problem '' but for dirac fermions . however , even for this seemingly simple case , there are subtleties which were overlooked in the initial treatment @xcite , and corrected in ref . . a number of papers appeared @xcite while our paper was being written , presenting additional results on this non - interacting problem . we shall review and extend the results of ref . for non - interacting electrons in section [ sec : nonint ] . we shall then proceed to the full treatment of the impurity problem , and allow for electron - electron coulomb interactions . in short , our results are as follows . for noninteracting electrons , the screening charge is a local delta - function in space to all orders in perturbation theory over the impurity charge . the sign of this screening charge is opposite to that of the impurity , as is usually the case . however , once interaction between electrons is turned on , the screening charge develops a long - range tail , even for small impurity charges . the tail follows approximately an @xmath0 law , with a coefficient which varies quite slowly with @xmath3 . notably , the sign of this tail is the same as that of the impurity . the long - range tail of the screening charge , thus , is a sensitive probe of the interaction between electrons , in particular to the renormalization of the fermion velocity and the `` quantum critical '' aspects @xcite of the interacting dirac fermion problem . many essential aspects of the theory above follow from its properties under the renormalization group ( rg ) transformation under which @xmath24 and @xmath25 . a standard analysis shows that all three couplings in @xmath26 , namely @xmath11 , @xmath1 , and @xmath27 , are invariant under this transformation at tree level . indeed , for two of the couplings , this invariance extends to all orders in perturbation theory : the coupling @xmath27 does not renormalize because of the non - analytic @xmath28 co - efficient , while @xmath1 remains invariant because it is protected by gauge invariance @xcite . so we need only examine the rg flow of a single coupling , the velocity @xmath11 . because @xmath11 is a bulk coupling , its flow can not be influenced in the thermodynamic limit by a single impurity , and so can be computed in the absence of the impurity . such a rg flow was initially examined in the more general context of theories with chern - simons couplings in ref . , but a complete presentation was given in the present context in ref . : we shall use the notation and results of the latter paper here , with the exception that we use two - component dirac fermions with @xmath4 while ref . uses four - component dirac fermions with @xmath29 . it will be useful for our analysis to introduce two combinations of the above couplings which also have engineering dimension zero , and hence are pure numbers . these are @xmath30 ( we have set @xmath31 elsewhere in the paper ) . as we will see , the coupling @xmath32 is a measure of the strength of the electron - electron coulomb interactions , while @xmath33 measures the strength of the electron - impurity coulomb interaction . we shall limit our explicit results here to the spatial form of the charge density @xmath34 ( where @xmath35 acts on the dirac space ) induced by the impurity . however , our rg strategy can be extended to other observables of experimental interest , such as the local density of states . as noted above , we will begin in section [ sec : nonint ] by considering only the electron - impurity coulomb interaction , while electron - electron coulomb interactions will be accounted for in section [ sec : int ] . this section will ignore the electron - electron coulomb interactions . formally , we work in the limit @xmath36 , but @xmath33 is kept fixed . the problem reduces to that of a single dirac electron in the attractive impurity potential @xmath37 this problem was originally studied in ref . . however , they introduced an arbitrary cutoff at high energy to regulate the problem at short distances , and this leads to spurious results @xcite . as we will demonstrate here , there is no dependence upon a cutoff energy scale at all orders in perturbation theory , provided the high energy behavior is regulated in a proper gauge - invariant manner . with no cutoff energy scale present , a number of results can be deduced by simple dimensional analysis . the fourier transform of the charge density @xmath38 is dimensionless , and therefore we can write @xmath39 where @xmath40 is a universal function of the dimensionless coupling @xmath33 . note that @xmath41 is required by this dimensional argument to be @xmath28-independent , and so @xmath42 . the arguments so far are perturbative , but non - perturbative effects can be deduced by solving the full dirac equation in the potential in eq . ( [ vr ] ) . this solution has appeared elsewhere @xcite , and so we will not reproduce it here . such an analysis shows that the perturbative arguments apply for @xmath43 , but new physics appears for @xmath44 . in particular , shytov _ et al . _ @xcite showed that @xmath45 for @xmath44 ( the sign of this tail is opposite to that of the impurity ) . we shall limit our discussion in this section to the @xmath46 case . one reason for doing so is that electron - electron coulomb interactions act to reduce the effective value of @xmath33 . this will become clearer in section [ sec : int ] , but we note here that a standard rpa screening of the potential @xmath47 in eq . ( [ vr ] ) can be simply accounted for by applying the mapping @xmath48 to the results of the present section . the value of @xmath32 in graphene is not small @xcite . we shall now establish the existence of the universal function @xmath40 in eq . ( [ n1 ] ) to all orders in @xmath33 . the existence of a universal @xmath40 is a consequence of the non - renormalization of the impurity charge @xmath1 @xcite . we compute @xmath41 diagrammatically , and the needed diagrams all have one fermion loop and are shown in fig . [ diags ] . . the filled square is the impurity site , the wavy line is the @xmath17 propagator , the line is the fermion propagator , and the filled circle is the charge density operator . , width=336 ] to first order in @xmath33 we have @xmath49 where @xmath50 is the bare polarization operator @xmath51 \nonumber \\ & = & \frac{g^2 n q}{16 v}\ , , \label{p0res}\end{aligned}\ ] ] and so we have @xmath52 . we will now consider the full problem defined in eq . ( [ zz ] ) , and account for both the electron - electron and electron - impurity coulomb interactions . the problem can be solved in two limits : in the weak coupling limit @xmath36 and the large @xmath66 limit , @xmath67 with fixed @xmath68 . in both cases @xmath69 , so one can limit oneself to linear response in which the induced charge is [ generalizing eq . ( [ p0 ] ) ] @xmath70 where @xmath71 is the full propagator of the coulomb potential @xmath17 , and @xmath72 is the polarization tensor . the connection between @xmath71 and @xmath72 is @xmath73 where @xmath74 is the bare propagator , @xmath75 to leading order ( either in coupling or @xmath76 ) , the polarization operator was given in eq . ( [ p0res ] ) , and we showed in section [ sec : nonint ] that this gives rise to a @xmath28-independent @xmath41 , or a screening charge localized at @xmath77 . however , if we compute corrections , we find logarithmically divergent diagrams , where the logarithms are cut off from above by the inverse lattice size and from below by @xmath28 . the leading logarithms are summed by a standard rg procedure . since the theory is renormalizable , we can eliminate the dependence on the cutoff by expressing the each diagram in terms of the renormalized parameters , instead of the bare parameters of the lagrangian . choosing the renormalization point to be @xmath78 , and denote @xmath79 as the fermion velocity at the scale @xmath11 , the polarization tensor can be schematically written as @xmath80 in @xmath81 there are logarithms of the ratio @xmath82 . we notice that @xmath83 is is invariant under a change of the renormalization @xmath78 , given that @xmath79 is changed correspondingly ( the particle density has no anomalous dimension ) . to eliminate the powers of @xmath84 we can choose @xmath85 , hence @xmath86 where in the perturbative expansion of the right hand side there is no large logarithms . thus to leading order it is given by a single diagram , which was computed previously [ eq . ( [ p0res ] ) ] , @xmath87 all the leadings logarithms are contained in the function @xmath88 , which satisfies the equation @xmath89 with the boundary condition @xmath90 . the screening charge is then @xmath91 the problem is now reduced to the problem of finding @xmath88 [ or , equivalently , @xmath92 . this problem has a long history @xcite ; most recently it has been revisited in ref . ( see also below ) . to find the spatial charge distribution @xmath93 one needs to take fourier transform of eq . ( [ nlambda ] ) . first one notice that if the velocity does not run then @xmath93 is proportional to @xmath94 . only when @xmath11 runs with the momentum scale does @xmath93 differ from @xmath95 away from the origin . when the running is slow ( as at weak coupling or at large @xmath66 ) , the amount of screening charge enclosed inside a circle of radius @xmath3 ( assumed to be much larger than the lattice spacing ) , to leading order , is @xmath96 the total screening charge is small if @xmath32 at the scale @xmath97 is small , and close to @xmath98 if @xmath32 is large . differentiating both sides of eq . ( [ nr - int ] ) with respect to @xmath3 , one finds @xmath99 ^ 2}\ , \frac{\beta(v(q))}{v(q)}\,.\ ] ] note that the beta function for @xmath11 is negative , therefore we arrive to a counterintuitive result the screening charge is _ positive_. to see what is happening , let us take the limit @xmath100 in eq.([nr - int ] ) . this limit corresponds to the infrared limit @xmath101 . we know that asymptotically @xmath88 grows to @xmath102 in this limit ( although only logarithmically ) , hence @xmath103 i.e. , the total screening charge is zero when integrated over the whole space ( although the integral goes to zero very slowly ) . the presence of an external ion , therefore , only leads to charge redistribution : a fraction of the unit charge is pushed from short distance ( of order of lattice spacing ) to longer distances , but none of the charge goes to infinity . therefore , there is a finite negative screening charge localized near @xmath77 . its value can be found by taking @xmath3 to be of order of inverse lattice spacing @xmath104 in eq . ( [ nr - int ] ) . the final result for the screening charge density can be written as @xmath105 ^ 2}\ , \frac{\beta(v(q))}{v(q)}\,.\ ] ] in the rest of the note we will concentrate our attention on the long - distance tail of @xmath38 , ignoring the delta function at the origin . at weak coupling ( @xmath106 ) , the beta function for @xmath88 is @xmath107 the solution to the rg equation , with the boundary condition @xmath108 at @xmath109 , is @xmath110 and the screening charge density is @xmath111 notice that the result is proportional to the square of the small coupling constant @xmath112 , although we have performed the calculation to leading order in the coupling . the reason is that for the charge density @xmath38 to be nonzero , it is necessary that the coupling constant runs . the density @xmath38 therefore contains the beta function @xmath113 , as seen in eq . ( [ nr - beta ] ) , and hence is second order in the coupling constant . in the @xmath76 expansion the beta function for @xmath88 was computed in ref . : @xmath114{0pt}{15pt}\\ -\displaystyle{\frac{8v}{\pi^2n}\left ( \frac{\arccos\lambda}{\lambda\sqrt{1-\lambda^2 } } + 1 -\frac\pi{2\lambda}\right ) } , & \qquad \lambda<1 . \end{array}\right.\ ] ] the two expressions smoothly match each other at @xmath115 . in is instructive to analyze two regimes where the rg equation can be solved analytically . the first regime is @xmath106 where the result is the same as in eq . ( [ n - weak ] ) . the second regime is the strong - coupling regime @xmath116 . this regime corresponds to a quantum critical point characterized by a dynamic critical exponent @xmath117 , whose value at large @xmath66 is @xcite @xmath118 in this regime @xmath119 . the solution to the rg equation , with the initial condition @xmath108 at @xmath109 , is @xmath120 in this regime @xmath121 i.e. , the charge density follows a power law behavior @xmath122 . the power is slightly different from @xmath123 . in real graphene @xmath32 is of order 1 , so one has to solve numerically the rg equation . we chose the scale @xmath78 to be comparable to the inverse lattice spacing , @xmath124 , and @xmath79 to be @xmath125 , a typical value found in experiments . we then run @xmath11 according to the leading ( in @xmath76 ) rg equation in two cases , in vacuum and when graphene is on a sio@xmath126 substrate with dielectric constant @xmath127 . we then plot @xmath128 as a function of the distance @xmath3 on figs . ( [ fig : scrcharge_vac ] ) and ( [ fig : scrcharge_sub ] ) . ( 0,0)(0,0 ) ( 150,-10)@xmath129 ( -50,100)@xmath130 on the distance @xmath3 for suspended graphene . note that coordinate @xmath3 is on a logarithmic scale.,width=384 ] ( 0,0)(0,0 ) ( 150,-10)@xmath129 ( -50,100)@xmath130 on the distance @xmath3 for graphene on a substrate with @xmath127 . note that coordinate @xmath3 is on a logarithmic scale.,width=384 ] as seen from the figures , the charge density @xmath38 roughly follows the @xmath0 law : when @xmath3 changes by two orders of magnitude , the product @xmath131 changes by a factor of less than 1.5 in both cases . in this paper we have considered the problem of screening of a coulomb impurity in graphene . we show that there is a qualitative difference between screening by non - interacting and interacting electrons . in the case of non - interacting electrons the induced charge density is localized at the position of the impurity when the impurity charge is small . the interaction between electrons lead to a long - distance tail in the induced charge distribution , with a counterintuitive sign which is the same as that of the impurity . an earlier version of this paper had a sign error in the @xmath54 term in eq . ( [ falpha ] ) ; we thank v. kotov for pointing this out to us , and for giving us a preview of the work of terekhov _ et al . _ @xcite which contains a closed form expression for the function @xmath40 . the authors thank a. v. andreev , m. i. katsnelson , v. n. kotov , and l. s. levitov for useful discussions . d.t.s . thanks the center for theoretical physics at mit , where part of this work was completed , for hospitality . this work was supported , in part , by doe grant no.de-fg02-00er41132 and nsf grant no .	we consider the problem of screening of an electrically charged impurity in a clean graphene sheet . when electron - electron interactions are neglected , the screening charge has a sign opposite to that of the impurity , and is localized near the impurity . interactions between electrons smear out the induced charge density to give a large - distance tail that follows approximately , but not exactly , an @xmath0 behavior and with a sign which is the _ same _ as that of the impurity .
mr 2251@xmath0178 was the first quasar detected by x - ray observations ( by _ ariel v _ and _ sas-3 _ ; cooke et al . 1978 , ricker et al . 1978 ) , and also the first quasar where a warm absorber ( wa ) was suggested to explain the x - ray spectrum based on _ einstein _ observations ( halpern 1984 ) . the x - ray flux of the source is variable on timescales of @xmath9 days , e.g. , the _ exosat _ observations reported by pan et al . these authors found the column density of the wa to vary and to correlate with the x - ray flux . mineo & stewart ( 1993 ) combined the earlier _ exosat _ observations with a _ ginga _ observation from 1989 and argued that the spectrum could be described by a power law with photon index @xmath10 and a wa model with a column density of @xmath11 . using this model they found the ionization parameter to be strongly correlated with the source luminosity . a deep _ rosat_/pspc observation from 1993 was reported by komossa ( 2001 ) who modeled the 0.12.4 kev spectrum using a wa with a column density of @xmath12 . mr 2251@xmath0178 was observed with _ asca _ ten times during 1993 and 1996 . these observations seem to be consistent with a wa with a column density of about @xmath13 ( e.g. , reynolds 1997 , otani et al . 1998 , reeves & turner 2000 , morales & fabian 2002 ) . mr 2251@xmath0178was also observed twice with _ bepposax _ during 1998 . the two observations are separated by 5 months and show identical spectral shape and flux . the wa column was found to be @xmath14 and the difference from previous observations was attributed to the motion of the absorber across our line - of - sight ( orr et al . 2001 ) . in summary , the observed 210 kev flux of mr 2251@xmath0178 covers the range of @xmath15@xmath16 ^-2s^-1 erg@xmath6s@xmath8 which translates to a 210 kev luminosity of @xmath15@xmath17 ^-2s^-1 erg@xmath6s@xmath8 ( h@xmath18 kms@xmath8mpc@xmath8 , @xmath19 , @xmath20 and assuming a @xmath1 power law ) . the galactic hydrogen column density towards mr 2251@xmath0178 has been derived from 21 cm measurements to be @xmath21 ( lockman & savage 1995 ) . ccccc & 71035000 & 1993 nov 11 & 3.6 & @xmath22 + _ asca _ & 71035010 & 1993 nov 16 & 6.9 & @xmath23 + _ asca _ & 71035040 & 1993 dec 12 & 9.9 & @xmath24 + _ asca _ & 71035060 & 1993 dec 14 & 5.8 & @xmath25 + _ asca _ & 71035020 & 1993 dec 19 & 6.7 & @xmath26 + _ asca _ & 71035050 & 1993 dec 24 & 7.5 & @xmath27 + _ asca _ & 74028000 & 1996 may 26 & 17.8 & @xmath28 + _ asca _ & 74028010 & 1996 jun 18 & 16.5 & @xmath29 + _ asca _ & 74028020 & 1996 nov 27 & 15.9 & @xmath30 + _ asca _ & 74028030 & 1996 dec 09 & 20.0 & @xmath31 + _ bepposax _ & 50556001 & 1998 jun 14 & 47.5 & @xmath32 + _ bepposax _ & 505560011 & 1998 nov 12 & 47.5 & @xmath33 + _ xmm - newton _ & 0112910301 & 2000 may 29 & 3.5&@xmath34 + _ xmm - newton _ & 0012940101 & 2002 may 18 & 44.7 & @xmath35 an fek@xmath3 line was first suggested in a _ ginga _ observation of mr 2251@xmath0178 , with an equivalent width ( ew ) of @xmath36 ev ( mineo & stewart 1993 ) . this was later confirmed by the _ asca _ and _ bepposax _ observations with ew of @xmath37 ev ( reynolds 1997 ; cf . , reeves & turner ( 2000 ) found ew of @xmath38 ev ) and @xmath39 ev ( orr et al . 2001 ) , respectively . the redshift of mr 2251@xmath0178 was determined by using nine optical narrow emission lines ( bergeron et al . 1983 ) to be @xmath40 ( we note that a few catalogs listed incorrect values , which has resulted in a variety of values quoted in the literature ) . the host galaxy has a gaseous component with temperature of @xmath41k ( derived from the [ ] line ratio ) and indications of low abundances of ne , o , and n ( bergeron et al . the galaxy is surrounded by a giant envelope which is observed via [ ] @xmath425007 emission . mr 2251@xmath0178 was observed by _ hst _ at three epochs ( monier et al . 2001 and references therein ) . the spectrum shows clear ly@xmath3 and absorption . ganguly , charlton , & eracleous ( 2001 ) found the doublet absorption to vary with time , suggesting an intrinsic origin for this absorption . the quasar is radio quiet with a radio flux of [email protected] mjy ( nvss catalog condon et al . 1998 ) . this paper presents new _ xmm - newton _ and _ far ultraviolet spectroscopic explorer ( fuse ) _ observations of mr 2251@xmath0178 . we also carry out an in - depth analysis of the 10 available _ asca _ observations and the two _ bepposax _ observations . we describe the data in 2 , perform an x - ray data temporal analysis in 3 , and discuss the x - ray spectral analysis in 4 and the uv spectral analysis in 5 . in 6 we elaborate on the implications of our results . this paper includes an extensive analysis of the historical uv and x - ray spectra of mr 2251@xmath0178 . the basic x - ray observations are listed in table [ ascalog ] and the data analysis is described in this section . mr 2251@xmath0178 was observed with _ xmm - newton _ during 2002 may 1819 for 64 ks . data were reduced using the science analysis system ( sas v5.3.0 ) in the standard processing chains as described in the data analysis threads and the abc guide to _ xmm - newton _ data analysis . source data were extracted from circular regions of radius 30@xmath44 and 40@xmath44 for the epic - pn and epic - mos , respectively . the epic - pn was operated in the small window mode resulting with good exposure time of 45 ks . the observed count rate ( @xmath457.3 countss@xmath8 before background subtraction ) was well below the pileup threshold ( 130 countss@xmath8 , see _ xmm - newton _ users handbook ) . for statistical purposes we binned the spectra to have at least 25 counts per bin . mos ccds were used in the large window mode and the observed count rate ( @xmath452.0 countss@xmath8 before background subtraction ) was just above the pileup threshold ( 1.8 countss@xmath8 ) . the mos observations will not be discussed here . the rgs1 and rgs2 were operated in the standard spectroscopy mode resulting in a good exposure time of 63 ks for each . background extraction is performed with the sas using regions adjacent to those containing the source in the spatial and spectral domains . the spectra were extracted into bins of @xmath450.04 in width ( 4 times the default bin width ) in order to increase the signal - to - noise ratio . to flux calibrate the rgs spectra we divided the counts by the exposure time and by the effective area at each wavelength . each flux - calibrated spectrum was also corrected for galactic absorption and the two spectra combined into an error - weighted mean . at wavelengths where the rgs2 bins did not match exactly the wavelength of the rgs1 bins , we interpolated the rgs2 data to enable the averaging . this final spectrum is shown in figure [ rgsspec ] . mr 2251@xmath0178 was also observed with _ xmm - newton _ during the validation and verification phase of the telescope during 2000 may 29 . we retrieved the data from the _ xmm - newton _ archive and reduced it in the same way as described above for the 2002 observation . unfortunately , due to the operating modes used , neither of the epic - mos detectors contain any useful data from mr 2251@xmath0178 during this observation . the epic - pn was operated in the small window mode resulting in good exposure time of 3.5 ks . the observed count rate ( @xmath4518.1 countss@xmath8 before background subtraction ) was well below the pileup threshold . the rgs1 and rgs2 were operated in the standard spectroscopy mode during two distinct exposures , each of about 6 ks ( which is the same as the good exposure time accumulated ) . as listed in table [ ascalog ] , the _ asca _ archive contains 10 observations of mr 2251@xmath0178 . the data were screened in the same manner as used for the _ tartarus database _ ( e.g. , turner et al . 1998 ) . we briefly describe this procedure here . ascascreen / xselect ( v0.45 ) was used for screening together with the criteria given in nandra et al . ( 1997 ) . in the case of the _ asca _ sis data , hot and flickering pixels were removed using the standard algorithm and only sis grades 0 , 2 , 3 , and 4 events were included in the analysis . the original pulse - height assignment for each event was converted to a pulse - invariant ( pi ) scale using sispi ( v1.1 ) . in the gis data , hard particle flares were rejected using the so - called ho2 count rate , and standard rise - time rejection criteria were employed . the effective exposure times resulting from these criteria are listed in table [ ascalog ] . the spectral analysis for each of the 10 _ asca _ observations was performed on the data from all four instruments simultaneously . different relative normalizations were allowed to account for ( small ) uncertainties in the determination of the effective area of each instrument . we also corrected the _ asca _ 1996 data for the sis degradation as indicated by yaqoob et al . individual spectra were binned in energy to contain a minimum of 20 counts per bin , and hence allowing meaningful @xmath46 minimization . fits to the data were carried out with xspec using our own models , as described below . quoted uncertainties on the parameters refer to the 90% confidence level . _ bepposax _ observed mr 2251@xmath0178 at two epochs : 1998 june 1418 and 1998 november 1216 . the observations and their analysis are described in orr et al . ( 2001 ) . in the following analysis we use the data and the calibrations supplied by the heasarc archive . we used only the data obtained with the low - energy concentrator spectrometer ( lecs ; 0.14 kev ) and the medium - energy concentrator spectrometer ( mecs ; 1.810 kev ) . again we use xspec , and allow different relative normalizations to account for uncertainties in the effective areas of the instruments . mr 2251@xmath0178 was observed with _ fuse _ during 2001 june 2021 . the observation was carried out using the lwrs aperture and is @xmath47 ks in duration . only a small part of the spectrum around the absorption has been published to date ( wakker et al . thus we have extracted the raw data from the _ fuse _ archive , and reduced it using the _ fuse _ software ( calfuse v2.2.2 and fuse idl tools version of 2002 july ) . the _ fuse _ spectrum is shown in figure [ fusespec ] . light curves were constructed for the source and background regions for all _ asca _ observations , in several different energy ranges . to increase the signal - to - noise ratio , the light curves from each pair of sis and gis detectors were combined . the light curves were then rebinned on a variety of timescales . 178 using a 128 s bin size . _ lower panels _ : the mean ` softness ratio ' of each observation using the 0.51.2 kev ( xm@xmath48 ) , 1.53.5 kev ( xm@xmath49 ) , and 4.010.0 kev ( xm@xmath50)bands . note that the spectrum is softer when the source is brighter . [ sislc ] , width=321 ] the combined sis light curves in the 0.510 kev band are shown in figure [ sislc ] . there is no evidence for short timescale variability within any of the observations . this is confirmed by a lack of significant ` excess variance ' ( turner et al . 1999 and references therein ) in any of the observations , with upper limits of typically @xmath51 ( at 90% confidence ) . such a lack of a short timescale variability is consistent with the anti - correlation between the excess variance and the luminosity found by nandra et al . clear variability is seen between observations , with a maximum flux change of a factor @xmath453 over 3 years . the smallest amplitude variation on the shorter timescale is a 20% flux decrease in 5 days between the fifth and the sixth _ asca _ observations . we have also constructed light curves in the 0.51.2 kev ( xm@xmath48 ) , 1.53.5 kev ( xm@xmath49 ) , and 4.010.0 ( xm@xmath50 ) bands of netzer , turner , & george ( 1994 ) . in the lower panels of figure [ sislc ] we plot the mean xm@xmath48/xm@xmath50 and xm@xmath49/xm@xmath50 ratios for each observation . we find no statistically significant variation in softness ratio during each of the three epochs ( the apparent decrease in the xm@xmath48/xm@xmath50 ratio during the 1993 nov dec epoch is significant at only the 75% confidence level ) . however , both ratios exhibit statistically significant variability ( 98% confidence ) _ between the epochs _ in a manner suggesting that the spectrum becomes softer when the source luminosity increases . these variations are discussed in [ history ] and [ historydiss ] . we examined the epic - pn and the two epic - mos background - subtracted light curves of mr 2251@xmath0178 . no significant flux variation is detected during the 64 ks observation . this is consistent with the _ asca _ observations which show no variations on timescales of less than a day . the epic - pn count rate from the 2000 observation ( @xmath52 countss@xmath8 ) is 2.5 times higher than the epic - pn data from 2002 ( @xmath53 countss@xmath8 ) . table [ pnsoftrat ] gives the softness ratios , as defined above , for the two epic - pn observations . we find clear variations in both softness ratios . orr et al . ( 2001 ) examined the light curves for the 1998 june and november _ bepposax _ observations as well as the hardness ratios . they find the count rates and hardness ratios to be similar at the two epochs . while the june light curve is well fitted with a constant count rate orr et al . ( 2001 ) find the november observation to better fit with a slowly decreasing linear function ( minus @xmath54% in 70 hours ) although a constant count rate can not be ruled out . ccc 2000 may 29 & @xmath55 & @xmath56 + 2002 may 18 & @xmath57 & @xmath58 kev . [ datamo2002 ] , width=321 ] we have carried out an extensive spectral analysis of the low resolution _ asca , bepposax _ and _ xmm - newton _ spectra of mr 2251@xmath0178as well as the high resolution rgs spectra . we first consider the high signal - to - noise , broad - band and rgs spectra obtained using _ xmm - newton _ in 2002 , and describe a multi - component model that is consistent with the data ( [ 2002-broad ] and [ rgsfit ] ) . this model is then compared to the _ xmm - newton _ data obtained during 2000 ( [ softexcess ] and [ 2000-rgs ] ) and to the historical x - ray datasets in [ history ] . we first fitted the 0.211 kev epic - pn spectrum of mr 2251@xmath0178 obtained in 2002 with a single galactic absorbed power law . this gives a poor description of the data ( @xmath59 ) in agreement with previous analysis of the source . the main discrepancy is at low energies , hence we repeated the fit for only the 311 kev band ( excluding the 4.57.5 kev rest - frame band where contamination from the fek@xmath3 line might be present ) . we find @xmath60 and normalization of @xmath61 photons@xmath6s@xmath8kev@xmath8 with @xmath62 . we show an extrapolation of this continuum to lower energies in figure [ datamo2002 ] . the presence of an excess absorption around the and edges , at 0.70.9 kev , is clear and indicates a wa component . excess absorption is also evident at energies below 0.5 kev , which is indicative of an additional , less - ionized absorber . cccccccc _ bepposax _ & 1998 jun 14 & 21.8 & @xmath63 & @xmath64 & @xmath65 & 2.04 & 1.25 + _ bepposax _ & 1998 nov 12 & 21.8 & @xmath66 & @xmath67 & @xmath68 & 2.22 & 1.24 + _ xmm - newton _ & 2000 may 29 & 21.5 & @xmath69 & @xmath70 & @xmath71 & 2.27 & 1.15 + _ xmm - newton _ & 2002 may 18 & 21.5 & @xmath72 & @xmath73 & @xmath74 & 1.28 & 1.21 + _ asca _ & 1993 nov 11 & 21.5 & @xmath75 & @xmath76 & @xmath77 & 3.39 & 0.96 + _ asca _ & 1993 nov 16 & 21.5 & @xmath78 & @xmath79 & @xmath80 & 2.72 & 0.99 + _ asca _ & 1993 dec 12 & 21.5 & @xmath81 & @xmath82 & @xmath83 & 3.17 & 0.98 + _ asca _ & 1993 dec 14 & 21.5 & @xmath84 & @xmath85 & @xmath86 & 2.85 & 1.02 + _ asca _ & 1993 dec 19 & 21.5 & @xmath87 & @xmath88 & @xmath89 & 2.65 & 0.92 + _ asca _ & 1993 dec 24 & 21.5 & @xmath90 & @xmath91 & @xmath92 & 2.20 & 0.96 + _ asca _ & 1996 may 26 & 21.8 & @xmath93 & @xmath94 & @xmath95 & 1.74 & 0.96 + _ asca _ & 1996 jun 18 & 21.8 & @xmath96 & @xmath97 & @xmath98 & 2.14 & 1.03 + _ asca _ & 1996 nov 27 & 22.1 & @xmath99 & @xmath94 & @xmath100 & 1.82 & 1.01 + _ asca _ & 1996 dec 09 & 22.1 & @xmath101 & @xmath102 & @xmath103 & 1.95 & 1.00 + _ bepposax _ & 1998 jun 14 & 21.8 & @xmath104 & @xmath105 & @xmath106 & 2.03 & 1.26 + _ bepposax _ & 1998 nov 12 & 21.8 & @xmath107 & @xmath108 & @xmath109 & 2.27 & 1.23 + _ xmm - newton _ & 2000 may 29 & 21.5 & @xmath110 & @xmath111 & @xmath112 & 2.27 & 1.12 + _ xmm - newton _ & 2002 may 18 & 21.5 & @xmath113 & @xmath114 & @xmath115 & 1.28 & 1.16 we used ion2003 , the 2003 version of the ion photoionization code ( netzer 1996 ; netzer et al . 2003 ) in order to model the wa . for simplicity , in all calculations we assume constant density gas with a density of @xmath116 , which is low enough to avoid complications due to collisional de - excitation and line transfer yet large enough to assume `` thin - shell '' geometry . the relevant parameters of the model are @xmath117 ( the oxygen ionization parameter defined over the range of 0.53810 kev ) , the column density @xmath118 ( in units of ) , the gas composition ( assumed to be solar and specified in netzer et al . 2003 table 2 ) , and the covering fraction . we used ion2003 to fit the 2002 epic - pn data assuming power law , with a slope fixed to the value found earlier ( @xmath119 ) , attenuated by the galactic column density , and two generic absorption components : one with a `` typical '' wa properties and one which is much less ionized . fitting the spectrum with this model yields some excess emission around 0.5 kev regardless of the exact values of @xmath117 and @xmath118 . we interpret this excess as due to emission of the triplet and the ly@xmath3 lines . hence , we added to the model an emission component constrained to have the wa ionization parameter and column density . having all these components , we obtained the following solution : for the wa component we find log(@xmath120 , @xmath121 and a line - of - sight covering factor of 0.8 . for the emission we find a global covering factor of 0.3 . for the less ionized absorber we find that it can be fitted by a neutral absorber ( in addition to the galactic one ) with @xmath122 . since mr 2251@xmath0178 is at low redshift this column can be interpret either as an additional galactic absorber or as neutral gas intrinsic to the source . the low ionization absorber can also be modeled as a combination of low ionization absorber with log(@xmath123 and @xmath124 and a neutral absorber of @xmath125 ( we show below that such low ionization component is required by the rgs data ) . both cases give equally good fits and the statistical analysis for all absorbers and emitters yield @xmath126 for both . the 2002 _ xmm - newton _ epic - pn data suggest a very weak fek@xmath3 line . using the continuum from table [ xrayfit ] , and fixing the line energy to 6.4 kev , we find a narrow line with a width of @xmath127 kev and a flux of @xmath128 ^-2s^-1 photons@xmath6s@xmath8 . the ew of the line is @xmath129 ev , consistent with previous studies of this source ( see 1 ) . we found no indication for a broad component to the fek@xmath3 line . this line will not be discussed any further . we note on a narrow absorption feature at around rest frame energy of @xmath130 kev ( see figure [ datamo2002 ] ) . we fitted this feature with a gaussian with a width fixed to the instrumental resolution and find its rest frame energy to be @xmath131 kev and a normalization of @xmath132 ^-2s^-1 photons@xmath6s@xmath8 . the ew of this gaussian is @xmath133 ev . the energy of this feature is consistent with the ly@xmath3 line . the 2002-rgs spectrum of mr 2251@xmath0178 shows several absorption and emission lines with wavelengths that are consistent with the systemic velocity , given the rgs resolution ( 0.04 , corresponding to 1000 ^-1 kms@xmath8 at 12 and 400 ^-1 kms@xmath8 at 30 ) . despite the low signal - to - noise ratio ( s / n ; of order 34 at around 20 ) evidence can be seen for emission lines from , , , , and . absorption lines are seen from the low ionization ions of , , and as well as many higher ionization species . the strongest of these lines are marked in figure [ rgsspec ] together with many other lines whose detection is less certain due to the low s / n . the wavelengths of the lines are those used by netzer et al . a full line list is given in table [ linetable ] where we differentiate between lines that are identified with high certainty and these that we regard as possible identification , due to the poor s / n . we also detect bound - free absorption due to the and edges and a noticeable curvature of the spectrum over the wavelength band of 1517 . the rgs spectrum also shows several features which we suspect to be artifacts . the strong emission - like feature at @xmath134 is probably an artifact caused by high background level and the proximity to the edge of the ccd . several absorption - like features , at around 1011 , are similar in shape and intensity to other absorption features and we suspect that some of those are due to . however , in this part of the spectrum there is only one ccd ( rgs2 ) and we can not confirm their reality by comparing the two rgs spectra . modeling of the 2002 rgs spectrum was done in two steps . first we experimented with a two component absorber , similar to the one discussed in [ 2002-broad ] . this involves a highly - ionized absorber and a second absorber of much lower - ionization . the highly - ionized wa has a large column density and is responsible for the bound - free absorption edges and the and emission lines . the less ionized component has a lower column density and is responsible for the and absorption lines . the absorption lines in both components are probably narrower than the instrumental resolution and we assumed that they can be characterized by a turbulent velocity of @xmath135 ^-1 kms@xmath8 . the modeling assumes that each of the components can be represented by a single cloud ( `` shell '' ) . thus , the gas on the line of sight produces the absorption features and the gas outside the line of sight produces the emission lines . ccc & 33.737 & 2 + & 29.747 & 2 + & 29.541 & 1 + & 28.787 & 2 + & 28.465 & 2 + & 26.990 & 2 + & 24.782 & 2 + & 23.771 & 1 + & 23.051 & 1 + & 22.729 & 1 + & 22.334 & 1 + & 22.101 & 1 + & 22.007 & 1 + & 21.788 & 1 + & 21.602 & 1 + & 19.826 & 2 + & 19.341 & 2 + & 19.135 & 1 + & 18.966 & 1 + & 18.627 & 2 + & 16.006 & 2 + & 14.208 & 1 + & 13.700 & 1 + & 13.447 & 1 + & 12.134 & 1 + & 10.579 & 1 + & 10.238 & 2 + & 10.025 & 1 + & 9.695 & 1 + & 9.231 & 2 + we first used the model parameters derived form the epic - pn data and calculated theoretical spectra for this gas . we note that according to kirsch ( 2003 ) the epic and rgs agree within @xmath136% in the normalization , and individual fitting shows a significant steeper slope for the epic ( see also den herder et al . 2003 and blustin et al . this effect seems to be present in our data and hence we do not require complete agreement between the slopes derived from fitting the rgs and the pn spectra . given those uncertainties , we found a good fit for the rgs continuum with our two - component model for a power law of photon index @xmath137 . we experimented with a range of parameters around the values found for the epic - pn . the parameters we found to fit best for the two absorbers in the rgs spectrum are : log(@xmath138 and @xmath118 in the range of @xmath4@xmath5 for the high - ionization wa component and log(@xmath139 and @xmath140 for the low - ionization component . we also assume , based on the uv measurement ( see [ hiresfuse ] ) and in agreement with the epic - pn model , a line of sight covering factor of 0.8 and an undetermined outflow velocity which is taken to be 300 ^-1 kms@xmath8 . we note that the less - ionized gas does not contribute anything to the observed x - ray emission lines and the required global ( @xmath141 ) covering factor for the emitting gas is @xmath142 . the two - component model fits the general continuum shape , the triplet , the ly@xmath3 , the forbidden line , and the absorption of and . a major discrepancy is the underestimation of the resonance emission line at 21.6 . a similar phenomenon has been observed in ngc3783 where the line is underpredicted in the best - fitting model of netzer et al ( 2003 ) . a possible explanation may be a complex optical depth structure for this optically thick line . for example , the lateral optical depth ( which can not be observed and is a function of the geometry ) may be smaller than the line - of - sight optical depth used in the calculations . as a results , line photons can escape more easily in some directions increasing , in this way , the emission line intensity . such a situation may arise in conical type flows where the cone lateral dimension is smaller than its height . the two - component fit is severely limited by the poor s / n of the grating observations . nevertheless , it shows that the ionized ( line - of - sight ) absorber and the ionized emitter are consistent with being the same gas . the next step includes a three component absorber . the main motivation for this is the fact that the rgs data around 1617 clearly falls below the two - component model . the excess absorption is probably caused by the unresolved transition array ( uta ) of iron m - shell lines ( behar , sako , & kahn 2001 ) which has been observed in several other agns ( see , e.g. , netzer et al . 2003 for the case of ngc3783 and netzer 2004 for a general discussion ) . our photoionization code includes all these lines but the two - component wa produces too shallow a feature at too short a wavelength . we find that an additional shell with a column density of @xmath143 and log@xmath144 can significantly improve the fit . this component produces a noticeable uta feature and contributes also to the observed emission . this requires lowering the emission from the high - ionization component by about 20% to produce an adequate fit to all emission lines . adding this component force us to increase the ionization parameter of the highly ionized gas ( the one with column density of @xmath5 ) to log@xmath145 . we note that the mean @xmath117 of these two wa components is the same as the one found earlier in the two - component model . the three - component model is compared with the rgs data in figure [ rgsspec ] on a wavelength scale where all the features can be seen . in figure [ threecompmo ] we show a comparison on a reduced wavelength scale to emphasize the uta range . the uta fit utilizes the improved dielectronic recombination rates of netzer ( 2004 ) and is in good agreement with the 1517 spectrum . ( red line ) . the three absorbers model produces a noticeable uta feature around 1617 and thus a better fit compared with the two absorber model . however , it fails to explain the epic - pn spectrum . [ threecompmo ] , width=321 ] there are two problems with the three - component model related to its agreement with the epic - pn data . first , we could not find a model which explains the uta feature and is also consistent with the epic - pn spectrum . in particular , while we are convinced in the presence of a uta feature , there is no way to assess the exact column density of the relevant ions ( and ) , in the intermediate @xmath117 component , given the s / n of the present data . second , the chosen column of @xmath5 for the high @xmath146 component is about the maximum which is still consistent with the epic - pn observation . yet , some strong features in the rgs spectrum seem to require even a larger absorbing column density . as argued above , a full model for the x - ray spectrum of mr 2251@xmath0178obtained by _ xmm - newton _ in 2002 requires a a power law continuum attenuated by ( neutral ) galactic absorption and at least two absorbers . in figure [ 2002scale2000 ] we plot the epic - pn 2000 data divided by the scaled 2002 model . the scaling is done by multiplying the power law continuum flux and the ionization parameter of the wa by the same factor of 1.9 ( which is the hard flux ratio between the two observations ) . the plot shows a large excess at low energies indicating the presence of an additional continuum component . we denote this continuum the `` soft excess '' . we have re - visited the 2002 observation in attempt to look for this component but the data do not require its presence in this observation . , width=321 ] next we attempted to determine the shape of the high energy continuum during the 2000 observation . for this we first fitted the 311 kev band ( excluding the 4.57.5 kev rest - frame band ) with a galactic absorbed power law . we find @xmath147 and normalization of @xmath148 photons@xmath6s@xmath8kev@xmath8 with @xmath149 . fixing the hard continuum at these values , we re - fitted the data adding this time the ionized and the neutral absorbers , and the ionized emission constrained to the ionized absorber . we have also included the additional soft excess component assuming it can be fitted by a second power law . the fit results with a @xmath150 and the following parameters : a soft excess component with @xmath151 and normalization of @xmath152 photons@xmath6s@xmath8kev@xmath8 , an ionized absorber with log(@xmath153 and @xmath154 , and a neutral absorber with a column density of @xmath155 . although the source flux during the 2000 _ xmm - newton _ observation was a factor of @xmath156 larger than the flux during the 2002 observation , the integration time is much shorter ( by a factor of 5 ) . thus , we were unable to obtain any useful constraints from the rgs data obtained in 2000 . more specifically , we could not identify any emission or absorption lines in this spectrum ( if any such lines are present in this spectrum and have the same ews as the lines in the 2002 observation , they are consistent with the noise level ) . cccc & 1993 nov 11 & @xmath157 & @xmath158 + _ asca _ & 1993 nov 16 & @xmath159 & @xmath160 + _ asca _ & 1993 dec 12 & @xmath161 & @xmath162 + _ asca _ & 1993 dec 14 & @xmath163 & @xmath164 + _ asca _ & 1993 dec 19 & @xmath165 & @xmath166 + _ asca _ & 1993 dec 24 & @xmath167 & @xmath168 + _ asca _ & 1996 may 26 & @xmath169 & @xmath170 + _ asca _ & 1996 jun 18 & @xmath171 & @xmath172 + _ asca _ & 1996 nov 27 & @xmath173 & @xmath170 + _ asca _ & 1996 dec 09 & @xmath174 & @xmath170 + _ bepposax _ & 1998 jun 14 & @xmath175 & @xmath176 + _ bepposax _ & 1998 nov 12 & @xmath177 & @xmath178 + _ xmm - newton _ & 2000 may 29 & @xmath179 & @xmath180 + _ xmm - newton _ & 2002 may 18 & @xmath181 & @xmath182 the 2000 and 2002 epic - pn observations of mr 2251@xmath0178 show that a full model for the x - ray spectrum must include a high energy power law continuum , a soft excess power law component and two absorbers . this combination was used to fit also the earlier _ bepposax _ and _ asca _ spectra of the source and the results are discussed in this section . we note that the low resolution _ bepposax _ and _ asca _ data are not sensitive to the inclusion of the emission component in our model . these data are also not sensitive to the differences between the two and the three absorption components discussed in [ rgsfit ] . thus in the fit below we include only one highly ionized wa component which represent an average of two such components . our initial assumption is that the absorbers column densities are constant and that the only changes are in the value of the ionization parameter which is proportional to the source luminosity . we fixed the power law slopes of the soft excess and the hard continuum to be @xmath2 and @xmath1 , respectively , and we only allow changes in their relative normalization . the @xmath1 value for the hard x - ray slope is consistent with all previous x - ray observations of mr 2251@xmath0178 and is in accord with earlier findings of reeves & turner ( 2000 ) , orr et al . ( 2001 ) , and morales & fabian ( 2002 ) . in table [ slopes ] we show power - law fits for the 310 kev band ( excluding the rest frame 5.07.5 kev band ) for all data sets in order to determine hard x - ray slope in each observation . as seen from the table , a @xmath1 continuum is in good agreement with all data sets . we also fixed the galactic absorption to the value found earlier . in the following fits we excluded the rest frame 5.07.5 kev band to avoid complications due to the fek@xmath3 line . the bound - free opacity of the low - ionization absorber is very similar to a totally neutral absorber , and can not be distinguished from such an absorber in the data collected with low spectral resolution . indeed no useful constraints regarding changes in the ionization state of this component can be obtained . thus , in this section we simply approximate the effects of the low - ionization absorber by a neutral absorber with a column density to @xmath7 ( as found above ) . constraints can , however , be obtained by the study of variations in the high - ionization absorber and underlying continuum , and these are considered in the remainder of this section . models show the soft excess , the wa , the emission line gas , and the hard power law , but do not include the iron - k@xmath3 line and the galactic absorption for clarity . note the disappearance of the soft excess in the 2002 _ xmm - newton _ observation and the difference in the absorption around 1 kev . [ softxmodels ] , width=321 ] we first fit the _ bepposax _ data , and re - fitted the _ xmm - newton _ data , with the same model and the fixed parameters as described above . the results of these fits are tabulated in the upper part of table [ xrayfit ] and are plotted in figure [ softxmodels ] . the fits to the _ xmm - newton _ observations indicate that the flux decrease from 2000 to 2002 was associated with the disappearance of the soft excess component . on the other hand , there is no change , within the data and model uncertainties , in the column density ( @xmath183 ) and the ionization parameter of the high - ionization wa . thus , there seems to be no connection between the ionization parameter and the x - ray luminosity of the source during this two year period . for the two _ bepposax _ observations , we found that the required wa column density is @xmath184 . this is consistent with the column density reported by orr et al . ( 2001 ) and gives a better fit ( @xmath185 is lower by 0.3 ) than the column of @xmath4 . this is significant at an f - test probability of @xmath186 . fitting the 0.111 kev _ bepposax _ observations we find that they also require the presence of a soft excess component . the two observations are entirely consistent with each other ( as reported also by orr et al . 2001 ) despite the 5 months separation . the high energy flux level of the _ bepposax _ and the 2000 _ xmm - newton _ observations are similar but the soft excess component contribution to the _ bepposax _ observation is lower by a factor of @xmath156 ( see figure [ softxmodels ] ) . the wa properties of the _ bepposax _ observations are not consistent with those of the _ xmm - newton _ observations , indicating again that the absorber in mr 2251@xmath0178 is changing on timescales of years . in general , we find a weakening in the soft excess as the source luminosity decreases , however , the dependence between the two is not simple . we now consider the _ asca _ data , which are limited to the 0.510.0 kev band . therefore we re - fitted the _ bepposax _ and _ xmm - newton _ data over the same energy range thus enabling a more meaningful comparison between the various observations . the lower part of table [ xrayfit ] summarizes the fit results and figure [ xmodels ] shows selected models . in figure [ ascadata ] we show selected _ asca _ sis0 data sets . models are showing the wa and power law and do not include the emission line gas or the iron - k@xmath3 line for clarity . the models parameters are detailed in the lower part of table [ xrayfit ] . the _ asca _ 1996 models are corrected for the sis0 degradation . [ xmodels ] , width=321 ] . [ ascadata ] , width=321 ] we were able to fit the first 6 _ asca _ observations ( taken during 6 weeks on 1993 nov dec ) with the same column density high - ionization wa ( @xmath4 ) changing only the ionization parameter ( see table [ xrayfit ] ) . however , the 1996 nov - dec observation requires a larger column density of @xmath187 which gives @xmath188 . when fixing the column density to @xmath4 the best fit gives @xmath189 , i.e. , the change in column density is highly significant ( an f - test probability of practically 0 ) . the best fit to the may - june 1996 observations also requires a larger column density absorber ( @xmath190 ) which gives @xmath191 ( when fixing the column density to @xmath4 or @xmath192 the @xmath185 is increased by @xmath193 over the 1020 degrees of freedom ) . this suggests that the properties of the wa have changed between 1993 and the end of 1996 . the four observations of 1996 can be divided into two groups : two observations dating 1996 may jun and two in 1996 nov dec . within each group the spectra are indistinguishable . however , we could not find a consistent model for the two epochs together . the differences between 1993 and 1996 , and the differences during 1996 , all indicate that there are real changes in the absorbing column on timescales of several months to several years and there are no differences on timescales of two months or less . the _ asca _ data are limited to energies above 0.5 kev and thus poorly constrain the soft excess component . this also introduces some uncertainty concerning the wa properties since we can not unambiguously determine the contribution of the soft excess component . like the _ xmm - newton _ and the _ bepposax _ results , the _ asca _ fits also indicate a general trend where the soft excess is stronger when the hard flux is higher . the fuse spectrum of mr 2251@xmath0178 ( figure [ fusespec ] ) shows broad emission lines of @xmath1941032,1038 and @xmath42977 . all these lines show significant blueshifted absorption . we also detect blueshifted absorption from at least 10 lines of the lyman series starting with ly@xmath195 and going up to the lyman edge , where the lines are blended together . the ly@xmath3 absorption is outside the _ fuse _ wavelength range and was observed , independently , by _ the fit parameters are detailed in table [ o6fittab ] . [ o6fit ] , width=321 ] cccc i & 1.240 & 0.782 & 530 + ii & 0.518 & 3.667 & 2500 + iii & 0.160 & 13.096 & 8950 in order to study the intrinsic absorption spectrum we first fitted the doublet emission lines . each of the two emission lines ( @xmath1961032,1038 ) was fitted with three kinematic components represented by gaussians with the same kinematic width , and had their flux ratio fixed at the ratio of oscillator strengths ( 2:1 ) . the results are shown in figure [ o6fit ] and are listed in table [ o6fittab ] . the observed spectrum was then divided by this emission model and the resulting normalized spectrum was used to obtain the absorption velocity spectra shown in figure [ velspec]a . we note that the continuum shown in figure [ o6fit ] does not match very well the continuum on the blue side of the ly@xmath195 line around 1021 . to produce the velocity spectrum of ly@xmath195 ( shown in figure [ velspec]a ) , we fitted the continuum on both sides of the line with a spline curve and divided the observed spectrum by this continuum . figure [ velspec]b shows the absorption and two more lyman absorption lines . the line region contains many galactic features which do not allow a proper continuum fit . therefore , this region was not normalized . nevertheless , there is a clear absorption which matches in its velocity range the blueshifted absorption seen in figure [ velspec]a . . ( b ) velocity spectra for , ly@xmath197 , and ly@xmath198 . airglow and galactic absorption and emission lines are marked with @xmath199 . ( c ) velocity spectra for ly@xmath3 observed with stis on 1998 december 19 ( solid line ) and for ly@xmath195 observed with _ fuse _ on 2001 june 20 ( dotted line ) . [ velspec ] , width=321 ] figures [ velspec]a and [ velspec]b suggest that the intrinsic absorption in mr 2251@xmath0178 is arising in at least 4 absorption systems : one at @xmath0580 ^-1 kms@xmath8 and at least 3 others which are blended together and form a wide trough covering the velocity range 0 to @xmath0500 ^-1 kms@xmath8 . the 3 centroid velocities in the trough are at about @xmath200 , and @xmath201 ^-1 kms@xmath8 . uv spectra of mr 2251@xmath0178 were taken by _ hst _ at three epochs ( monier et al . 2001 and references therein ) : 1996 august 2 with the fos , 1998 december 19 and 2000 november 5 with stis . to produce the velocity spectrum of the ly@xmath3 absorption line , we fitted the 1998 stis spectrum with a spline curve and divided the observed spectrum by this continuum . the ly@xmath3 absorption line shows a similar profile to the ly@xmath195 profile observed with _ fuse _ 2.5 years later ( see figure [ velspec]c ) . the similarity is mainly in the center of the lines while the later spectrum suggesting the absorption got narrower ( though this could be an artifact of the continuum fitting ) . the similarity in the lines centers indicates they are saturated . however , the lines are not completely black indicating the absorber does not completely cover the continuum source . the line profile suggests a covering factor of @xmath4590% for the absorbing material . this value is consistent with the covering factor fitted to the x - ray data . the exception is the narrow trough at @xmath202 which has a different depth in the two lines . one possibility is that , in this system , the lines are not saturated . the alternative explanation is that the system is saturated but it lies under the blue wing of the main broad absorption trough and this broader absorption is not saturated , resulting in a different depth to the blue wing . ganguly , charlton , & eracleous ( 2001 ) suggested that the doublet absorption line that was detected in the fos observation , in 1996 with ew of [email protected] , was not detected in the stis spectrum taken 4 years later ( down to a 3@xmath203 limit corresponding to ew=0.19 ) . the absorption system detected in the fos spectrum has a width of @xmath204 ^-1 kms@xmath8 ( monier et al . 2001 ) , consistent with the large trough seen in the _ fuse _ spectrum . the resolutions of the two spectra are too poor ( 230 and 600 ^-1 kms@xmath8 for the fos and stis , respectively ) to detect the narrow absorption system at @xmath205 ^-1 kms@xmath8 ( which has a width of order 100 ^-1 kms@xmath8 ) . the new _ xmm - newton _ data presented in this paper clearly show the presence of ionized gas seen in both emission and absorption in mr 2251@xmath0178 . the spectral analysis is severely limited by the poor s / n of the grating observations but several interesting results clearly emerge . in the grating data we identified x - ray emission lines from , , , , and . we also identify , with high certainty , absorption lines from the low ionization ions of , , and , as well as the signature of absorption edges due to and . many other absorption lines are probably detected ( table [ linetable ] ) but their reality and intensities are highly uncertain , because of the limited s / n . for the 2002 rgs data we suggest one of two possible models . the first model consists of two absorbers : a highly ionized absorber with a column density of @xmath206 @xmath6 and log(@xmath138 , and a low ionization absorber with a column density of @xmath7 @xmath6 and log(@xmath207 . the second possibility is a three - component model where we split the highly ionized absorber from the above model into two components : one with log(@xmath208 and the other with log(@xmath209 . the ionized ( line - of - sight ) absorber and the ionized emitter in both cases are consistent with being the same gas with a global covering factor of 0.4 . the highly ionized absorption lines are probably narrower than 200 ^-1 kms@xmath8 , a limit which is imposed by the equivalent width of the strongest predicted lines , given the column density , the ionization parameter , and the s / n of the observations . 1 . all x - ray observations are consistent with a two - component continuum : a high energy power law of slope @xmath210 and a low energy soft excess component with @xmath211 . both components are absorbed by the wa and by the intrinsic neutral gas . the wa observed during the 6 weeks of _ asca _ observations in 1993 is consistent with being a single absorber with a column density of @xmath4 . less conclusive results are obtained for the short timescales behavior due to the poor s / n . the data are consistent with a scenario in which the decrease in flux caused a corresponding decrease in the ionization parameter . such a behavior has been suggested in the past for several other sources ( e.g. , mcg@xmath06@xmath030@xmath212 the _ asca _ observation of otani et al . 1996 ; ngc3516 the _ chandra _ observation of netzer et al . this interpretation is not unique and the data also supports a more complex case where the source luminosity is not simply correlated with the ionization parameter ( e.g. , mcg@xmath06@xmath030@xmath212 orr et al . 1997 ; ngc3783 behar et al . 2003 ; netzer et al . 2003 ) 3 . on timescales of years , the wa properties are different and our model requires that the absorbing gas properties are changing in time . for example , the _ asca _ 1996 observations clearly indicate a larger column density ( @xmath5 vs. @xmath4 ) and a smaller ionization parameter ( log@xmath213 vs. @xmath214 ) absorber compared with the one observed in 1993 . this could indicate new material entering our line - of - sight , between 1993 and 1996 , adding to or replacing the earlier gas . the two groups of observations taken in 1996 , that are separated by 5 months , are also not consistent with the notion of having the same wa . comparing these two periods we find that the luminosity is about the same while the ionization parameter dropped by about a factor of 3 . we suggest , again , a physical motion of the gas which resulted in a higher column density of material at about the same distance . thus , a real change in the absorber properties can take place over time scales of only a few months . + the comparison of the 1993 _ asca _ observations and the 2002 _ xmm - newton _ observation suggest a different change . in 2002 , the agn flux is smaller by a factor of 2 compared with 1993 , yet the column density and the derived ionization parameter are about the same as in 1993 . similarly , a comparison between the two _ xmm - newton _ observations indicates that the source luminosity decreased significantly from 2000 to 2002 yet the derived wa properties remained about the same . this might mean that the absorbing material properties have changed between the two epochs ( the sed in both is very similar but the luminosity decrease between 2000 and 2002 was not accompanied by a corresponding decrease in ionization parameter ) . an alternative explanation is that the absorbing material is very far from the central source and of low enough density such that it did not respond to the continuum luminosity variations . the soft excess continuum luminosity is positively correlate with the hard continuum luminosity . the overall picture which emerges from this study is of a changing absorber made of material that enters and disappear from the line - of - sight on timescales of several months . on shorter timescales , of several weeks , the models are consistent with a picture in which the absorbing material responds instantly to the continuum luminosity variations . due to the data quality and the model complexity ( two power laws and several absorbers ) we can not unambiguously determine those properties . the 2002 _ xmm - newton _ observation show the presence of a low - ionization / neutral absorber intrinsic to mr 2251@xmath0178 . we derived the column density of this gas ( @xmath7 ) from the shape of the soft x - ray continuum below 0.6 kev . the rgs data show evidence for and absorption lines which allow us to constrain its level of ionization ( log@xmath215 ) . macchetto et al . ( 1990 ) find evidence for circumnuclear gas on distances between 3 and 6 kpc and of gaseous filaments farther out at distances of 3050 kpc . their lower limits on the [ ] density ( e.g. , their table 4 ) implies a column densities of @xmath216 . this is entirely consistent with the properties of the low ionization absorber found in our analysis . thus , it is possible that the same gas responsible for the [ ] emission is detected in absorption via and x - ray absorption lines . this means that parts of the emission line nebula observed in mr 2251@xmath0178 lie in our line of sight to the central source . this paper presents the first fuv spectrum of mr 2251@xmath0178 . we detect emission from , , and . we also detect at least 4 absorption systems in , , and , three of which are blended together . the three blended systems are best seen in ( figure [ velspec]b ) and are definitely suggested in the other absorption lines ( the main absorption system at @xmath217 ^ -1 kms@xmath8 looks like 3 systems blended together , see figure [ velspec]a ) . the lyman absorption lines are seen all the way to the lyman edge ( figure [ fusespec ] ) . the lower series lines have similar ews which suggests saturation . however , the lines are not completely black which means incomplete line of sight coverage . figure [ velspec]a demonstrate that the absorption system with the largest blueshift might not be saturated since the depth of the 1038 line is about half the depth of 1032 , as expected from their oscillator strength ratios . on the other hand , the three blended absorption systems have the same depth in both lines of the doublet , indicating that these systems are saturated . the similarity of the and the absorption profiles suggests that all blended lines of the two ions are saturated and have the same covering fraction . our spectral analysis indicates a covering factor between @xmath4560% and @xmath4590% for these systems . the total depth in all the fuv and uv absorption lines is larger than the underlying continuum , thus the broad emission lines are absorbed by the uv absorber . this indicates that the uv absorber lies outside of the blr . using mr 2251@xmath0178 continuum flux at 5100 ( @xmath218 ^-2s^-1^-1 erg@xmath6s@xmath8@xmath8 ; bergeron et al . 1983 ) and the relation between the blr size and the object s luminosity ( equation 6 of kaspi et al . 2000 ) we find the blr size to be @xmath219 cm . we take this to be a lower limit on the distance of the uv absorber from the central black hole . several studies suggested a link between the uv and x - ray absorber in agns ( e.g. , mathur et al 1994 ; mathur , elvis , & wilkes 1995 ; shields & hamann 1997 ; crenshaw et al . the data presented in this paper enables us to study this connection in mr 2251@xmath0178 . as explained , the resolution and s / n of the rgs spectrum does not allow the exact measurement of the x - ray absorption systems . we can only confirm that the velocity shifts and the fwhm of the uv absorption systems are consistent with the ones observed in the x - ray absorber . we found that the fuv absorption lines ( in , , and ) are blends of at least 4 absorption systems , all blueshifted with respect to the emission line by 0 to -600 ^-1 kms@xmath8 . the _ hst _ spectra ( [ hst ] ) show a ly@xmath3 absorption line which is consistent with the _ fuse _ absorption lines . it also shows a absorption which has a width of 400 ^-1 kms@xmath8 and about the same blueshift range as seen in the fuv . the resolution of the rgs spectrum at 22 is @xmath45550 ^-1 kms@xmath8hence any absorption lines which are similar in width to the uv absorption lines are predicted to be one pixel wide . in the x - ray spectrum we identify , , and which are consistent with the widths and blueshifts of the uv absorption lines , as well as hints for absorption from . we identify few absorption lines from highly ionized species with lines at wavelength shorter than 19 ( see table [ linetable ] ) . given the poor s / n , this is consistent with the absorption seen in the uv spectra . the consistency between the uv and x - ray absorption suggests that they could arise in the same gas . we have set a lower limit of @xmath219 cm on the distance of the uv absorber from the source using the blr distance . if the uv and x - ray absorptions are the same , this is also a lower limit on the distance of the x - ray absorber . we thank e. behar for helpful discussions . we also thank the anonymous referee for constructive comments . we acknowledge a financial support by the israel science foundation grant no . . h. n. thanks the columbia university astrophysics group for their hospitality and support during part of this investigation . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration ; the abstract service of nasa s astrophysics data system ; data obtained through the heasarc on - line service , provided by nasa / gsfc ; and data from the tartarus database , which is supported by jane turner and kirpal nandra under nasa grants nag5 - 7385 and nag5 - 7067 .	we present the first _ xmm - newton _ observations of the radio - quiet quasar mr 2251@xmath0178 obtained in 2000 and 2002 . the epic - pn spectra show a power - law continuum with a slope of @xmath1 at high energies absorbed by at least two warm absorbers ( was ) intrinsic to the source . the underlying continuum in the earlier observation shows a `` soft excess '' at low x - ray energies which can be modeled as an additional power - law with @xmath2 . the spectra also show a weak narrow iron k@xmath3 emission line . the high - resolution grating spectrum obtained in 2002 shows emission lines from , , , , and , as well as absorption lines from the low - ionization ions of , , and , and other confirmed and suspected weaker absorption lines . the lines are consistent with the properties of the emission line gas observed as extended optical [ ] emission in this source . the signal - to - noise of the 2000 grating data is too low to detect any lines . we suggest a model for the high - resolution spectrum which consist of two or three warm - absorber ( wa ) components . the two - components model has a high - ionization wa with a column density of @xmath4@xmath5 @xmath6 and a low - ionization absorber with a column density of @xmath7 @xmath6 . in the three - components model we add a lower ionization component that produce the observed iron m - shell absorption lines . we investigate the spectral variations in mr 2251@xmath0178 over a period of 8.5 years using data from _ asca _ , _ bepposax _ , and _ xmm - newton_. all x - ray observations can be fitted with the above two power laws and the two absorbers . the observed luminosity variations seems to correlate with variations in the soft x - ray continuum . the 8.5 year history of the source suggests a changing x - ray absorber due to material that enters and disappears from the line - of - sight on timescales of several months . we also present , for the first time , the entire _ fuse _ spectrum of mr 2251@xmath0178 . we detect emission from , , and and at least 4 absorption systems in , , and , one at @xmath0580 ^-1 kms@xmath8and at least 3 others which are blended together and form a wide trough covering the velocity range of 0 to @xmath0500 ^-1 kms@xmath8 . the general characteristics of the uv and x - ray absorbers are consistent with an origin in the same gas .
our understanding of the intergalactic medium ( igm ) has undergone a paradigm shift in recent years , largely as a result of powerful hydrodynamical simulations . when the ly@xmath0 forest was first observed in the late 1960s , the rich field of absorption blueward of the qso s ly@xmath0 emission was interpreted as the detection of discrete intergalactic clouds ( e.g. lynds & stockton 1966 ; lynds 1971 ; sargent et al . 1980 ) . in order to understand the existence of such isolated absorbers , various theories of cloud confinement , including self - gravity ( melott 1980 ) , cold dark matter minihaloes ( rees 1986 ) and the presence of an inter - cloud medium ( e.g. sargent et al . 1980 ; ostriker & ikeuchi 1983 ) were proposed . however , the advent of hydrodynamical simulations , which model the growth of structure in the high redshift universe , provided a significant revision to our picture of the igm ( see the recent review by efstathiou , schaye and theuns 2000 ) . it has been found that in the presence of a uv ionizing background , the ` bottom - up ' hierarchy of structure formation knitted a complex , but smoothly fluctuating ` cosmic web ' in the igm ( e.g. cen et al . 1994 ; hernquist et al . 1996 ; bi & davidsen 1997 ) . the absorption in the ly@xmath0 forest is caused not by individual , confined clouds , but by a gradually varying density field characterized by overdense sheets and filaments and extensive , underdense voids . the advance in theoretical simulations has been matched by increasingly high quality data , as comprehensively reviewed by rauch ( 1998 ) . one of the major discoveries concerning the igm has been the identification of metal absorption lines associated with many of the ly@xmath0 forest clouds ( cowie et al . 1995 ; tytler et al . thus , whilst the ly@xmath0 forest was once thought to be chemically pristine , it has now been well - established that a large fraction of the high column density ly@xmath0 clouds ( @xmath3 ( ) @xmath6 14.5 ) , associated with collapsing , over - dense structures , contain metals ( most notably ) the signature of enrichment by the products of stellar nucleosynthesis ( songaila & cowie 1996 ) . the presence of metals in the ly@xmath0 forest may be reasonably explained either by in - situ enrichment ( local star formation in the cloud itself or in a nearby galaxy ) or by early pre - enrichment by a high redshift episode of population iii stars . whilst the effects of supernova feedback are still not fully understood and therefore the spatial extent of wind - driven ejecta is poorly constrained , enrichment by galactic winds and superbubbles is unlikely to be an efficient way to distribute metals over distances large in comparison with the mean separation between galaxies ( maclow & ferrara 1999 ) . instead , models have appealed to larger scale processes such as merging and turbulent diffusion as the dominant mixing mechanisms ( gnedin & ostriker 1997 ; gnedin 1998 ; ferrara , pettini & shchekinov 2000 ) . such processes would take time to smooth out the metallicity of the igm so that at @xmath7 the metal enrichment is still expected to be very patchy . whilst the deep potential wells of galaxies inhibit efficient , widespread distribution of metals far from their sites of formations , small regions of star formation at high redshift may be able to eject their nucleosynthetic products for more homogeneous mixing ( e.g. nath & trentham 1997 and references therein ) . an episode of population iii star formation may therefore have spread metals far from their sites of formation , seeding ` sterile ' regions of the igm with metals ( ostriker & gnedin 1996 ) . clearly , distinguishing between in - situ and pop iii scenarios has important implications for understanding not only the first generation of stars , but also the mechanisms by which metals are mixed and distributed from their stellar birthplaces . several papers ( cowie & songaila 1998 ; lu et al . 1998 ; ellison et al . 1999a , hereafter paper i ) have addressed this question by attempting to probe the low density regions of the igm where the difference in metallicity predicted by in - situ and pop iii enrichment may be most marked . the detection of @xmath81548,1550 associated with low column density ly@xmath0 clouds ( log @xmath3 ( ) @xmath9 14.0 ) is observationally challenging , even with the capabilities of keck , due to the extreme weakness of the metal lines . analysis techniques have therefore been developed to effectively enhance the sensitivity of the data beyond the normal equivalent width limits of the spectra . two in particular have received recent attention , namely the production of a stacked spectrum ( lu et al . 1998 ) and the use of individual pixel optical depths of ly @xmath0 and ( cowie & songaila 1998 ) . in paper i we attempted to reconcile the apparently conflicting results obtained from these two techniques with an analysis of a very high signal - to - noise ratio ( s / n ) spectrum of the ultra - luminous bal quasar apm 08279 + 5255 . rigorous testing of the analysis procedures revealed that both methods suffered from hitherto unrecognised limitations and it was concluded that the question of whether or not the low density regions of the igm have been enriched remains unanswered . in order to probe deeper into the low density igm , we have obtained an exceptionally high s / n spectrum of the well - known lensed qso , q1422 + 231 , using the hires instrument ( vogt 1994 ) on the kecki telescope . although this qso has been well studied in the past ( e.g. songaila & cowie 1996 ; songaila 1998 ) , we have roughly doubled the exposure time of earlier spectra , allowing us to probe the metallicity of the ly@xmath0forest to more sensitive levels than has previously been achieved . the spectrum is much better suited to the present work than the the much more complex bal quasar apm 08279 + 5255 ( ellison et al . we present a careful and extensive analysis of the systems in order to determine the extent of metal enrichment in the igm . this paper is organised as follows . in 2 we describe the observations , the data reduction procedures and the voigt profile fitting process used to determine column densities . we briefly discuss in 3 the suitability of q1422 + 231 for this work and define the redshift interval over which we will perform the analysis . in 4 , we determine the column densities of the 34 detected absorption systems in the spectrum and investigate the column density distribution of these absorbers . finally , we critically assess the two methods of analysis , described in sections [ stacksec ] and [ tausec ] , which have been developed to probe low column density ly@xmath0 clouds to very sensitive levels . we utilize a suite of simulation techniques to fully test these methods and quantify the potential inaccuracies in our analysis . we adopt @xmath10 = 1.0 throughout . observations of q1422 + 231 were made with the hires spectrograph ( vogt 1994 ) on the kecki telescope in february 1999 , using the @xmath11 arcsec slit . the resultant resolution is @xmath12 . individual exposure times varied between 30 min and 40 min , for a total exposure of 630 min in two ( overlapping ) grating settings to provide total wavelength coverage between the quasar s ly@xmath0 and emission lines . the data were reduced as described in songaila ( 1998 ) and added to the spectrum of q1422 + 231 obtained under similar conditions by songaila and cowie ( 1996 ) . the total integration time for the two datasets is 1130 minutes and has a s / n of 200 300 redward of the qso s ly@xmath0 emission , a quality superior to all previously published data . after extraction and sky subtraction , we noted that the cores of the saturated absorption lines contain a very small systematic residual flux . this error in the zero level is possibly due to light from a foreground source or a small underestimate of the background sky . the correction required to bring the line cores to zero was found to vary slightly with wavelength , ranging from 0.8% for the bluest absorption lines ( @xmath13 ) to 0.2% at @xmath14 . no correction could be estimated redward of the qso s ly@xmath0 emission since the absence of saturated lines gave us no basis for estimating the adjustment required . however , since the correction factor required appears to diminish with increasing wavelength and is already very small at 5500 , and given that such an adjustment will make little difference to the relatively weak lines studied in the red , we consider any residual eror to be unimportant for the analysis presented here . the continuum fit was achieved using the starlink package dipso with a cubic spline polynomial applied to windows of spectrum judged to be free from absorption . voigt profiles were fitted to the entire normalized ly@xmath0 forest using vpfit ( webb 1987 ) to decompose the complex absorption into individual components defined by a column density ( @xmath3 ( ) ) , a @xmath1-value ( doppler width ) and a redshift . this task , though time consuming , is an important feature of our analysis since synthetic spectra can be simulated based on the line list of voigt profile parameters . for example , additional associated with the fitted can be included to these synthetic spectra according to any desired enrichment recipe . as will be discussed later in this paper , this is an essential step for testing the analysis techniques used here . q1422 + 231 ( @xmath15 ) is a well - studied quasar and actually consists of four closely spaced lensed images with separations of 0.5 1.3 arcsec ; the lensing galaxy is at @xmath16= 0.34 ( patnaik et al . 1992 ; kundic et al 1997 ; tonry 1998 ) . gravitational lenses enhance the emission from high redshift qsos making them more powerful probes of the intergalactic and interstellar medium , but obviously the sight lines will sample different spatial regions of the intervening material . its redshift and luminosity ( v=16.5 ) have made q1422 + 231 an ideal candidate with which to probe intervening material through closely spaced sightlines ( petry et al . 1998 ; rauch et al . 1999 ) . the observations reported here are of the closely spaced a and b components . in our observations the sightlines are unresolved . it is well established that multiple lines of sight through quasar pairs separated by several arcsecs show coherence between ly@xmath0 clouds on scales @xmath6 100 kpc ( e.g. bechtold et al . 1994 ; dinshaw et al . this is consistent with the scenario that has emerged from hydrodynamical simulations that portrays structure in the igm not as discrete localized clouds but as a smoothly fluctuating medium . for metal line systems , whilst there may be slight differences in the individual components that constitute complexes , rauch et al . ( 1998 ) have shown that the total system column density remains largely unchanged over @xmath17 10 kpc . the small linear separations probed by q1422 + 231 ( @xmath18 kpc h@xmath5 for the systems in the @xmath19 range considered here ) are therefore unlikely to give rise to by line of sight differences so large as to compromise our column density determinations . in order to avoid confusion between ly@xmath0 and higher order lyman lines , we restrict our analysis of the ly@xmath0 forest in q1422 + 231 to the interval 4740 @xmath20(ly@xmath0)@xmath21 which corresponds to a redshift range of 2.90 @xmath22 3.54 . the lower limit of this interval is determined by the onset of ly@xmath23 absorption and the upper limit is enforced in order to avoid effects due to quasar proximity ( e.g. lu , wolfe & turnshek 1991 ) and corresponds to a velocity separation of @xmath24 km s@xmath5 relative to the emission redshift , @xmath25 = 3.625 . in order to improve the statistics and s / n of stacked data ( 5 ) , we have , in some of the work presented here , included the spectrum of apm 08279 + 5255 , recently analysed in paper i. apm 08279 + 5255 is an ultra - luminous broad absorption line ( bal ) quasar with a systemic redshift @xmath25 = 3.911 and a broad band magnitude r = 15.2 . the data obtained for this quasar ( as presented in paper i ) are also of excellent quality , though the broad absorption line makes some portions of the spectrum less ideal for this problem than the spectrum of q1422 + 231 , and taken together with q1422 + 231 they represent a premium data set for this work . the wavelength region used for apm 08279 + 5255 is 4995 @xmath20(ly@xmath0)@xmath26 ( 3.11 @xmath27 3.70 ) which in addition to the selection criteria applied to q1422 + 231 , takes into account the bal nature of this quasar . finally , a region contaminated by atmospheric absorption from 6865 @xmath20 ( ) @xmath9 6940 ( corresponding to @xmath28 for ) was excluded in both spectra . q1422 + 231 has been the target of several observing campaigns over the years , which have resulted in spectra of various s / n ratios . whilst the primary motivation for obtaining such a high s / n spectrum was our scientific goal of probing the low column density ly@xmath0 forest , continued focus on this target has produced important results for igm enrichment on a range of column density scales . in this section we present all of the detected systems within the redshift range defined in 3 , several of which have not been detected in previous spectra and we show that the power law column density distribution established for strong absorbers with @xmath3()@xmath6 12.75 ( songaila 1997 ) continues to significantly lower column densities . we detect 34 systems within our defined wavelength interval , 29 of which are associated with saturated ly@xmath0 clouds ( log @xmath3 ( ) @xmath29 14.5 ) . these systems exhibit a wide variety of column densities and complexity ( i.e. number of components ) as can be seen in figure [ civsys ] . the column densities , @xmath1-values and redshifts determined for each component using the voigt profile fitting program vpfit , are presented in table [ civlist ] . examples of systems detected in our spectrum which have not been observed in other published spectra ( songaila & cowie 1996 ; songaila 1998 ) are the weak systems at @xmath30 and 3.518 ( c24 , c25 and c33 in figure [ civsys ] ) . previous studies of absorbers in a variety of quasar sightlines has established a power law column density distribution ( with index @xmath31 , see eqn . 1 ) complete down to log @xmath3 ( ) @xmath32 12.75 at @xmath33 and 12.25 at @xmath34.(songaila 1997 ) . the column density distribution function is defined as @xmath35 where , @xmath36 is the number of systems per column density interval per unit redshift path . the redshift path ( used instead of @xmath19 in order to account for co - moving distances ) is given by @xmath37 $ ] for our adopted cosmology . an important question is how much of the iceberg has been exposed ? to what limit does this power law distribution continue ? as spectral data have improved , early determinations of the column density distribution of absorbers have been shown to be incomplete at low column densities ( compare for example petitjean & bergeron 1994 and songaila 1997 ) . the superb quality of this single spectrum can address whether the established power law continues to lower @xmath3 ( ) , in which case the apparent fall - off towards low @xmath3 ( ) seen previously is due to incompleteness , or whether there is a real turnover in number density . in figure [ f_n ] we show the column density distribution of derived from the systems in q1422 + 231 , assumed to be a power law of the form of eqn . 1 . a maximum likelihood fit to the data ( binned in figure [ f_n ] for display purposes only ) gives a power law index of @xmath2 , consistent with other recent estimates . this high quality spectrum , however , clearly uncovers more of the ` iceberg ' than previous studies and the power law continues down to log @xmath3 ( ) @xmath17 12.3 ( figure [ f_n ] , solid points ) . below this column density , @xmath36 shows an apparent departure from the power law which may be due to incompleteness or alternatively reflect a real turnover in the @xmath3 ( ) distribution . the formal 5@xmath38 detection limit for in our spectrum is log @xmath3 ( ) = 11.6 , for the median @xmath1-value of the observed absorption lines , @xmath39 km s@xmath5 . this detection limit is for _ one line _ and is based on the @xmath401548 doublet component . however , identification of a suspected system is dependent on confirmation from the weaker @xmath401550 line whose oscillator strength is only half of the @xmath41-value of the 1548 line . this effectively reduces our sensitivity for detecting _ systems _ by a factor of two to a 5@xmath38 detection limit of log @xmath3 ( ) = 11.9 . moreover , lines with @xmath1-values significantly larger than the median @xmath1=13 km s@xmath5 , may not be detected . in order to estimate the incompleteness for systems with log @xmath3 ( ) @xmath9 12.3 due to large @xmath1 , 40 doublets were simulated for the two bins in figure [ f_n ] which show a departure from the established power law , i.e. @xmath3 ( ) = 12.05 and @xmath3 ( ) = 11.75 . the @xmath1-value for each line was drawn at random from the real distribution of doppler widths and noise was added at the appropriate level . each simulated line was then inspected to assess whether it would have been identified in the original spectrum so that a correction factor could be determined to estimate incompleteness . for @xmath3 ( ) = 12.05 a correction factor of 2.4 was determined ( 17/40 test systems detected in the incompleteness trial ) , with the largest @xmath1-value in the lines detected being 13 km s@xmath5 . in the lowest column density bin , @xmath3 ( ) = 11.75 only 9 out of 40 lines ( with @xmath42 8 km s@xmath5 ) were identified , corresponding to a correction factor of 4.4 . clearly , these correction factors assume that the @xmath1-value distribution does not significantly change with decreasing column density . the two data points adjusted for incompleteness are represented by open circles in figure [ f_n ] . thus , it appears that the power law distribution of column densities continues down to at least log @xmath3 ( ) = 11.75 . recently , hydrodynamical simulations have been used to predict the expected scatter in individual / ratios within strong ly@xmath0 absorbers for a fixed [ c / h ] ( hellsten et al . 1997 ; rauch , haehnelt & steinmetz 1997 , dav et al . 1998 ) . when compared with observations , these models show that the data are consistent with a mean igm metallicity [ c / h ] = @xmath43 with variations of up to a factor of 10 around this average value . in these models , the systems associated with high column density ly@xmath0 clouds are dominated by metals from in - situ star formation , since simulations show that these metal enriched clouds are found within a few tens of kpc from collapsed , dense clumps at @xmath7 ( e.g. haehnelt 1998 and references therein ) . we would then expect the metallicity of such clouds to be variable , being dependent on the local star formation history , and therefore the scatter amongst individual / ratios at a given @xmath3 ( ) to be larger than that predicted for a homogeneous [ c / h ] . we can calculate / ratios for 19 systems in our spectrum of q1422 + 231 . of the 34 detected systems in the spectrum presented here , 29 are associated with saturated ly@xmath0 lines . although it is not possible to determine accurate column densities from these lines alone , for 14 of the systems higher order lyman lines are both accessible and suitably uncontaminated by blends so that an accurate @xmath3 ( ) can in fact be derived . in addition , 5 of the systems in q1422 + 231 are associated with clouds with log @xmath3 ( ) @xmath9 14.5 which are not saturated so that the column densities can be determined from the voigt profile fit of ly@xmath0only . the and column densities determined for these 19 systems are presented in table [ hifits ] . dav et al . ( 1998 ) have also extensively studied the / ratios of q1422 + 231 , as determined from the spectrum of songaila & cowie ( 1996 ) . they constructed a mock spectrum of q1422 + 231 from hydrodynamic simulations for comparison with the data . having both simulated and observed spectra at their disposal , they measured / ratios in both datasets in a _ consistent manner _ ( using the autovp voigt profile fitter , dav et al . 1997 ) , and found that an intrinsic scatter of approximately 0.5 dex in metallicity is required to fit the data . however , the observed range of / ratios is the result of many complex effects , which include not only spatial variations in the temperature - density relationship , but also , for example , fluctuations in the ionizing background . these complex effects have not yet been fully incorporated into models and therefore we should be mindful that the scatter in measured values of / is an upper limit to the variations in [ c / h ] when compared with homogeneously enriched simulations with a uniform ionizing background . dav et al . ( 1998 ) also found that the most robust diagnostic for determining the mean carbon abundance in detected systems is @xmath44log @xmath3()@xmath45 , although this statistic is clearly dependent on the sensitivity of the data . as a consistency check , we determine the @xmath44log @xmath3()@xmath45 for those systems identified in our spectrum of q1422 + 231 whose column densities are above the detection limit of the songaila & cowie ( 1996 ) spectrum ( estimated to be log @xmath3 ( @xmath6 12.0 ) . we calculate @xmath44log @xmath3()@xmath45 = 12.77 which compares well with the value of 12.72 determined by dav et al . ( 1998 ) , considering the errors associated with column density determinations of individual systems . this demonstrates that the different line finding and fitting procedures used on different spectra reproduce the same answer , even when the same systems are observed with a higher s / n . for voigt profile fitting , this is an important point to make in a process that is sometimes not clear - cut . it is also an illustration that the detection limit of the spectrum used by dav et al . ( 1998 ) was well determined and did not suffer from serious incompleteness . ideally , one would like to see how the mean [ c / h ] varies as we detect progressively weaker systems . however , a simple extrapolation to lower column densities is not possible , since the relation between @xmath44log @xmath3()@xmath45 and [ c / h ] presented in dav et al . ( 1998 ) is tailored specifically for the detection limits of their data . in order to apply this technique to the full sample of systems detected in the new q1422 + 231 spectrum presented here , it would be necessary to re - create a model spectrum to match our data . however , we remind the reader that the caveats mentioned in relation to the scatter of / are also applicable here , a fact of which one should be aware when comparing observational and simulated results and interpreting the conversion to metallicity . we can summarise the results from this section with the following conclusions . by obtaining an ultra - high s / n spectrum of the @xmath46 quasar q1422 + 231 , we have detected 34 absorption systems with 2.91 @xmath22 3.54 . some hitherto undetected weak systems are reported whose column densities are consistent with the established @xmath36 distribution showing that this power law ( @xmath2 ) continues down to at least log @xmath3 ( ) = 11.75 , a factor of 10 deeper than the previous determination at @xmath33 in songaila ( 1997 ) . by fitting down the lyman series , we are able to determine accurate / ratios for 19 of the identified systems , rather than relying on a statistical estimate of the median metallicity as has often been done . dav et al . ( 1998 ) have previously determined that the scatter in / in the q1422 + 231 spectrum of songaila & cowie ( 1996 ) requires an intrinsic scatter in the [ c / h ] of a factor of @xmath17 3 . finally , by considering only the systems above the detection limit , we obtain the same @xmath44log @xmath3()@xmath45 as dav et al . ( 1998 ) , a statistic used to infer that the mean [ c / h ] = @xmath43 . however , we stress that these simulations do not take into account the full complexity of the physical processes determining the / ratio in the igm so that both the mean [ c / h ] and its scatter may change with future more detailed work . as discussed in previous sections , the / ratio in low @xmath3 ( ) systems may hold the key to understanding metal enrichment mechanisms in the igm . since one of the major limitations in detecting weak lines is s / n , stacking many sections of the spectrum is one way to circumvent this problem ( e.g. norris , peterson & hartwick 1983 ; tytler et al . 1995 ) . in our application of this technique , we select ly@xmath0 lines with 13.5 @xmath9 log @xmath3 ( ) @xmath9 14.0 whose corresponding spectrum shows no obvious metal lines or contamination from other absorption features . each section is de - redshifted to the rest frame , re - binned to the dispersion of the lowest @xmath19 system , weighted according to its s / n , stacked and finally re - normalized . the optimal weighting used to co - add the sections is given by @xmath47 where @xmath48 is the variance of the data . to achieve maximum sensitivity , we use low column density lines from both the apm 08279 + 5255 and q1422 + 231 spectra . within the ranges defined in 3 , a total of 67 low @xmath3 ( ) lines were identified in the two qso sightlines . _ the resulting composite spectrum has a s / n = 1250 and shows no absorption at the rest wavelength of , @xmath49 _ , as can be seen from the top panel of figure [ data_stack ] . in order to place a significance limit on this non - detection , a synthetic spectrum was created , re - producing the ly@xmath0 forest of the 2 quasars from the fitted line lists . was included for lines with log @xmath3 ( ) @xmath9 14.5 assuming a fixed / ratio and @xmath1 ( ) = @xmath50(ly@xmath0 ) ( representing a combination of thermal and bulk motion , as found in paper i ) . figure [ data_stack ] shows the results of stacking synthetic spectra with log / = @xmath51 . the resultant absorption in this stack has an equivalent width of 0.15 m and represents a 4@xmath38 feature which we therefore adopt as the detection limit for our stacked data . several analyses ( e.g. songaila & cowie 1996 ; paper i ) have determined the / ratio in high column density ly@xmath0 clouds to be in the range from log / = @xmath52 to @xmath53 . the detection limit of the synthetic stack is almost factor of two lower than the metal - poor limit of this range , and thus it would appear to indicate a drop in the / ratio at lower column densities . it must be understood , however , that this technique has several potential problems which may compromise its efficiency in detecting weak absorption features . we now discuss these problems in turn . * in order to co - add each section , the data must be re - binned , usually to the dispersion of the lowest redshift system . this could smooth out a weak feature , although the scale of smoothing is very small compared with the width of the expected line so that this is not likely to be a major effect . * even for a fixed carbon abundance there will be a scatter in the values of / ( as discussed in the previous section ) . therefore , if absorption is detected in the composite spectrum it will be averaged over a range of column densities that can not be recovered individually . moreover , depending on the scatter of / values , the absorption may be dominated by the strong tail end of this metallicity range . in fact , it is conceivable that residual absorption could be caused by only a few relatively strong lines , since this method relies on an average which is very sensitive to a non - gaussian tail . interpreting a residual signal in the composite spectrum is therefore not straightforward , although in the analysis presented here we find that there is no absorption in the stacked data and so we can determine a useful detection limit . more serious problems for the present analysis are : * the sensitivity of this method to errors in continuum fitting , anomalous pixels and other forms of contamination . the usual procedure is to visually inspect each section before adding it to the stack in order to ensure that it is ` clean ' . this will filter out major contamination by , for example , uncorrected cosmic ray events or absorption due to systems other than . however , small errors in the continuum fit or deviant pixels could seriously compromise the efficiency of the stack . in addition , a re - normalization of the composite spectrum is usually required , since the small errors in the original continuum fit have now been compounded by stacking . * stacking the individual sections of spectrum in order to build up a signal is pivotal upon centering the absorption feature in the composite . if there is a significant error in the stack center , i.e. if an offset exists between the redshift of the parent ly@xmath0 line and its associated complex , the composite signal would be smeared out . depending on the magnitude of the offset , this effect may lead us to under - estimate the amount of absorbing material . this problem could be exacerbated by the afore - mentioned need for a re - normalization because it may be difficult , if not impossible , to distinguish a weak smeared signal from the compound continuum errors . in paper i , we showed that there is indeed a random redshift offset ( @xmath54 ) between the measured position of the ly@xmath0 and corresponding lines . in that work it was found that the redshift offset had a distribution with @xmath55 4 @xmath56 equivalent to a velocity difference of 27 km s@xmath5 , although this statistic was based on a relatively small number of systems . the addition of a second quasar to the analysis has improved the statistics and for a total of 56 systems in the two quasars we now determine a @xmath57=2.6 @xmath56 ( @xmath5817 km s@xmath5 for an average redshift of 3.45 ) . this redshift offset was determined in two different ways . in both cases the redshift of the was estimated by taking the centroid of the system since it is often complex , consisting of several blended components . for most of the corresponding ly@xmath0 lines , the absorption is well represented by a single component and so in the first instance the redshift of the cloud was obtained from the voigt profile fit . for comparison , the redshift of ly@xmath0 was also determined using the same centroid method applied to the systems and it was found that both determinations yielded almost identical results for the distribution of @xmath54 , shown in figure [ offsets ] . since it is the strongest component in each complex that first emerges from the noise in the weak systems , this will be the feature enhanced by the stacking procedure . therefore , we also investigated the distribution of offsets between the fitted @xmath19 of ly@xmath0 and the redshift of the deepest trough and once again the value @xmath58 17 km s@xmath5 was found . we investigated how seriously the composite spectrum would be affected by this offset which effectively shifts the expected positions of the lines by a random amount . again , synthetic spectra were produced , this time including a redshift offset applied to the position of the line with @xmath54 drawn at random from a gaussian distribution with @xmath57=2.6@xmath59 . we find that in order to reproduce a 4@xmath38 detection in the presence of a redshift offset , twice as much must be included in the ly@xmath0 forest clouds , i.e. log ( / ) = @xmath60 , see the bottom panel of figure [ data_stack ] . this is consistent with the measured / in log @xmath3()@xmath614.5 lines and is therefore not a sufficiently sensitive limit to establish whether the low column density ly@xmath0 clouds contain significantly less that the stronger lines . we conclude that a factor of at least two improvement in sensitivity is required in order to show conclusively whether the low column density absorbers are more metal deficient than their high column density counterparts . alternatively , it must be shown that the redshift offset in these lines is @xmath61 17 km s@xmath5 and dilution of absorption from smearing the stack is unimportant . the redshift offset in saturated lines could have two possible explanations , one physical and the other an observational artefact . there could be an intrinsic redshift difference between the and ly@xmath0 absorbers , caused by , for example , ionization effects or outflows . an additional effect could be a redshift offset caused by the blending of strong ly@xmath0 lines which , when saturated , can not be distinguished into separate components . two examples of this are shown in figure [ lya_breakdown ] . plotted in velocity space are two examples of ly@xmath0 absorbers ( black solid line ) and corresponding ( dashed line ) . both saturated ( systems ` b ' and ` c ' ) and weak ( systems ` a ' and ` d ' ) ly@xmath0 clouds are shown and in the top panel the ly@xmath23 ( in gray ) is also included . the system ` c ' is associated with an apparently monolithic ly@xmath0 absorber centered at @xmath62 = 0 and exhibits a redshift offset of approximately 25 km s@xmath5 . however , as seen from the ly@xmath23 absorption , this system is made up of more than one component and the clearly has a redshift that more closely matches the strongest of these . the unsaturated ly@xmath0 absorber associated with the system ` a ' does not break down into a multi - component system in ly@xmath23 and the redshift offset is correspondingly small ( @xmath63 1 km s@xmath5 ) . however , there are other examples of systems associated with weak ly@xmath0 that _ do _ show a significant offset , for example system ` d ' in the lower panel of figure [ lya_breakdown ] . unfortunately , higher order lyman lines are not available for this particular case . we re - measured the redshift offset for each of the 19 systems in table [ hifits ] for which an accurate @xmath3 ( ) had been obtained by tracing down the lyman series . for 16 of these systems , the appeared to be associated with a single component ( i.e. not obviously blended ) , as seen from ly@xmath23 and/or ly@xmath64 . the offsets associated with the sub - cloud whose redshift most closely matches that of the are shown in the lower panel of figure [ offsets ] . given the small number of systems for which the lyman series can be traced , at least to ly@xmath23 , the statistics of the offset distribution are not very meaningful . however , the scatter is now clearly smaller , although the high @xmath3 ( ) ly@xmath0 lines at the top end of the column density distribution may still have several sub - components that are unresolved in our data and the offset may be further reduced if the lyman series could be traced down to subsequent transitions . we investigated a possible solution to the problem of an unknown offset from the predicted position of systems discussed above . the technique involves scanning for the maximum absorption around the predicted position of absorption and re - centering the stack at this wavelength . in practice , each ( de - redshifted ) data section is scanned @xmath65 from the predicted line center ( i.e. for @xmath54 = 0 ) and the section re - centered on the pixel with the maximum optical depth ( @xmath66 ) prior to stacking . figure [ preshift ] shows an example of how this technique is clearly able to improve upon a direct stack in the presence of a @xmath54 for a synthetic spectrum containing lines with log / = @xmath672.0 . it can be seen that without re - centering , whilst the overall level of the continuum is below unity for the ` smeared ' stack , all profile information is lost such a broad depression may be mistaken for compound errors in the continuum level and lost in the subsequent renormalization of the stacked spectrum . executing a re - center on the @xmath66 pixel before stacking recovers almost all of the original signal in this simulation , the characteristic line profile is prominent and will not be lost when the post - stacking continuum is re - fitted . however , as one attempts to detect progressively weaker absorption , the likelihood of centering on a noise pixel becomes higher and this technique is no longer efficient . in order to determine whether a ` re - center ' is a viable improvement to the stacking method , given the s / n of the data and the low column density of the targeted features , we performed a feasibility study in which test absorption lines were created over a range of / ratios . the results of this study are shown in figure [ test_shift ] where we plot the distribution of @xmath57 caused by line center misidentification ( i.e. where @xmath66 is a noise feature rather than line trough ) as a function of / ratio . clearly , when the @xmath38 of the offset caused by trough misidentification exceeds the value of the intrinsic offset we are trying to overcome , this re - centering technique is no longer useful . however , these are conservative estimates for how well the centering would perform on real data , since we would expect some trough misidentification from contaminating ( i.e. anything but ) lines in addition to the effects of noise investigated here . since we need to determine whether the low column density ly@xmath0 lines have the same metallicity as their high @xmath3 ( ) counterparts , we must ideally reach levels of sensitivity deeper than log /=@xmath52 . figure [ test_shift ] shows that for log / @xmath682.6 , this technique no longer compensates adequately for the intrinsic offset and is therefore unable to improve the stacking technique in the search for weak lines . smoothing the spectrum over several pixels before locating @xmath66 improves the pre - shifting slightly as shown by the gray squares in figure [ test_shift ] . however , even with smoothing , this re - centering procedure can not compensate for a @xmath57 = 2.6 @xmath56 below a log / = @xmath69 . in summary , by stacking together the regions associated with low column density ly@xmath0 lines in q1422 + 231 and apm 08279 + 5255 , we have obtained a s / n = 1250 composite spectrum which shows no residual absorption . using simulated spectra we find that for log / = @xmath51 in the weak ly@xmath0 lines we would expect a 4@xmath38 detection of the composite absorption . the shortcomings of this technique are discussed and we focus on an observed redshift offset between the position of ly@xmath0 and its associated . the redshift offset determined from the detected systems has a dispersion @xmath57 = 2.6@xmath70 which corresponds to a @xmath4 = 17 km s@xmath5 . we have investigated how this offset would affect the stacking procedure by using simulated spectra and find that a @xmath4 = 17 km s@xmath5 will reduce the sensitivity of this method by a factor of two and that a metallicity of log / = @xmath60 is now required to achieve a 4@xmath38 detection . we are unable to know whether the same redshift offset persists in the low @xmath3 ( ) clouds targeted by the stacking technique , but a large range of offsets are measured over the full column density range of detected systems , so one must clearly take into account the possible repercussions when interpreting the stacked data . a potentially more robust way of measuring weak absorption in high s / n spectra is to analyse the optical depths in each ly@xmath0 forest pixel and its corresponding pixel ( cowie & songaila 1998 ) . this technique also has the advantage that by tracing down the lyman series , one can determine the / ratio in each pixel over a very large range of @xmath71(ly@xmath0 ) . in paper i we investigated the potential of this method with apm 08279 + 5255 and critically analysed its performance on synthetic spectra . specifically , we investigated the effect of including a redshift offset and how the results depended on the @xmath1-value of . we concluded that , despite the excellent quality of the spectrum , the results from the spectrum of apm 08279 + 5255 were inconclusive and could not determine whether the metallicity of the igm is constant or diminished at low values of , because at the lowest values of @xmath71(ly@xmath0 ) both scenarios were consistent with the observations . the spectrum of q1422 + 231 presented here not only has a significantly higher s / n than that of apm 08279 + 5255 , but has many of its higher order lyman lines accessible for analysis and considerably less contaminating absorption by , for example , systems ( ellison et al . this spectrum therefore represents the best data yet obtained for this analysis of the low column density ly@xmath0 forest . briefly , the optical depth technique consists of stepping through the ly@xmath0 forest measuring @xmath71(ly@xmath0 ) for each pixel . the noise ( @xmath38 ) array is used to determine which pixels are included in the analysis via a series of optical depth criteria to account for effects such as saturation . using the values in the noise arrays rather than fixing the rejection criteria provides maximum flexibility for this technique so that it can be readily applied to spectra of different s / n ratios . for pixels with a residual flux ( @xmath72 for a normalised spectrum ) @xmath73 above the zero level , we trace down the lyman series since there is too little residual flux in these saturated pixels to determine an accurate optical depth from ly@xmath0 alone . however , the danger here is that higher order lines may be contaminated by lower redshift ly@xmath0 . we therefore use @xmath71(ly@xmath0 ) = minimum(@xmath71(ly@xmath74)@xmath75 / @xmath76 ) over all observed higher order lines using the higher order pixel if @xmath77(ly@xmath78 . this minimizes the effect of contamination and maximizes the number of usable pixels and range of @xmath71(ly@xmath0 ) which can be considered for analysis . if no @xmath77(ly@xmath79 pixels are found , then the pixel is discarded . the position of the associated @xmath81548 , 1550 lines are calculated and , again to avoid contamination , we use @xmath71(1548)= minimum(@xmath80,@xmath81(1550))-values ( oscillator strengths ) in the doublet is 2:1 ] if the flux in the 1550 component @xmath82 , otherwise only @xmath71(1548 ) is considered . the optical depths are then binned according to their corresponding @xmath71(ly@xmath0 ) and the median determined for each interval . taking the median of a large number of pixel optical depths not only provides a statistical advantage over considering a relatively small number of lines ( as for the stacking method ) , but also is much less susceptible to non - gaussian effects . in order to estimate 1 @xmath38 errors for the optical depth determinations , we used bootstrap re - sampling with 2sections of the ly@xmath0 forest and corresponding , i.e. we drew @xmath74 random sections from the complete set of @xmath74 sections ( in this case @xmath83 ) that comprise the original data , with replacement . this procedure was repeated 250 times and the 1@xmath38 error was taken to be the dispersion of these 250 realizations . figure [ 1422_tau ] shows the results obtained for the optical depth analysis of q1422 + 231 . the shape of the optical depth distribution appears consistent with a constant level of / ( i.e. parallel to the dashed line ) over optical depths from @xmath71(ly@xmath0 ) @xmath17 100 down to @xmath17 2 3 , below which @xmath71 ( ) flattens off to an approximately constant value . each of the low optical depth bins ( @xmath71(ly@xmath0 ) @xmath9 3 ) contains approximately 10% of the pixels . this percentage decreases with increasing @xmath71(ly@xmath0 ) with only approximately 2% of pixels in the @xmath71(ly@xmath0 ) = 10 bin . in addition , we determine optical depths in pixel pairs separated by the @xmath81548 , 1550 doublet ratio _ regardless of @xmath71(ly@xmath0 ) _ over the entire range considered for absorption . this is done using the same method of doublet comparison as before to eliminate contamination . the median of this reference distribution is plotted in figure [ 1422_tau ] as a dotted line and represents the median absorption for an effectively random set of pixel pairs separated by doublet ratio and will include the effects of noise and low level contamination expected to affect our results . this median optical depth is higher than the observed @xmath71 ( ) for all bins with @xmath71(ly@xmath0)@xmath84 1 . this is a significant result which indicates that the absorption for these optical depths is less than that expected by selecting random pixel pairs . the key question here is whether the signal in the low optical depth pixels is due to absorption in low density regions of the igm or whether the flattening of the data is caused by some limiting factor in our analysis such as contamination or noise . as we saw in the previous section , thorough simulations of the analysis technique are vital for interpreting our results . here , we have taken the ly@xmath0 forest directly from the data and artificially enriched it with to test our methods and address several questions . in paper i we explored the effect of @xmath1-values , redshift offset and noise on synthetic spectra in order to determine whether a break in the / distribution could be distinguished from a constant ratio in the spectrum of apm 08279 + 5255 . to review these findings and to give a visual impression of the optical depth analysis , we show the results of this technique on four synthetic spectra in figure [ 2panel ] and compare them with the results from q1422 + 231 ( solid points ) . the top panel ( ` a ' ) of this figure shows the optical depth analysis of 2 spectra , one of which has a constant log / = @xmath53 in all ly@xmath0 lines ( shown as a dotted line ) and a second spectrum which has log / = @xmath53 only in log @xmath3 ( ) @xmath6 14.5 lines ( dot - dashed line ) , with no noise added to either spectrum redward of ly@xmath0 . the bottom panel shows the same spectra as panel ` a ' except that noise has now been added to both spectra , based on the error array of the actual data ( typically s / n = 200 redward of ly@xmath0 ) . all four spectra use the real ly@xmath0 forest of q1422 + 231 and have had added to the synthetic spectrum based on the fitted linelist . in addition , all four synthetic spectra in figure [ 2panel ] have a @xmath54 = 2.6 @xmath59 and @xmath1 ( ) = @xmath50(ly@xmath0 ) . the dashed line indicates constant log / = @xmath53 . there are several points to note here . firstly , in the absence of noise , there is a clear drop in @xmath71 ( ) below @xmath71(ly@xmath0 ) @xmath85 if no is added in ly@xmath0 lines with log @xmath3 ( ) @xmath9 14.5 . however , this steep decline is much less drastic when noise is included , and although @xmath71 ( ) shows a steady decrease down to @xmath71(ly@xmath0 ) @xmath17 1 , below this value it flattens off to an approximately constant value . the inclusion of noise also causes the same apparent flattening in the spectrum of constant / , although the @xmath71 ( ) in this spectrum is consistently above the value measured for the dot - dashed line . there is also a small contribution to this flattening from line blending which changes the overall slope of the expected / , an effect that is exacerbated for larger @xmath1 ( ) . this shows that even in relatively high s / n spectra , distinguishing a break in the / distribution is very difficult . instead , as we shall see later in this section , one of the main uses of this method is to determine whether the optical depths measured with this technique can be adequately accounted for with the detected systems or whether there must be significant amount of still below our current detection limit . we also note that for large @xmath71(ly@xmath0 ) the measured @xmath71 ( ) is always less than expected from the dashed line , given the input ratio of /. this is due to two effects . firstly , since this technique considers @xmath71(ly@xmath0 ) as the minimum value obtained by tracing down the lyman series when a line is saturated , if contamination is successfully removed we will tend to under - estimate @xmath71(ly@xmath0 ) at these optical depths due to noise ( remember that the real ly@xmath0 forest is used in the simulated spectra so that this will be an effect even in panel ` a ' of figure [ 2panel ] ) . secondly , the that is included in the synthetic spectra is based on a linelist of fitted values which will probably not be accurate for saturated lines . these effects also account for the drop of @xmath71 ( ) at very large @xmath71(ly@xmath0 ) which contain @xmath9 1% of the total pixels and are therefore very uncertain . clearly , the exact results of the optical depth analysis are sensitive to the combination of several factors including blending , @xmath1-values and noise , which we will discuss further later in this section . therefore , rather than attempting to directly fit the observed distribution of optical depths in q1422 + 231 we aim to determine whether the optical depths measured in the spectrum can be explained solely by the relatively strong absorbers detected directly and presented in 4 or whether the results from this analysis are indicative of additional . if the latter is true , are these metals in the low density igm ? we must also investigate whether our results can be explained in terms of contamination , noise or some other limiting factor in the data . finally , we consider the effect of scatter and redshift offset ( @xmath54 ) , which have been found to compromise the efficiency of the stacking method . three synthetic spectra were produced with ly@xmath0 forest absorption taken directly from the data ( i.e. not reconstructed from a linelist ) and the 34 detected systems re - produced from the voigt profile parameters in table [ civlist ] . spectrum ` a ' has had no further metals added and therefore shows the results expected of an optical depth analysis if we had already uncovered all of the in the spectrum . spectra ` b ' and ` c ' have both been enriched with additional metals . in spectrum ` b ' , is included in all strong ( log @xmath3 ( ) @xmath6 14.5 ) lines with @xmath3 ( ) = 12.0 , i.e. the detection limit for directly identifiable systems . this spectrum therefore represents the maximum amount of that could be ` hidden ' in high column density ly@xmath0 lines . in addition to this ` hidden ' in strong lines and the fitted in table [ civlist ] , spectrum ` c ' has been enriched with a constant log / = @xmath53 in weak ly@xmath0 lines ( @xmath3 ( ) @xmath9 14.5 ) . the results from analysis of these three synthetic spectra are compared with the data in figure [ testabc ] . for all three spectra , we include noise taken from the @xmath38 error array , a random redshift offset in the position of ( @xmath86 ) and take @xmath1 ( ) = @xmath87 @xmath1(ly@xmath0 ) . the conclusions that can be drawn from figure [ testabc ] are as follows . first , there is clearly more in the data than we have directly identified in 4 , since the dotted line of the synthetic spectrum in the top panel is well below the solid line at all but the very highest @xmath71(ly@xmath0 ) points . for @xmath71(ly@xmath0 ) @xmath29 3 , the optical depths can be recovered when additional metals are added into strong ly@xmath0absorbers , but below our current detection limit ( spectrum ` b ' ) . this supports the results of 4 where we determine that @xmath36 is consistent with a power law distribution that continues down to log @xmath3 ( ) = 11.75 , with no evidence for a turnover in column density distribution . however , adding at the limit of our detection in log @xmath3()@xmath6 14.5 lines with no extra metals in weaker ly@xmath0 clouds can not reproduce the measured @xmath71 ( ) in @xmath71(ly@xmath0)@xmath9 3 pixels . the results from spectrum ` c ' in the bottom panel of figure [ testabc ] which include log / = @xmath53 in weak ly@xmath0 lines show that the observed optical depths in the low @xmath71(ly@xmath0 ) pixels are consistent with the presence of some metals in significantly lower column density clouds , although for @xmath71(ly@xmath0 ) @xmath6 1 @xmath71 ( ) exceeds the observed value . from figure [ 2panel ] we have also seen that a constant log / = @xmath53 in all ly@xmath0 lines is a good approximation to the data . we stress here that these simulations are not an attempt to fit the observed distribution of optical depths , rather they are tests to determine whether the systems in table [ civlist ] can account for the measured absorption . if not , then the objective is to investigate in which column density regime additional metals could be added in order to achieve the observed quota . the results from these simulations show that the analysis of optical depths in the spectrum of q1422 + 231 is consistent with hitherto undetected in both the strong and weak ly@xmath0 lines . whilst the results from our determination of the column density distribution from detected systems is consistent with a power law function @xmath36 that continues down to at least log @xmath3 ( ) = 11.7 , these optical depth results provide _ direct evidence _ that there are more metals lurking below our current detection limit . we investigate the effect that contaminating lines may have on this result . the analysis may be affected not only by low level contamination from other metal lines such as or telluric absorption ( strong absorbers will have been rejected by the doublet strength comparison discussed in 6.1 ) , but also effects due the non - uniformity of ionizing sources and noise / fluctuations in the continuum level due to fitting errors . this latter effect will cause fluctuations around the true continuum which will be important only if they are greater than the level of the noise . an additional ( possibly systematic ) error may be present if the low order polynomial fit to the continuum is consistently over or under - estimated . to test the continuum fit of the regions , sections of the data redward of ly@xmath0 deemed to contain no obvious absorption were examined . from a total of @xmath17 5000 pixels , a mean flux of 1.0002 and a median of 1.0001 were determined , vaules one order of magnitude lower than @xmath71 ( ) @xmath88 . it therefore seems unlikely that a systematic error in the continuum fit is the cause of the observed flattening of optical depths seen in figure [ 1422_tau ] . the distribution of noise pixels would also suggest that there is no significant effect from weak telluric lines . errors in the continuum fit blueward of ly@xmath0 emission , however , are likely to be a more serious effect , firstly because selection of absorption - free zones is difficult and secondly because the s / n is lower ( 50 150 ) . the result of continuum fitting errors in the forest in this analysis will be to classify pixels with small @xmath71(ly@xmath0 ) into the wrong optical depth bin . this will effectively associate with the wrong @xmath71(ly@xmath0 ) and averaging out the metal absorption by this ` mis - binning ' could explain the observed flattening of the measured optical depths . the effect of a small continuum error is therefore similar to the effect of noise , affecting only those pixels with optical depths smaller than the fitting error . we also find that including a large number of weak contaminating lines such as or could increase the measured @xmath71 ( ) at low @xmath71(ly@xmath0 ) to the constant level determined . however , whilst realistic column densities do have a small effect on the optical depth result , it is unlikely that weak contaminating lines account for all of the observed absorption . as discussed previously , a scatter is expected in the observed / values even if [ c / h ] and the ionizing background remain uniform . to investigate the importance of this effect , we simulated a ` control ' spectrum by re - producing directly the ly@xmath0forest of q1422 + 231 and using @xmath1 ( ) = @xmath50(ly@xmath0 ) and an input log / = @xmath53 with no scatter and no redshift offsets . for comparison , we then produced a second spectrum in the same way , but rather than adopting a constant / , a gaussian scatter was introduced with @xmath38(/ ) = @xmath89 . the comparison between the two simulated spectra is presented in figure [ scatter ] where the ` control ' spectrum is shown as open squares connected with a solid line and the spectrum containing a scatter of metallicities is shown as solid diamonds connected with a dotted line . the points have been offset from one another in the figure for clarity . the points at high @xmath71(ly@xmath0 ) are again very uncertain due to the reasons previously discussed with regards to figure [ 2panel ] . the error bars are estimated using the same bootstrap technique as employed for the data . the results from the 2 spectra are entirely consistent with one another and indicate that a gaussian scatter will therefore not affect the overall median of @xmath71 ( ) . whilst the extent and magnitude of the scatter in the distribution of / values has not yet been fully investigated in simulations down to column densities below log @xmath3 ( ) @xmath17 14.0 13.5 , this result should not be significantly affected unless the scatter becomes highly non - gaussian at low @xmath3 ( ) . also plotted in figure [ scatter ] are the results of the optical depth analysis if a redshift offset is included in the position of the line . as before , metals are added to all ly@xmath0 lines with a metallicity log / = @xmath53 . there is no scatter in these values , but an offset in the position of has been included , drawn at random from a gaussian distribution with @xmath90 ( @xmath4 = 17 km s@xmath5 ) . these points are also consistent with the control spectrum , so we can conclude from this simulation batch that neither scatter in the metallicity nor redshift offset will have a large effect on the outcome of our optical depth analysis . finally we note that more spectra are required in order to provide a representative analysis of the high redshift igm , since our view of the enrichment of the ly@xmath0 forest probed with a single sightline is clearly blinkered . from the list of systems in table [ civlist ] it can be seen that these absorbers are not uniformly distributed in redshift . for example , splitting this spectrum in two by redshift produces vastly different optical depth distributions due to a relative dearth of systems that spans over 300 in this spectrum in the range @xmath91 . with more spectra , there is the possibility of not only developing a more representative study of these high redshift absorbers but also investigating the redshift evolution of systems . in this paper , we have addressed the enrichment history of the igm by studying the ly@xmath0 forest and its associated systems in a very high s / n ( @xmath92 , high resolution ( @xmath17 8 km s@xmath5 ) spectrum of the well - known lensed quasar , q1422 + 231 obtained with keck / hires . the numerous systems associated with high column density ly@xmath0 absorbers are fitted with voigt profiles defined by a redshift , @xmath1-value and column density for each component line . we investigate the column density distribution , @xmath36 , to very sensitive levels and detect several weak systems which had not been previously identified in lower quality spectra of the same object . we determine a power law index @xmath2 which continues down to log @xmath3 ( ) = 12.2 before starting to turnover . by simulating synthetic absorption lines with @xmath1-values taken at random from the observed distribution , we estimate a correction factor to account for incompleteness and find that the corrected data points now indicate that the power law continues down to at least log @xmath3 ( ) = 11.75 , a factor of ten more sensitive than previous measurements ( e.g. songaila 1997 ) . this shows that even at these low column densities there is no evidence for a flattening of the power law and therefore there are probably many more systems that lie below the current detection limit . we investigate two methods with which it may be possible to recover these weak systems . firstly , we select 67 ly@xmath0lines with 13.5 @xmath9 @xmath3 ( ) @xmath9 14.0 in q1422 + 231 and apm 08279 + 5255 and produce a stacked spectrum centered on the predicted position of @xmath401548 . the composite stack has a s / n = 1250 and shows no residual absorption ; we use synthetic stacked spectra to determine a 4@xmath38 upper limit of log / = @xmath51 . we critically assess the accuracy of this method by performing the stacking procedure on a suite of simulated , synthetic spectra and identify several associated problems . we investigate , in particular , the effect of a redshift offset between the position of the ly@xmath0 line and its associated . with improved statistics , we refine the redshift offset determined in paper i and find that a @xmath4=17 km s@xmath5 is present in the systems which we detect directly . by including a random redshift offset drawn from a gaussian distribution with @xmath4=17 km s@xmath5 in our stacking simulations , we find that log / = @xmath60 is now required to achieve a 4@xmath38 detection . this limit is still consistent with current measured metallicities in higher column density ly@xmath0 clouds and is therefore not sufficiently sensitive to determine whether the / ratio drops in low @xmath3 ( ) lines . a feasibility study is performed to assess the effectiveness of a ` re - center ' on the maximum optical depth pixel prior to stacking for removing the effect of an unknown offset . we find that this technique can not improve the quality of the stack result in the / regime that we are targeting . it is not yet clear whether the observed redshift offset persists in the low column density ly@xmath0 clouds , but it must be considered a factor . moreover , the effects of contamination , continuum fitting errors and anomolous pixels also pose problems for this technique , although in simulations the redshift offset appears to be a major effect . therefore , if this technique is to be pursued further to reach a meaningful detection limit , an improvement in s / n by at least a factor of two is required . the optical depth technique introduced by cowie & songaila ( 1998 ) is considered as an alternative approach . this technique is shown to exhibit several advantages over the stacking method such as its insensitivity to redshift offsets and its ability to exclude contamination from other absorption features . we develop this technique as a method that can be used to test whether the detected systems represent the full tally of absorbers . the data have optical depths consistent with an almost constant log / @xmath93 down to @xmath71(ly@xmath0)@xmath17 23 , below which @xmath71 ( ) flattens off to an approximately constant value . it is unlikely that this flattening is real and is most probably caused by the effect of noise and/or continuum errors , even at such high s / n ratios as have been achieved in this spectrum . given the many effects that may alter the measured @xmath71 values , such as blending and noise , we do not attempt to fit the observed distribution of optical depths . instead , our strategy is to test whether the detected systems are sufficient to reproduce the measured @xmath71 ( ) and if not , determine how much additonal may be present below our current detection limit . by simulating synthetic spectra with different enrichment recipes , we have shown that the systems detected directly in the spectrum are not sufficient to reproduce the results of the optical depth analysis of q1422 + 231 . this is in agreement with the conclusions drawn from the column density distribution of , i.e. that the data are consistent with a continuous power law @xmath36 down to at least @xmath3 ( ) = 11.75 and that there is therefore likely to be a large number of weak metal lines not yet directly detected . this agrees with the conclusions of cowie & songaila ( 1998 ) . in order to interpret the results from the optical depth method , we have simulated synthetic spectra with a range of input / ratios . we find that including associated with strong ly@xmath0 lines ( @xmath3 ( ) @xmath614.5 ) but below the current detection limit , in addition to the 34 identified systems , can reproduce the optical depths measured in the observed spectrum for @xmath71(ly@xmath0 ) @xmath6 3 . for smaller values of @xmath71(ly@xmath0 ) , some additional metals are required and we find that including in low column density lines ( @xmath3 ( ) @xmath9 14.5 ) with log / @xmath94 produces optical depth results consistent with those measured in the data . however , determining the precise / in the low @xmath3 ( ) ly@xmath0 clouds and the density to which the enrichment persists is still uncertain due to effects such as noise and continuum fluctuations and it is therefore not possible to say whether the low column density forest is pristine . nevertheless , we find that even in the high optical depth pixels ( which will not be seriously affected by noise or small continuum errors ) the detected systems are not sufficient to cause all the measured absorption and clearly there are more metals in the igm than we can currently detect . the authors would like to thank bob carswell , alberto fernandez - soto , matthias steinmetz and the anonymous referee for discussions and useful suggestions . we are grateful to len cowie for his support and enthusiasm towards this project . sle and js are supported by a pparc postgraduate award and js acknowledges additional funding from the isaac newton trust . we are also grateful to the expert assistance of the staff at the keck i telescope for their help in obtaining the spectra presented here . haehnelt , m. g. , 1998 , the young universe : galaxy formation and evolution at intermediate and high redshift . edited by s. dodorico , a. fontana , and e. giallongo . asp conference series ; vol . 146 , p.249 cccr c1 & 2.90969 & 12.44 & 8.8 + & 2.91005 & 11.90 & 11.4 + c2 & 2.94522 & 11.74 & 4.2 + & 2.94551 & 12.32 & 12.6 + c3 & 2.94754 & 12.47 & 7.2 + c4 & 2.96065 & 12.50 & 13.4 + & 2.96110 & 12.68 & 13.6 + & 2.96146 & 12.61 & 7.6 + & 2.96197 & 13.28 & 19.3 + & 2.96235 & 12.80 & 9.7 + c5 & 2.97143 & 12.23 & 13.3 + c6 & 2.97584 & 12.46 & 35.3 + & 2.97622 & 12.76 & 9.2 + c7 & 2.99922 & 12.66 & 16.7 + & 2.99959 & 11.61 & 6.2 + c8 & 3.03505 & 12.14 & 8.3 + c9 & 3.03672 & 12.35 & 28.4 + c10 & 3.06338 & 12.98 & 24.5 + & 3.06433 & 12.66 & 11.0 + & 3.06383 & 12.01 & 5.6 + c11 & 3.07101 & 12.43 & 7.9 + c12 & 3.08666 & 12.95 & 12.8 + c13 & 3.08990 & 12.59 & 7.9 + & 3.09020 & 13.33 & 30.5 + & 3.09051 & 13.10 & 47.8 + & 3.09108 & 12.91 & 8.8 + c14 & 3.09468 & 12.17 & 14.1 + c15 & 3.11930 & 11.87 & 11.1 + & 3.11973 & 12.12 & 14.4 + c16 & 3.13252 & 12.46 & 26.0 + c17 & 3.13379 & 12.80 & 17.5 + & 3.13409 & 12.27 & 6.4 + & 3.13448 & 13.00 & 16.2 + c18 & 3.13712 & 12.89 & 16.9 + & 3.13799 & 12.33 & 27.5 + c19 & 3.19143 & 12.09 & 6.5 + c20 & 3.23330 & 11.77 & 2.6 + c21 & 3.24047 & 12.53 & 40.4 + c22 & 3.25716 & 12.50 & 42.5 + c23 & 3.26564 & 12.65 & 26.9 + & 3.26584 & 11.95 & 9.8 + c24 & 3.27596 & 12.02 & 13.0 + c25 & 3.31710 & 12.35 & 13.0 + c26 & 3.33410 & 11.84 & 7.7 + c27 & 3.37994 & 12.55 & 15.5 + & 3.38045 & 12.30 & 12.3 + & 3.38135 & 11.87 & 5.0 + & 3.38167 & 13.27 & 11.6 + & 3.38223 & 13.23 & 14.0 + & 3.38271 & 12.61 & 8.1 + & 3.38316 & 12.22 & 20.4 + c28 & 3.41080 & 12.20 & 10.7 + & 3.41149 & 12.90 & 21.2 + c29 & 3.44691 & 12.97 & 8.5 + & 3.44736 & 13.46 & 13.4 + c30 & 3.47963 & 12.85 & 31.7 + [ civlist ] cccr c31 & 3.49488 & 12.46 & 11.4 + c32 & 3.51465 & 12.83 & 7.5 + & 3.51497 & 12.59 & 15.3 + c33 & 3.51770 & 11.87 & 10.2 + c34 & 3.53490 & 12.57 & 34.7 + & 3.53505 & 12.69 & 16.0 + & 3.53549 & 12.43 & 7.7 + & 3.53599 & 13.69 & 19.9 + & 3.53659 & 13.12 & 21.7 + & 3.53738 & 13.22 & 17.2 + & 3.53872 & 13.58 & 9.2 + & 3.53937 & 13.58 & 14.1 + & 3.54005 & 12.82 & 25.6 + & 3.53848 & 13.33 & 25.0 + & 3.54140 & 12.10 & 4.4 + cccc 2.947 & [email protected] & 12.47 & -2.71 + 2.999 & [email protected] & 12.70 & -3.07 + 3.035 & [email protected] & 12.14 & -2.50 + 3.037 & [email protected] & 12.34 & -2.29 + 3.063 & [email protected] & 13.18 & -2.18 + 3.071 & [email protected] & 12.43 & -1.42 + 3.132 & [email protected] & 12.46 & -1.28 + 3.134 & [email protected] & 13.26 & -2.45 + 3.137 & [email protected] & 13.00 & -2.93 + 3.191 & [email protected] & 12.09 & -2.80 + 3.276 & [email protected] & 12.02 & -2.05 + 3.318 & [email protected] & 12.35 & -1.62 + 3.334 & [email protected] & 11.84 & -2.94 + 3.382 & [email protected] & 13.68 & -3.02 + 3.411 & [email protected] & 12.98 & -2.21 + 3.479 & [email protected] & 12.85 & -2.44 + 3.515 & [email protected] & 13.02 & -2.32 + 3.518 & [email protected] & 11.87 & -1.42 + 3.539 & [email protected] & 14.29 & -1.88 + [ hifits ]	we have obtained an exceptionally high s / n , high resolution spectrum of the gravitationally lensed quasar q1422 + 231 . a total of 34 systems are identified , several of which had not been seen in previous spectra . voigt profiles are fitted to these systems and to the entire ly@xmath0 forest in order to determine column densities , @xmath1-values and redshifts for each absorption component . the column density distribution for is found to be a power law with index @xmath2 , down to at least log @xmath3 ( ) = 12.3 . we use simulations to estimate the incompleteness correction and find that there is in fact no evidence for flattening of the power law down to log @xmath3 ( ) = 11.7 a factor of ten lower than previous measurements . in order to determine whether the enrichment extends to even lower column density clouds , we utilize two analysis techniques to probe the low column density regime in the ly@xmath0forest . firstly , a composite stacked spectrum is produced by combining the data for q1422 + 231 and another bright qso , apm 08279 + 5255 . the s / n of the stacked spectrum is 1250 and yet no resultant absorption is detected . we discuss the various problems that affect the stacking technique and focus in particular on a random velocity offset between and its associated which we measure to have a dispersion of @xmath4 = 17 km s@xmath5 . it is concluded that , in our data , this offset results in an underestimate of the amount of present by a factor of about two and this technique is therefore not sufficiently sensitive to probe the low column density ly@xmath0 clouds to meaningful metallicities . secondly , we use measurements of individual pixel optical depths of ly@xmath0 and corresponding lines . we compare the results obtained from this optical depth method with analyses of simulated spectra enriched with varying enrichment recipes . from these simulations , we conclude that more than is currently directly detected in q1422 + 231 is required to reproduce the optical depths determined from the data , consistent with the conclusions drawn from consideration of the power law distribution .
the scaling of small quantum processors @xcite into large qubit arrays capable of fault - tolerant quantum computation ( ftqc ) @xcite is an outstanding challenge for leading experimental quantum information platforms @xcite . modular @xcite and monolithic @xcite approaches require a systems approach that simultaneously and compatibly addresses challenges in all layers of the quantum computer stack @xcite : from the quantum hardware at the low level , through classical control electronics in the middle , to software at the high level ( i.e. , micro - instruction sets , compilers , and high - level programming languages ) . currently , the surface code @xcite provides an experimentally attractive paradigm for ftqc owing to its modest requirements on the quantum hardware : only nearest - neighbor coupling is needed between qubits , and the error threshold falls robustly close to @xmath2 across a range of error models and error - decoding strategies , signficantly higher than those of steane and shor codes @xcite . in superconducting quantum integrated circuits based on circuit qed ( cqed ) @xcite , the error rate of single - qubit gates has reached @xmath3 @xcite , while those of two - qubit conditional - phase ( @xmath4 ) gates and measurement are @xmath5 @xcite and @xmath6 @xcite , respectively . the scalability of monolithic systems hinges on the ability to copy - paste a unit cell in the quantum plane , with suitable quantum interconnect between cells , and suitable classical interconnect to and from the control plane . the latter pursuit is very active , with several groups developing vertical ( rather than the traditional lateral ) interconnection of input / output ( i / o ) signals using through - the - wafer coaxial lines @xcite , electro - mechanical sockets @xcite , and bump - bonding in flip - chip configuration @xcite . for true scalability , it is crucial that the unit cell also extend into the classical control plane . a unit cell for control signals opens the door to hardware simplification through spatial multiplexing , i.e. , the selective routing of control signals ( with minimal customization ) to spatially separated components . while frequency - division multiplexing is already heavily exploited in cqed @xcite , spatial multiplexing is in its infancy . precision control of same - frequency qubits using a microwave - frequency vector switch matrix ( vsm ) for pulse multicasting has only recently been demonstrated @xcite . in this paper , we propose a scalable scheme for the qec cycle of a monolithic superconducting surface code by defining a concrete unit cell for both the quantum hardware and the control signals . we focus on a fabric of fast - flux - tunable transmon qubits interacting with nearest neighbors via flux - controlled conditional - phase @xmath7 gates @xcite realized by pulsing into the resonator - mediated @xmath8 avoided crossing of the interacting transmon pair ( numbers indicate excitation level ) . our approach is compatible with adiabatic @xcite , sudden @xcite and fast - adiabatic @xcite use of these crossings . our eight - qubit unit cell uses three fixed frequencies for all single - qubit control and eight detuning sequences for two - qubit gates . this approach to classical control allows significant control hardware savings via spatial multiplexing . by pipelining the measurement of the two types of stabilizers of the surface code , we engineer detuning patterns avoiding all second - order transmon - transmon interactions except those exploited in @xmath9 gates , regardless of fabric size . our scheme allows changing the weight of stabilizer measurements by simple on / off masking of detuning pulses , making it applicable to both defect - based and planar logical qubits @xcite , including lattice surgery @xcite . layout of a surface - code fabric . red circles with @xmath10 labels represent data qubits . blue ( green ) circles with @xmath0 ( @xmath1 ) labels represent ancillas performing @xmath0-type ( @xmath1-type ) quantum parity checks of their nearest - neighbor data qubits . each check is realized as an indirect quantum measurement , consisting of a coherent step involving pairwise interactions ( dashed lines ) followed by ancilla measurement . the delineated fabric of nine data qubits ( @xmath11 through @xmath12 ) and eight ancillas ( @xmath13 through @xmath14 and @xmath15 through @xmath16 ) constitutes the distance-3 planar logical qubit named surface-17 . ] a surface - code fabric consists of the two - dimensional square lattice of data - carrying qubits shown in fig . [ fig : large_sc ] . the stabilizers of this code are the @xmath0-type ( @xmath1-type ) parity operators @xmath17 @xmath18 , where @xmath19 denotes data qubits on the corners of the blue ( green ) plaquettes . conventionally , these stabilizers are measured indirectly using ancilla qubits positioned at the center of the plaquettes , forming a second square lattice . standard circuits for measuring @xmath0- and @xmath1-type stabilizers , shown in fig . [ fig : plaquettes ] , combine a sequence of coherent interactions of the ancilla with its nearest - neighbor data qubits , followed by projective ancilla measurement . using controlled - not ( @xmath20 ) gates as the fundamental interaction , @xmath0-type and @xmath1-type stabilizer measurements can be fully parallelized with circuit depth seven . we define circuit depth as the number of operations on each ancilla per qec cycle , counting in measurement but excluding ancilla initialization [ we assume pauli frame updating ( pfu ) @xcite is used for data and ancilla qubits ] . the order of two - qubit gates in fig . [ fig : plaquettes ] is important for two reasons @xcite . first , data qubits common to adjacent plaquettes must do all their interactions with one ancilla before the other . second , the s ( n ) pattern for @xmath0-type ( @xmath1-type ) stabilizers provides resilience to single ancilla - qubit errors even in small distance - three surface codes such as surface-17 . this circuit consists of the patch delineated in fig . [ fig : large_sc ] , with nine data qubits ( labelled @xmath11 to @xmath12 ) , four ancillas ( @xmath13 to @xmath14 ) for @xmath0-type stabilizer measurements , and four ancillas ( @xmath15 to @xmath16 ) for @xmath1-type stabilizer measurements . @xmath0-type ( a ) and @xmath1-type ( b ) plaquettes . data qubits are labelled according to their position relative to the ancilla ( ne = northeast , nw = northwest , se = southeast , and sw = southwest ) . standard circuits for measuring @xmath0-type ( c , e ) and @xmath1-type ( d , f ) stabilizers indirectly using ancillas , using @xmath20 ( c , d ) or @xmath4 ( e , f ) as the primitive data - ancilla interaction . the order of two - qubit gates , ne - nw - se - sw ( ne - se - nw - sw ) for @xmath0-type ( @xmath1-type ) , ensures that all data qubits common to adjacent plaquettes do their interactions with one ancilla before the other , and also provides resilience to ancilla errors in surface-17 @xcite . using the relations @xmath21 , one can see that the opening and closing @xmath22 gates can be replaced by @xmath23 and @xmath24 rotations , respectively . ] when the two - qubit gate is @xmath4 , parallelizing the stabilizer measurements of surface-17 requires depth nine because of non - commutation between hadamard @xmath25 gates and @xmath4 gates . the full circuit for the parallelized qec cycle of surface-17 using @xmath4 gates is shown in fig . [ fig : parallelized ] . using gate and measurement times from recent experiments ( @xmath26 for single - qubit gates , @xmath27 for @xmath9 gates , and @xmath28 for ancilla readout and photon depletion in readout resonator ) , the qec cycle will complete in @xmath29 . depth - nine quantum circuit for parallelized @xmath0- and @xmath1-stabilizer measurements in surface-17 using @xmath4 gates . the six @xmath4 gates inside each gray box are executed simultaneously . typical values of gate and readout times are indicated at the top . the bottom arrow represents the looping of qec cycles . qubits are labelled as in fig . [ fig : large_sc ] . ] on paper , it is straightforward to compose a depth - nine quantum circuit for the fully parallelized qec cycle of a surface - code fabric of arbitrary size . however , to the best of our knowledge following numerous failed attempts , the full parallelization of @xmath0- and @xmath1-type stabilizer measurements makes it impossible to realize a scalable implementation with @xmath9 gates that satisfies all of the following desirable properties : * microwave pulses for single - qubit gates should be applied at a fixed , small number of frequencies . * transmons should maximally exploit their coherence sweetspot @xcite . * flux - pulsed transmons should not cross any other interaction zones on their way to or from the intended @xmath8 avoided crossings realizing the @xmath9 gate . * the flux - pulsing schemes should be extensible to a surface code of arbitrary size using a fixed number of detuning sequences and a fixed detuning range . * the implementation should be compatible with logical qubit operations . we have found frequency arrangements and flux - pulse sequences that meet the first three criteria . however , all of these solutions require a growing number of detuning sequences and detuning ranges as the fabric expands , in order to avert all other interactions on the way to and from the @xmath8 avoided crossings of @xmath9 gates . furthermore , these solutions seem practically infeasible already for distance five ( surface-49 @xcite ) . to our knowledge , no fully parallel solution exists with a fixed number of detuning sequences and a fixed detuning range . in the next section , we introduce a pipelined ( rather than parallelized ) version of the qec cycle that simultaneously meets the five desirable properties for a fabric of arbitrary size . our scalable scheme , which we term _ pipelined qec cycle _ , combines four key elements : 1 . repeating unit cells of eight qubits ; 2 . pipelined @xmath0- and @xmath1-type stabilizer measurements ; 3 . three frequencies for single - qubit control ; 4 . eight detuning sequences implementing the requisite @xmath9 gates , realizable by on / off masking of three flux - pulse primitives . we now introduce these elements in detail . the first element is a unit cell ( fig . [ fig : sc_layout ] ) from which a surface code of arbitrary size can be assembled by repetition ( and truncation at boundaries ) . a unit cell contains four data qubits ( @xmath30 to @xmath31 ) and four ancillas ( @xmath32 , @xmath33 , @xmath34 , and @xmath35 ) . crucially , the cell is the fundamental unit of repetition not just for the quantum hardware . it is also the unit of repetition for all coherent control . composing the surface - code fabric by repetition of 8-qubit unit cells . red and pink circles represent data qubits , blue ( green ) circles represent ancillas for @xmath0-type ( @xmath1-type ) stabilizer measurements , and dashed lines represent nearest - neighbor couplings . dot colors also indicate the frequency for single - qubit microwave control ( red for @xmath36 , greed and blue for @xmath37 , and pink for @xmath38 ) . contours delineate unit cells ( with qubits named @xmath30 to @xmath31 , @xmath32 , @xmath34 , @xmath33 , and @xmath35 ) . ] the second element is the pipelined execution of the @xmath0- and @xmath1-type stabilizer measurements . the pipelining concept is illustrated in fig . [ fig : pipelined](a ) . while stabilizer measurements of one type always run simultaneously , the coherent and readout steps of ancillas of the other type are interleaved . in other words , ancillas of one type undergo coherent steps while ancillas of the other type are measured . time slots a and b ( d and e ) are for single - qubit gates pertinent to the @xmath0-type ( @xmath1-type ) stabilizer measurements , while slots 1 to 4 ( 5 to 8) are for two - qubit gates . note that nine of the @xmath4 gates involve two qubits within the cell , while fourteen involve one qubit from a neighboring unit cell . generally , ancilla measurement ( including any photon depletion of the readout resonator ) will take longer than the coherent steps , leaving time to perform operations on the data qubits in steps c and f while all ancillas are measured . possible operations include logical gate operations , refocusing pulses , or single - qubit gates performing error correction . clearly , performing such operations during steps c or f would not increase the qec cycle time . pipelining offers several advantages . first , it compresses the stabilizer measurements to depth seven , two single - qubit - gate steps less than fully - parallelized quantum circuits ( such as fig . [ fig : parallelized ] for surface-17 ) . a second and more crucial advantage is the ability to scale without increasing the number of frequencies for single - qubit control or qubit detuning sequences , as explained next . the third and fourth elements are best described together . figure [ fig : pipelined](b ) presents our choice of frequencies for single - qubit control and the qubit - specific detuning sequences for realizing the two - qubit qec cycle interactions . single - qubit gates on data qubits ( steps a , b , d , and e ) are performed at frequencies @xmath36 and @xmath38 ( alternating in data - qubit rows ) , while those on ancillas are performed at intermediate frequency @xmath37 . note that with only nearest - neighbor coupling , two distinct frequencies ( one for ancilla qubits and one for data qubits ) reduce the exchange coupling between same - frequency qubits to fourth order ( qubit - resonators , resonator - qubit , qubit - resonator , resonator - qubit ) . when extending to the proposed three frequencies , this also allows engineering the detuning sequences so that no transmon crosses any other second - order interaction zone on the way to or from the @xmath8 avoided crossings exploited in the @xmath9 gates . during steps 1 - 4 and 5 - 8 , transmons are flux pulsed to a discrete set of frequencies , depending on whether they interact , idle , or are measured : @xmath30 and @xmath39 to @xmath36 or @xmath40 ; ancillas to @xmath37 , @xmath41 or @xmath42 ; and @xmath43 and @xmath31 to @xmath38 or @xmath44 . @xmath4 gates occur between transmons at @xmath45 and @xmath37 , and between transmons at @xmath46 and @xmath38 . here , @xmath47 is the transmon anharmonicity @xcite . transmons at @xmath36 , @xmath41 , and @xmath44 are hidden away and do not interact . the frequency detuning patterns during interaction steps 1 through 4 and 5 through 8 can be synthesized by on / off masking of three flux - pulse primitives using a switch matrix : a first primitive detuning data qubits of type @xmath30 and @xmath39 from @xmath36 to @xmath40 , a second one detuning ancillas from @xmath37 to @xmath42 , and a third one detuning data qubits of type @xmath43 and @xmath31 from @xmath38 to @xmath44 . for example , the detuning sequence for @xmath39 in fig . [ fig : pipelined](b ) can be synthesized by masking the pulse primitive on ( off ) at steps 1 , 4 , 6 , and 7 ( 2 , 3 , 5 , and 8) . two types of logical qubits can be envisioned for surface code : defect - based @xcite and planar @xcite . defect - based logical qubits are introduced by stopping the measurement of one or two stabilizers ( @xmath0-type for _ rough _ logical qubits , and @xmath1-type for _ smooth _ ones @xcite ) . in our scheme , turning stabilizer measurements fully off can be accomplished in either of two ways . one is to mask off the @xmath22 gates of the corresponding ancilla using the vsm , without changing the detuning sequence or stopping the ancilla measurement . if the ancilla is in @xmath48 , all its @xmath9 gates are inactive and there is no net action on the logical qubit . if it starts in @xmath49 , the stabilizer operator ( not its measurement ) is applied . this performs a logical @xmath50 @xmath51 gate on a rough ( smooth ) qubit . the ancilla measurements provide the key input allowing the decoder to keep track of the action by pfu . a second way to turn a stabilizer fully off is to mask off all the flux - pulse primitives in the interaction step . logical operations , such as move and braiding operations on defect - based qubits @xcite , and lattice surgery on planar ones @xcite , also require dynamically changing the weight of specific stabilizer measurements , i.e. , selectively removing specific data qubits from the quantum parity checks . in our scheme , this can easily be achieved by selective on / off masking of flux - pulse primitives . for example , removing a qubit of type @xmath39 from the @xmath0-type stabilizer measurement below it simply requires masking off the pulse primitive at step 1 . the order of the two - qubit gates can also be changed by masking . ideally , @xmath36 ( @xmath38 ) would match the sweetspot frequency of @xmath30 and @xmath39 ( @xmath43 and @xmath31 ) , and @xmath37 would match that of all ancillas , to minimize dephasing from @xmath52 flux noise . in practice , @xmath36 would be set to the lowest sweetspot frequency among all transmons labelled @xmath30 or @xmath39 , and so on . it is straightforward to expand the circuit of fig . [ fig : pipelined](a ) with refocusing single - qubit gates to minimize dephasing in the transmons that are not at their sweetspot during single - qubit control @xcite . the frequencies @xmath36 , @xmath37 , @xmath38 , @xmath41 , and @xmath44 must be chosen so that residual interactions during single - qubit gates can be tolerated . for simplicity , we consider a uniform detuning scale @xmath53 . the single - qubit gate error is @xmath54 , where @xmath55 is the @xmath8 coupling strength @xcite . for fast - adiabatic @xmath9 gates , @xmath56 . thus , @xmath57 for @xmath26 and @xmath27 requires @xmath58 . alternative frequency arrangement for qubits in the unit cell , with ancillas @xmath32 and @xmath34 ( @xmath33 and @xmath35 ) at the outer frequency @xmath36 ( @xmath38 ) and data qubits at the inner frequency @xmath37 . ] there exist other possible frequency arrangements than that shown in fig . [ fig : pipelined](b ) . for example , consider the inverted arrangement with all data qubits at @xmath37 and the ancillas at the outer frequencies . figure [ fig : ancilla_out ] shows one of these configurations , with @xmath32 and @xmath34 ( @xmath33 and @xmath35 ) at @xmath36 ( @xmath38 ) . it is straightforward to modify the detuning sequences for this arrangement to also avert all unwanted interactions . however , upon comparing this alternative to the original arrangement , we observe a key difference making the original preferable for a cqed implementation with flux - tuneable transmons . specifically , the original exactly balances the number of interaction steps in which qubits can remain at their upper frequency ( i.e. , at or closest to their coherence sweetspot ) , while the flipped arrangement allows this on just two ( out of eight ) steps for data qubits and zero or four ( out of four ) steps for ancilla qubits . the reduced data - qubit dephasing during the coherent steps will lead to a lower logical error rate . note that this advantage of the original arrangement is made possible by lowering the ancillas to @xmath41 for their measurement , at which the additional dephasing is innocuous in view of the measurement - induced projection . residual single - qubit gate cross - talk between @xmath30 and @xmath39 ( @xmath43 and @xmath31 ) can be reduced by breaking the degeneracy in frequency @xmath36 ( @xmath38 ) , which requires increasing the number of primitive pulses from three to five , or even in @xmath37 , further increasing the number of primitive pulses to eight . a digitally addressable switch matrix and primitive - flux - pulse - synthesizer operating at room temperature ( cryogenically ) are exciting challenges for the near ( long ) term . the switch matrix should allow qubit - specific customization of the flux - pulse primitives , including fine adjustment of delay ( to compensate cable - length mismatch ) , amplitude ( for fine tuning of two - qubit and single - qubit phase ) , and dc offset ( for tuning to @xmath36 , @xmath37 and @xmath38 ) . ( a ) . note that a similar cqed chip design is envisioned in ref . our scheme reduces the number of feedlines by bridging these over bus resonators . ] . ( b ) transmission - line crossovers bridge readout feedlines over bus resonators . ( c ) vertical coaxial lines couple all input and output signals . ] we have presented a concrete scheme for the qec cycle of an arbitary - size surface code implemented with flux - tuneable transmons . the scheme combines four key concepts : an eight - qubit unit cell as the basis for repetition of quantum hardware and control signals ; pipelining of @xmath0- and @xmath1-type stabilizer measurements ; a fixed set of three frequencies for single - qubit control ; and a fixed set of eight detuning sequences implementing the requisite controlled - phase gates . these eight detuning sequences can be composed by on - off masking of three flux - pulse primitives . experimental efforts are underway to implement and evaluate the pipelined qec cycle in surface-17 . we pursue a realization of the surface-17 quantum hardware with transmons in a planar cqed architecture @xcite made extensible using vertical interconnect for all input and output signals ( fig . [ fig : cqeds17 ] ) . each transmon will have a dedicated flux line @xcite allowing control of its transition frequency on nanosecond timescales , a dedicated microwave - drive line , and a dedicated dispersively - coupled resonator for readout . we opt for coupling nearest - neighbor data and ancilla transmons via bus resonators ( rather than direct capacitance @xcite ) . figure [ fig : implementation](a ) shows a prototype seven - port transmon developed in our lab , nicknamed _ starmon _ the vertical i / o will be realised either using through - silicon - vias [ fig . [ fig : implementation](c ) ] or bump bonding in a flip - chip arrangement . experimental efforts are also underway to demonstrate the scalability of the classical control plane . diagonally running feedlines coupled to readout resonators will allow simultaneous readout by frequency - division multiplexing @xcite , reducing the need for cryogenic amplifiers and microwave electronics ( circulators , etc . ) , as well as homodyne detection systems at room temperature . while frequency multiplexing for readout is common in cqed , the simultaneous readout of nine qubits using one feedline as required by surface-17 is not yet achieved . finally , demonstrating the control - hardware savings achievable by spatial multiplexing is an immediate priority . single - qubit control at @xmath36 , @xmath37 and @xmath38 will make use of a next - generation vsm ( follow - up to @xcite ) to independently route precision drag @xcite pulses to same - frequency qubits , with significant savings in microwave sources . finally , a switch matrix for constructing frequency detuning sequences by on / off masking of flux - pulse primitives is envisioned for room temperature . a cryogenic implementation remains highly attractive for the longer term . we thank c. c. bultink , m. a. rol , x. fu , c. garca - almudever , l. riesebos , d. deurloo , b. criger , t. e. obrien , and b. tarasinski for helpful discussions . this research is supported by the dutch organisation for scientific research on matter ( fom ) , intel corporation , and by the office of the director of national intelligence ( odni ) , intelligence advanced research projects activity ( iarpa ) , via the u.s . army research office grant w911nf-16 - 1 - 0071 . the views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements , either expressed or implied , of the odni , iarpa , or the u.s . government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation thereon . 36ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop * * ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop ( ) link:\doibase 10.1103/revmodphys.87.307 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop ( ) @noop * * , ( ) @noop * * , ( ) @noop `` , '' @noop ( ) @noop `` , '' @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) http://www.nature.com/nature/journal/v460/n7252/abs/nature08121.html [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1038/nature03350 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) http://journals.aps.org/pra/abstract/10.1103/physreva.76.042319 [ * * , ( ) ] @noop @noop @noop * * , ( ) @noop * * , ( )	we present a scalable scheme for executing the error - correction cycle of a monolithic surface - code fabric composed of fast - flux - tuneable transmon qubits with nearest - neighbor coupling . an eight - qubit unit cell forms the basis for repeating both the quantum hardware and coherent control , enabling spatial multiplexing . this control uses three fixed frequencies for all single - qubit gates and a unique frequency detuning pattern for each qubit in the cell . by pipelining the interaction and readout steps of ancilla - based @xmath0- and @xmath1-type stabilizer measurements , we can engineer detuning patterns that avoid all second - order transmon - transmon interactions except those exploited in controlled - phase gates , regardless of fabric size . our scheme is applicable to defect - based and planar logical qubits , including lattice surgery .
recently a population of neurons in the cat visual cortex has been reported to exhibit synchronized firings in a stimulus dependent manner@xcite . the occurrence of correlations in firing times of neurons seems to be a ubiquitous phenomenon in real nervous systems . the role of such synchronized firings for information processing in the brain has been attracting growing interest of researchers , and several authors have suggested neural network models based on the concept of temporal coding , where information of a neuron is represented by its firing times . indeed , to explain the experimental findings that visual information of an external object is processed with being divided into several pieces of information , several authors@xcite have suggested that the synchronized firings of neurons may serve as a linker for those pieces of information . the problem of investigating how an associative memory is realized in real nervous systems as well as of constructing biologically relevant models is of central concern of neuroscientists . since the establishment of systematic theories of associative memory for networks with an energy function that is ensured by assuming symmetric synaptic couplings@xcite , several attempts have been made to make models as biologically plausible as possible@xcite . previously we investigated the effects of asymmetric couplings@xcite for memorizing pre - synaptic and post - synaptic activities , which are incorporated into the standard symmetric hebb learning rule@xcite , by studying networks of analog neurons@xcite , whose continuous - time dynamics involves a positive - valued transfer function representing the mean firing rates of a neuron as a function of membrane potential . instead of working with the concept of rate coding based on the idea that neuronal information is represented by mean firing rate of a single neuron , one may be concerned with the concept of temporal coding , when considering that spatio - temporal patterns of neuronal firings will make information carried by a population of neurons much richer than spatial patterns alone . a spiking neuron is considered to be one of the candidates for implementation of temporal coding@xcite . the time evolution of membrane potential that is generated in response to an injected synaptic electric current of a spiking neuron is described by such nonlinear dynamics as hodgkin - huxley equation@xcite , fitzhugh - nagumo equation@xcite , or the equation of an integrated - and - fire neuron . spiking neurons in a network are generally supposed to interact with each other via pulses generated in the firing events occurring in the pre - synaptic neuron . we have shown previously that even in the presence of time delays in transmission of the pulses an associative memory based on a network of spiking neurons can be realized by assuming a simple hebb - type learning rule alone , and that the memory retrieval accompanies synchronized firings of neurons . the dynamics of such associative memory was analyzed by means of sublattice method in our previous paper@xcite . in the previous analysis@xcite we assumed that every neuron shares identical characteristics to exhibit the same reaction in response to the same injected current . in real nervous systems , however , neurons in a network may have their own individual characteristics . the problem of whether temporal coding functions robustly in the presence of certain heterogeneties of neurons will be of particular interest . for the purpose of investigating such a problem for the phenomenon of synchronized oscillations in associative memory neural networks , we consider it appropriate to deal with simple models of coupled oscillators with distributed natural frequencies and external noise . it is well known that , under certain conditions , a population of oscillators with distributed natural frequencies is allowed to get partially entrained in such a way that oscillators with a natural frequency near the central frequency become to oscillate synchronously at identical frequency as a result of cooperative interactions@xcite . kuramoto@xcite showed that the dynamics of this kind of network of oscillators with sufficiently week interactions can be reduced to a simple phase dynamics . supposing that neurons in a network are treated as phase oscillators , associative memory has been shown to be realized under a simple learning rule of the hebb - type either in the case of a finite number of stored patterns@xcite or in the case with a single natural frequency@xcite . satisfactory analysis of the case with a distribution of natural frequency and extensively many stored patterns has been far less conducted . quite recently we have reported the study of a deterministic phase oscillator network with a distribution of natural frequencies where an extensive number of binary patterns ( @xmath1 ) are stored with use of the hebb learning rule@xcite . the main purpose of the present study is the theoretical analysis of associative memory based on temporal coding with use of networks of phase oscillators in the more general case where the number of stored patterns that are given by uniformly distributed random numbers on @xmath0 is extensive , natural frequencies of the oscillators are distributed according to a certain distribution function , and furthermore external white noise is added to the system . while one can analyze a phase oscillator network with a single natural frequency by means of the replica method that makes full use of its associated energy function , one can not resort any more to the standard method of statistical physics based on the existence of an energy function in the case of networks with a distribution of natural frequencies . one can , however , use the self - consistent signal - to - noise analysis ( scsna)@xcite to deal with general cases without energy functions . to apply the scsna it is necessary to know fixed - point equations describing the equilibrium states of the network . when considering such equations in stochastic systems , we may take advantage of the concept of the tap equation@xcite . the tap equation is known to exist for the sk model of spin glasses@xcite and the hopfield model of an ising spin neural networks@xcite , and to represent a functional relationship between the thermal or time average of each spin in equilibrium and its corresponding effective local field that involves the so called onsager reaction term@xcite . usefulness of the tap equation in deriving the order parameter equations of associative memory networks is attributed to the fact that the resulting equations of the replica calculations by ags is recovered by the result of application of the scsna to the tap equation@xcite , where the onsager reaction term is canceled exactly by the appearance of the renormalized output term of the scsna@xcite . we note that the tap equation of the naive mean field model with the interactions given by the hebb learning rule defines to an analog network equation with the transfer function @xmath2@xcite . we first evaluate an analogue of the tap equation of the naive mean field type for our model with a distribution of natural frequencies by dealing with the fokker - plank equation . in order to obtain the onsager reaction term we compute the free energy of the network without a distribution of natural frequencies to derive the tap equation by following the method of plefka@xcite and nakanishi@xcite . then we assume that the tap - like equation also exists with the onsager reaction term remaining the same even for networks with a distribution of natural frequencies , and that such tap - like equation defines an effective transfer function to which the scsna is applicable to obtain the order parameter equations . the present paper is organized as follows . in section [ model ] we introduce a neural network of simple phase oscillators and describe how the network functions as an associative memory based on a simple learning rule of the hebb - type . in section [ derivation ] we give a theoretical analysis to derive the macroscopic order parameter equations describing the long time behavior of the system . on the basis of the order parameter equations , in section [ application ] we investigate properties of memory retrieval accompanying synchronization in the networks by assuming a discrete symmetric natural frequency distribution with three frequency components . results of numerical simulations are presented showing good agreement with those of theoretical analysis . in section [ discussion ] , comparing our work with those of other researchers conducted previously , we summarize the results of the present study . [ model ] the system under consideration is a network of @xmath3 phase oscillators subjected to external white noise , whose dynamics is expressed as @xmath4 where @xmath5 and @xmath6 represent the phase and the natural frequency of oscillator @xmath7 respectively . @xmath8 and @xmath9 represent a certain phase shift and the strength of coupling between oscillator @xmath7 and @xmath10 respectively . the gaussian white noise @xmath11 is assumed to satisfy @xmath12 and @xmath13 . when @xmath14 , @xmath15 satisfy @xmath16 with @xmath17 denoting complex conjugation and @xmath9 take real values , the system ( [ system ] ) has the energy function : @xmath18 then one has an equilibrium probability distribution @xmath19 proportional to @xmath20 . in the case of @xmath21 , the function ( [ hamiltonian ] ) becomes a lyapunov function of the system and hence the state of the system eventually settles into a certain fixed point attractor after a long time . in the present study we assume natural frequencies to be distributed accordingly to an even distribution function @xmath22 so that the average of natural frequencies become zero without loss of generality . to store @xmath23 quenched random patterns @xmath24 chosen from the uniform distribution on the interval @xmath25 , we assume the hebb type learning rule , and set the parameter @xmath8 and real valued @xmath9 such that @xmath26 where @xmath27 . this definition of couplings gives networks the following properties : * in successful retrieval an entrainment occurs , where synchronized oscillators satisfy @xmath28 with a target pattern @xmath29 recalled . ( note that if @xmath5 is the solution of dynamics ( [ system ] ) , uniformly shifted phase @xmath30 also becomes its solution . what matters is not the phase itself but their difference @xmath31 . ) * in the case of unsuccessful retrieval , all the oscillators fail to synchronize , running at their own natural frequencies . to measure the distance between the pattern @xmath29 and the state of the system , we introduce the overlap for pattern @xmath29 : @xmath32 where we denote @xmath33 by @xmath34 . then by use of the local field : @xmath35 the dynamics ( [ system ] ) is rewritten as @xmath36 where @xmath37 denotes the loading rate @xmath38 . from eq . ( [ nativedynamics ] ) it is easy to see that the learning rule ( [ hebb ] ) indeed realizes the above mentioned properties if the number of stored patterns is finite(@xmath39 ) and @xmath14 . the memory retrieval accompanying synchronization can also occur for @xmath40 even in the presence of a distribution of natural frequencies . [ derivation ] behaviors of associative memory networks depend crucially on the nature of the local fields or the neurons of oscillators , because the updating rule for the time evolution of the system is based on the local fields as is seen in eq . ( [ nativedynamics ] ) . when the long time behavior of a network is described by fixed point type attractors , the relation between the resulting output state of a neuron and the corresponding local fields becomes essential for determining equilibrium properties of the networks . such a relation is naturally introduced in the case of deterministic analog networks where neurons are characterized by transfer functions describing the input - output relation . for such stochastic systems ( [ system ] ) as ising spin networks equilibrium fixed - point equations called the tap equations are known to exist as expressing the relation between time average of each output of a neuron and its corresponding effective local fields involving the so called onsager reaction term that is proportional to the time - averaged output . once the tap - like equation level description is available , the scsna , in which one compute the variance of the cross - talk noises in the local field as a result of storing an extensive number of patterns , can be applied to obtain the order parameter equations in the limit @xmath41 . if @xmath14 and @xmath21 , the state of the network eventually settles into an equilibrium state given by a fixed point attractor owing to the existence of the lyapunov function ( [ hamiltonian ] ) , and then the local fields do not fluctuate in time . even in the presence of external white noise ( @xmath42 ) , the local fields also get fixed in time after a long time , provided @xmath43 . when the local fields are fixed over the time change due to the law of large numbers , theoretical treatment becomes simple because one can reduce the many body problem to a single body problem . in more general case where @xmath40 and/or natural frequencies are distributed , the local fields may fluctuate even with a large number of oscillators as can be shown in the numerical simulation illustrated in fig . [ fluctuation ] . the fluctuations seem to be aperiodic and rigorous analysis of such fluctuations is quite difficult . to deal this situation we are forced to resort to a certain approximation by confining ourselves only to the near equilibrium behavior of the system : _ we replace the time - dependent local fields by their time - averaged ones : _ @xmath44 where the over bar represents the time average at near equilibrium . once we apply this approximation , we are allowed to treat each interconnected oscillator as an element obeying the dynamics of one degree : @xmath45 which is expected to describe the behavior of the oscillator @xmath7 near equilibrium . it turns out that the time average @xmath46 at equilibrium can be expressed as a function of @xmath6 and @xmath47 : @xmath48 . we call that the transfer function in the case of network of analog neurons . it is easy to show that the transfer function @xmath49 satisfies @xmath50 hence it suffices to calculate @xmath51 to obtain @xmath52 . in the absence of white noise ( @xmath21 ) , we can easily obtain the transfer function for real - valued local field @xmath53@xcite : @xmath54 in deriving eq . ( [ nod ] ) in the case of @xmath55 we used @xmath56 together with @xmath57 . in the case of @xmath58 or @xmath59 we computed the time average of @xmath60 over the period @xmath61 of the periodic oscillations of @xmath62 : @xmath63 in fig . [ field ] we illustrate the shape of the transfer function @xmath64 that is obtained from eqs . ( [ circulate ] ) and ( [ nod ] ) . in the presence of white noise ( @xmath65 ) , from the langevin equation ( [ replace ] ) we obtain the fokker - plank equation : @xmath66 where @xmath67 is the probability distribution of phase @xmath68 at time @xmath69 , and periodic boundary conditions @xmath70 are imposed . since we are concerned with the probability distribution @xmath71 attained after a long time , we set @xmath72 to obtain@xcite @xmath73 with @xmath74 where @xmath75 and @xmath76 . noting the ergodic property on the fokker plank eq . ( [ fokkerplank ] ) , which holds when @xmath77 is viewed as a given parameter , we obtain the time average of @xmath78 by computing the average over the equilibrium distribution @xmath71 : @xmath79 the transfer function we have obtained here can be considered to be an analogue of the so called tap equation by the naive mean field model , because the time average of the output @xmath80 is represented as a function of time - averaged local fields @xmath81 . we note , however , that the genuine tap equation , which is defined for systems with an energy function , should describe the functional relation between the time averaged output and the effective local field that differs in general from the time - averaged one . the difference between the two types of local fields is known to be the onsager reaction term in the theory of random spin systems . we assume that the tap - like equation may hold even for the present system without an energy function , and suppose it to be given by the following equations : @xmath82 in the case with @xmath14 , by evaluating the free energy of the system , we can derive an explicit expression of the coefficient @xmath83 taking the form ( see appendix [ tap ] ) : @xmath84 where @xmath85 it will , however , be difficult to rigorously derive an expression of @xmath83 for the general case with a natural frequency distribution . thus we are led to make an assumption that the legitimate expression ( [ gtap ] ) for the case with @xmath14 can naturally be extended to the general case . describing the desired @xmath83 requires the introduction of the order parameter @xmath86 that appears in the scsna . we consider the case with @xmath87 and @xmath88 , where we choose pattern 1 as the target . assuming , without loss of generality , that @xmath89 for all @xmath7 and @xmath90 is real owing to the rotational symmetry , the local field eq . ( [ taplocalfield ] ) is rewritten in the form : @xmath91 where we have used @xmath92 and @xmath93 to represent respectively @xmath94 and @xmath95 for brevity . when we consider the case with a finite number of stored patterns , the analysis is straightforward since we already know the form of the transfer functions ( [ nod ] ) ( @xmath21 ) and ( [ withd ] ) ( @xmath42 ) . since @xmath96 and @xmath97 in this case , we have in the limit @xmath41 @xmath98 where @xmath99 . solving this equation numerically , the size of the overlap is evaluated as a function of various parameters including @xmath100 . in the case of an extensive number of stored patterns ( @xmath40 ) , however , cross - talk noise ( the second term of eq . ( [ plainlocalfield ] ) ) in the local fields becomes to an appreciable extent , then one has to employ the method of scsna . the crux of the scsna is the evaluation of the variance of cross - talk noise that interferes occurrence of retrieval state . according to the prescription of the scsna@xcite , the local fields ( [ plainlocalfield ] ) is assumed to be in the form : @xmath101 where @xmath102 , whose explicit expression is given later , is a quantity that is very close to @xmath103 and has negligible correlation in the limit @xmath41 , and @xmath104 and @xmath105 are to be self - consistently determined in the course of analysis . defining @xmath106 , eq . ( [ tapfixedpoint ] ) reads @xmath107 where @xmath108 . considering a general case with @xmath109 , we solve eq . ( [ fixedpoint ] ) for @xmath110 to obtain the renormalized output @xmath111 . performing a taylor expansion of @xmath112 about @xmath113 , we have @xmath114 with @xmath115 substituting eq . ( [ taylor ] ) into eq . ( [ plainlocalfield ] ) and comparing the result with eq . ( [ assumption ] ) ( see appendix [ evaluation ] for details ) we obtain @xmath116 where @xmath117 since this equation holds for every site @xmath7 , @xmath104 and @xmath105 are self - consistently determined as @xmath118 the variance of renormalized noise @xmath119 in the right hand side of eq . ( [ assumption ] ) can be evaluated by noting that @xmath120 and @xmath121 distribute over sites obeying an identical gaussian distribution independently and the site average of @xmath122 can be replaced by the pattern average : one has @xmath123 to represent the noise as @xmath124 , where @xmath125 and @xmath126 are real and obey a normal gaussian , we define @xmath127 summarizing eqs . ( [ overlap]),([q]),([assumption]),([fixedpoint]),([u]),([lambda]),([gamma ] ) , and ( [ r ] ) , we have a set of macroscopic order parameter equations @xmath128 where @xmath129 represents @xmath130 . detailed derivation of eqs . ( [ scu ] ) and ( [ scr ] ) is given in appendix [ miscellaneous ] . to discuss the generalized expression for @xmath83 for the case with a distribution of natural frequencies , we consider for the moment the case with @xmath14 , where eq . ( [ gtap ] ) exactly holds . in this case it turns out that @xmath131 by a rough argument given below . note that @xmath131 implies @xmath132 , and that the effective transfer function ( [ withd ] ) becomes @xmath133 using eqs . ( [ circulate ] ) and ( [ bessel ] ) and performing the average over the gaussian distribution with unit variance for eq . ( [ scu ] ) , we obtain @xmath134 then , from eq . ( [ gtap ] ) , we have @xmath135 which immediately reconfirms @xmath136 now we observe that _ in the case with @xmath14 the onsager reaction term @xmath137 cancels out with the term @xmath138 that emerges as a result of the evaluation of the correlation between the state of oscillators and the stored patterns . _ then _ we assume eq . ( [ generalgtap ] ) to hold generally so that @xmath131 . _ as will be shown later the results obtained based on this assumption show good agreement with the results of numerical simulations . [ application ] for the sake of simplicity we focus on the behavior of the oscillator network with a discrete natural frequency distribution @xmath139 : @xmath140 where @xmath141 represents the ratio of the number of oscillators with @xmath142 to the total number of oscillators @xmath3 . [ window ] to roughly sketch the effects of the natural frequency distribution with three frequency components , we investigate the behavior of the overlap with change of @xmath143 in the case of @xmath21 . in fig . [ break ] , we give @xmath143-dependence of the overlap calculated from eqs . ( [ sctrf])-([scgamma ] ) and the result of numerical simulations with @xmath144 for the case with @xmath145 and @xmath21 . good agreement between the theory and numerical simulations implies the validity of the present analysis . as is expected , an entrainment indeed occurs in the case of small @xmath143 ( @xmath146 ) , where one has successful retrieval accompanying a large size of overlap . even in the case of large @xmath143 ( @xmath147 ) successful retrieval can be realized with small size of overlap , since natural frequency of @xmath148 oscillators remains 0 . to give an qualitative explanation for the occurrence of a window for break down of the retrieval states , we consider the case with @xmath39 for the moment . we can easily obtain the values of order parameters @xmath149 and @xmath86 as a function of @xmath143 ( see appendix [ a0 ] ) as shown in fig . [ reason](a ) . we see a phase transition to occur at @xmath150 , and @xmath86 is seen to increase as @xmath143 approaches @xmath151 : @xmath152 as @xmath153 , while @xmath154 for @xmath155 owing to the entrainment . even when @xmath156 , such an enhancement of @xmath86 around @xmath151 remains unchanged as can be seen in fig . [ reason](b). noting eq . ( [ scr ] ) we can easily understand the noise variance @xmath157 may be enhanced accordingly in the interval @xmath158 , where retrieval states disappears . in fig . [ breakpd ] we draw the @xmath159 phase diagram to show the behavior of the storage capacity as a function of @xmath143 . the window observed in fig . [ break ] turns to arise from the valley of @xmath160 curves . behaviors of synchronization in the networks of coupled oscillators with white noise differ from those of the deterministic networks with @xmath21 . in the absence of white noise ( @xmath21 ) , one can in general divide the oscillators with @xmath161 into two groups of synchronized and desynchronized oscillators accordingly to the criterion of whether the phase velocity of an oscillation vanishes or not . noting eq . ( [ scm ] ) and the form of effective transfer function with @xmath21 ( eq . ( [ nod ] ) ) illustrated in fig . [ field ] , we find that desynchronized oscillators do not contribute to the value of overlap @xmath162 , though they contribute to the value of other order parameters such as @xmath163 and @xmath86 . this is because we are concerned with the time averaged behavior of the local fields , where the time - averaged phase difference between the desynchronized oscillators with natural frequencies @xmath164 and @xmath165 should be @xmath166 in the absence of white noise ( i e. @xmath167 for @xmath168 ) . in the case with white noise ( @xmath42 ) , however , the phase of each oscillator with @xmath161 evolves with a certain non - zero time - averaged phase velocity , since the action of white noise prevents any oscillators from settling into fixed points . hence it becomes impossible to distinguish between the synchronized and desynchronized oscillators any more . nevertheless a finite value of the overlap is realized because of the existence of the equilibrium probability distribution on @xmath25 that is achieved after a long time with @xmath169 . in fig . [ intensity ] we display the behavior of the overlap as a function of the noise intensity @xmath100 obtained by the theoretical analysis and numerical simulations . as expected , the size of the overlap decreases as the noise intensity increases until the system undergoes a discontinuous transition at a critical noise intensity @xmath170 , above which a disordered state with @xmath171 is realized . good agreement between the two results implies the validity of our treat ment based on the time averaged local fields together with the assumption that the tap - like equation also holds in the case with a distribution of natural frequencies . in fig . [ dpd ] , we give @xmath172 phase - diagram that represents the storage capacity plotted as a function of @xmath143 for various values of @xmath100 . in most of the region of @xmath143 , the storage capacity @xmath173 decreases as the noise intensity increases . we see that for @xmath100 smaller than a certain critical @xmath174 , @xmath160 exhibits non - monotonic behavior with change in @xmath143 , while for @xmath175 @xmath160 is monotonically decreasing with @xmath143 . the occurrence of a window for the break down of the retrieval states with fixed @xmath37 for @xmath176 turns out to be attributed to the appearance of valley caused by the non - monotonic behavior of the @xmath160 curve as in the case of @xmath21 ( fig . [ break ] ) . [ discussion ] we have investigated properties of an associative memory model of oscillator neural networks based on simple phase oscillators , where the influence of white noise together with a natural frequency distribution is considered in the case of an extensive number of stored patterns . in the presence of white noise every oscillator as well as its local field undergoes fluctuating motions even in the stationary state after a long time . to deal with such a situation we have taken an approach based on the concept of the tap - like equation . to approximately derive the tap - like equation for the system without an energy function we have taken the time average for the fluctuating local field of each oscillator neuron to make it constant in time . on the basis of the time - averaged local field we have dealt with the single - body fokker - plank equation to obtain the time averaged outputs of the oscillators in the stationary state , from which we have evaluated the effective transfer function . the relation between the time - averaged output and the local fields involving such a transfer function can be viewed as an analogue of the naive tap - like equation without considering the so called onsager reaction term . we have supposed the proper form of tap - like equation to be given by appropriately adjusting the onsager reaction term such that setting @xmath14 naturally leads to the legitimate tap equation , which we have obtained from the evaluation of the gibbs free energy . applying the scsna to this tap - like equation , we have obtained the macroscopic order parameter equations , based on which the properties of associative memory of the network has been studied . assuming a discrete symmetric natural frequency distribution with three frequency components for the sake of simplicity , we have presented the phase diagram showing the behavior of storage capacity as a function of the parameter @xmath143 representing the width of the natural frequency distribution . in the case of @xmath21 the storage capacity @xmath173 has been found to exhibit non - monotonic behavior as @xmath143 is varied and to attain a minimum at a certain @xmath143 . as a result of the occurrence of the valley in the @xmath177 curve the break down of the retrieval state with fixed @xmath37 occurs for intermediate values of @xmath143 . when noise is present such a behavior has been found to be somewhat relaxed , and only for small values of the noise intensity @xmath100 the phenomenon of the break down of the retrieval state can be observed . our analytical result has shown excellent agreement with the results of numerical simulations . our results show that associative memory based on temporal coding can be realized in the network of simple phase oscillators even in the presence of not only a distribution of natural frequencies but also external white noise . the result that temporal coding is robust against the existence of environmental noise is remarkable . memory retrieval occurs in such a way that the oscillators undergo synchronized motions with the phase difference @xmath31 between any two of the oscillators @xmath7 and @xmath10 settling into , for long times , the difference @xmath178 of the memory pattern @xmath29 . in our model @xmath179 is chosen from uniformly distributed random numbers on @xmath25 . the resultant behavior , however , is qualitatively the same as that for our previous work ( @xmath21)@xcite on the special case where @xmath180 or @xmath166 and hence @xmath9 is given by the well - known form @xmath181 with @xmath182 . a characteristic feature of memory retrieval accompanying synchronization is that , in contrast to networks with fixed point type attractors , each neuron exhibit oscillations in the local field or the membrane potential that are easily detected by other neurons in a certain network to determine whether memory retrieval is successful or not . also worth noting is the appearance of two types of retrieval states with respect to the degree of synchronization : a high degree of synchronization that occurs for small @xmath143 with overlap @xmath162 large and a low degree of synchronization that occurs for large @xmath143 with @xmath162 small . the fundamental assumption we have used in the present study is the existence of the tap - like equation ( [ tapfixedpoint ] ) and ( [ taplocalfield ] ) for our system together with the expression of @xmath83(eq . ( [ generalgtap ] ) ) . in the case of @xmath14 there occurs no problem because the genuine tap equation exists as has been shown . in this case we have found that @xmath83 ( eq . ( [ gtap ] ) ) is exactly canceled out by @xmath105 , as in the case of the network of ags , to yield @xmath131 together with the order parameter equations , that recover the ones by cook@xcite , who analyzed @xmath183-state spin model including the case with @xmath184 for arbitrary temperatures by means of the replica symmetric approximation . in the case with distributed natural frequencies , to obtain the form of the effective transfer function of the tap - like equation , we replace time - dependent local fields @xmath185 in eq . ( [ nativedynamics ] ) by their time - averaged ones , and assume that the onsager reaction term of the form : @xmath186 appears in effective local field as a result of fluctuation of local fields . the assumption of this form of the onsager reaction term yields @xmath131 , which leads to @xmath187 in eqs . ( [ scm])-([scu ] ) . meanwhile the theoretical result we have obtained here is recovered by the tap - like equation evaluated by simply replacing the local fields @xmath185 in eq . ( [ nativedynamics ] ) by the time - dependent local field of the form : @xmath188 where the renormalized local field : @xmath189 is constant in time . note that , by use of this assumption , we can derive the order parameter equations without knowing the form of @xmath83 . in the present case both procedures give the same result , however , in some cases it seems that the later procedure gives the more accurate form of the transfer function . we will discuss details on this point somewhere else . some special cases of the present model have also been investigated by several authors other than cook . arenas et al . @xcite have investigated the case with @xmath39 , where the natural frequency distribution is assumed to obey a gaussian distribution . the result of this case can also be recovered by the present analysis . to our knowledge the case with distributed natural frequencies and @xmath40 was first studied by park et al . @xcite for different synaptic couplings by means of replica calculations based on the energy that is defined so as to satisfy @xmath190 . in our case the @xmath191 takes the form @xmath192 . however this energy does not make any sense because the equilibrium distribution @xmath193 does not satisfy the periodical boundary condition @xmath194 . aonishi et al . @xcite studied the case with @xmath21 and a gaussian distribution for natural frequencies by considering that @xmath154 holds in the set of scsna eqs . based on a different scheme from ours even in the presence of the group of the desynchronized oscillators . yamana et al . also studied the deterministic oscillator network ( @xmath21 ) with a discrete distribution of natural frequencies that stores binary patterns by making an approximation that the motions of the group of desynchronized oscillators do not exert an influence on the behavior of the synchronized oscillators . discarding the effect of desynchronized oscillators corresponds to considering the transfer function that takes the value zero inside the circle with radius @xmath164 ( see fig . [ field ] ) . for a wide class of natural frequency distributions this scheme seems to work to a good approximation in the case with @xmath21 , because the contribution of the desynchronized oscillators to such order parameters as @xmath149 and @xmath86 is small . it is noted , however , that , in the case of @xmath42 , the phase of every oscillator with @xmath161 evolves with a certain non - zero time - averaged velocity and hence one can not distinguish between synchronized and desynchronized oscillators . accordingly for stochastic networks with @xmath42 methods based on neglecting the effect of the desynchronized oscillators will not make sense and one has to deal with all of the oscillators equally as in the present analysis . finally , we briefly discuss the relevance of our results to biologically related models of associative memory . biologically relevant models@xcite should be based on such spiking neurons as the hodgkin - huxley type@xcite and integrate - and - fire type neurons@xcite . a simple integrate - and - fire neuron that is defined by 1-dimensional linear equation except for firing event can be described in terms of phase that is obtained by properly scaling the 1-dimensional output variable . synaptic couplings implimented into spiking neural networks are often assumed to incorporate the so called alpha function@xcite or its variant represented by the dynamics of a certain gating variable@xcite . so , major differences between the simple phase oscillator model we have dealt with on the basis of the diffusive couplings among the oscillators and the spiking model will be the form of the synaptic couplings together with symmetry of an individual oscillator with respect to rotation of the phase variable . while the present model is assumed to take a sinusoidal phase interaction for simplicity , a spiking model with a synaptic interaction based on the alpha function takes the form of pulse like couplings@xcite , which will lead to considering higher harmonics in the phase interaction . a spiking neural network model of associative memory we previously studied using fitzhugh - nagumo neurons exhibits a nearly comparable size of the storage capacity to that of the standard analog network with the transfer function @xmath195 that is larger than the storage capacity of the present model@xcite . it will then be of interest to observe the outcome of introducing higher harmonics in the phase interaction of the simple phase oscillator model . we expect the storage capacity of the phase oscillator network to increase when the higher harmonics is taken into account . such analysis is now under way . the problem of investigating properties of neurons synchronizing the envelope of a burst of spikes is also of interest , but is beyond the scope of the present paper , which aims studying the effects of such heterogeneities as a natural frequency distribution and external noise on the robustness of temporal coding in the oscillator network of associative memory . we consider that taking not only phase but also amplitude as variables for oscillatory neurons will provide a solvable model suitable for studying the case with such synchronization in networks of bursting neurons , which is also under way . one of the authors ( m. yoshioka ) would like to acknowledge the support grant - in - aid for encouragement of young scientists ( no . 4415 ) from the ministry of education . [ tap ] to obtain the tap equation for the present model with the energy function ( [ hamiltonian ] ) we follow the method of plefka@xcite and nakanishi@xcite used for the sk model and neural networks of ising spins . the hamiltonian ( [ hamiltonian ] ) with a complex - valued external field @xmath196 included reads @xmath197 where @xmath141 is introduced for the analysis below . applying legendre transformation to the free energy corresponding to the hamiltonian @xmath198 , one has @xmath199 where @xmath200 and @xmath201 . @xmath202 denotes expectation with respect to the hamiltonian @xmath198 . we perform a taylor expansion with respect to @xmath141 @xmath203 where @xmath204 . noting @xmath205 , we rewrite eq . ( [ gfunc ] ) in the form @xmath206 with @xmath207 where @xmath208 and @xmath209 denotes expectation with respect to the hamiltonian @xmath198 with @xmath210 . noting @xmath211 , from eq . ( [ firstrewrite ] ) , it follows @xmath212 then , from this equation and eq . ( [ firstrewrite ] ) , one has @xmath213 where @xmath214 evaluating @xmath215 by expanding this equation upto third order in @xmath141 yields @xmath216 where it is noted that @xmath217 for every integer @xmath218 . then , noting @xmath219 , we have @xmath220 } a\nonumber\\ & & + { \bigg [ } -\frac{\beta}{16}\sum_{i\neq j}{\big \{}{e_{{i}}{\left ( { { 2},{0 } } \right ) } } { e_{{j}}{\left ( { { 0},{2 } } \right ) } } { c_{{ij}}^{\ast { 2}}}+{e_{{i}}{\left ( { { 0},{2 } } \right ) } } { e_{{j}}{\left ( { { 2},{0 } } \right ) } } { c_{{ji}}^{\ast { 2}}}\nonumber\\ & & + 2{e_{{i}}{\left ( { { 1},{1 } } \right ) } } { e_{{j}}{\left ( { { 1},{1 } } \right ) } } { c_{{ij}}^{\ast { } } } { c_{{ji}}^{\ast { } } } { \big \}}{\bigg ] } a^2\nonumber\\ & & + { \bigg [ } -\frac{\beta^2}{96}\sum_{i\neq j}{\big \ { } { e_{{i}}{\left ( { { 3},{0 } } \right ) } } { e_{{j}}{\left ( { { 0},{3 } } \right ) } } { c_{{ij}}^{\ast { 3}}}+{e_{{i}}{\left ( { { 0},{3 } } \right ) } } { e_{{j}}{\left ( { { 3},{0 } } \right ) } } { c_{{ji}}^{\ast { 3 } } } \nonumber\\ & & + 3{e_{{i}}{\left ( { { 2},{1 } } \right ) } } { e_{{j}}{\left ( { { 1},{2 } } \right ) } } { c_{{ij}}^{\ast { 2}}}{c_{{ji}}^{\ast { } } } + 3{e_{{i}}{\left ( { { 1},{2 } } \right ) } } { e_{{j}}{\left ( { { 2},{1 } } \right ) } } { c_{{ij}}^{\ast { } } } { c_{{ji}}^{\ast { 2}}}{\big \}}\nonumber\\ & & -\frac{\beta^2}{48}\sum_{{\left ( { ijk } \right ) } } { \big \{}{e_{{i}}{\left ( { { 2},{0 } } \right ) } } { e_{{j}}{\left ( { { 1},{1 } } \right ) } } { e_{{k}}{\left ( { { 0},{2 } } \right ) } } { c_{{ij}}^{\ast { } } } { c_{{ik}}^{\ast { } } } { c_{{jk}}^{\ast { } } } + { e_{{i}}{\left ( { { 1},{1 } } \right ) } } { e_{{j}}{\left ( { { 2},{0 } } \right ) } } { e_{{k}}{\left ( { { 0},{2 } } \right ) } } { c_{{ik}}^{\ast { } } } { c_{{ji}}^{\ast { } } } { c_{{jk}}^{\ast { } } } \nonumber\\ & & + { e_{{i}}{\left ( { { 1},{1 } } \right ) } } { e_{{j}}{\left ( { { 1},{1 } } \right ) } } { e_{{k}}{\left ( { { 1},{1 } } \right ) } } { c_{{ij}}^{\ast { } } } { c_{{jk}}^{\ast { } } } { c_{{ki}}^{\ast { } } } + { e_{{i}}{\left ( { { 0},{2 } } \right ) } } { e_{{j}}{\left ( { { 2},{0 } } \right ) } } { e_{{k}}{\left ( { { 1},{1 } } \right ) } } { c_{{ji}}^{\ast { } } } { c_{{jk}}^{\ast { } } } { c_{{ki}}^{\ast { } } } \nonumber\\ & & + { e_{{i}}{\left ( { { 2},{0 } } \right ) } } { e_{{j}}{\left ( { { 0},{2 } } \right ) } } { e_{{k}}{\left ( { { 1},{1 } } \right ) } } { c_{{ij}}^{\ast { } } } { c_{{ik}}^{\ast { } } } { c_{{kj}}^{\ast { } } } + { e_{{i}}{\left ( { { 1},{1 } } \right ) } } { e_{{j}}{\left ( { { 1},{1 } } \right ) } } { e_{{k}}{\left ( { { 1},{1 } } \right ) } } { c_{{ik}}^{\ast { } } } { c_{{ji}}^{\ast { } } } { c_{{kj}}^{\ast { } } } \nonumber\\ & & + { e_{{i}}{\left ( { { 1},{1 } } \right ) } } { e_{{j}}{\left ( { { 0},{2 } } \right ) } } { e_{{k}}{\left ( { { 2},{0 } } \right ) } } { c_{{ij}}^{\ast { } } } { c_{{ki}}^{\ast { } } } { c_{{kj}}^{\ast { } } } + { e_{{i}}{\left ( { { 0},{2 } } \right ) } } { e_{{j}}{\left ( { { 1},{1 } } \right ) } } { e_{{k}}{\left ( { { 2},{0 } } \right ) } } { c_{{ji}}^{\ast { } } } { c_{{ki}}^{\ast { } } } { c_{{kj}}^{\ast { } } } { \big \}}{\bigg ] } a^3\nonumber\\ & & + { \cal o}{\left ( { a^4 } \right ) } , { \label{complex}}\end{aligned}\ ] ] where @xmath221 , and @xmath222 denotes all combination to be taken so that either two of the indexes do not coincide ( note that @xmath223 implies @xmath224 ) . then substituting eq . ( [ hebb ] ) into eq . ( [ complex ] ) yields , in the limit @xmath41 , @xmath225 where @xmath226 . note that all the relevant terms higher than the term of first order in @xmath141 under the limit @xmath41 comes from the following in eq . ( [ complex ] ) @xmath227 since every higher order term than the first order one contains @xmath228 , one may expect that it yields terms of the form of eq . ( [ remain ] ) . summarizing those terms we will have @xmath229 then , noting @xmath230 , we obtain @xmath231 where @xmath232 . in the case of @xmath210 @xmath198 becomes @xmath233 thus we have @xmath234 where @xmath235 , and @xmath236 is just the effective transfer function we introduced in eqs . ( [ circulate]),([density ] ) , and ( [ withd ] ) . considering the case with @xmath237 , from eqs . ( [ aexpansion ] ) and ( [ azero ] ) we finally obtain the tap equation : @xmath238 [ evaluation ] to derive eq . ( [ compare ] ) , we substitute eq . ( [ plainlocalfield ] ) into eq . ( [ assumption ] ) to obtain @xmath239 utilizing the relations @xmath240 , and so on , the fourth term of eq . ( [ substitution ] ) becomes , in the limit @xmath41 , @xmath241 following the almost same scheme the fifth term of the right hand side of eq . ( [ substitution ] ) is shown to vanish in the limit @xmath41 . substituting eq . ( [ assumption ] ) into the left hand side of eq . ( [ substitution ] ) we obtain eq . ( [ compare ] ) . [ miscellaneous ] the eq . ( [ scu ] ) is straightforwardly derived from the definition of @xmath86 by noting @xmath242}}{\bigg \rangle}{\bigg \rangle}}$ ] and performing integration by parts . to show eq . ( [ scr ] ) from eq . ( [ r ] ) it is suffice to prove that @xmath86 is real . to show @xmath86 is real , note the rotationary symmetry structure of the form of transfer function ( [ circulate ] ) as is illustrated in fig . [ field ] . because of this symmetry structure of @xmath243 we also have @xmath244 in the presence of non - zero complex @xmath245 . one also immediately finds @xmath246 and @xmath247 . then it follows that @xmath248 and @xmath249 . on the other hand noting @xmath250 and changing the variables for integration , we have , from eq . ( [ scu ] ) , @xmath251 accordingly we have @xmath252 to conclude that @xmath86 is real . [ a0 ] in the case with @xmath21 and @xmath39 , substituting eq . ( [ nod ] ) into eq . ( [ finite ] ) , we have @xmath253 using eq . ( [ nod ] ) we also obtain , from eqs . ( [ scq ] ) and ( [ scr ] ) , @xmath163 and @xmath86 as a function of @xmath162 : @xmath254 where we have noted @xmath255 , that is obtained by representing the local field with the polar coordinate . i e. @xmath256 .	we study associative memory based on temporal coding in which successful retrieval is realized as an entrainment in a network of simple phase oscillators with distributed natural frequencies under the influence of white noise . the memory patterns are assumed to be given by uniformly distributed random numbers on @xmath0 so that the patterns encode the phase differences of the oscillators . to derive the macroscopic order parameter equations for the network with an extensive number of stored patterns , we introduce the effective transfer function by assuming the fixed - point equation of the form of the tap equation , which describes the time - averaged output as a function of the effective time - averaged local field . properties of the networks associated with synchronization phenomena for a discrete symmetric natural frequency distribution with three frequency components are studied based on the order parameter equations , and are shown to be in good agreement with the results of numerical simulations . two types of retrieval states are found to occur with respect to the degree of synchronization , when the size of the width of the natural frequency distribution is changed .
the interactions of black holes with matter fields are fundamental and important in observational astrophysics . it is well - known that super - radiant phenomenon ( outgoing intensity of matter fields becomes greater than ingoing intensity ) can occur in rotating black hole spacetime . the successive occurrence of super - radiant phenomena leads to instability , which is called `` black hole bomb '' . especially kerr - anti de sitter ( ads ) spacetime leads to successive super - radiant phenomena because it plays the role of the reflection mirror @xcite . also the stability of bh is related to the bh thermodynamics @xcite , which should be defined in equilibrium states . g. t hoot introduced the brick wall model in order to study bh thermodynamics @xcite by imposing dirichlet boundary condition on the event horizon of bh . the boundary condition defines normal modes ( i.e. , bound states ) of matter fields and the sum of normal modes determines the partition function and the entropy of bh . the boltzmann factor of the matter fields in the brick wall model becomes ill - defined , if the super - radiance for rotating bh occurs @xcite . the super - radiant phenomenon is also important relating to the recent works on the star motion @xcite and the radiation from axion @xcite . here we list up important features of the super - radiant phenomena in rotating bh : 1 . the general condition of super - radiant phenomena for scalar fields is @xmath3 where @xmath4 , @xmath5 , @xmath6 and @xmath7 ( defined in sec . 2.3 ) denote the frequency , momentum near the horizon , azimuthal angular momentum for scalar fields and angular velocity of bh , respectively @xcite . usually the super - radiant condition is @xmath8 under the implicit understanding that the frequency is positive @xmath9 . the super - radiance occurring under this condition is classified as type 1 @xcite . 2 . the super - radiant phenomena for massless neutrinos and massive spinor fields do not occur , because the momenta near horizon is treated as positive definite @xmath1 due to the masslessness of neutrinos as well as the positive frequency condition @xmath9 @xcite . we studied the super - radiant phenomena of scalar and spinor fields as scattering problem in kerr spacetime using the bargmenn - wigner formulation @xcite . we derived the consistency condition for the current conservation relation among scalar and spinor fields , in accordance with the general super - radiant condition in equation ( 1.1 ) . * the super - radiance with @xmath10 does not occur for neither scalar nor spinor fields , which is `` type 1 '' in our classification . * the super - radiance with @xmath11 does occur for both scalar and spinor fields , which is `` type 2 '' , in our classification . based on our previous works @xcite , we study in this paper the bound state as normal modes for scalar and spinor fields in four dimensional kerr - ads spacetime and derive the physical spectrum condition for normal modes : @xmath12 . key points are non - existence of zero modes @xmath13 and the analyticity of matter fields with respect to rotation parameter @xmath2 of bh . the organization of this paper is as follows . in section 2 , zero and normal modes are studied for scalar fields in kerr - ads spacetime . the spectrum condition for normal modes is derived . we shall show that the partition function in the brick wall model is well - defined and the super - radiant modes is type 2 . in section 3 , zero and normal modes are studied for spinor fields in kerr - ads spacetime . the spectrum condition for spinor fields is also derived , which turn out to be the same as that for scalar fields . in section 4 , numerical analysis of normal modes for scalar fields in case of @xmath14 ( mass of scalar fields ) will be presented . summary is given in section 5 . in this section we study zero mode and normal modes as bound state problem for scalar fields in four - dimensional kerr - ads spacetime . let us consider the kerr - ads metric in boyer - lindquist coordinates : @xmath15 where @xmath16 , @xmath17 , and @xmath18 denote the mass , rotation parameters of bh , and cosmological parameter , respectively . the event horizon @xmath19 is defined as the outer zero of factor @xmath20 . the explicit expression of metrics @xmath21 are obtained from the line element in eq . ( 2.1 ) given in appendix a. the lagrangian and field equations for complex scalar fields @xmath22 in kerr - ads spacetime is given by : @xmath23 where @xmath24 denotes the mass of scalar fields . if we write @xmath22 as @xmath25 then the scalar field equations are separated into angular and radial field equations respectively : @xmath26 where @xmath27 , and @xmath28 denote the frequency , azimuthal angular momentum of scalar fields , and the separation parameter , respectively . now we study the normal modes as bound states with dirichlet or neumann boundary conditions on the horizon : @xmath29 with asymptotic condition : @xmath30 in eq . ( 2.7 ) as @xmath31 for regularization in case of necessity . ] the following two identities can be obtained from eqs . ( 2.5 ) and ( 2.6 ) : @xmath32 for angular part and @xmath33 for radial part . let us define @xmath34 as follows @xmath35 then from eqs . ( 2.9 ) and ( 2.10 ) we can obtain the following equations : @xmath36 which reduce to the real values of @xmath37 and the orthonormal relations : @xmath38 let us define the inner product notation of @xmath39 and @xmath40 as follows @xmath41 the orthonormal relations are expressed : @xmath42 with the eigenfunctions @xmath43 where @xmath44 and @xmath45 denote set of quantum numbers . let us expand the scalar field @xmath46 and its conjugate momentum @xmath47 in eigenfunctions @xmath48 as follows @xmath49 the inverse relations can be found to @xmath50 using the completeness relations : @xmath51 note that any states , including quasi - normal modes and super - radiant modes , can express in terms of eigenfunctions due to the completeness relation ( 2.18 ) . canonical quantization relations are given by @xmath52 = \left[\phi^{\dagger}(t , x ) , \pi^{\dagger}(t , x')\right]=\frac{i}{\sqrt{-g}}\delta^{(3)}(x - x ' ) , \end{aligned}\ ] ] which lead to the quantization of creation and annihilation operators : @xmath53 = [ b_{\alpha},b_{\alpha'}^{\dagger } ] = \delta^{(3)}_{\alpha , \alpha'}\ .\ ] ] energy and angular momentum of scalar fields are given by @xmath54 which can be also written as @xmath55 the effective energy of scalar field , as measured by the co - rotating observer , is given by @xmath56 where @xmath57 denotes the angular velocity of bh on the horizon @xmath58 . note that for normal modes @xmath59 near the horizon , we have ingoing and outgoing solutions with the momentum @xmath5 with respect to a radial variable @xmath60 as @xmath61 and @xmath62 . the effective frequency @xmath63 is the same in magnitude with the momentum @xmath5 because scalar field is treated as massless near the horizon : @xmath64 now we consider the existence or non - existence of zero mode defined by @xmath65 . radial equation for zero mode becomes : @xmath66 where @xmath67 denotes the frequency of zero mode . the zero mode equation near horizon becomes from eq . ( 2.24 ) : @xmath68 where @xmath69 with @xmath70 . general solution of eq . ( 2.24 ) is given by , using the frobenius method : @xmath71 where @xmath72 denote integration constants . if we impose the dirichlet boundary condition , then from eq . ( 2.26 ) , we find that @xmath73 . therefore non - trivial solution to eq . ( 2.25 ) which satisfies the dirichlet boundary condition does not exist . we can also show the zero mode solution to eq . ( 2.25 ) does not satisfy the neumann boundary condition either . from this result we can derive the spectrum condition of normal modes for scalar fields with dirichlet or neumann boundary conditions as follows : * when the specific rotation parameter @xmath74 , the zero mode line is on @xmath75 and the allowed physical modes are @xmath9 in @xmath76 plane as in the standard scalar field theory in flat minkowski spacetime . * when @xmath77 , the zero mode is defined by the line @xmath65 in @xmath76 plane and the allowed physical modes should satisfy the spectrum condition : @xmath78 the reason is that any normal modes can not across the zero mode line where the radial functions can not satisfying the dirichlet or neumann boundary conditions . in this argument we take account of the analyticity with respect to rotation parameter @xmath2 under conditions that the outer horizon @xmath58 should be well separated zero of @xmath20 from other zeros and the rotation parameter should be less than the cosmological parameter @xmath79 for regularity of @xmath80 and @xmath81 . let us comment on super - radiant modes and the brick wall model . * we can apply the spectrum condition for normal modes to super - radiant modes because any modes can express in terms of eigenfunctions due to the completeness relation in eq . then allowed types of super - radiant modes taking account of general super - radiant condition given in equation ( 1.1 ) are the following two : 1 . @xmath82 is unphysical and does not occur because this type contradicts with the spectrum condition of ( 2.27 ) . 2 . @xmath83 are physical which coincides with both equations ( 1.1 ) and ( 2.27 ) . + in type 2 super - radiance , annihilation operators of particles with @xmath84 are understood as creation operators of antiparticles with @xmath85 as the interpretation in quantized field theory . also , we mention that the results obtained here are consistent with our previous work basen on the bargmann - wigner formulation for scattering problem in kerr spacetime @xcite . * the partition function in the kerr - ads black hole background spacetime becomes well - defined : @xmath86 considering the expression of effective energy ( 2.21 ) and the spectrum condition ( 2.27 ) . next we consider zero and normal modes as bound states of massive spinor fields in kerr - ads spacetime . we study the spectrum condition and investigate the allowed type of super - radiant modes for spinor case , which can be compared with that for scaler case studied in the previous section . in order to study the dirac equation in curved spacetime , the local tetrads @xmath87 are introduced at each point of the general curved spacetime . the line element expressed in equation ( 2.1 ) is given as @xmath88 the relation between the local tetrad and the general curved coordinate defines viervein @xmath89 as @xmath90 , where the greek letters ( @xmath91 ) denote curved spacetime indices , and the latin letters ( i , j , ... ) denote local tetrads . the explicit expressions of vierbeins for kerr - ads spacetime are given in appendix b. the dirac equation in general curved spacetime is given by @xmath92 where @xmath93 , @xmath94 , and @xmath95 denote the spin connection , the anti - symmetric product of dirac gamma matrices , and the mass of spinor fields , respectively . the spin connections are divided into two terms : @xmath96 whose derivation is in appendix c. the first term of the right - hand side of eq . ( 3.3 ) is geometric in origin and the second term has spinorial origin . the totally anti - symmetric product of dirac matrices is denoted by @xmath97 . each term is calculated as @xmath98 then the explicit form of dirac equation in kerr - ads spacetime in the boyer - lindquist coordinate becomes : @xmath99 the representation of dirac matrices are : @xmath100 where @xmath101 are pauli matrices . with the ansatz @xmath102 where @xmath27 denote the frequency and azimuthal quantum number , respectively , the dirac equation becomes @xmath103 where the first term on the right - hand side of eq . ( 3.3 ) is eliminated . furthermore , let us expand the spinor wave function @xmath104 in chiral eigenfunctions @xmath105 as follows : @xmath106 with @xmath107 . then the dirac equation becomes @xmath108 where the second term on the right - hand side of eq . ( 3.3 ) is eliminated . let @xmath109 be such that @xmath110 then the dirac equation reduces to the following set of the ordinary differential equations : @xmath111 where @xmath112 is the separation parameter . the dirac equations of the form ( 3.13)-(3.16 ) are consistent with other previous works for schwarzschild and kerr spacetime @xcite . now we study zero mode for spinor field . radial equations for zero mode @xmath113 become @xmath114 where @xmath115 . near the horizon , zero mode equations ( 3.17 ) become @xmath116 where @xmath117 with @xmath70 . general solutions to eq . ( 3.18 ) are given by @xmath118 where @xmath119 and @xmath120 are integration constants . the dirichlet and neumann boundary conditions for spinor fields require the spacial component ( in the present case , the radial component ) of conserved current on the horizon to vanish : @xmath121 where @xmath122 . the explicit expression for @xmath123 is given by @xmath124 where we used eqs . ( 3.8 ) , ( 3.10 ) and ( 3.12 ) . there are two solutions of @xmath125 , @xmath126 which correspond to the dirichlet and neumann boundary conditions , respectively . here a constant phase factor @xmath127 is introduced , where @xmath128 is in eq . ( note that the additional phase factor reduces to unity in the massless limit : @xmath129 as @xmath130 ) . the zero mode solution in equation ( 3.19 ) satisfies neither ( i ) nor ( ii ) in eq . this means that the zero mode of spinor fields does not exist as physical states . from the non - existence of zero mode for spinor fields we can derive the spectrum condition of normal modes for spinor fields satisfying @xmath12 as in the scalar field case . as a consequence of the spectrum condition , type 1 super - radiance ( @xmath131 ) can not occur whereas type 2 super - radiance ( @xmath132 ) does occur for spinor fields as well as scalar fields . now we present numerical analysis of normal modes for scalar fields in kerr spacetime with the dirichlet boundary condition and with special choice of parameter @xmath14 . in this case the separation parameter is set to the value @xmath133 . parameters in the theory are chosen for the cosmological parameter and the black hole mass as @xmath134 and @xmath135 , and for the frequency , azimuthal quantum number of spinor fields as @xmath136 , @xmath137 . in figure 1 , the zero mode line , the ground state mode with respect to radial component denoted as @xmath138 and the first excited state modes as @xmath139 are shown . we can recognize that the normal modes with dirichlet boundary condition on the horizon lie above the zero mode line and the spectrum condition is realized . vs. m , width=302,height=188 ] zero and normal modes are studied for scalar and spinor fields in kerr - ads spacetime in case of the dirichlet and neumann boundary conditions . * for zero mode ( @xmath65 ) , radial eigenfunctions that satisfy the dirichlet or neumann boundary condition does not exist for both scalar and spinor fields . * the spectrum condition for physical normal modes are in the range of @xmath140 , because ( i ) normal modes are in the positive omega region @xmath9 in no rotating cases @xmath74 and ( ii ) wave functions must be analytic with respect to the rotation parameter @xmath2 . the spectrum condition holds for both scalar and spinor fields . * we apply the spectrum condition for normal modes ( i.e. , bound state problem ) to the super - radiant phenomena , we find that the type 2 super - radiances ( @xmath141 and @xmath142 ) are possible for both scalar and spinor fields . note that the type 1 super - radiances @xmath143 does not occur . * the preliminary numerical analysis with the special choice of parameters : @xmath14 and @xmath144 supports the previous theoretical analysis discussed in sections 2 and 3 . * the partition function in brick wall model is well - defined because of the spectrum condition @xmath140 . note that the result on the super - radiance is consistent with our previous works where the bargmann - wigner formulation was applied to scattering problem for both bosons and fermions in kerr spacetime @xcite , and normal modes for scalar fields in btz spacetime was studied analitically and numerically @xcite . notice also that as to type 2 super - radiance , the annihilation operators of particles with negative frequency and negative azimuthal quantum number correspond to the creation operators of antiparticles with positive frequency and positive azimuthal quantum number as in standard quantum fields theory . this interpretation supports the idea proposed by penrose @xcite . type 2 super - radiance is natural in the classical picture of super - radiant phenomena in rotating spacetime . the metric are obtained from the line element of kerr - ads spacetime in equation ( 2.1 ) as @xmath145 : @xmath146 the determinant of metric is given as @xmath147 the vierbein in kerr - ads spacetime are obtained through the relation @xmath90 with the coordinate relation in equation ( 3.1 ) : @xmath148 the inverse vierbein are obtained by the relation @xmath149 as @xmath150 the definition of the spin connection @xmath151 is obtained from the vierbein hypothesis : @xmath152 as @xmath153 where @xmath154 denotes the christoffel s 3-index symbol . using the identity @xmath155 , the spin connection term in the dirac equation ( 3.2 ) is written as @xmath156 the first term on the right - hand side in eq . ( c.3 ) is written in a compact form as @xmath157 and the second term is written as @xmath158 the sum of eqs . 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one major advantage of soft matter systems in comparison to their atomic counterparts is the ability one has to engineer the constituent particles at the molecular level . in this way , an enormous variety in architectures can be achieved , leading to a corresponding richness in the structural and phase behaviour of such systems . polymers play a prominent example within this class of materials . they come in a variety of forms , such as , e.g. , as linear chains , branched , star - shaped , dendritic , copolymer , as well as functions , e.g. , steric stabilisers , additives , depletants etc . additional flexibility arises from the possibility of influencing the structural and phase behaviour of polymer solutions by changing the solvent quality . from the theoretical point of view , the task of bridging the gap between the microscopic and macroscopic length scales of polymeric systems is a formidable one . indeed , the constituent macromolecules may contain thousands or even millions of atoms , interconnected to one another in complicated ways . the program of starting with the individual interactions between monomers and proceeding to the calculation of the free energy of the system rapidly becomes intractable . however , considerable progress can be made if one invokes a `` coarse - graining '' procedure , a point of view that has been proved very fruitful in many areas of research in condensed matter physics . here we follow a two - step procedure . first , instead of attempting to carry out the statistical trace over all the individual monomers in one step , a certain generalised coordinate of the macromolecule as an effective point particle is invoked . examples include the centre of mass of the polymer or some suitably selected monomer , as we explain below . all monomers belonging to the macromolecules are then traced out for a given , fixed configuration of the effective coordinates , which defines an _ effective interaction _ between these coordinates @xcite . once this is achieved , the second step consists of viewing the the macroscopic system as collection of point particles interacting by means of the effective interaction . now all known tools from the theory of atomic and molecular fluids can be employed to derive structural and thermodynamic quantities for the system under consideration . in the recent years , the program sketched above has been carried out with success for various polymeric systems . it has been found that the effective interactions obtained belong to a new class of ultrasoft potentials which have very unusual properties when compared with the hard - sphere ( hs ) system , the prototype of atomic liquids . in this work , we first present a concise review of these novel properties for the fluid phases and then some new results regarding the rich variety of crystal structures that can be stabilised by ultrasoft potentials . the rest of the paper is organised as follows . in section [ effective.section ] we give the general definition of the effective interaction and present specific examples for a number of systems that have been worked out recently . in section [ mff.section ] we show that the systems described by ultrasoft interactions are well described by a simple mean - field picture in the fluid phase for a wide range of thermodynamic conditions . to further delineate these systems we define two categories of mean - field fluids , the _ strong _ and the _ weak _ ones . the former are characterised by a direct correlation function @xmath0 that satisfies to excellent accuracy the condition @xmath1 in broad regions of the thermodynamics space , where @xmath2 is the interparticle pair potential . the latter only satisfy an approximate mean - field picture for their thermodynamic properties , and not for their structure . in section [ cluster.section ] we demonstrate that strong mean - field fluids can be further divided into two categories : those displaying reentrant melting and those displaying a cascade of clustering transitions , the criterion being set by the positivity of the fourier transform of the effective interaction . in section [ exotic.section ] we discuss the richness of the phase diagrams of weak mean - field fluids , taking the case of star polymers as a concrete example . reentrant melting as well as a wealth of structural phase transitions and exotic crystal phases are all shown to be stemming from the ultrasoftness of the effective interactions . finally , in section [ summary.section ] we summarise and conclude . the effective interaction between flexible , fluctuating aggregates can be given a precise statistical - mechanical definition . let us consider a solution containing @xmath3 polymeric macromolecules , each one of them composed of @xmath4 monomers . the total number of particles in the system is @xmath5 . one starts from the full hamiltonian @xmath6 of the problem , assumed to be known . then , out of the @xmath7 particles in the problem ( in our case all monomers ) , one selects the @xmath3 ones that are to be considered as `` effective particles '' and holds them fixed in some prescribed configuration @xmath8 , where @xmath9 is the position of the @xmath10-th effective particle . afterwards , the @xmath11 remaining particles are canonically traced out and the result of this integration is a constrained partition function @xmath12 . the effective hamiltonian @xmath13 is defined as : @xmath14 where @xmath15 , with the absolute temperature @xmath16 and boltzmann s constant @xmath17 . it can be shown @xcite that such an effective hamiltonian has two useful properties : it preserves the overall thermodynamics of the system and it guarantees that the correlation functions of any order between any of the @xmath3 remaining particles remain invariant , regardless of whether the expectation values are calculated with the original hamiltonian @xmath6 or with the effective one @xmath13 . though the procedure of tracing out the @xmath11 degrees of freedom necessarily generates interactions of all orders between the @xmath3 particles @xcite in many cases it is sufficient to truncate those at the pair level , introducing thereby the _ pair - potential approximation_. the great advantage of employing this point of view is that , in comparison with the original problem , the numbers of particles , @xmath7 , has been reduced by a factor @xmath4 . in addition , whereas in the original problem the pair interactions between the monomers are quite complicated , due to the need of taking into account the connectivity and architecture of the molecule , in the effective description the pair potential is spherically symmetric , depending only on the magnitude of the separation vector between the two effective coordinates . a new picture of the polymers emerges thereby in which the latter can be seen as ultrasoft colloids having a dimension of the order of their radius of gyration @xmath18 . this sets at the same time the characteristic length scale of the effective interaction between them . when the polymers are neutral , @xmath18 is the only length scale appearing in this colloidal description , whereas if they carry charge , additional scales set , e.g. , by the concentration of the solution , the counterions and/or the salt ions also come into play . we examine some characteristic cases below . _ polymer chains . _ two possible choices for the effective coordinates have been investigated thus far . one possibility is to consider the effective interaction @xmath19 between the _ centres of mass _ of the linear chains , when these are kept fixed at a distance @xmath20 from one another . this was first done in the pioneering work of flory and krigbaum @xcite , who found a gaussian interaction between the centers of mass . although the functional form of the flory - krigbaum potential is correct , the dependence of the prefactor on the degree of polymerisation @xmath4 is not : whereas the flory - krigbaum mean - field approach predicts a @xmath21-dependence the prefactor turns out to be @xmath4-independent for sufficiently large @xmath4 , as can be easily seen from examining the polymer second virial coefficient . standard scaling arguments show that the radius of gyration @xmath18 is the only relevant length scale for the dilute and semi - dilute regimes of polymers in a good solvent @xcite . this immediately implies that the second - virial coefficient should scale as @xmath22 , which , in turn , implies that effective interaction must have an amplitude that is independent of @xmath18 @xcite , at least in the scaling limit . a number of simulational @xcite as well as theoretical approaches @xcite involving two self - avoiding chains , reached the conclusion that the aforementioned interaction has a gaussian form . the lack of divergence of this effective interaction at zero separation should not be surprising . indeed , the centres of mass of two polymer chains can coincide without any of the monomers violating the excluded volume conditions . in addition , it can be seen that the effective `` particles '' one chooses for the coarse - grained description of the system do not need to be real particles of the physical system . recently , louis _ et al . _ @xcite have independently carried out state - of - the art simulations involving not just two but @xmath23 chains and varying the number of chains to cover a very broad range of concentrations , ranging from dilute solutions up to nine times the overlap concentration . they confirmed that the effective potential has a gaussian - like form which at zero density can be well approximated by @xmath24 , \label{vcom.eq}\ ] ] where @xmath25 and @xmath26 . for higher densities a superposition of three gaussians provides a very accurate fit @xcite , but the basic shape does not deviate much from the low - density gaussian form . the rather weak density dependence can be shown to arise from the density independent many - body forces @xcite . moreover , the same authors have shown that employing this effective interaction leads to a very accurate description of the thermodynamics ( equation of state ) of polymer solutions for a wide range of concentrations , thus confirming the validity of the idea that polymer chains can be viewed as soft colloids @xcite . an alternative is to consider the end - monomers or the central monomers of the chains as effective coordinates @xcite . general scaling arguments establish that in this case the effective interaction diverges logarithmically with the monomer - monomer separation @xmath20 @xcite . when the central monomers are chosen , linear chains are equivalent to star polymers with @xmath27 arms . motivated by this analogy , jusufi _ et al . @xcite derived the effective interaction @xmath28 by combining monomer - resolved , off - lattice simulations with theoretical arguments . the sought - for interaction features in this case a logarithmic divergence for small separations , in full agreement with the scaling arguments , and crosses over to a gaussian decay for larger ones : @xmath29 } & { \rm for $ r > \sigma_{\rm s}$ } , } \label{pot_ss2}\end{aligned}\ ] ] where @xmath30 and @xmath31 is a free parameter of the order of @xmath32 and is obtained by fitting to computer simulation results . for @xmath27 the value @xmath33 has been obtained @xcite , which , together with the potential in eq . ( [ pot_ss2 ] ) above yields for the second virial coefficient of polymer solutions the value @xmath34 , in agreement with the estimate @xmath35 from renormalisation group and simulations @xcite . _ dendrimers . _ by employing a simple , mean - field theory based on the measured monomer density profiles of fourth - generation dendrimers , a gaussian function of the form ( [ vcom.eq ] ) has been shown to accurately describe the effective interaction between the centres of mass of these dendrimers @xcite . the prefactor @xmath36 has in this case a higher value then for linear polymers , @xmath37 . small - angle neutron scattering ( sans ) profiles from concentrated dendrimer solutions are reproduced very well theoretically , at least below the overlap concentration @xmath38 . _ star polymers . _ by chemically anchoring @xmath39 linear chains on a common core , star polymers with functionality @xmath39 are constructed . in the theoretical analysis of the conformations and the effective interactions of stars , the finite size of the core particle is ignored , an excellent approximation when the chains are long . the natural choice for the effective coordinates is now the position of the central particle , i.e. , of the star centre . for small functionalities , @xmath40 , jusufi _ et al . _ @xcite have shown that a logarithmic - gauss potential of eq . ( [ pot_ss2 ] ) accurately describes the effective interaction . the decay parameter @xmath41 of the gaussian is @xmath39-dependent , for details see ref . @xcite . for larger functionalities , @xmath42 , the daoud - cotton @xcite blob picture of the stars is valid and the star - star interaction potential @xmath43 reads as @xcite : @xmath44 } & { \rm for $ r > \sigma_{\rm s}$ } , } \label{pot_ss}\end{aligned}\ ] ] with the `` corona diameter '' @xmath45 . both star - star potentials , the one valid for @xmath40 , eq . ( [ pot_ss2 ] ) and the one valid for @xmath42 , eq . ( [ pot_ss ] ) , show an ultrasoft logarithmic divergence as @xmath46 . the strength of the divergence is controlled by the functionality @xmath39 , so that at the formal limit @xmath47 the interaction ( [ pot_ss ] ) tends to the hs - potential . _ polyelectrolyte stars . _ if the polymer chains of a star polymer contain ionisable groups , the latter dissociate upon solution in a polar ( aqueous ) solvent , leaving behind charged monomers and resulting in a solution consisting of charged star polymers and counterions . the resulting macromolecules are called polyelectrolyte ( pe ) stars . in pe stars the chains are stretched due to the coulomb repulsion of the charged monomers . the degree of stretching and condensation of counterions on the rods depends on the amount of charge and on the bjerrum length . for moderate to high charging fractions , the effective interaction between the centres of the pe - stars have been analysed recently by means of computer simulations and theory @xcite . this interaction , @xmath48 , is dominated by the entropic contribution of the counterions that remain trapped within the star corona . for a broad range of functionalities and charge fractions , it can be accurately described by the fit : @xmath49 + \frac{2}{5}\left[\left(\frac{r}{\sigma}\right)^{2-\zeta } -1\right ] + \frac{3(1-\zeta)}{5(1+\kappa \sigma ) } & { \rm { for $ r\leq \sigma$ } } ; \\ \frac{3(1-\zeta)}{5(1+\kappa \sigma ) } \left(\frac{\sigma}{r}\right ) \exp[-\kappa(r-\sigma ) ] & { \rm { for $ r\geq \sigma$ } } , } % \\ \label{vpes.eq}\end{aligned}\ ] ] where @xmath23 is the number of counterions , @xmath50 the corona diameter of the pe - star , and @xmath51 the inverse debye screening length due to the free counterions . finally , @xmath52 and @xmath53 are fit parameters , where @xmath54 . the last condition ensures that the potential of eq . ( [ vpes.eq ] ) above tends to a finite value as @xmath46 . hence , once more we are dealing with an ultrasoft interaction that varies slowly as the particle centres approach one another . ultrasoft interactions therefore describe quite a number of different systems . their common characteristic is that the constituent particles are polymers of various architectures that dominate the spatial extent of the aggregates . in other words , one expects similar ultrasoft interactions to show up also when one deals with core - shell particles , consisting of a solid core and a polymeric shell , whenever the thickness of the latter greatly exceeds the radius of the former . in addition , the ultrasoft interactions can be tuned by controlling the number of arms , the charge , the length of the chains , the generation number ( in the case of dendrimers ) etc . hence , it is useful to explore the general characteristics of this family of potentials and the ramifications they have on the structural and thermodynamic properties of the fluid and crystal phases of such systems . motivated by the fact that effective interactions between polymeric colloids can be _ bounded _ ( i.e. , finite at all separations @xmath20 ) , we examine here in general the properties of systems characterised by pair potentials of the form @xmath55 with @xmath56 for all @xmath57 . in eq . ( [ bounded.eq ] ) above , @xmath36 is an energy scale and @xmath50 a length scale . moreover , @xmath2 is non - attractive , i.e. , @xmath58 everywhere . we introduce dimensionless measures of temperature and density as @xmath59 where @xmath17 is boltzmann s constant and @xmath60 is the density of a system of @xmath4 particles in the volume @xmath61 . we will refer to @xmath62 as the ` packing fraction ' of the system . the key idea for examining the high temperature and/or high density limit of such model systems in three and higher dimensions is the following . we consider in general a spatially modulated density profile @xmath63 which does not vary too rapidly on the scale @xmath50 set by the interaction . at high densities , @xmath64 , the average interparticle distance @xmath65 becomes vanishingly small , and it holds @xmath66 , i.e. , the potential is extremely long - range . every particle is simultaneously interacting with an enormous number of neighboring molecules and in the absence of short - range excluded volume interactions the excess free energy of the system @xcite can be cast in the mean - field approximation ( mfa ) to be equal to the internal energy of the system @xcite : @xmath67 \cong { { 1}\over{2 } } \int\int { \rm d}^3 r { \rm d}^3 r ' v(\|{\bf r } - { \bf r'}\| ) \rho({\bf r } ) \rho({\bf r ' } ) , \label{dft.mfa}\ ] ] with the approximation becoming more accurate with increasing density . then , eq . ( [ dft.mfa ] ) immediately implies that in this limit the direct correlation function @xmath0 of the system , defined as @xcite @xmath68}\over { \delta \rho({\bf r } ) \delta \rho({\bf r ' } ) } } , \label{dcf.dft}\ ] ] becomes independent of the density and is simply proportional to the interaction , namely @xmath69 using the last equation , together with the ornstein - zernike relation @xcite , we readily obtain an analytic expression for the structure factor @xmath70 of the system as @xmath71 where @xmath72 is the fourier transform of @xmath73 . bounded and positive - definite interactions have been studied in the late 1970s by grewe and klein @xcite . the authors considered a kac potential of the form : @xmath74 where @xmath75 is the dimension of the space and @xmath76 is a parameter controlling the range _ and _ strength of the potential . moreover , @xmath77 is a nonnegative , bounded and integrable function . grewe and klein showed rigorously that at the limit @xmath78 , the direct correlation function of a system interacting by means of the potential ( [ kac ] ) is given by eq . ( [ mfa ] ) above . the connection with the case we are discussing here is straightforward : as there are no hard cores in the system or a lattice constant to impose a length scale , the only relevant length is set by the density and is equal to @xmath79 in our model and by the parameter @xmath80 in model ( [ kac ] ) . in this respect , the limit @xmath78 in the kac model of grewe and klein is equivalent to the limit @xmath81 considered here . although the limit of grewe and klein corresponds to @xmath82 and @xmath83 , the relation ( [ mfa ] ) has been shown to be an excellent approximation at arbitrarily low temperatures for high enough densities @xcite and for temperatures @xmath84 practically at _ all densities _ @xcite . hence , the mean - field approximation is valid in a vast range of the thermodynamic space of such systems , which has led to their characterisation as _ mean - field fluids _ ( mff ) @xcite . associated with the structural relation ( [ mfa ] ) are scaling relations of thermodynamic quantities , arising from the compressibility sum rule @xcite : @xmath85 where @xmath86 , and the primes denote the second derivative . from eqs . ( [ mfa ] ) and ( [ compress.sum.rule ] ) it then follows @xmath87 with @xmath88 . this simple scaling is not at all equivalent to a second virial theory . in fact , simple virial expansions have a rather small radius of convergence for mean field fluids @xcite . it then follows that the excess pressure , chemical potential and compressibility satisfy the scaling relations @xmath89 , @xmath90 and @xmath91 @xcite . in the semi - dilute regime the density dependence of the pair - potentials is necessary to properly describe this correction to simple mff behaviour @xcite . but this in turn implies that the extra factor @xmath92 in the scaling of the pressure arises from the many - body interactions , since these are what cause the density - dependence in the first place @xcite . ] all these stem from the validity of the strong _ structural_-mfa relation ( [ mfa ] ) which guarantees the validity of the weaker _ thermodynamical _ mfa relation ( [ mfa.fren ] ) . in what follows , we will argue that many ultrasoft systems can still satisfy the thermodynamic relation ( [ mfa.fren ] ) approximately , _ without _ satisfying eq . ( [ mfa ] ) . to distinguish between the two classes , we now qualify the term `` mean - field fluids '' and call the ones for which both the structural and thermodynamic mfa work well _ strong mean - field fluids_. for such systems , for which the potentials are typically also bounded , the density functional of eq . ( [ dft.mfa ] ) has been extended to mixtures @xcite allowing a straightforward and transparent analysis of the phase separation in the bulk , interfacial @xcite and wetting properties @xcite of such mixtures . let us now then turn our attention to interaction potentials such as those given in eqs . ( [ pot_ss2 ] ) and ( [ pot_ss ] ) . these diverge at the origin slowly enough , so that the three - dimensional integral @xmath93 is finite and equal ( by definition ) to the value of the fourier transform of the potential @xmath94 at @xmath95 . it is now impossible to satisfy the strong mean - field condition of eq . ( [ mfa ] ) everywhere . indeed , the direct correlation function @xmath0 has to remain finite at @xmath96 , whereas the pair potential diverges . hence , as shown in fig . [ cofr.plot](a ) , there will always exist a region in the neighborhood of the origin in which eq . ( [ mfa ] ) is violated . at the same time , it can be seen in this figure that the extent of this region shrinks with increasing density , hence the fluid becomes more ` strong mean - field'-like as it gets denser . the smaller @xmath97 , the lower the density at which the mfa for @xmath70 , eq . ( [ sofq.analytic ] ) , becomes a reasonable approximation . star polymer fluid at various densities , obtained by solving the rogers - young closure , with the mean - field result , @xmath98 . ( b ) same as in ( a ) but for the quantities @xmath99 and @xmath100.,width=226,height=230 ] star polymer fluid at various densities , obtained by solving the rogers - young closure , with the mean - field result , @xmath98 . ( b ) same as in ( a ) but for the quantities @xmath99 and @xmath100.,width=226,height=230 ] the discrepancies between @xmath0 and @xmath98 become much less important when we turn our attention to the thermodynamics . to obtain the excess helmholtz free energy , one needs only the _ integral _ of @xmath99 , see eq . ( [ compress.sum.rule ] ) . as demonstrated in fig . [ cofr.plot](b ) , upon multiplication with the geometrical factor @xmath101 , the deviations of @xmath0 from @xmath98 become suppressed , so that we can write , to a very good approximation : @xmath102 eq . ( [ appr.integral ] ) together with eq . ( [ compress.sum.rule ] ) yield an approximate scaling of the excess free energy of the weak mean - field fluids with density that is identical to that of the strong mean - field fluids , eq . ( [ mfa.fren ] ) . the accuracy of the approximation for the star polymer fluid with @xmath103 arms [ eq . ( [ pot_ss ] ) ] is shown in fig . [ fxliq.plot ] . the line labeled as exact free energy there was obtained by solving the rogers - young closure @xcite for the fluid at a wide density range and subsequently utilizing the compressibility sum rule [ eq . ( [ compress.sum.rule ] ) ] to obtain the excess free energy . comparisons with simulations @xcite have indeed demonstrated that this procedure delivers an essentially exact numerical result . star fluid . the slope of the straight line is 2 , indicating the quadratic dependence of the excess free energy density on particle density . the arrow indicates the location of the crossover density @xmath104 , above which the scaling of eq . ( [ mfa.fren ] ) holds with a relative error of less than @xmath105,width=264,height=264 ] clearly , the mean - field approximation improves with increasing density , as the number of particles effectively interacting with one another grows . the crossover density @xmath104 above which the quadratic scaling of the free energy holds is @xmath39-dependent and grows with increasing @xmath39 . indeed , the functionality acts as a prefactor that controls the strength of the logarithmic divergence of the potential at the origin . formally , the mean - field approximation also becomes better with growing spatial dimension @xmath75 , as the geometrical prefactor @xmath106 multiplying @xmath0 and @xmath98 suppresses the small-@xmath20 discrepancies of the two more efficiently . we call systems for which the mean - field idea holds only for the thermodynamics _ weak mean - field fluids_. if one navely applies the _ strong _ mean - field relation , eq . ( [ mfa ] ) , to _ weak _ mean - field fluids , one obtains results for the structure factor @xmath107^{-1}$ ] that can be seriously in error for finite @xmath108-values . _ only _ at @xmath109 and at sufficiently high densities is it a reasonable approximation to set @xmath110^{-1}$ ] . in this section , we turn our attention to the phase behaviour of strong mean - field fluids . two representatives of this class whose phase behaviour has been studied in detail are the gaussian core model ( gcm ) of eq . ( [ vcom.eq ] ) and the ` penetrable sphere model ' ( psm ) characterised by the interaction potential @xmath111 , with the heaviside step function @xmath112 . clearly , the psm reduces to the hard sphere model for @xmath113 . the gcm has been the subject of extensive investigations by stillinger _ et al . _ in the late 1970s @xcite . the @xmath113 phase diagram of the model was calculated , showing the existence of two stable crystal structures , fcc for low densities and bcc for high ones . in addition , a host of mathematical relations for the gcm has been established and on the basis of free energy estimates is has been postulated that the system displays reentrant melting behaviour at low temperatures . on the basis of simulation studies at selected thermodynamic points , a rough phase diagram of the gcm has been drawn @xcite . a detailed study of the structural and phase behaviour of the gcm was carried out recently by lang _ there , it was indeed shown that for temperatures @xmath114 the system remains fluid at all densities , whereas for @xmath115 reentrant melting is observed : increasing the density , the system first undergoes a fluid @xmath116 fcc transition , followed by a structural fcc @xmath116 bcc transition and at higher densities the bcc solid remelts , i.e. , a bcc @xmath116 fluid transition takes place . the width of the solid - phase region grows with decreasing temperature . a structural signature of this unusual phase diagram in the fluid phase above @xmath117 is an anomaly in the behaviour of the liquid structure factor @xmath70 . the height of its main peak first grows with increasing density and after achieving a maximum , it decreases again , reflecting the stability of the fluid beyond the reentrant melting . the hansen - verlet freezing criterion @xcite was shown to be satisfied at _ both _ the freezing and the reentrant melting lines @xcite . moreover , it was found that at high densities not only the mean - field relation , eq . ( [ mfa ] ) is satisfied to excellent accuracy but also that the hypernetted chain closure ( hnc ) becomes quasi - exact @xcite . the system becomes ` quasi - ideal ' at those densities , meaning that the radial distribution function has the limiting behaviour @xmath118 . this , in conjunction with the mean - field property @xmath1 and the exact relation @xmath119 $ ] , forces the bridge function to obey the limit @xmath120 , hence rendering the hnc exact . the psm was first studied in detail by by means of integral equation theories , computer simulations and cell - model calculations at small temperatures , @xmath121 @xcite . in contrast to the gcm , no reentrant melting was found . instead , the freezing line of the system appeared to persist at all temperatures , and cascades of clustering transitions in the solid were found , in which solids with multiple site occupancies are stable with increasing temperature and density . these findings were independently confirmed in a density - functional study of the low - temperature phase behaviour of the psm @xcite . sophisticated integral - equation approaches at arbitrarily high temperatures revealed a loss of the solution along the ` diagonal ' @xmath122 of the phase diagram @xcite , again a feature pointing to an instability of the liquid by increasing density at arbitrarily high temperatures . thus , the psm and the gcm show completely different phase behaviours , although they are both bounded and non - attractive potentials . there is a cascade of clustering transitions for the former , enabling freezing at all temperatures , and a reentrant melting for the latter , associated with the inability to stabilise crystals above a certain critical temperature . the key in understanding these two very different types of behaviour lies in the strong mean - field character of these fluids and the associated expression for the fluid structure factor , eq . ( [ sofq.analytic ] ) . if the fourier transform of the pair interaction @xmath123 has oscillatory behaviour ( i.e. , if @xmath124 becomes negative for some @xmath108-values ) , then at the wavenumber @xmath125 where @xmath126 attains its most negative value , @xmath127 , the liquid structure factor @xmath128 will display a maximum . for any given temperature @xmath129 , there exists then a density @xmath130 such that @xmath131 , causing a divergence of the fluid structure factor at @xmath132 and marking a ` spinodal line ' at this finite wavenumber . thus , the fluid can not be stable at all densities . as a matter of fact , freezing will take place before the spinodal line is reached . the psm clearly belongs to this category , since the abrupt jump of the pair interaction @xmath133 at @xmath134 causes long - range oscillations of the potential in fourier space . by employing the hansen - verlet criterion , @xmath135 , it has been found @xcite that in the psm freezing takes place at the ` diagonal ' @xmath122 on which the integral equation approach of fernaud _ et al . _ @xcite breaks down . if , on the other hand , the fourier transform of the potential , @xmath124 is a positive - definite , monotonically decreasing function of @xmath108 , eq . ( [ sofq.analytic ] ) assures that at sufficiently high temperatures , where the mean - field approximation is valid at all densities @xcite , @xmath70 is a monotonic function of @xmath108 approaching rapidly the value @xmath136 with increasing @xmath108 and being deprived of any peaks . the lack of peaks in the structure factor implies the lack of any tendency within the liquid towards spontaneous formation of spatially modulated patterns . thus , the fluid remains stable for all densities at sufficiently high temperatures . this , combined with the observation that at low temperatures and densities bounded potentials all become hard - sphere like and hence they must cause a freezing transition there , leads to a reentrant - melting scenario for such systems . clearly , the gcm belongs to this category . representative results for a particular family of strong mean - field fluids and schematic phase diagrams can be found in ref . @xcite . we now focus on weak - mean field fluids , for which no simple criterion for their freezing behaviour can be established , since eqs . ( [ mfa ] ) and ( [ sofq.analytic ] ) are not satisfied any more . the star - polymer fluid characterised by the pair potential of eq . ( [ pot_ss ] ) is a case in point . , for which indeed the interaction of eq . ( [ pot_ss ] ) holds , and not the low - functionality case for which the interaction of eq . ( [ pot_ss2 ] ) is valid . the reason is that at low functionalities the stars do not freeze at any density and hence they are not an appropriate system for considering thermodynamically stable crystals . ] the physical system of star polymers provides an excellent testbed for the investigation of the thermodynamic stability of more complicated crystals than usual fcc- and bcc - lattice arrangements . the fcc - lattice is the one favoured by hard interactions , since it has the property of maximising the available volume and hence the entropy of the particles for a given particle density @xcite . on the other hand , the presence of ` soft tails ' in the potential has the effect of favouring the more open bcc - lattice , as was convincingly demonstrated for the case of the screened coulomb potential ( yukawa interaction ) arising in charge - stabilised colloidal suspensions @xcite . these two common lattices were considered for a long time to be the only ` candidates ' in a search for stable crystals for given interatomic potentials . however , in modern colloidal science , new possibilities open up . it is technically possible to manufacture micelle - like particles featuring a hard core and a soft , fluffy corona of grafted or adsorbed polymer chains , with the thickness @xmath137 of the latter being much larger than the radius @xmath138 of the former . star polymers correspond to the case @xmath139 ; the theoretical arguments leading to the effective potential of eq . ( [ pot_ss ] ) are in fact based on the assumption @xmath140 . under these physical circumstances , the effective interaction between the micelles is dominated by the ultrasoft repulsion between the overlapping , flexible coronas and _ not _ by the excluded volume interactions of the hard cores . thereby , the requirement of maximising of the volume available to each particle does not play the decisive role any more . these properties manifest themselves in the phase diagram of star - polymer solutions . due to the irrelevance of the temperature for this entropic interactions , the phase diagram was drawn in the ( @xmath141)-plane by watzlawek _ et al . _ @xcite , where @xmath142 and @xmath60 is the number density of @xmath4 stars in the volume @xmath61 . the phase diagram is shown in fig . [ stars.phdg.fig ] . the fluid phase remains stable at all concentrations for @xmath143 , a result that confirms and makes precise early scaling - argument predictions of witten _ et al . _ @xcite . for @xmath144 and at packing fractions @xmath145 , the usual fcc- and bcc - crystals are seen to be stable , the former for larger and the latter for smaller functionalities . this is consistent with the fact that the effective potential of eq . ( [ pot_ss ] ) has a yukawa decay length scaling as @xmath146 , hence large @xmath39 is analogous to the strongly screened charge - stabilised colloidal suspensions . however , for @xmath147 , unusual crystal structures appear . first , in the domain @xmath148 , a body - center - orthogonal ( bco ) crystal is thermodynamically stable . the bco - lattice is characterised by a body - centered , orthogonal conventional unit cell with three unequal sides and reduces to the bcc - lattice for ratios @xmath149 between the sides and to the fcc for ratios @xmath150 @xcite . the bco - lattices appearing in this region of the phase diagram feature strongly anisotropic unit cells with typical size ratios @xmath151 . thus , these are crystals with coordination number 2 . for packing fractions @xmath152 , the diamond lattice with coordination number 4 turns out to be stable . thus , we see that _ very open _ structures with their characteristically low coordination numbers are stabilised by the ultrasoft star - star potential . this feature has been attributed to the very slow divergence of the interaction as @xmath46 , combined with its crossover to a yukawa - form for @xmath153 @xcite . indeed , in such circumstances it may be energetically preferable for the system to have a small number of nearest neighbours at a small distance from any given lattice point than a large number of neighbours at a greater distance , as is the case for the optimally - packed fcc - lattice . in the fluid there exist clear structural signatures both for the topology of the phase diagram of fig . [ stars.phdg.fig ] and for the variety of the crystal phases featured there . the reentrant transition from a fluid to a bcc - lattice and then again to a fluid , occurring for @xmath154 , is manifested in fluid structure factors @xmath155 that show a main peak that first grows with increasing density and then drops again @xcite , as in the case of the gaussian core model mentioned above . moreover , once more the hansen - verlet freezing criterion @xcite was found to be satisfied on both sides of the freezing and reentrant melting line . the radial distribution function @xmath156 of the fluid at various densities , on the other hand , carries the signature of a local coordination that resembles that of the thermodynamically neighbouring solids . to demonstrate this , we show in fig . [ gofr.fig ] the function @xmath156 for star polymer fluids at @xmath103 , which are thermodynamically stable , at packing fractions @xmath157 and @xmath158 . comparison with fig . [ stars.phdg.fig ] shows that the former corresponds to a state at the vicinity of the bco - phase and the latter to one at the vicinity of the diamond phase . consider now the average coordination number @xmath159 in the fluid phase , defined as @xmath160 where @xmath161 is the position for which @xmath156 has its first minimum and is indicated by the arrows in fig . [ gofr.fig ] . for the two packing fractions shown we obtain @xmath162 at @xmath157 and @xmath163 at @xmath158 . the first is very close to the coordination number @xmath164 of the neighboring bco - lattice and the latter to @xmath165 of the diamond lattice . the fluid distribution functions contain local correlations that point to the ordering of the incipient crystal phases . at two different packing fractions , @xmath157 [ ( a ) ] and @xmath158 [ ( b ) ] . the arrows indicate the positions @xmath161 that define the borderline of the first coordination shell in the fluid phase.,width=226,height=230 ] at two different packing fractions , @xmath157 [ ( a ) ] and @xmath158 [ ( b ) ] . the arrows indicate the positions @xmath161 that define the borderline of the first coordination shell in the fluid phase.,width=226,height=230 ] the @xmath156 s of the fluid above the overlap density , @xmath166 , show in addition anomalous behaviour featuring two distinct length scales , as is clear from fig . [ gofr.fig](b ) . as analysed in detail in ref . @xcite , two characteristics of the interaction are responsible for this behaviour : on the one hand , the existence of the crossover of the interaction of eq . ( [ pot_ss ] ) at @xmath134 from a logarithmic to an exponentially decaying form . and on the other , the ultrasoftness of the logarithmic potential , allowing the existence of fluids at arbitrarily large densities ( for @xmath167 ) , a feature unknown for the usual interactions appearing in liquid - state physics and which are all ` perturbations ' of the hard - sphere potential ( e.g. , lennard - jones , inverse - powers etc . ) thus , ultrasoft potentials carry unique structural signatures which should in principle be visible in scattering experiments from soft , polymeric fluids . a great deal of insight into the general physical mechanisms driving the stability of open structures in soft systems was gained through the recent work of ziherl and kamien @xcite . they considered in full generality systems with particles composed of a hard core and long , deformable coronas and argued as follows . at any given density above the overlap concentration , the coronas are forced to overlap and compress , which gives rise to an entropic free energy cost . the volume available to the coronas is fixed and equal to the difference of the total volume minus that occupied by the hard cores . denoting by @xmath75 the thickness of the coronal layer and @xmath168 the total area of an imaginary membrane separating the compressed coronas , it then turns out that the product @xmath169 is constant . as the free energy cost for the compression of the chains increases with decreasing thickness @xmath75 , it turns out that favourable phases are those for which the interfacial area @xmath168 is minimal . thereby , the problem reduces to that of determining the ordered arrangement of point particles that generates wigner - seitz ( ws ) cells having the smallest possible area for a given density . it is then conceivable that the fcc - lattice will be unfavoured , since its ws cell has a larger area than that of the bcc , for instance . in this way , ziherl and kamien established a beautiful connection of this problem with lord kelvin s celebrated question of determining the area - minimising partition of space for an arrangement of soap bubbles of equal volume @xcite . following these arguments , it then turns out that there exists yet another candidate phase that has an area even smaller than the bcc - lattice @xcite , namely the a15-lattice @xcite . self - assembled micelles of dendritic molecules with a particular architecture have been experimentally seen to crystallise into this phase @xcite . the conventional unit cell of the a15-lattice is shown in fig . [ a15.fig ] . it can be thought of as the cell of a bcc - lattice ( dark points ) decorated with ` dimers ' ( light points ) running along the middle of the faces . the dimers are oriented parallel on opposite phases , forming thus columns through space . the orientation of the dimers lying on intersecting faces is perpendicular to one another , so that one third of all dimers lies along each of the three cartesian directions in space . the dimer length is @xmath170 , where @xmath171 is the edge length of the cube , and it is placed symmetrically along the face , i.e. , the distance of any monomer to its nearest edge is @xmath172 . the a15-lattice is not a bravais lattice ; it can be constructed as a simple cubic ( sc ) arrangement with an eight - member basis , thus it contains 8 sites per conventional cell . its ws - cell is a goldberg decatetrahedron consisting of two hexagonal and 12 pentagonal faces @xcite . semi - quantitative calculations on the stability of the a15-lattice were carried out by ziherl and kamien @xcite , using a model ` square - shoulder ' potential within a simplified cell model . narrow regions in thermodynamic phase space were found , in which the a15-lattice was stable but this finding is uncertain in view of the approximations involved and the limited extent of this region . to investigate this question in more detail , we have employed extensive lattice - sum calculations for the star - polymer system , using the pair potential of eq . ( [ pot_ss ] ) and extending both the set of candidate lattices and the region of densities we looked at . we compared between the sc , diamond , bco ( which includes the fcc- and bcc - lattices as special cases ) and a15-lattices in the regions @xmath173 and @xmath174 , at selected arm numbers @xmath39 to be shown below . the lattice sums were performed by fixing the particles at the prescribed lattice positions and keeping them frozen there , i.e. , no thermal fluctuations ( harmonic corrections ) were taken into account . this approach reproduces very well the solid part of the phase diagram of fig . [ stars.phdg.fig ] : the free energy of the star - polymer crystals turns out to be dominated by the lattice - sum term , the corrections to it from the particle oscillations as well as the entropic contribution from the same playing only a minor role . in this way , we are of course unable to compare the solid free energies with those of the fluid , therefore no prediction about melting can be made . however , for large enough @xmath39 , the interaction is steep enough , so that the system will be definitely be in a crystalline phase for which the lattice sums provide an accurate prediction of the most stable structure among the candidates . for the bco - phases we minimised the lattice sums with respect to the two size ratios , @xmath175 and @xmath176 between the edge lengths , @xmath171 , @xmath177 and @xmath178 at any given density . without loss of generality , we assume in what follows that @xmath171 is the longest of the three edges , thus @xmath179 . the minimised bco - energy was then compared with the energies of all other lattices ; the one with the smallest lattice energy per particle wins . ) . the phase denoted bco@xmath180 in this figure is the same as the one denoted bco in fig . [ stars.phdg.fig ] but now it has to be distinguished from the additional bco - phases showing up at higher densities and named bco@xmath181 , bco@xmath182 and bco@xmath183.,width=377 ] the ` zero - temperature ' phase diagram of the system . the latter is an irrelevant thermodynamic variable because the entropic interaction of eq . ( [ pot_ss ] ) is proportional to @xmath184 and the thermal energy is the _ only _ energy scale of the problem . ] obtained this way is shown in fig . [ latticesums.fig ] . first , we note that the phases being stable up to @xmath185 are precisely those also seen in the finite - temperature phase diagram of fig . [ stars.phdg.fig ] and also that the phase boundaries based on the lattice sums agree very well with the ones at finite temperatures . the a15-phase does not alter the hitherto explored part of the phase diagram . however , for @xmath186 , a host of new phases and of transitions between those show up . the a15-phase is stable at the high - density part of the phase diagram , confirming thus explicitly the prediction of refs . @xcite and @xcite that this phase is a suitable candidate at the high concentrations of ultrasoft particles . nested between the stability domain of the a15-lattice and the diamond lattice , four new bco - phases and ( iso)structural transitions between those show up , having the following characteristics . the phase denoted bco@xmath181 has size ratios @xmath187 and @xmath188 , the former being constant at all densities and functionalities and the latter showing very weak variation , see also fig . [ ratios.fig ] . hence , the unit cell of the bco@xmath181-phase is anisotropic only in one cartesian direction and has a wide basis and height that is smaller than the base edge length . accordingly , the coordination number of this phase is 2 . the bco@xmath182-phase , dominating at high functionalities , has size ratios @xmath189 which also show very little variation with @xmath62 and @xmath39 , see fig . [ ratios.fig ] . similarly to the bco@xmath181-phase , therefore , the cell of the bco@xmath182-phase also has anisotropy in only one cartesian direction . contrary to it , however , the height of the cell is now _ larger _ than the edge length of the base and therefore the number of nearest neighbours is 4 . the phase transition bco@xmath181 @xmath116 bco@xmath182 is first order : as can be seen in figs . [ ratios.fig](c ) and ( d ) , the size ratios jump abruptly from the values ( @xmath190 ) of bco@xmath181 to the values ( @xmath191 ) of bco@xmath182 . these are the only stable bco - phases in this part of the phase diagram as long as @xmath192 . between the diamond and the a15-phase , within their domains of stability . also shown is the characterisation of those phases . the four different functionalities are indicated on the plots.,width=226,height=230 ] between the diamond and the a15-phase , within their domains of stability . also shown is the characterisation of those phases . the four different functionalities are indicated on the plots.,width=226,height=230 ] between the diamond and the a15-phase , within their domains of stability . also shown is the characterisation of those phases . the four different functionalities are indicated on the plots.,width=226,height=230 ] between the diamond and the a15-phase , within their domains of stability . also shown is the characterisation of those phases . the four different functionalities are indicated on the plots.,width=226,height=230 ] for functionalities @xmath193 , two more bco - phases show up . first , there is a narrow region occupied by the bco@xmath183-phase , which has all three edge lengths of its unit cell different and is thus similar to the bco@xmath180-phase discovered before . as seen in figs . [ ratios.fig](a ) , ( b ) and ( c ) , these ratios evolve from the values @xmath187 and @xmath188 of the bco@xmath181-phase towards the values @xmath194 and @xmath195 of the fcc - phase . the order of the transitions between those phases has been investigated numerically . within the limits of accuracy of our numerical code , we have found that for @xmath196 and @xmath197 the transition bco@xmath183 @xmath116 fcc is first order , i.e. , the size ratios of the bco@xmath183-phase jump with density to the ratios @xmath198 of the fcc - phase abruptly . this is seen in figs . [ ratios.fig](a ) and fig . [ ratios.fig](b ) . for @xmath199 , the transition is second - order with the bco@xmath183-size ratios evolving to the fcc - ones smoothly , see fig . [ ratios.fig](c ) . the nature of the transition bco@xmath181 @xmath116 bco@xmath183 is also mixed . for @xmath196 , [ fig . [ ratios.fig](a ) ] , a finite jump of the values of the ratios was found , although the step size in changing the packing fraction was made as small as @xmath200 . thus , we characterise this transition at @xmath196 as first - order . for @xmath197 , [ fig . [ ratios.fig](b ) ] this transition appears to be second - order . hence , we conclude that there must be a line of second - order transitions terminating at a tricritical point between @xmath197 and @xmath196 , ( for the bco@xmath181 @xmath116 bco@xmath183 transition ) and similarly a tricritical point between @xmath199 and @xmath197 ( for the bco@xmath183 @xmath116 fcc - transition ) to be succeeded by lines of first - order transitions . all four bco - phases and the three transitions among them , bco@xmath181 @xmath116 bco@xmath182 @xmath116 bco@xmath183 @xmath116 fcc , appear only in a very narrow @xmath39-range around @xmath199 , see fig . [ ratios.fig](c ) . the line of second - order transitions bco@xmath181 @xmath116 bco@xmath183 meets the lines of first - order transitions bco@xmath181 @xmath116 bco@xmath182 and bco@xmath182 @xmath116 bco@xmath183 at a critical endpoint located at @xmath39 slightly less than 48 . similarly , the line of second - order transitions bco@xmath183 @xmath116 fcc also terminates at a critical endpoint , meeting the first - order lines bco@xmath182 @xmath116 a15 and fcc @xmath116 a15 . it is an intriguing phenomenon that such a richness in the stable crystal structures and in the nature of the transitions among them occurs as the result of a simple , spherically symmetric interaction and this points to the many surprises of ultrasoft potentials and their tendency to produce open , exotic structures . at the same time , it must be pointed out that we do not expect the solid phases occurring for @xmath201 to survive the competition with a fluid . indeed , as can be seen from fig . [ stars.phdg.fig ] , the critical functionality @xmath202 below which no solids are stable increases with increasing density . the bcc - phase is extinguished by the fluid for @xmath203 , whereas the bco- and diamond phases for @xmath204 . it is therefore anticipated that the bco@xmath181- , bco@xmath183- and fcc - phases seen in fig . [ latticesums.fig ] for @xmath201 will be wiped out by the fluid there . however , by arbitrarily increasing @xmath39 one can always reach a domain where the fluid will be beaten by the crystal and hence the bco@xmath181 @xmath116 bco@xmath182 @xmath116 a15-transitions will be there also after a finite - temperature calculation . in particular , the a15-phase , on whose stability it has been speculated for some time , has now been proved to be indeed the most stable phase at sufficiently high densities among the candidates considered . the logarithmic - yukawa potential employed in our study has been shown to describe well the effective interactions between star polymers in a good solvent for a wide range of concentrations . however , at the region of stability of the a15-crystal , the pair potential description is not expected to be particularly accurate . many - body contributions are expected to become important there @xcite . moreover , the yukawa tail of the potential , describing the interactions of the outermost daoud - cotton blobs , should be absent since the compressed coronas there are deprived of the outermost blob structure of the isolated stars . thus , in this respect , the logarithmic - yukawa potential has to be looked upon rather as a toy model . nevertheless , the physical characteristic driving the transitions discovered above is the ultraslow divergence of the logarithm which , in the neighborhood of the average particle distance can be locally expanded as a ramp - like potential and the finer details of the interaction should become irrelevant . we have shown that `` ultrasoft interactions '' arise naturally from coarse - graining procedures for a broad range of soft matter systems . besides greatly simplifying the statistical mechanics of these complex systems once the interactions are derived , all the well known tools of liquid state theory can be applied to calculate correlations and phase behaviour they also lead to new phenomenology . signatures of these ultra soft interactions include anomalous fluid correlations , reentrant melting as well as the stabilisation of exotic , open crystal structures . in contrast to their atomic counterparts , soft matter systems can therefore stabilise such crystals without the presence of angle - dependent , anisotropic potentials : radially symmetric , ultrasoft interactions are quite sufficient . thus , a new _ mean field fluid _ paradigm can be established , which goes beyond the usual prototype for classical fluids , the hard - sphere model . the latter , being always dominated by packing effects , tends to favour close packed structures . exotic crystals with unusual ordering have been observed in hard - sphere - like suspensions @xcite but that case refers to _ binary mixtures _ whose phase behaviour is indeed much richer than that of their one - component counterparts . it is with great pleasure that we dedicate this paper to professor peter pusey on the occasion of his 60th birthday . we thank martin watzlawek and primoz ziherl for helpful discussions . aal thanks the isaac newton trust , cambridge , for financial support .	we discuss recent developments and present new findings in the colloidal description of soft polymeric macromolecular aggregates . for various macromolecular architectures , such as linear chains , star polymers , dendrimers and polyelectrolyte stars , the effective interactions between suitably chosen coordinates are shown to be ultrasoft , i.e. , they either remain finite or diverge very slowly at zero separation . as a consequence , the fluid phases have unusual characteristics , including anomalous pair correlations and mean - field like thermodynamic behaviour . the solid phases can exhibit exotic , strongly anisotropic as well as open crystal structures . for example , the diamond and the a15-phase are shown to be stable at sufficiently high concentrations . reentrant melting and clustering transitions are additional features displayed by such systems , resulting in phase diagrams with a very rich topology . we emphasise that many of these effects are fundamentally different from the usual archetypal hard sphere paradigm . instead , we propose that these fluids fall into the class of mean - field fluids .
the celebrated camassa - holm equation taking the form @xmath0 was derived as a shallow water wave equation by camassa and holm @xcite . this equation firstly appeared in the work of fuchssteiner and fokas @xcite as an abstract bi - hamiltonian equation with infinitely many conservation laws . but it attracted no special attention until its rediscovery by camassa and holm . the most attractive character is that it admits n - peakon solutions , which are also called peakons , taking the form : @xmath1 it is remarked that , for the camassa - holm equation , the peakons are proved to be stable @xcite . the peakons " have been of great research interest . in the literature , besides the camassa - holm equation , there exist many integrable systems with peaked solutions such as the degasperis - procesi equation @xcite , novikov equation @xcite , geng - xue equation @xcite , hunter - saxton equation @xcite and some generalizations of camassa - holm equation @xcite etc .. ( there are plenty of papers on this topic . it was not our purpose trying to be exhaustive . thus , we beg indulgence for the numerous omissions it certainly contains . ) the explicit expressions of multipeakon solutions to several integrable systems have been obtained . for example , in @xcite , beals , sattinger and szmigielski used inverse spectral method and the stieltjes theorem on continued fractions to get multipeakon solutions to the camassa - holm equation . the closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem . as for the cases of the degasperis - procesi equation , the novikov equation and the geng - xue equation , please consult @xcite . motivated by the work of beals , sattinger and szmigielski , we have managed to confirm the multipeakon solutions from another way the determinant technique and succeeded . with the help of determinant identities , we propose an extended camassa - holm equation , which also admits n - peakon solutions . the proposed equation is @xmath2_x+m[(s+r)u_x+ru]=0 , \ \ \ \ 2m=4u - u_{xx},\ ] ] where @xmath3 . obviously , this equation reduces to the camassa - holm equation when @xmath4 . unexpectedly , we find that this equation possesses a nonisospectral lax pair , in other words , the spectrum in the lax pair is dependent on time @xmath5 instead of a constant . therefore , we shall call it generalized nonisospectral camassa - holm ( gnch ) equation . as for the derivation of the gnch equation , we have made an inverse calculation . assume that the form of the explicit solution is still the same as that of the camassa - holm equation . we firstly alter the evolution with respect to time @xmath5 for the moments of hankel determinant . then we deduce the dynamical system by using the determinant identities . at last , the corresponding partial differential equation is obtained . to the best of our knowledge , this equation is novel . the paper is organized as follows . in section 2 , the work of beals et al . is reviewed . as our results are obtained by use of determinant technique , we show some determinant identities in section 3 . the explicit formulas of n - peakon solutions to the gnch equation are presented in section 4 and the lax pair is also given there . in section 5 , we consider three special cases of the gnch equation . section 6 is devoted to discussions . in @xcite , in order to obtain the explicit n - peakon solutions to the camassa - holm equation , beals et al . used inverse spectral method , the theory of continued fractions and formulas of stieltjes . let us sketch the idea . they considered the finite dimension case of the continuum equations when @xmath6 is taken to be a discrete measure with weights @xmath7 at locations @xmath8 : @xmath9 it is clear that the form @xmath10 can be calculated from and the second equation in . in this case , the camassa - holm equation may be written as a hamiltonian system for @xmath11 , which is the same as that in @xcite : @xmath12 where @xmath13 and the notation @xmath14 denotes @xmath15 at the jump point @xmath8 of @xmath16 . the amplitudes @xmath7 and the locations @xmath8 of the peaks were calculated explicitly by using the inverse scattering approach . the spectral data for this spectral problem consists of the eigenvalues @xmath17 and the coupling coefficients @xmath18 . it is known that the camassa - holm equation possesses the lax pair @xcite : @xmath19 where @xmath20 is a constant . from the above equation , they found that the time evolution for the eigenfunctions @xmath21 and the coupling coefficients satisfy @xmath22 in order to make the problem simpler , they mapped the spectral problem into @xmath23 with @xmath24 , by using the liouville transformation @xmath25 in this case , the eigenvalues and coupling coefficients are preserved . and the measure @xmath6 is transformed into @xmath26 where @xmath27 then they solved the inverse spectral problem for the finite string on @xmath28 and they obtained @xmath29 where @xmath30 denotes the determinant @xmath31 with the convention @xmath32 and @xmath33 for @xmath34 . and here @xmath35 are defined by @xmath36 with @xmath37 and the spectrum @xmath17 satisfying @xmath38 . here we mention that we shall use the convention @xmath39 in the full text . thus the explicit formulas of n - peakon solutions were obtained by combining - . we summarize the result as follows . [ th : beals ] the camassa - holm equation admits the n - peakon solution @xmath40 with @xmath41 and @xmath42 here @xmath43 and the moments @xmath35 are restricted by @xmath44 with @xmath45 and nonzero real constants @xmath46 . * remark : * in order to make @xmath8 and @xmath7 exist , @xmath47 are required to be distinct , @xmath48 are positive , and the determinant @xmath49 for @xmath50 do not vanish . and one sufficient condition for @xmath51 is that all the @xmath47 have the same sign . noting that the explicit formulae for the n - peakon solutions to the camassa - holm equation can be expressed by hankel determinants , it is necessary to seek for some potential properties behind them . in this section , we shall present some interesting results for the hankel determinants , some of which have appeared in @xcite . the object in this section is the hankel determinant behaving as @xmath52 with the convention @xmath53 and @xmath54 for @xmath34 . besides , we also introduce the determinant taking the form of @xmath55 with the convention @xmath56 . [ lemma : linear ] if the elements @xmath57 subject to @xmath58 then there hold @xmath59\prod_{1\leq i < j\leq n}(\mu_j-\mu_i)^2,\\ & & \gamma_k^l=0,\ \ k > n \end{aligned}\ ] ] for any @xmath60 . noting that any @xmath61 is a linear combination of @xmath62 , the matrix @xmath63 may be written as a product of two matrices , namely , @xmath64 based on the above decomposition , we see that the rank of the matrix @xmath63 is not more than @xmath65 . thus the determinant @xmath66 is equal to zero if @xmath67 . when @xmath68 , the result can be obtained by noting the vandermonde matrices on the right . [ coro : linear ] if the elements @xmath69 subject to @xmath70 with @xmath71 and @xmath72 for @xmath73 , then there hold @xmath74\prod_{1\leq i < j\leq n}(\mu_j-\mu_i)^2,\ \ l\geq1,\\ & & \gamma_k^0-\frac{1}{2}\gamma_{k-1}^2=0,\ \ k > n,\\ & & \gamma_k^l=0,\ \ k > n,\ \ l\geq1,\\ & & \gamma_{n+1}^{-1}+f_n^0-\frac{1}{4}\gamma_{n-1}^0=0 . \end{aligned}\ ] ] if we let @xmath75 then @xmath76 the five formulae are the consequences of applying lemma [ lemma : linear ] to different determinants with the elements @xmath77 . the proofs can be achieved by noting that the left sides in the above formulae are @xmath78 , @xmath79 , @xmath80 , @xmath81 and @xmath82 , respectively . for any determinant @xmath83 , the well known jacobi determinant identity @xcite reads @xmath84 where @xmath85 denotes the determinant of the matrix obtained from @xmath83 by removing the rows with indices @xmath86 and the columns with indices @xmath87 . applying the jacobi determinant identity , we have the following identities . [ lemma : bilinear ] for any @xmath88 , @xmath60 , @xmath89 the proofs for the case of @xmath90 are obvious by noting the convention @xmath91 and @xmath56 . now we consider the case of @xmath92 . taking @xmath93 @xmath94 and setting @xmath95 the jacobi identity yields to the first two equalities , respectively . the last two equalities are the consequences of applying the jacobi identity to @xmath96 and @xmath97 with @xmath98 respectively . applying lemma [ lemma : bilinear ] , the following corollaries are easily obtained . [ coro : sum ] for @xmath99 , @xmath100 we shall take the proof to the last equality for example and the others are omitted because the proofs are similar . the detail of the proof to the last one is @xmath101\\ & = & \frac{\gamma_{k}^{3}}{\gamma_{k+1}^1 } , \end{aligned}\ ] ] where we have used the convention @xmath102 . [ coro : multiple - identity ] for @xmath99 , there hold @xmath103\nonumber\\ & = & \gamma_{k}^1\gamma_{k}^2(4\gamma_{k+2}^{-1}+4f_{k+1}^0-\gamma_{k}^{3})-\gamma_{k+1}^1\gamma_{k-1}^{3}(2\gamma_{k+1}^0-\gamma_{k}^2)\nonumber\\ & & + ( \gamma_{k+1}^0-\gamma_{k}^2)[2\gamma_{k+1}^0\gamma_{k}^2-(\gamma_{k}^2)^2].\label{multi - identity2}\end{aligned}\ ] ] the first equality can be confirmed by applying lemma [ lemma : bilinear ] and eliminating @xmath104 . more exactly , the proofs can be completed by noting that @xmath105 the second equality may also be verified by employing lemma [ lemma : bilinear ] and eliminating @xmath106 . more precisely , it is proved by noting that @xmath107 we mention that theorem [ th : beals ] , which describes the n - peakon solutions to the camassa - holm equation , can be proved by using determinant technique . assume that the form of the n - peakon solution is the same as . by modifying the time evolution in , we obtain a generalized equation after some technical inverse operations , which is the gnch equation . obviously , it also admits the n - peakon solution . the result is described as below . [ th : gnch - solution ] the gnch equation admits the n - peakon solution having the form of @xmath108 the explicit expressions for @xmath7 and @xmath8 are as follows : @xmath109 with @xmath110 here @xmath43 and the moments @xmath35 are restricted by @xmath111a_j(t)},\ \ \ a_0(t)=\frac{\log\frac{1}{2}}{r+2s},\ \ \ \lambda_0=0 \end{aligned}\ ] ] with @xmath112 and nonzero real constants @xmath47 . we shall present our result by the way of proving theorem [ th : gnch - solution ] rather than how to derive the gnch equation . as the camassa - holm equation is the special case of the generalized nonisospectral camassa - holm equation , it means that we give an alternative proof to the case of the camassa - holm equation . in order to prove theorem [ th : gnch - solution ] , we need some lemmas . we remark that corollaries [ coro : linear],[coro : sum],[coro : multiple - identity ] and lemma [ lemma : bilinear ] in section 3 still hold in the case of replacing @xmath113 by @xmath114 , respectively . here @xmath115 is defined by @xmath116 with the convention @xmath117 . besides , as for the derivative of @xmath30 with respect to @xmath5 , we also have the following properties . [ lemma : derivative2 ] for @xmath92 , @xmath118g_k^{l-1 } , \ \ l\geq2.\end{aligned}\ ] ] here we shall give a detailed proof of the first formula . the proofs of the rest two can be achieved by following the steps of the proof of the first one . it is easy to see that the element @xmath35 satisfy the differential equation @xmath119a_{k-1 } , \ \ \ k=0 \ { \text{and}}\ k\geq2 , \ \ \ \dot{a}_1=(2r+2s)a_{0}-(r+s),\ ] ] where we have used @xmath38 and the convention @xmath39 . by using the differential rule for determinants , we have @xmath120{a}_{j-1}&a_{j+1}&\cdots&a_{k-1}\\ a_1&a_2&\cdots&a_{j}&[r(j+2)+2s]{a}_{j}&a_{j+2}&\cdots&a_{k}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\ a_{k-1}&a_{k}&\cdots&a_{k+j-2}&[r(j+k)+2s]{a}_{k+j-2}&a_{k+j}&\cdots&a_{2k-2 } \end{array}\right\|\\ & = & ( 2r+2s)g_{k-1}^{1}\\ & & + \sum_{j=0}^{k-1}\left\|\begin{array}{cccccccc } a_0&a_1&\cdots&a_{j-1}&[r+2s]{a}_{j-1}&a_{j+1}&\cdots&a_{k-1}\\ a_1&a_2&\cdots&a_{j}&[2r+2s]{a}_{j}&a_{j+2}&\cdots&a_{k}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\ a_{k-1}&a_{k}&\cdots&a_{k+j-2}&[rk+2s]{a}_{k+j-2}&a_{k+j}&\cdots&a_{2k-2 } \end{array}\right\|\\ & = & ( 2r+2s)g_{k-1}^{1}+\sum_{j=0}^{k-1}\sum_{i=0}^{k-1}(-1)^{i+j}[r(i+1)+2s]a_{i+j-1}\delta_k^0\binom{i+1}{j+1}\\ & = & ( 2r+2s)g_{k-1}^{1}+\sum_{i=0}^{k-1}(-1)^i[r(i+1)+2s]\sum_{j=0}^{k-1}(-1)^{j}a_{i+j-1}\delta_k^0\binom{i+1}{j+1}.\end{aligned}\ ] ] noting that @xmath121 and @xmath122 we get @xmath123 thus the proof is completed . * remark : * it is noted that we have introduced the term @xmath124 , while beals et al . @xcite did nt . that s because it will be helpful for proving the conclusion by use of the determinant technique . now we shall present the proof to theorem [ th : gnch - solution ] by employing corollaries [ coro : linear ] , [ coro : sum ] , [ coro : multiple - identity ] and lemma [ lemma : bilinear ] , [ lemma : derivative2 ] . assume that @xmath6 is taken to be a discrete measure as . the gnch equation is reduced to @xmath125m_j.\ ] ] note that @xmath126 and @xmath127 if we substitute into , then is written as @xmath128-(\frac{3r}{2}+s)\sum_{i = j+1}^ng_i(1-y_i^2)\frac{(1+y_j)(1-y_i)}{(1-y_j)(1+y_i)}.\\\end{aligned}\ ] ] the above system can be simplified as @xmath129 where the @xmath130 in the second equation has been eliminated by inserting the first equation to the second one . therefore , in order to confirm theorem [ th : gnch - solution ] , it is sufficient to prove that @xmath131 satisfy the system and . substituting the expressions into eq . , we see that is equivalent to @xmath132\sum_{i = j}^n[2\frac{\delta_{n - i+1}^0\delta_{n - i}^2}{\delta_{n - i+1}^1\delta_{n - i}^1}-\frac{(\delta_{n - i}^2)^2}{\delta_{n - i+1}^1\delta_{n - i}^1 } ] \end{aligned}\ ] ] where we have simplified it . applying lemma [ lemma : derivative2 ] and corollary [ coro : sum ] and replacing @xmath133 by @xmath134 , the above equation can be written as @xmath135\delta_{k}^2-(3r+2s)g_{k}^1\delta_{k+1}^0\\ & = & ( r+2s)(\delta_{k}^2)^2(\frac{\delta_{k+2}^{-1}}{\delta_{k+1}^1}-\frac{\delta_{n+1}^{-1}}{\delta_{n}^1})-(r+2s)(\delta_{k}^2)^2(\frac{g_{n}^{0}}{\delta_{n}^1}-\frac{g_{k+1}^{0}}{\delta_{k+1}^1})\\ & & + ( \frac{r}{4}+\frac{s}{2})\frac{1}{4}(\delta_k^2)^2(\frac{\delta_{n-1}^{3}}{\delta_{n}^1}-\frac{\delta_{k}^{3}}{\delta_{k+1}^1})+(\frac{3r}{4}+\frac{s}{2})(2\delta_{k+1}^0-\delta_{k}^2)^2\frac{\delta_{k}^{3}}{\delta_{k+1}^1}\\ & & -\frac{r}{2}[2\delta_k^2\delta_{k+1}^0-(\delta_k^2)^2][2\frac{g_{k+1}^0}{\delta_{k+1}^1}-\frac{\delta_{k}^3}{\delta_{k+1}^1}],\end{aligned}\ ] ] which can be reduced to @xmath135\delta_{k}^2-(3r+2s)g_{k}^1\delta_{k+1}^0\nonumber\\ & = & ( r+2s)(\delta_{k}^2)^2(\frac{\delta_{k+2}^{-1}}{\delta_{k+1}^1}+\frac{g_{k+1}^{0}}{\delta_{k+1}^1})+(r+2s)((\delta_{k+1}^0)^2-\delta_{k+1}^0\delta_{k}^2)\frac{\delta_{k}^{3}}{\delta_{k+1}^1}\nonumber\\ & & + r[2\delta_{k+1}^0-\delta_k^2][\frac{\delta_{k+1}^0\delta_{k}^3}{\delta_{k+1}^1}-\frac{\delta_k^2g_{k+1}^0}{\delta_{k+1}^1}],\label{formula1}\end{aligned}\ ] ] by using the fifth relation of corollary [ coro : linear ] . we recall that @xmath136 holds as is described in corollary [ coro : multiple - identity ] . substituting this equality into and rearranging it , we are left to prove @xmath137[g_k^1+\frac{\delta_{k+1}^0\delta_{k}^3}{\delta_{k+1}^1}-\frac{\delta_k^2g_{k+1}^0}{\delta_{k+1}^1}]=0.\end{aligned}\ ] ] notice that @xmath138 is nothing but an identity in lemma [ lemma : bilinear ] . thus the proof of is completed . next we proceed to the proof of . similar to , is equivalent to @xmath139\\ & & + ( \frac{r}{2}+s)\delta_{k+1}^0(\delta_{k+1}^0-\delta_{k}^2)[2\delta_{k+1}^0\delta_{k}^2-(\delta_{k}^2)^2]-\frac{r}{2}(\delta_{k+1}^0)^2[2\delta_{k+1}^0\delta_{k}^2-(\delta_{k}^2)^2],\end{aligned}\ ] ] where @xmath133 is replaced by @xmath134 for simplicity . by using lemma [ lemma : derivative2 ] and corollary [ coro : sum ] and the fifth relation of corollary [ coro : linear ] , the above equation can be written as @xmath140-\delta_{k+1}^0\{[(2r+2s)g_{k+1}^0-(r+s)\delta_{k}^3]\delta_{k}^1\nonumber\\ & & + \delta_{k+1}^1[(2r+2s)g_{k}^0-(r+s)\delta_{k-1}^3]\}\nonumber\\ & = & ( \frac{r}{2}+s)\delta_{k}^1\delta_{k}^2(4\delta_{k+2}^{-1}+4g_{k+1}^0-\delta_{k}^{3})-(\frac{3r}{2}+s)\delta_{k+1}^1\delta_{k-1}^{3}(2\delta_{k+1}^0-\delta_{k}^2)\nonumber\\ & & -r\delta_{k+1}^1(\delta_{k+1}^0-\delta_{k}^2)[2g_{k}^0-\delta_{k-1}^3]+(\frac{r}{2}+s)(\delta_{k+1}^0-\delta_{k}^2)[2\delta_{k+1}^0\delta_{k}^2-(\delta_{k}^2)^2]\nonumber\\ & & -\frac{r}{2}\delta_{k+1}^0[2\delta_{k+1}^0\delta_{k}^2-(\delta_{k}^2)^2].\label{formula2}\end{aligned}\ ] ] we recall that @xmath141\\ & = & \delta_{k}^1\delta_{k}^2(4\delta_{k+2}^{-1}+4g_{k+1}^0-\delta_{k}^{3})-\delta_{k+1}^1\delta_{k-1}^{3}(2\delta_{k+1}^0-\delta_{k}^2)\\ & & + ( \delta_{k+1}^0-\delta_{k}^2)[2\delta_{k+1}^0\delta_{k}^2-(\delta_{k}^2)^2],\end{aligned}\ ] ] which is the second identity in corollary [ coro : multiple - identity ] . subtracting this equality multiplied by @xmath142 from , we are left to prove @xmath143+\frac{1}{2}\delta_{k}^2[2\delta_{k+1}^0\delta_{k}^2-(\delta_{k}^2)^2]\\ & & -\delta_{k+1}^0[2\delta_{k+1}^0\delta_{k}^2-(\delta_{k}^2)^2].\end{aligned}\ ] ] by employing identities in lemma [ lemma : bilinear ] @xmath144 and eliminating @xmath145 and @xmath146 , what we need to prove is reduced to @xmath147 this is valid because the identity @xmath148 holds , which appears in lemma [ lemma : bilinear ] . thus is verified and we complete the proof of theorem [ th : gnch - solution ] . we end this section by a lax pair of the gnch equation . it is noted that , unlike the case of camassa - holm equation , @xmath20 in the lax pair of the gnch equation is dependent on time @xmath5 . [ th : gnch - lax ] the gnch equation may be obtained by the compatibility condition of the following system @xmath149f_x+(\frac{r+s}{2}u_x(x , t)+\frac{r}{2}u(x , t))f,\end{aligned}\ ] ] where @xmath3 and @xmath150 is a function satisfying the differential equation @xmath151 the proof can be achieved by differentiating the first equation with respect to @xmath5 and the second twice with respect to @xmath152 and setting @xmath153 . in this section we shall present three special cases of gnch equation . taking @xmath154 in the gnch equation , we derive a variant of the camassa - holm equation . @xmath155 where @xmath156 . actually , this equation is just the camassa - holm equation ( under the scale transformation @xmath157 ) . consider the case of @xmath158 ( i.e. the camassa - holm equation ) . it is worth noting that any @xmath159 in theorem [ th : gnch - solution ] is a linear combination of @xmath160 , which is a little different from that in theorem [ th : beals ] . we recall that , in theorem [ th : beals ] , @xmath47 are required to be distinct , @xmath48 are positive , and the determinant @xmath49 for @xmath50 do not vanish so that @xmath8 and @xmath7 exist . as for theorem [ th : gnch - solution ] , we only require that @xmath47 are distinct , and the determinant @xmath49 for @xmath50 do not vanish because @xmath160 are always positive . when @xmath161 , the gnch equation reduces to @xmath162_x+m(u_x+u)=0 , \ \ \ \ 2m=4u - u_{xx}.\ ] ] here we call it the nonisospectral camassa - holm equation because its lax pair is as below . the nonisospectral camassa - holm equation may be obtained by the compatibility condition of the following system @xmath163 where @xmath150 is a function satisfying the differential equation @xmath164 the following result for its n - peakon solutions holds . [ coro : ch2-solution ] the nonisospectral camassa - holm equation admits the n - peakon solution having the form of @xmath165 the explicit expressions for @xmath7 and @xmath8 are as follows : @xmath41 with @xmath42 here @xmath43 and the moments @xmath35 are restricted by @xmath166 with @xmath167 and nonzero real constants @xmath47 . taking @xmath168 in the gnch equation , we have a mixed type camassa - holm equation @xmath169_x+m(6u_x+4u)=0 , \ \ \ \ 2m=4u - u_{xx},\ ] ] which also admits the n - peakon solution . [ coro : ch3-solution ] the mixed type camassa - holm equation admits the n - peakon solution having the form of @xmath165 the explicit expressions for @xmath7 and @xmath8 are as follows : @xmath41 with @xmath42 here @xmath43 and the moments @xmath35 are restricted by @xmath170 with @xmath171 and nonzero real constants @xmath47 . the mixed type camassa - holm equation may be obtained by the compatibility condition of the following system @xmath172 where @xmath150 is a function satisfying the differential equation @xmath173 at the end of this section , we illustrate the 1-peakon solution and 2-peakon solution to eq . , respectively . 1 . _ 1-peakon . _ + take @xmath174 and @xmath175 . + from corollary [ coro : ch3-solution ] , we easily obtain @xmath176,\ \ \ \ m_1=\frac{2}{4t-1}\ ] ] so that admits the 1-peakon solution @xmath177 from this expression , we see that @xmath178 is a turning point of the 1-peakon solution . that is , the amplitude @xmath179 satisfys @xmath180 for @xmath181 and @xmath182 for @xmath183 , which means the solution is from antipeakon to peakon along with @xmath5 . a simple simulation ( see fig.1 ) using matlab is given below by computing the explicit formula . 2-peakon . _ + take @xmath174 , @xmath184 and @xmath185 . + we get the 2-peakon solution ( see fig.2 ) @xmath186 with @xmath187,\\ & & m_1=\frac{2[(4t-1)^4+(4t+1)^4]}{(4t+1)(4t-1)[(4t+1)^3+(4t-1)^3]},\\ & & m_2=\frac{2[(4t-1)^2+(4t+1)^2]}{(4t+1)^3+(4t-1)^3 } \end{aligned}\ ] ] from corollary [ coro : ch3-solution ] . + it is easy to work out the turning points @xmath188 and there hold for the amplitudes @xmath189 which interpret the performance of fig.2 . we have derived one generalized nonisospectral camassa - holm equation , which admits n - peakon solutions taking as similar explicit formulae as the camassa - holm equation . some special cases of this system are also studied . the approach is mainly by use of the determinant technique . after we derived our result , we accidentally found that estvez et al . have investigated a nonisospectral 2 + 1 camassa - holm hierarchy in @xcite , where 1 + 1 hierarchies are derived by using lie symmetries reduction . the proposed equation in this paper is different from those . and what we concern about is the n - peakon solutions . actually , we investigated one kind of nonisospectral camassa - holm equation with n - peakon solutions . when @xmath190 in eq . are dependent on time @xmath5 instead of constants , the corresponding equation is still integrable . in other words , if we consider the more generalized determined system @xmath149f_x+(\frac{r+s}{2}u_x(x , t)+\frac{r}{2}u(x , t))f,\end{aligned}\ ] ] where @xmath191 , @xmath192 are two given functions dependent on time @xmath5 and @xmath150 is a function satisfying the differential equation @xmath193 then the compatibility condition leads to an equation with variable coefficients @xmath194_x+m[(s+r)u_x+ru]=0 , \ \ \ \ 2m=4u - u_{xx},\ ] ] where @xmath191 , @xmath192 are two given functions of @xmath5 . however , we did nt derive its n - peakon solutions except the special cases with @xmath154 ( the camassa - holm equation with variable coefficients ) and @xmath195 ( the nonisospectral camassa - holm equation with variable coefficients ) . it still needs further studies . when we consider the determined system @xmath19 where @xmath150 satisfy the differential equation @xmath196 with a given function @xmath197 . the compatibility condition yields another nonisospectral camassa - holm equation @xmath198 however , this equation is not novel because it can be transformed into the camassa - holm equation by the variable transformation @xmath199 therefore , it is natural to ask whether there exist more novel nonisospectral camassa - holm equations . do they admit multipeakon solutions ? we shall consider these problems in the future . this work was partially supported by the national natural science foundation of china ( grant no . 11331008 , 11371251 ) , the knowledge innovation program of lsec , icmsec , academy of mathematics and systems science , chinese academy of sciences .	motivated by the paper ( beals , sattinger and szmigielski , adv . math . 154 ( 2000 ) 229257 ) , we propose an extension of the camassa - holm equation , which also admits the multipeakon solutions . the novel aspect is that our approach is mainly based on classic determinant technique . furthermore , the proposed equation is shown to possess a nonisospectral lax pair . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * keywords : * nonisospectral camassa - holm equation ; multipeakon solutions ; determinant technique _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
within the minimal supersymmetric extension of the standard model ( mssm ) the gauge couplings unify nearly perfectly around an energy scale of approximately @xmath3 gev , if susy particles exist with masses of the order of @xmath4 tev . extending the mssm with non - singlet superfields tends to destroy this attractive feature , unless ( a ) the additional fields come in complete @xmath5 multiplets or ( b ) the standard model gauge group is extended too . here we study a model in which the sm group is enlarged to @xmath6 . it is a variant of the models first proposed in @xcite and later discussed in more detail in @xcite . our main motivation for studying this model can be summarized as : ( i ) it unifies , in the same way the mssm does , even if the scale of @xmath0 breaking is as low as the electro - weak scale ; ( ii ) it can be easily embedded into an @xmath1 grand unified theory ; ( iii ) it has the right ingredients to explain neutrino masses ( and angles ) by either an inverse @xcite or a linear @xcite seesaw ; ( iv ) it allows for higgs masses significantly larger than the mssm without the need for a very heavy susy spectrum @xcite and ( v ) it potentially leads to rich phenomenology at the lhc . with the data accumulated in 2011 both atlas @xcite and cms @xcite have seen some indications for a higgs boson with a mass of roughly @xmath7 gev . this result , perhaps unsurprisingly , has triggered an avalanche of papers studying the impact of such a relatively hefty higgs on the supersymmetric parameter space @xcite . the general consensus seems to be , that the mssm can generate @xmath7 gev only if squarks and gluinos have masses in the multi - tev range . while this is , of course , perfectly consistent with the lower bounds on susy masses obtained from searches at the lhc @xcite , such a heavy spectrum could make it quite difficult indeed for the lhc to find direct signals for susy . there are , of course , several possibilities to circumvent this conclusion . first of all , it is well - known that the loop corrections to @xmath8 are dominated by the top quark - squark loops . thus , little or no constraints on sleptons and on squarks of the first two generations can in fact be derived from higgs mass measurements , once the assumption of universal boundary conditions for the soft susy parameters is abandoned . second , in the next - to - minimal ssm ( nmssm ) the @xmath8 can be heavier than in the mssm due to the presence of new f - terms from the additional singlet higgs @xcite , especially in models with non - universal boundary conditions for the ( soft ) higgs mass terms @xcite or in the generalized nmssm @xcite . and , third , in models with an extended gauge group additional @xmath9-terms contribute to the higgs mass matrices , relaxing the mssm upper limit considerably @xcite . this latter possibility is the case we have studied in a previous paper @xcite using the minimal @xmath0 models of @xcite . here , we extend the analysis of @xcite , including both higgs and susy phenomenology . due to the extended gauge structure the model necessarily has more higgses than the mssm . near d - flatness of the @xmath10 breaking then results in one additional light higgs , @xmath11 @xcite . mixing between the mssm @xmath12 and @xmath11 enhances the mass of the mostly mssm higgs and , potentially , affects its decays . this is reminiscent to the situation in the nmssm , where an additional light and mostly singlet higgs state seems to be preferred @xcite if the signals found by atlas @xcite and cms @xcite are indeed due to a 125 gev higgs . the mssm - like @xmath13 in our model can have some exotic decays . for example , the @xmath13 will decay to two lighter higgses , if kinematically possible , although this decay can never be dominant due to constraints coming from lep . the model also includes right - handed neutrinos with electro - weak scale masses and there is a small but interesting part in parameter space where @xmath14 , where the higgs decays to two neutrinos . these decays always lead to one light and one heavy neutrino , with the latter decaying promptly to either @xmath15 or @xmath16 . ( mostly right ) sneutrinos can be lighter than the @xmath13 , in which case the higgs can have invisible decays . the susy spectrum of the model is also richer than the mssm : it has seven neutralinos and nine sneutrino states . these additional sneutrinos can easily be the lightest supersymmetric particle ( lsp ) and thus change all the constraints on susy parameter space , usually derived from the requirement that the neutralino be a good dark matter candidate with the correct relic density @xcite . even though the lightest sneutrino can also be the lsp in the mssm , direct detection experiments have ruled out this possibility a long time ago @xcite . in susy decays , within the mssm right squarks decay directly to the bino - like neutralino , leading to the standard missing momentum signature of supersymmetry . due to the extended gauge group , right squarks can decay also to heavier neutralinos , leading to longer decay chains and potentially to multiple lepton edges . decays of the heavier neutralinos also produce higgses , both the @xmath13 and the @xmath11 appear , with ratios depending on the right higgsino content of the neutralinos in the decay chains . the rest of this paper is organized as follows . in the next section we discuss the setup of the model , its particle content , superpotential and soft terms and the symmetry breaking . the phenomenologically most interesting mass matrices of the spectrum are given in section [ sec : masses ] where we also discuss numerical results on the susy and higgs mass eigenstates . here , we focus on higgs and slepton / sneutrino masses , which are the phenomenologically most interesting . in section [ sec : decays ] we define some benchmark points for the model and discuss their phenomenologically most interesting decay chains . we then close with a short summary . in the appendix we give mass matrices not presented in the text , formulas for the 1-loop corrections in the higgs sector and more information about the calculation of the rges , including anomalous dimensions as well as the 1-loop @xmath17 functions for gauge couplings and gauginos . in this section we present the particle content of the model , its superpotential and discuss the symmetry breaking . we consider the simplest model based on the gauge group @xmath19 . we will call this the mblr model below . as has been shown in @xcite it can emerge as the low - energy limit of a certain class of @xmath1 guts broken along the `` minimal '' left - right symmetric chain @xmath20 the main virtue of this setting is that an mssm - like gauge coupling unification is achieved with a sliding @xmath21 breaking scale , i.e. this last stage can stretch down even to the electro - weak scale . different from the previous works @xcite , we assume that the first two breaking steps down to @xmath21 happen _ both _ at ( or sufficiently close to ) the gut scale . this assumption is used only for simplifying our setup , it does not lead to any interesting changes in phenomenology . .[tab : fc]the matter and higgs sector field content of the @xmath22 model . generation indices have been suppressed . the @xmath23 superfields are included to generate neutrino masses via the inverse seesaw mechanism . under matter parity , the matter fields are odd while the higgses are even . [ cols="^,^,^,^,^",options="header " , ] note , that in the couplings to the @xmath24-quarks a partial cancellation occurs in contrast to the ones to @xmath25-quarks , which get enhanced . moreover , the same feature appears in the vertex @xmath26-@xmath27-@xmath28 which leads to some interesting consequences discussed in section [ sec : susycascades ] . we find that the decays into the heavy neutrino states are always possible and have a sizable branching ratio provided tr@xmath29 . in table [ tab : zpbrs ] we summarize the most important final states of the @xmath2 for the different scenarios . as can be seen the heavy neutrino final states have always a sizable branching ratios with up to about 30 per - cent when summing over the generations . but even for rather heavy neutrinos as in blrsp5 own finds for this channel a 15 per - cent branching ratio . in several cases also channels into susy particles are open , in particular in scenarios with sneutrino lsps . in case of supersymmetric particles the final states containing sleptons or sneutrinos have the largest branching ratios . channels into neutralinos or charginos are suppressed . they proceed either via the mixing with the @xmath30 which is rather small or via the projection of the higgsino - right onto the corresponding neutralino state . the appearance of additional final states leads to a reduction of the event numbers in the most sensitive search channels , i.e. reducing cross section times branching ratio , and , thus , the bounds obtained by the lhc collaborations @xcite are less constraining in the blr model . this is depicted in fig . [ fig : ppmumu ] where we show the production cross section @xmath31 arround the @xmath2 resonance . in case that the width of the @xmath2 is calculated using only sm final states the cross section is increased roughly by a factor 1.6 in comparison to the case where also right handed neutrinos and susy particle contribute to the width of @xmath2 . with this choice of parameters , the main effect is due to r - neutrinos . we attribute the remaining difference to the official atlas result to slightly different values in the couplings and slightly different branching ratios of the final states . our results agree also with the ones of ref . we conclude that , although in our benchmark points we take always @xmath32 tev , a significantly lower mass is possible consistent with data . c as discussed above , the heavy neutrino states can be produced via the @xmath2 with a considerable branching ratio of about 30 per - cent when summing over all heavy neutrinos . moreover , see below , they can also be produced in the cascade decays of supersymmetric particles . these heavy neutrinos mix with the light neutrino states implying a reduction of the couplings of the light neutrinos to the @xmath30-boson and , thus , also a reduction of the invisible width of the @xmath30-boson . taking the data from ref . @xcite this can be translated into the following condition on the @xmath33 sub - block @xmath34 , @xmath35 , of the neutrino mixing matrix : @xmath36 at the 3-@xmath37 level . we have checked that all our benchmark points fullfill this condition . the main decay modes of the heavy neutrinos are @xmath38 where @xmath39 , @xmath40 , @xmath41 and @xmath42 , provided they are kinemtically allowed . if there is no kinematical suppression we find in general the branching ratios scale like @xmath43 where we have summed over the light higgs bosons , the light neutrinos and leptons , respectively . we stress that these states are quasi - dirac neutrinos and , thus , for six heavy neutrinos at lhc the existence of up to three new particles could be established . note , that the final states containing a @xmath44-boson allow for a direct mass measurement . beside the above decay modes , also decays into susy particle are possible if kinematics allow for it . for example we find that for blrsp4 the decay into @xmath45 are possible and have branching ratios of about 3 per - cent . in scenarios like blrsp3 , blrsp4 and blrsp5 the main production of the heavy neutrinos is via the @xmath2 and , thus , a high luminosity will be required to observe such final states . in this section we point out several features which distinguish the blr model from the usual mssm . for the sake of preparing the ground , let us first summarize the main features of the mssm relevant for the lhc , focusing for the time being on scenarios where the gluino is heavier than the squarks : ( i ) the gluino decays dominantly into squarks and quarks . ( ii ) l - squarks and l - sleptons decay dominantly into the chargino and the neutralino which are mainly @xmath46-gauginos . apart from kinematical effects the branching ratio for decays into the charged wino divided by the branching ratio into the neutral wino is about 2:1 . ( iii ) r - squarks and r - sleptons decay dominantly into the bino - like neutralino with a branching ratio often quite close to 100 per - cent . ( iv ) in case of third generation sfermions also decays into higgsinos are important . in the blr model one has two main new features : ( i ) there are additional neutralinos and ( ii ) the sneutrino sector is enlarged as well . the latter implies that sneutrino lsps are possible consistent with all astrophysical constraints and direct dark matter searches @xcite . this feature is for example realised in study points blrsp1 and blrsp3 from 1.05 to 1.0475 without changing the collider features of blrsp1 . ] . let us start the discussion with blrsp1 . in this point the four lighter neutralinos are the usual mssm neutralinos with the standard hierarchy . the fifth state corresponds to the additional @xmath47-gaugino , which we call @xmath48 , whereas the two additional states are the additional higgsinos . note that the lighest neutralino is not stable anymore but decays into final states containing all nine neutrinos as well as the three lightest sneutrinos . of the latter ones the second lightest is so long lived that it will lead to a displaced vertex in a typical collider detector . the third sneutrino decays dominatly via three - body decays into @xmath49 and @xmath50 with @xmath51 and @xmath52 . as discussed in section [ sec : heavyneutrinos ] the heavy neutrinos decay dominantly into @xmath44-bosons and charged leptons , thus the decays of the lightest neutralino are _ not _ invisible . @xmath48 appears for example in the decays of @xmath53 and @xmath54 with branching ratios br(@xmath55 and br(@xmath56 . for completeness we remark that the decays of @xmath57 and @xmath58 into @xmath59 is suppressed as the corresponding coupling is supressed as are the couplings of @xmath2 to @xmath24-type quarks in this model . @xmath59 decays dominantly into sleptons and sneutrinos . combining all the above together one gets a much richer structure for the decays of the r - squarks , e.g. the following decay chains : @xmath60 with @xmath61 and @xmath62 . of course , several other combinations are possible as well . from equations ( [ eq : rsquarktoz ] ) to ( [ eq : rsquarktoz2 ] ) one sees immediatly that the standard signature of r - squarks , namely jet and missing energy , is only realized in a few cases in this study point , e.g. if in eq . ( [ eq : rsquarktoz ] ) the @xmath30 decays into neutrinos . interestingly , the chain via @xmath59 into sleptons leads to a characteristic edge in the invariant mass of the lepton which can be used to determine the corresponding masses once combined with information from other decay chains . also in the study points blrsp2 and blrsp5 @xmath53 and @xmath54 decay into heavy neutralinos , which contain sizable content of the extra @xmath47 gaugino , with a sizable branching ratio . however , there the situation is somewhat less involved as in these study points the lightest bino - like neutralino is the lsp . another interesting feature is , that @xmath59 decays also into the heavier sneutrinos which themselves decay into the lsp plus @xmath63 . similarely @xmath63 can be produced in the decays of the heavy neutrinos implying that this state can be produced with sizable rate in susy cascade decays . however , as the corresponding final states are quite complicated a dedicated monte carlo study will be necessary to decide if this is indeed a discovery channel for @xmath63 . from the point of view of susy cascade decays blrsp2 looks essentially like a standard mssm point . inspection of the spectrum shows that @xmath64 is essentially a higgsino corresponding to the extended @xmath47 sector but it shows hardly up in the cascade decays . its main production channel is via an @xmath65-channel @xmath2 but even in this case the corresponding cross section is so low that it will not be dedected at the lhc even with an integrated luminosity of 300 fb@xmath66 . another interesting feature shows up in the decays of @xmath67 which is mainly the neutral wino and gets copiously produced in the decays of the l - squarks : it decays with about 77 ( 15 ) per - cent into @xmath63 ( @xmath68 ) , implying that the cascade decays are an important source of higgs bosons . in case of blrsp3 one has sneutrino lsps like in blrsp1 but with a different hierarchy in the spectrum , as the three lightest sleptons are lighter then the lighest neutralino . therefore the @xmath69 has also sizable decay rates into charged sleptons which sum to about 30 per - cent . the sleptons decay then further into @xmath70 and @xmath71 via 3-body decays into @xmath72-pairs . the latter , however , are rather soft due to the small mass difference . in addition we have the decay channel into a light neutrino and one of the heavier sneutrinos which themselves decay into a lighter sneutrino and either one of the higgs boson or the @xmath30-boson . putting again all these decays together one obtains for the @xmath73 decays @xmath74 with @xmath75 and @xmath76 . this implies that the decays of the r - squarks show again a more complicated structure compared to the usual cmssm expectations . channels ( [ eq : h1part1 ] ) and ( [ eq : h1part2 ] ) give @xmath63 in about 15 per - cent of the final states of @xmath73 . moreover , @xmath64 and @xmath77 decay dominantly into sleptons and sneutrinos . here a new feature is found for @xmath77 , as also the following chains @xmath78 gives rise to sharp edge structures . however , as the main final states of @xmath30 and @xmath79 are two jets , the feasability still needs to be investigated . in blrsp4 we have chosen @xmath80 gev in order to construct an lsp which is essentially a @xmath81 . here , the @xmath82-sleptons are lighter than @xmath64 , which is essentially bino - like in this point , giving rise to the following decay chain of the down - type @xmath82-squarks @xmath83 nearly all cascade decays end in a @xmath64 or one of the lighter sleptons . due to the fact , that in this particular case the additional sneutrino states are hardly produced , it might be difficult to disthinguish it from the nmssm , at least as long as the @xmath2 is not discovered . the heavier @xmath84-sleptons do not show up in the cascade decays of squarks and gluinos but can be produced via the @xmath2 as discussed in section [ tab : zpbrs ] . blrsp5 is similar to blrsp1 but compatible with pure gut conditions , e.g. @xmath85 and @xmath86 are not input in this case put derived quantities . to fullfill the tadpole equations we have to choose @xmath87 and @xmath88 if we want a relatively low @xmath89 gev while @xmath90 gev . the choice of @xmath91 leads automatically to large masses for the heavy neutrinos such that the lightest higgs can not decay into those states . as in blrsp1 the down - type @xmath82-squarks decay not only into @xmath73 but also into @xmath92 with a branching ratio of about 13 per - cent . for completeness , we note that here @xmath93 . however this state gets hardly produced in any of the susy decays or via the @xmath2 . therefore , it is likely that lhc will miss it and also at a linear collider such as ilc or clic it will be difficult to study , due to the small production cross section . we have studied the minimal supersymmetric @xmath0 extension of the standard model . the model is minimal in the sense that the extended gauge symmetry is broken with the minimal number of higgs fields . in the matter sector the model contains ( three copies of ) a superfield @xmath94 , to cancel anomalies . adding three singlet superfields @xmath95 allows to generate small neutrino masses with an inverse seesaw mechanism . the phenomenology of the model differs from the mssm in a number of interesting aspects . we have foccused on the higgs phenomenology and discussed changes in susy spectra and decays with respect to the mssm . the model is less constrained then the cmssm from the possible measurement of a higgs with a mass of the order of 125 gev . if the hints found in lhc data @xcite is indeed correct our model predicts two relatively light states should exist , with the second @xmath8 corresponding ( mostly ) to the lightest of the `` right '' higgses , added to break the extended gauge group . it is interesting , as we have discussed , that very often a right sneutrino is found to be the lsp . this will affect all constraints on cmssm parameter space derived from constraints on the dark matter abundance . in fact , if the right sneutrino is indeed the lsp in our model , no constraint on any cmssm parameters can be derived from dm constraints . the model has new d - terms in all scalar mass matrices , which can lead to sizeable changes in the susy spectra , of potential phenomenological interest . we have discussed a few benchmark points , covering a number of features which could allow to distinguish the model from the cmssm . obviously this includes the discovery of a @xmath2 at the lhc where we have shown that the current bounds from lhc data depend on the details of the particle spectrum . also the cascade decays of supersymmetric particles can be significantly more involved than in the usual cmssm as the additional neutralinos , neutrinos and sneutrinos lead to enhancement of the multiplicities in the final states . this implies that the existing limits on the cmssm parameter space get modified as standard final states have reduced branching ratios and at the same time additional final states are present . in case that the mblr model is indeed realized these new cascade decays will offer additional kinematical information on the particle spectrum . w.p . thanks the ific for hospitality during an extended stay . m.h . and l.r . acknowledge support from the spanish micinn grants fpa2011 - 22975 , multidark csd2009 - 00064 and by the generalitat valenciana grant prometeo/2009/091 and the eu network grant unilhc pitn - ga-2009 - 237920 . l.r thanks the instituto superior tcnico , lisbon , for kind hospitality during an extended stay . w.p . has been supported by the alexander von humboldt foundation and in part by the dfg , project no . po-1337/2 - 1 and the helmholtz alliance `` physics at the terascale '' . * * mass matrix for down - squarks * , basis : @xmath96 @xmath97 @xmath98 * * mass matrix for up - squarks * , basis : @xmath99 @xmath100 @xmath101 * * mass of the charged higgs boson * : one obtains the same expression as in the mssm : @xmath102 * * mass matrix for charginos * , basis : @xmath103 @xmath104 we are going to present now the basic steps to calculate the mass spectrum . as starting point we use electroweak precision data to get the gauge and yukawa couplings : the sm - like yukawa couplings are calculated from the fermion masses and the one - loop relations of ref . @xcite which have been adjusted to our model . similarly , also the standard model gauge couplings are calculated by the same procedure presented in ref . @xcite , but again , including all new contributions of the mode under consideration . since the entire rge running is performed in the basis @xmath105 , the value of the gut normalized @xmath106 and @xmath107 are matched to the gut normalized hypercharge coupling @xmath108 by @xmath109 this is nothing else then an inversion of the well known relation between the gauge couplings for @xmath110 including the off - diagonal gauge couplings given in eq . ( [ eq : gy_from_gr_gbl ] ) . we are using the @xmath1 gut normalization of @xmath111 for @xmath112 and @xmath113 for @xmath114 . to get the correct values of @xmath115 as well as @xmath116 and @xmath117 an iterative procedure is used : @xmath115 is calculated as ratio of the @xmath108 and @xmath106 when running down from the gut scale and applying @xmath118 when the gauge and yukawa couplings are derived , the rges are then evaluated up to the gut scale where the corresponding boundary conditions of eqs . ( [ eq : msugra ] ) , ( [ eq : msugra2 ] ) and ( [ eq : gutoffdiagonal ] ) are applied . afterwards a rge running of the full set of parameters to the susy scale is performed . we use always 2-loop rges which include the full effect of kinetic mixing @xcite . the running parameters are then used to calculate the tree level mass spectrum . however , it is well known that the one - loop corrections can be very important for particular particles and have to be taken into account . the best known example is the light mssm higgs boson which get shifted by up to 50% per - cent in case of heavy stops . similar effects can be expected in the extended higgs sector especially since these can be very light at tree - level . similarly , the gauginos arising in an extended gauge sector can be potentially light and receive important corrections at one - loop @xcite . to take these and all other possible effects into account we use a complete one - loop correction of the entire mass spectrum . our procedure to calculate the one - loop masses is based on the method proposed in ref.@xcite : first , all running @xmath119 parameters are calculated at the susy scale and the susy masses at tree - level are derived . the ew vevs @xmath120 and @xmath121 are afterwards re - calculated using the one - loop corrected @xmath30 mass and demanding @xmath122 in addition with the running value of @xmath123 . note that @xmath124 as well as all other self - energies include the corrections originated by all particles present in the mblr . these calculations are performed in @xmath119 scheme and t hooft gauge . also the complete dependence on the external momenta are taken into account . the re - calculated vevs are afterwards used to solve the tree - level tadpole equations again and to re - calculate the tree - level mass spectrum as well as all vertices entering the one - loop corrections . using these vertices and masses , the one - loop corrections @xmath125 to the tadpole equations are derived and we use as renormalization condition @xmath126 these one - loop corrected tadpole equations are solved with respect to the same parameter as at tree level resulting in new parameters @xmath127 , @xmath128 respectively @xmath129 , @xmath130 , @xmath131 . the final step is to calculate all self - energies for different particles and to use those to get the one - loop corrected mass spectrum . 1 . * real scalars * : for a real scalar @xmath132 , the one - loop corrections are included by calculating the real part of the poles of the corresponding propagator matrices @xcite @xmath133 = 0 , \label{eq : propagator}\ ] ] where @xmath134 equation ( [ eq : propagator ] ) has to be solved for each eigenvalue @xmath135 which can be achieved in an iterative procedure . this has to be done also for charged scalars as well as the fermions . note , @xmath136 is the tree - level mass matrix but for the parameters fixed by the tadpole equations the one - loop corrected values @xmath137 are used . * complex scalars * : for a complex scalar @xmath138 field we use at one - loop level @xmath139 while in case of sfermions @xmath140 agrees exactly with the tree - level mass matrix , for charged higgs bosons @xmath141 and @xmath142 or @xmath143 and @xmath143 has to be used depending on the set of parameters the tadpole equations are solved for . majorana fermions * : the one - loop mass matrix of a majorana @xmath144 fermion is related to the tree - level mass matrix by @xmath145 , \end{aligned}\ ] ] where we have denoted the wave - function corrections by @xmath146 , @xmath147 and the direct one - loop contribution to the mass by @xmath148 . * dirac fermions * : for a dirac fermion @xmath149 one has to add the self - energies as @xmath150 note , this procedure agrees with the method implemented in spheno 3.1.10 to calculate the loop masses in the mssm as well as with the code produced by sarah 3.0.39 or later . however , there are small differences to earlier versions of spheno as well as other spectrum calculators : the mssm equivalent of condition eq ( [ eq : conditionmz ] ) is often solved in an iterative way using the one - loop corrected parameters from the tadpole equations to calculate @xmath124 until @xmath151 has converged . in this context also @xmath141 and @xmath152 are used in the vertices entering the one - loop corrections . however , these steps mix tree- and one - loop level and break therefore gauge invariance : when we tried this approach the relation between goldstone and gauge bosons mass is violated . however , the numerical differences in case of the mssm turned out to be rather small . as discussed in section [ sec : loopcorrections ] we have calculated the entire mass spectrum at one - loop . for that purpose it is necessary to calculate all possible 1-loop diagrams for the one- and two - point functions . as example we here give the corresponding expressions for the one - loop corrections of the tadpoles as well as the self - energy for the scalar higgs fields . for all other self - energies we refer to the output of sarah . the results are expressed via passarino veltman integrals @xcite . the basic integrals are @xmath153\biggl [ ( q - p)^2-m_2 ^ 2+i\varepsilon\biggr ] } } \thickspace , \label{b0 def}\end{aligned}\ ] ] with the renormalization scale @xmath154 . all the other , necessary functions can be expressed by @xmath155 and @xmath156 . for instance , @xmath157~,\ ] ] and @xmath158 the numerical evalution of all loop - integrals is performed by spheno . with this conventions we can write the one - loop tadpoles as @xmath159 with @xmath160 . @xmath161 denotes the vertex of the three particles @xmath162 , @xmath163 @xmath164 , while @xmath165 will be used for four - point interactions . for chiral couplings we use @xmath166 as coefficient of the left and @xmath167 as coefficient of the right polarization operator . for instance , @xmath168 is the coupling of a pure down - type higgs to a @xmath30 boson while @xmath169 corresponds to the left - chiral part of the interaction of a @xmath82-higgs to a neutralino of the second generation . the expressions for all vertices can be obtained with sarah . + using these conventions the self - 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even higgs has a mass reaching easily 125 gev , with no constraints on the susy spectrum . right sneutrinos can be the lightest supersymmetric particle , changing all dark matter constraints on susy parameter space . the model has seven neutralinos and squark / gluino decay chains involve more complicated cascades than in the mssm . we also discuss briefly low - energy and accelerator constraints on the model , where the most important limits come from recent @xmath2 searches at the lhc and upper limits on lepton flavour violation .
although much can be learned by studying stellar nurseries and the fascinating process of stellar birth , we have much yet to learn in the field of stellar geriatrics . stars that do not proceed to explosive ends , the low- and intermediate - mass stars , undergo a period of mass loss , often extreme , in which 50% or more of the star s initial mass is transferred back to the ism . the rates of this mass loss vary widely from @xmath0 per year to as much as @xmath1 per year . such prodigious mass loss and the large number of low- and intermediate mass stars results in the fact that most of the interstellar medium perhaps as much as 80 to 90 percent has been cycled through a star and ejected via this process . flash - in - the - pan supernovae do have a significant impact , especially enriching the heavier metals in the ism , but the bulk of the material is provided by the aging process of common stars similar to the sun . understanding how the mass loss process proceeds and its implications on the chemical modification of the ism in our own galaxy has obvious implications for the study of more distant galaxies as well as being of interest itself . the mass loss process proceeds from the formation of dust in the upper atmospheres of evolved stars , a few stellar radii from the optical photosphere . this process has long been thought to be driven by the pulsations inherent in these kinds of stars , but it now appears likely that it is driven by dramatic temperature fluctuations caused by the formation of tio in the stellar atmosphere , which changes the physical conditions in the dust formation region @xcite . although the details of dust formation remain an unknown factor , we know roughly that when the temperature and density conditions are appropriate , nucleation can occur , leading to the formation of dust . this dust , exposed to the radiation field of the evolved star , absorbs outward momentum and begins to accelerate . gas not incorporated into dust grains is carried along with the dust through momentum coupling . as conditions allow , molecules can form from the gas that is carried along with the outward - moving dust . using existing centimeter and millimeter wave interferometers , studies of the molecules formed in these winds have been completed showing more or less spherically symmetric mass loss @xcite with some interesting results such as rotation @xcite and perfect spherical symmetry of an apparently single ejection event @xcite . certain molecules are capable of maser emission ( e.g. sio , h@xmath2o and oh ) . when such masers are found in the winds of evolved o - rich stars , they are powerful probes of the mass loss kinematics . some c - rich stars do exhibit hcn masers at high frequencies , but the o - dominated species are absent . however , they can provide only rough information about the physical conditions of the wind itself , provided by the physical conditions required for maser emission . using vlbi techniques , which provide resolutions as fine as 100 @xmath3arcseconds , the motions of masing gas can be tracked with high accuracy and the kinematics of the wind modelled . in practice , this has proven to be a challenging undertaking . the non - linear emission process and apparently complex distributions of the masers make modelling difficult . only rough models have yet been made placing the masers in ellipsoidal distributions undergoing a variety of kinematic motions . the maser observations do indicate more - or - less spherically or elliptically symmetric mass loss with acceleration occurring to the outermost regions of the wind where acceleration ceases due to decoupling of the gas from the dust . a mild controversy about the relative angular scales of the oh and h@xmath4o maser distributions ( e.g. the oh masers , although predicted to be at large radii , appear at about the same angular scale as the h@xmath4o masers ) is likely due to beaming effects . water masers are preferentially tangentially beamed as they reside in an accelerating portion of the wind while the oh masers are radially beamed as they reside in a constant velocity region of the wind @xcite . as difficult as the physics and geometries are , our current understanding is limited due to lack of adequate modelling in my opinion . other results indicate non - negligible rotation @xcite of the envelopes and the influence of magnetic fields on the shape of the shell @xcite . we have yet to understand the dust formation process in these objects . although infrared interferometric observations @xcite hint at a very clumpy and dynamic process , we have few tools available to probe this process in detail . we have only a very rough picture of the structure of the extended photosphere and wind of evolved stars . the role of magnetic fields in agb stars has not been explored in any detail , though they must impact the dust formation process , the wind itself and obviously provide information on the star itself . although detailed studies have been made using millimeter interferometers of the chemical structure of the nearest and largest evolved stars , much of the chemical structure in these objects remains a mystery . alma will help here but will miss the low - frequency line transitions . a detailed understanding of the structure of these objects awaits the square kilometer array . the ska , with its high resolution , sensitivity to a range of emission mechanisms and low - frequency observing capability will allow studies of evolved stars that have not been possible before and provide complementary observations to those provided by alma and other instruments . i discuss the anticipated observations ska can provide in the sections below . it has been shown @xcite that certain agb stars , the mira variables , undergo temperature changes of 30% and luminosity changes of a factor of two during their visual fluctuation period ( @xmath5 ) . as shown in @xcite , these changes should result in stellar radius fluctuations of about 40% . such dramatic fluctuations would lead to both dramatic shock waves that propagate from the star into its extended photosphere and also measurable changes in light curves at radio , infrared and optical wavelengths . the visual light curve fluctuates dramatically ( extreme cases show fluctuations of 8 magnitudes ) while the infrared light curves rarely fluctuate by more than a magnitude and the radio light curves fluctuate only by a few percent at most . observed light curves do not match those predicted by the radial fluctuations implied by the temperature and luminosity fluctuations @xcite . figure [ reidmira ] shows the model from @xcite that ( to first order ) reproduces the radio , infrared and visual light curves . the model predicts that tio is formed in the upper atmosphere as the star approaches minimum light and this additional opacity source can greatly decrease the observed light at visual wavelengths while having less impact at infrared wavelengths and almost no impact at radio wavelengths . this new discovery shows that much remains to be learned about evolved stars . after all , mira variables are one of the oldest astronomical phenomena studied and only now has an adequate first - order model been developed to explain their fluctuations . the ska will allow further testing of these models at far greater sensitivity . observations of the flux from the photospheres of agb stars are exceedingly difficult . typical fluxes are on the order of 200 microjy and require special calibration techniques with current interferometers . the ska , with a sensitivity of about 0.1 microjy at 20 ghz will provide the most accurate agb star light curves across all wavelengths . such measurements will allow improved modelling of the opacity source fluctuations in the star . the discovery of particularly large extrasolar planets orbiting close to their host star opens up the possibility for the observation of eclipses using the ska , as has been observed in the star hd 209458 . however , the eclipse type that is potentially observable would be an active radio - emitting planet similar to jupiter being eclipsed by its host star rather than the more typical eclipse . as pointed out in @xcite , jupiter - like planets will produce detectable radio emission out to distances of 10 pc . the passage of a planet of this type behind its host agb star would be detectable , since the emission from the planet would be of order 10 microjy , compared to the photospheric flux of 200 microjy . the diameter of the radio photosphere of a typical agb star is of order 5 au . at 1kpc , such a source would have a maximum angular diameter of 6 milliarcseconds . at 22 ghz and with 1000 km baselines , the ska will have a resolution of roughly 3 milliarcseconds . this resolution corresponds to linear resolution of 3 au at 1 kpc . for agb star diameters of 3 - 5 au , they can be moderately resolved with ska . thus , for only the nearest agb stars will any degree of imaging be possible . the number of agb stars closer than 1kpc is limited . without a substantial increase in the highest frequency observed by the ska or the maximum baselines , imaging of only the nearest agb stars will be possible . that said , some very interesting imaging projects can be undertaken for large agb stars not further than 1 kpc from the earth . for example , the well - known and nearby ( 150 pc ) carbon star irc+10216 has a photospheric size of 35 milliarcseconds and an extended envelope diameter of nearly 1. with a resolution element of 3 mas , the surface of the star would be imaged well and the overall envelope , especially in spectral lines ( see below ) would be highly resolved . the imaging design goal of 0.1 arcsecond resolution at 1.4 ghz over a 1 degree field ( and scaled with frequency ) is sufficient to provide high spatial dynamic range imaging at high sensitivity for objects of this type . the science the ska will allow is the direct imaging of the dust formation process and connection with stellar pulsation for the nearest and largest agb stars . maser emission from gas in the outflowing winds of agb stars is a common phenomenon in o - rich agb stars . masers are regions of gas in the stellar wind that have sufficient velocity coherence to amplify background photons via amplified emission of radiation . such amplification is possible due to a population inversion of the molecular species in question and a fortuitous alignment of molecular rotational , vibrational or ro - vibrational energy levels . several species are found . sio masers are located close to the star ( within a few stellar radii and below the dust formation zone ) . masers are located at intermediate distances from a few tens of stellar radii to a few hundred . the remote oh masers are located up to several thousand stellar radii from the host star . in addition to knowing the location of the various maser species , we have a good understanding of the overall shell structure around these stars @xcite . figure [ shell ] shows graphically our current understanding of the circumstellar region around an agb star . the star itself is from between 1 and 5 au in size . above this surface is a chromospheric region followed by a molecular photosphere ending between 1 and 2 au above the optical photosphere . the radio photosphere ( about 0.5 to 1au in thickness ) is located near the sio maser formation region . beyond this zone , wind acceleration begins as dust forms in a region from roughly 5 au to 10 au ( depending on the properties of the star and pulsation phase ) . the h@xmath4o and oh masers begin to appear at radii of 15 au or more and the oh masers are found further out from the water maser shell . the exact sizes of the various regions , their exact locations and how they interact remain rough measurements . , width=283 ] masers are imaged using vlbi techniques and typically have resolved sizes of a milliarcsecond or so , but observations with merlin show that a weak diffuse emission can also be present @xcite . depending on the upper frequency cutoff for the ska , the oh ( 1.6 ghz ) , methanol ( 6.7 ghz ) and water masers ( 22 ghz ) could be observable . however , it is not the detection of maser emission with the ska that is of greatest interest ( though the sensitivity of the instrument would allow detection of extragalactic masers to a much greater distance than currently available ) . it is the sensitivity to both the stellar photospheric emission and the dust continuum in the wind combined with the vlbi observations that will be of prime interest . vlbi imaging techniques are sensitive only to very high brightness temperatures and the smallest angular sizes ( 1 - 5 milliarcseconds ) and therefore only the maser spots themselves and not the environment in which they are located can be imaged . with the angular resolution of the ska at 1.4 ghz ( 0.1 `` ) , the thermal emission across a typical oh maser distribution 1''-2 " in diameter could be mapped with sufficient resolution and sensitivity to allow alignment of the vlbi maser observations with the overall dust distribution and star itself . combined with infrared interferometric observations , which are now beginning to show the details of the dust distribution at high angular resolutions ( see figure [ monnierfigure ] ) @xcite ( @xmath6 milliarseconds , but over limited fields of view ) , the ska will play a critical role in providing information on the largest scales . outstanding problems to be addressed include the details of dust formation , such as whether the process proceeds uniformly as a function of pulsation cycle or at particular times , the degree of clumpiness of the dust formation and the exact physical conditions that lead to dust formation . the transition of agb stars from roughly spherically symmetric mass - losing objects to the asymmetric planetary nebulae has yet to be understood completely and the combination of the kinematic information provided by vlbi maser observations and the dust distribution will hopefully shed new light on this area . in the frequency range of the ska are 634 molecular line transitions , many of which have not been well studied , only detected , and some of which have still not been identified @xcite . for convenience , these transitions are provided in table [ lovastable ] . many of these species are expected to be present in the winds of evolved stars . as pointed out by zijlstra ( 2003 ) , both the dust and gas created in the stellar winds of agb stars survive in the ism . the dust , as indicated by reddening and particles found in meteorites or as micrometeoroid particulates in our own upper atmosphere @xcite , survives in interstellar space . the presence of the diffuse interstellar bands are the main piece of evidence for the existence of rather complex molecules in interstellar space . as yet , we do not know if the molecules were formed within an envelope of an agb star and survived ejection into interstellar space or if they were incorporated onto dust grains and later evaporated from their surfaces upon exposure to interstellar uv radiation . it is likely that the true situation will be a mixture of both of these cases . a number of molecules with transitions in the frequency range of the ska are of particular interest for astrobiology . these include the building block molecule for simple sugars such as ribose and deoxyribose , furan ( @xmath7 ; e.g. with a transition at 10.6 ghz ) @xcite . the same authors detected c - c@xmath4h@xmath8o , one of the few cyclic molecules in space . they note that the presence of these molecular species in cold dark clouds suggests that rather complex organic molecules may have been present in the solar system before the planets formed , a first step toward explaining the origin of life on the early earth . magnetic fields of agb stars are now thought to be fairly strong from observations of sio masers @xcite . depending on the exact models used , the field strengths seem to be between 5 - 10 g at radii of 3 au or so . such strong magnetic fields will have obvious impacts on both the molecular gas ( zeeman splitting for a number of species such as ccs and so ) and possibly produce circularly polarized radio emission from the star itself ( analogous to the emission observed from the sun ) . although requiring careful instrumental polarization characterization , observations of these effects will provide confirmation of the magnetic field strength implied by the sio maser polarization observations and further constraints on agb stars themselves . potential movies of regions of magnetic field enhancements on the surfaces of nearby agb stars could be tracked with time , testing maser observations of rotation . 9 j. h. bieging and rieu , n.q . , astronomical journal 112 , ( 1996 ) , 706 . d. a. boboltz & k. b. marvel , astrophysical journal 545 , ( 2000 ) , 149 . dickens , w. m. irvine , a. nummelin , h. mollendal , s. saito , s. thorwith , a. hjalmarsoni , m. ohishi , spectrochimica acta part a : molecular and biomolecular spectroscopy 57 , ( 2001 ) , 643 . a. j. kemball & p.j . diamond , astrophysical journal , 481l ( 1997 ) , 111 . lovas , nist recommended rest frequencies for observed interstellar molecular microwave transitions : 2002 revision , ( 2002 ) . s. messenger , l.p . keller , f.j . stadermann , r.m . walker and e. zinner , science 300 , ( 2003 ) , 105 . monnier , r. millan - gabet , p.g . tuthill , w.a . traub , n.p . carleton , v. coud du foresto , w. c. danchi , m.g . lacasse , s. morel , g. perrin , i.l . porro , f.p . schloerb , c.h . townes , astrophysical journal 605 , ( 2004 ) , 436 . monnier , p.g . tuthill , b. lopez , p. cruzalebes , w. c. danchi , c.a . haniff , astrophysical journal 512 , ( 1999 ) , 351 . k. murakawa , j.a . yates , a.m.s . richards , r.j . cohen monthly notices of the royal astronomical society 344 , ( 2003 ) , 1 . h. olofsson , p. bergman , r. lucas , k. eriksson , b. gustafsson and j.h . bieging , astronomy and astrophysics 330 , ( 1998 ) , l1 . e. petit and s. b. nicholson , astrophysical journal 78 , ( 1933 ) , 320 . reid and k.m . menten , astrophysical journal 456 , ( 1997 ) 327 . reid and j.e . goldston , astrophysical journal 568 , ( 2002 ) 931 . reid in _ cosmic masers : from protostars to blackholes _ , astronomical society of the pacific , 2002 . richards , j. a. yates and r. j. cohen , monthly notices of the royal astronomical society 306 , ( 1999 ) 954 . richards , j. a. yates and r. j. cohen , monthly notices of the royal astronomical society 299 , ( 1998 ) 319 . rieu and j.h . bieging , astrophysical journal 359 , ( 1990 ) , 131 . taylor and r. braun , _ science with the square kilometer array _ , 1999 . zijlstra , the scientific promise of the ska : ska workshop oxford , 2003 , kramer & rawlings , eds . web reference http://www.skatelescope.org/documents/workshop_oxford2002.pdf lll\|lll + frequency ( mhz ) & formula & quantum numbers & frequency ( mhz ) & formula & quantum numbers + frequency ( mhz ) & formula & quantum numbers & frequency ( mhz ) & formula & quantum numbers + + 701.679 & ch & 23/2 j=3/2 f=2 - 2 & 19418.796 & c - c@xmath9hd & 1(1,0)-1(0,1 ) f=1 - 0 + 704.175 & ch & 23/2 j=3/2 f=2 + -1- & 19426.679 & ch@xmath4chcn & 2(1,1)-1(1,0 ) f=2 - 1 + 722.303 & ch & 23/2 j=3/2 f=1 + -2- & 19427.851 & ch@xmath4chcn & 2(1,1)-1(1,0 ) f=3 - 2 + 724.791 & ch & 23/2 j=3/2 f=1 - 1 & 19429.098 & ch@xmath4chcn & 2(1,1)-1(1,0 ) f=1 - 0 + 834.285 & ch@xmath9oh & 1(1,0)-1(1,1 ) a-+ & 19430.85 & unidentified & + 1065.076 & ch@xmath9cho & 1(1,0 ) - 1(1,1 ) a-+ & 19609.78 & unidentified & + 1371.722 & ch@xmath4chcn & 2(1,1)-2(1,2 ) f=1 - 1 & 19682.50 & unidentified & + 1371.797 & ch@xmath4chcn & 2(1,1)-2(1,2 ) f=3 - 3 & 19692.50 & unidentified & + 1371.934 & ch@xmath4chcn & 2(1,1)-2(1,2 ) f=2 - 2 & 19755.111 & hc@xmath10n & 34 - 33 + 1538.108 & nh@xmath4cho & 1(1,0)-1(1,1 ) f=1 - 1 & 19757.538 & nh@xmath9 & 6(3)-6(3 ) + 1538.676 & nh@xmath4cho & 1(1,0)-1(1,1 ) f=1 - 2 & 19771.50 & unidentified & + 1539.264 & nh@xmath4cho & 1(1,0)-1(1,1 ) f=2 - 1 & 19780.800 & cccn & 2 - 1 j=5/2 - 3/2 f=5/2 - 3/2 + 1539.527 & nh@xmath4cho & 1(1,0)-1(1,1 ) f=1 - 0 & 19780.826 & cccn & 2 - 1 j=5/2 - 3/2 f=3/2 - 1/2 + 1539.832 & nh@xmath4cho & 1(1,0)-1(1,1 ) f=2 - 2 & 19781.094 & cccn & 2 - 1 j=5/2 - 3/2 f=7/2 - 5/2 + 1540.998 & nh@xmath4cho & 1(1,0)-1(1,1 ) f=0 - 1 & 19799.951 & cccn & 2 - 1 j=5/2 - 3/2 f=3/2 - 1/2 + 1570.805 & nh@xmath11cho & 1(1,0)-1(1,1 ) f=2 - 2 & 19800.121 & cccn & 2 - 1 j=5/2 - 3/2 f=5/2 - 3/2 + 1584.274 & @xmath12oh & 23/2 j=3/2 f=1 - 2 & 19838.346 & nh@xmath9 & 5(1)-5(1 ) + 1610.247 & ch@xmath9ocho & 1(1,0)-1(1,1 ) a & 19871.344 & hccnc & 2 - 1 + 1610.900 & ch@xmath9ocho & 1(1,0)-1(1,1 ) e & 19967.396 & ch@xmath9oh & 2(1,1)-3(0,3 ) e + 1612.2310 & oh & 23/2 j=3/2 f=1 - 2 & 19974.50 & unidentified & + 1624.518 & @xmath13oh & 23/2 j=3/2 f , f1=7/2,4 - 7/2,4 & 20064.21 & unidentified & + 1626.161 & @xmath13oh & 23/2 j=3/2 f , f1=9/2,4 - 9/2,4 & 20109.547 & ch@xmath4cn & 1 - 0 3/2 - 1/2 5/2 - 3/2 5/2 - 5/2 + 1637.564 & @xmath12oh & 23/2 j=3/2 f=1 - 1 & 20115.77 & ch@xmath4cn & 1 - 0 1/2 - 1/2 3/2 - 3/2 5/2 - 5/2 + 1638.805 & hcooh & 1(1,0)-1(1,1 ) & 20117.43 & ch@xmath4cn & 1 - 0 3/2 - 1/2 5/2 - 3/2 3/2 - 1/2 + 1639.503 & @xmath12oh & 23/2 j-3/2 f=2 - 2 & 20118.014 & ch@xmath4cn & 1 - 0 3/2 - 1/2 5/2 - 3/2 5/2 - 3/2 + 1665.4018 & oh & 23/2 j=3/2 f=1 - 1 & 20118.16 & ch@xmath4cn & 1 - 0 3/2 - 1/2 1/2 - 1/2 3/2 - 3/2 + 1667.3590 & oh & 23/2 j=3/2 f=2 - 2 & 20119.606 & ch@xmath4cn & 1 - 0 3/2 - 1/2 5/3 - 3/2 7/2 - 5/2 + 1692.795 & @xmath12oh & 23/2 j=3/2 f=2 - 1 & 20121.61 & ch@xmath4cn & 1 - 0 3/2 - 1/2 3/2 - 3/2 3/2 - 3/2 + 1720.5300 & oh & 23/2 j=3/2 f=2 - 1 & 20123.96 & ch@xmath4cn & 1 - 0 3/2 - 1/2 1/2 - 1/2 3/2 - 3/2 + 2661.61 & hc@xmath14n & 1 - 0 f=1 - 1 & 20124.22 & ch@xmath4cn & 1 - 0 1/2 - 1/2 3/2 - 1/2 3/2 - 1/2 + 2662.87 & hc@xmath14n & 1 - 0 f=2 - 1 & 20124.22 & ch@xmath4cn & 1 - 0 3/2 - 1/2 3/2 - 3/2 1/2 - 1/2 + 2664.76 & hc@xmath14n & 1 - 0 f=0 - 1 & 20124.45 & ch@xmath4cn & 1 - 0 3/2 - 1/2 3/2 - 1/2 3/2 - 3/2 + 3139.404 & h@xmath4cs & 2(1,1)-2(1,2 ) & 20124.49 & ch@xmath4cn & 1 - 0 1/2 - 1/2 3/2 - 3/2 5/2 - 3/2 + 3195.162 & ch@xmath9cho & 2(1,1 ) - 2(1,2 ) a-+ & 20126.031 & ch@xmath4cn & 1 - 0 3/2 - 1/2 3/2 - 3/2 3/2 - 1/2 + 3263.794 & ch & 21/2 j=1/2 f=0 - 1 & 20128.770 & ch@xmath4cn & 1 - 0 1/2 - 1/2 3/2 - 1/2 3/2 - 3/2 + 3335.481 & ch & 21/2 j=1/2 f=1 - 1 & 20139.76 & ch@xmath4cn & 1 - 0 1/2 - 1/2 1/2 - 3/2 3/2 - 5/2 + 3349.193 & ch & 21/2 j=1/2 f=1 - 0 & 20168.48 & unidentified & + 4388.7786 & h@xmath4c@xmath12o & 1(1,0)-1(1,1 ) f=1 - 0 & 20171.089 & ch@xmath9oh & 11(1,11)-10(2,8 ) a+ + 4388.7960 & h@xmath4c@xmath12o & 1(1,0)-1(1,1 ) f=0 - 1 & 20203.31 & unidentified & + 4388.7963 & h@xmath4c@xmath12o & 1(1,0)-1(1,1 ) f=2 - 2 & 20209.209 & ch@xmath15co & 1(0,1)-0(0,0 ) + 4388.8011 & h@xmath4c@xmath12o & 1(1,0)-1(1,1 ) f=2 - 1 & 20281.00 & unidentified & + 4388.8035 & h@xmath4c@xmath12o & 1(1,0)-1(1,1 ) f=1 - 2 & 20303.946 & hc@xmath16n & 18 - 17 + 4388.8084 & h@xmath4c@xmath12o & 1(1,0)-1(1,1 ) f=1 - 1 & 20336.135 & hc@xmath10n & 35 - 34 + 4592.9563 & h@xmath11co & 1(1,0)-1(1,1)1/2,1/2 - 1/2,3/2 & 20357.226 & ch3c@xmath8h & 5(1)-4(1 ) + 4592.9738 & h@xmath11co & 1(1,0)-1(1,1)1/2,1/2 - 3/2,3/2 & 20357.423 & ch3c@xmath8h & 5(0)-4(0 ) + 4592.9759 & h@xmath11co & 1(1,0)-1(1,1)3/2,1/2 - 1/2,3/2 & 20371.45 & nh@xmath9 & 5(2)-5(2 ) + 4592.9857 & h@xmath11co & 1(1,0)-1(1,1)3/2,1/2 - 5/2,3/2 & 20460.01 & hdo & 3(2,1)-4(1,4 ) + 4592.9934 & h@xmath11co & 1(1,0)-1(1,1)3/2,1/2 - 3/2,3/2 & 20501.5 & unidentified & + 4593.0494 & h@xmath11co & 1(1,0)-1(1,1)1/2,1/2 - 1/2,1/2 & 20533.235 & unidentified & + 4593.0690 & h@xmath11co & 1(1,0)-1(1,1)3/2,1/2 - 1/2,1/2 & 20533.289 & c@xmath17h & 23/2 17.5 - 16.5 + 4593.0800 & h@xmath11co & 1(1,0)-1(1,1)1/2,1/2 - 3/2,1/2 & 20723.5 & unidentified & + 4593.0812 & h@xmath11co & 1(1,0)-1(1,1)1/2,3/2 - 1/2,3/2 & 20728.67 & unidentified & + 4593.0864 & h@xmath11co & 1(1,0)-1(1,1)3/2,3/2 - 1/2,3/2 & 20735.452 & nh@xmath9 & 9(7)-9(7 ) + 4593.0865 & h@xmath11co & 1(1,0)-1(1,1)5/2,3/2 - 5/2,3/2 & 20765.80 & unidentified & + 4593.0942 & h@xmath11co & 1(1,0)-1(1,1)5/2,3/2 - 3/2,3/2 & 20790.00 & unidentified & + 4593.0961 & h@xmath11co & 1(1,0)-1(1,1)3/2,3/2 - 5/2,3/2 & 20792.563 & h@xmath4ccc & 1(0,1)-0(0,0 ) + 4593.0985 & h@xmath11co & 1(1,0)-1(1,1)1/2,3/2 - 3/2,3/2 & 20792.872 & c@xmath18h & 23/2 j=15/2 - 13/2 f=8 - 7 e + 4593.0994 & h@xmath11co & 1(1,0)-1(1,1)3/2,1/2 - 3/2,1/2 & 20792.945 & c@xmath18h & 23/2 j=15/2 - 13/2 f=7 - 6 e + 4593.1039 & h@xmath11co & 1(1,0)-1(1,1)3/2,3/2 - 3/2,3/2 & 20794.444 & c@xmath18h & 23/2 j=15/2 - 13/2 f=8 - 7 f + 4593.1741 & h@xmath11co & 1(1,0)-1(1,1)1/2,3/2 - 1/2,1/2 & 20794.512 & c@xmath18h & 23/2 j=15/2 - 13/2 f=7 - 6 f + 4593.1795 & h@xmath11co & 1(1,0)-1(1,1)3/2,3/2 - 1/2,1/2 & 20804.830 & nh@xmath9 & 7(5)-7(5 ) + 4593.2003 & h@xmath11co & 1(1,0)-1(1,1)5/2,3/2 - 3/2,1/2 & 20838.20 & unidentified & + 4593.2046 & h@xmath11co & 1(1,0)-1(1,1)1/2,3/2 - 3/2,1/2 & 20847.50 & unidentified & + 4593.2099 & h@xmath11co & 1(1,0)-1(1,1)3/2,3/2 - 3/2,1/2 & 20852.527 & nh@xmath9 & 10(8)-10(8 ) + 4617.121 & nh@xmath4cho & 2(1,1)-2(1,2 ) f=2 - 2 & 20878.00 & unidentified & + 4618.967 & nh@xmath4cho & 2(1,1)-2(1,2 ) f=3 - 3 & 20908.848 & ch@xmath9oh & 16(-4,13)-15(-5,10 ) e + 4619.993 & nh@xmath4cho & 2(1,1)-2(1,2 ) f=1 - 1 & 20917.157 & hc@xmath10n & 36 - 35 + 4660.242 & oh & 21/2 j=1/2 f=0 - 1 & 20970.658 & ch@xmath9oh & 10(1,10)-11(,9 ) a+ t=1 + 4750.656 & oh & 21/2 j=1/2 f=1 - 1 & 20994.617 & nh@xmath9 & 6(4)-6(4 ) + 4765.562 & oh & 21/2 j=1/2 f=1 - 0 & 20999.79 & unidentified & + 4829.6412 & h@xmath15co & 1(1,0)-1(1,1 ) f=1 - 0 & 21070.739 & nh@xmath9 & 11(9)-11(9 ) + 4829.6587 & h@xmath15co & 1(1,0)-1(1,1 ) f=0 - 1 & 21134.311 & nh@xmath9 & 4(1)-4(1 ) + 4829.6594 & h@xmath15co & 1(1,0)-1(1,1 ) f=2 - 2 & 21143.18 & unidentified & + 4829.6639 & h@xmath15co & 1(1,0)-1(1,1 ) f=2 - 1 & 21231.00 & unidentified & + 4829.6664 & h@xmath15co & 1(1,0)-1(1,1 ) f=1 - 2 & 21285.275 & nh@xmath9 & 5(3)-5(3 ) + 4829.6710 & h@xmath15co & 1(1,0)-1(1,1 ) f=1 - 1 & 21301.261 & hc@xmath14n & 8 - 7 + 4916.312 & hcooh & 2(1,1)-2(1,2 ) & 21322.50 & unidentified & + 5005.3208 & ch@xmath9oh & 3(1,2)-3(1,3 ) a-+ & 21431.932 & hc@xmath16n & 19 - 18 + 5289.015 & ch@xmath4nh & 1(1,0)-1(1,1 ) f=0 - 1 & 21447.8 & unidentified & + 5289.678 & ch@xmath4nh & 1(1,0)-1(1,1 ) f=1 - 0 & 21453.93 & unidentified & + 5289.813 & ch@xmath4nh & 1(1,0)-1(1,1 ) f=2 - 2 & 21470.4 & unidentified & + 5290.614 & ch@xmath4nh & 1(1,0)-1(1,1 ) f=2 - 1 & 21480.809 & c@xmath14h & 21/2 j=9/2 - 7/2 f=5 - 4 e + 5290.879 & ch@xmath4nh & 1(1,0)-1(1,1 ) f=1 - 2 & 21481.299 & c@xmath14h & 21/2 j=9/2 - 7/2 f=4 - 3 e + 5291.680 & ch@xmath4nh & 1(1,0)-1(1,1 ) f=1 - 1 & 21484.695 & c@xmath14h & 21/2 j=9/2 - 7/2 f=5 - 4 f + 5324.058 & hc@xmath14n & 2 - 1 f=2 - 2 & 21485.248 & c@xmath14h & 21/2 j=9/2 - 7/2 f=4 - 3 f + 5324.270 & hc@xmath14n & 2 - 1 f=1 - 0 & 21488.255 & h@xmath4cccccc & 8(1,8)-7(1,7 ) + 5325.330 & hc@xmath14n & 2 - 1 f=2 - 1 & 21498.182 & hc@xmath10n & 37 - 36 + 5325.421 & hc@xmath14n & 2 - 1 f=3 - 2 & 21546.94 & unidentified & + 5327.451 & hc@xmath14n & 2 - 1 f=1 - 1 & 21550.342 & ch@xmath9oh & 12(2,11)-11(1,11 ) a+ t=1 + 6016.746 & oh & 23/2 j=5/2 f=2 - 3 & 21569.5 & unidentified & + 6030.747 & oh & 23/2 j=5/2 f=2 - 2 & 21576.5 & unidentified & + 6035.092 & oh & 23/2 j-5/2 f=3 - 3 & 21582.6 & unidentified & + 6049.084 & oh & 23/2 j=5/2 f=3 - 2 & 21587.400 & c - c@xmath9h@xmath4 & 2(2,0)-2(1,1 ) + 6278.628 & h@xmath4cs & 3(1,2)-3(1,3 ) & 21592.1 & unidentified & + 6389.933 & ch@xmath9cho & 3(1,2 ) - 3(1,3 ) a-+ & 21595.8 & unidentified & + 6668.5192 & ch@xmath9oh & 5(1,6)-6(0,6 ) a++ & 21598.4 & unidentified & + 7761.747 & oh & 21/2 j=3/2 f=1 - 1 & 21606.30 & unidentified & + 7820.125 & oh & 21/2 j=3/2 f=2 - 2 & 21615.5 & unidentified & + 7895.989 & hc@xmath16n & 7 - 6 f=6 - 5 & 21703.3580 & nh@xmath9 & 4(2)-4(2 ) + 7896.010 & hc@xmath16n & 7 - 6 f=7 - 6 & 21715.8 & unidentified & + 7896.023 & hc@xmath16n & 7 - 6 f=8 - 7 & 21930.476 & cc34s & 2,1 - 1,0 + 7987.782 & hc@xmath14n & 3 - 2 f=2 - 1 & 21980.5453 & hnco & 1(0,1)-0(0,0 ) f=0 - 1 + 7987.994 & hc@xmath14n & 3 - 2 f=3 - 2 & 21981.4706 & hnco & 1(0,1)-0(0,0 ) f=2 - 1 + 7988.044 & hc@xmath14n & 3 - 2 f=4 - 3 & 21982.0854 & hnco & 1(0,1)-0(0,0 ) f=1 - 1 + 8135.870 & oh & 21/2 j=5/2 f=2 - 2 & 22079.204 & hc@xmath10n & 38 - 37 + 8189.587 & oh & 21/2 j=5/2 f=3 - 3 & 22235.044 & h@xmath4o & 6(1,6)-5(2,3 ) f=7 - 6 + 8775.088 & ch@xmath9nh@xmath4 & 2(0,2)-1(0,1 ) f=1 - 0 aa & 22235.077 & h@xmath4o & 6(1,6)-5(2,3 ) f=6 - 5 + 8777.442 & ch@xmath9nh@xmath4 & 2(0,2)-1(0,1 ) f=3 - 2 aa & 22235.120 & h@xmath4o & 6(1,6)-5(2,3 ) f=5 - 4 + 8778.200 & ch@xmath9nh@xmath4 & 2(0,2)-1(0,1 ) f=2 - 2 aa & 22235.253 & h@xmath4o & 6(1,6)-5(2,3 ) f=6 - 6 + 8778.260 & ch@xmath9nh@xmath4 & 2(0,2)-1(0,1 ) f=1 - 1 aa & 22235.298 & h@xmath4o & 6(1,6)-5(2,3 ) f=5 - 5 + 8779.496 & ch@xmath9nh@xmath4 & 2(0,2)-1(0,1 ) f=2 - 1 aa & 22258.173 & cco & 2,1 - 1,0 + 8815.814 & h@xmath19cccn & 1 - 0 f=1 - 1 & 22307.670 & hdo & 5(3,2)-5(3,3 ) + 8817.096 & h@xmath19cccn & 1 - 0 f=2 - 1 & 22344.030 & ccs & 2,1 - 1,0 + 8819.019 & h@xmath19cccn & 1 - 0 f=0 - 1 & 22471.180 & hcooh & 1(0,1)-0(0,0 ) + 9024.009 & hc@xmath16n & 8 - 7 & 22559.915 & hc@xmath16n & 20 - 19 + 9058.447 & hc@xmath19ccn & 1 - 0 f=1 - 1 & 22624.8892 & 15nh@xmath9 & 1(1)-1(1 ) f , f1=1.5,1 - 1.3,1 + 9059.318 & hcc@xmath19cn & 1 - 0 f=1 - 1 & 22624.9331 & 15nh@xmath9 & 1(1)-1(1 ) f , f1=1.5,1 - 0.8,1 + 9059.736 & hc@xmath19ccn & 1 - 0 f=2 - 1 & 22624.9410 & 15nh@xmath9 & 1(1)-1(1 ) f , f1=0.5,1 - 0.8,1 + 9060.6080 & hcc@xmath19cn & 1 - 0 f=2 - 1 & 22624.9469 & 15nh@xmath9 & 1(1)-1(1 ) f , f1=1.5,2 - 1.5,2 + 9097.0346 & hcccn & 1 - 0 f=1 - 1 & 22639.3 & unidentified & + 9098.3321 & hcccn & 1 - 0 f=2 - 1 & 22644.3 & unidentified & + 9100.2727 & hcccn & 1 - 0 f=0 - 1 & 22649.843 & 15nh@xmath9 & 2(2)-2(2 ) + 9118.823 & ch@xmath9och@xmath9 & 2(0,2)-1(1,1 ) aa & 22653.022 & nh@xmath9 & 5(4)-5(4 ) + 9119.671 & ch@xmath9och@xmath9 & 2(0,2)-1(1,1 ) ee & 22660.225 & hc@xmath10n & 39 - 38 + 9120.509 & ch@xmath9och@xmath9 & 2(0,2)-1(1,1 ) ae & 22678.6 & unidentified & + 9120.527 & ch@xmath9och@xmath9 & 2(0,2)-1(1,1 ) ea & 22688.312 & nh@xmath9 & 4(3)-4(3 ) + 9235.119 & nh@xmath4cho & 3(1,2)-3(1,3 ) f=3 - 3 & 22732.429 & nh@xmath9 & 6(5)-6(5 ) + 9237.034 & nh@xmath4cho & 3(1,2)-3(1,3 ) f=4 - 4 & 22789.421 & 15nh@xmath9 & 3(3)-3(3 ) + 9237.704 & nh@xmath4cho & 3(1,2)-3(1,3 ) f=2 - 2 & 22827.741 & ch@xmath9ocho & 2(1,2)-1(1,1 ) e + 9486.71 & unidentified & & 22828.134 & ch@xmath9ocho & 2(1,2)-1(1,1 ) a + 9493.061 & c@xmath8h & 3/2 - 1/2 f=1 - 0 & 22834.1851 & nh@xmath9 & 3(2)-3(2 ) + 9496.4 & unidentified & & 22878.949 & dc5n & 9 - 8 + 9497.616 & c@xmath8h & 3/2 - 1/2 f=2 - 1 & 22924.940 & nh@xmath9 & 7(6)-7(6 ) + 9508.005 & c@xmath8h & 3/2 - 1/2 f=1 - 1 & 23046.0158 & 15nh@xmath9 & 4(4)-4(4 ) + 9547.953 & c@xmath8h & 1/2 - 1/2 f=1 - 0 & 23098.8190 & nh@xmath9 & 2(1)-2(1 ) + 9551.717 & c@xmath8h & 1/2 - 1/2 f=0 - 1 & 23121.024 & ch@xmath9oh & 9(2,7)-10(1,10 ) a+ + 9562.904 & c@xmath8h & 1/2 - 1/2 f=1 - 1 & 23122.983 & cccs & 4 - 3 + 9703.508 & c@xmath18h & 23/2 j=3.5 - 2.5 f=4 - 3 e & 23142.2 & unidentified & + 9703.600 & c@xmath18h & 23/2 j=3.5 - 2.5 f=3 - 2 e & 23228.0 & unidentified & + 9703.835 & c@xmath18h & 23/2 j=3.5 - 2.5 f=4 - 3 f & 23232.238 & nh@xmath9 & 8(7)-8(7 ) + 9703.936 & c@xmath18h & 23/2 j=3.5 - 2.5 f=3 - 2 f & 23241.246 & hc@xmath10n & 40 - 39 + 9877.606 & hc@xmath10n & 17 - 16 & 23421.9823 & 15nh@xmath9 & 5(5)-5(5 ) + 9885.89 & cccn & 1 - 0 j=3/2 - 1/2 f=5/2 - 3/2 & 23444.778 & ch@xmath9oh & 10(1,9)-9(2,8 ) a- + 9936.202 & ch@xmath9oh & 9(-1,9)-8(-2,7 ) e & 23565.160 & c@xmath18h & 23/2 j=17/2 - 15/2 f=9 - 8 e + 9978.686 & ch@xmath9oh & 4(3,2)-5(2,3 ) e & 23565.226 & c@xmath18h & 23/2 j=17/2 - 15/2 f=8 - 7 e + 10058.257 & ch@xmath9oh & 4(3,1)-5(2,4 ) e & 23567.169 & c@xmath18h & 23/2 j=17/2 - 15/2 f=9 - 8 f + 10152.008 & hc@xmath16n & 9 - 8 & 23567.238 & c@xmath18h & 23/2 j=17/2 - 15/2 f=8 - 7 f + 10278.246 & hdo & 2(2,0)-2(2,1 ) & 23600.242 & sic2 & 1(0,1)-0(0,0 ) + 10458.639 & hc@xmath10n & 18 - 17 & 23657.471 & nh@xmath9 & 9(8)-9(8 ) + 10463.962 & h@xmath4cs & 4(1,3)-4(1,4 ) & 23687.898 & hc@xmath16n & 21 - 20 + 10648.419 & ch@xmath9cho & 4(1,3 ) - 4(1,4 ) a-+ & 23692.9265 & nh@xmath9 & 1(1)-1(1 ) f , f1=1/2,1 - 1/2,0 + 10650.563 & hc@xmath14n & 4 - 3 f=3 - 2 & 23692.9688 & nh@xmath9 & 1(1)-1(1 ) f , f1=3/2,1 - 1/2,0 + 10650.654 & hc@xmath14n & 4 - 3 f=4 - 3 & 23693.8722 & nh@xmath9 & 1(1)-1(1 ) f , f1=1/2,1 - 3/2,2 + 10650.686 & hc@xmath14n & 4 - 3 f=5 - 4 & 23693.9051 & nh@xmath9 & 1(1)-1(1 ) f , f1=3/2,1 - 5/2,2 + 11119.445 & ccs & 1,0 - 0,1 & 23693.9145 & nh@xmath9 & 1(1)-1(1 ) f , f1=3/2,1 - 3/2,2 + 11280.006 & hc@xmath16n & 10 - 9 & 23694.4591 & nh@xmath9 & 1(1)-1(1 ) f , f1=1/2,1 - 1/2,1 + 11561.513 & cccs & 2 - 1 & 23694.4700 & nh@xmath9 & 1(1)-1(1 ) f , f1=1/2,1 - 3/2,1 + 12162.979 & ocs & 1 - 0 & 23694.4709 & nh@xmath9 & 1(1)-1(1 ) f , f1=3/2,2 - 5/2,2 + 12178.593 & ch@xmath9oh & 2(0,2)-3(-1,3 ) e & 23694.4803 & nh@xmath9 & 1(1)-1(1 ) f , f1=3/2,2 - 3/2,2 + 12408.003 & hc@xmath16n & 11 - 10 & 23694.5014 & nh@xmath9 & 1(1)-1(1 ) f , f1=3/2,1 - 1/2,1 + 12782.769 & hc@xmath10n & 22 - 21 & 23694.5060 & nh@xmath9 & 1(1)-1(1 ) f , f1=5/2,2 - 5/2,2 + 12848.48 & unidentified & & 23694.5123 & nh@xmath9 & 1(1)-1(1 ) f , f1=3/2,1 - 3/2,1 + 12848.731 & hc@xmath20n & 38 - 37 & 23694.5153 & nh@xmath9 & 1(1)-1(1 ) f , f1=5/2,2 - 3/2,2 + 13043.814 & so & 1(2)-1(1 ) & 23695.0672 & nh@xmath9 & 1(1)-1(1 ) f , f1=3/2,2 - 3/2,1 + 13116.451 & unidentified & & 23695.0782 & nh@xmath9 & 1(1)-1(1 ) f , f1=3/2,2 - 3/2,1 + 13116.569 & unidentified & & 23695.1132 & nh@xmath9 & 1(1)-1(1 ) f , f1=5/2,2 - 3/2,1 + 13186.46 & unidentified & & 23696.0297 & nh@xmath9 & 1(1)-1(1 ) f , f1=1/2,0 - 1/2,1 + 13186.853 & hc@xmath20n & 39 - 38 & 23696.0406 & nh@xmath9 & 1(1)-1(1 ) f , f1=1/2,0 - 3/2,1 + 13186.98 & unidentified & & 23697.9 & unidentified & + 13313.312 & hc@xmath14n & 5 - 4 & 23718.325 & hc13ccccn & 9 - 8 + 13363.801 & hc@xmath10n & 23 - 22 & 23720.575 & nh@xmath9 & 2(2)-2(2 ) f1=1 - 2 + 13434.596 & oh & 23/2 j=7/2 f=3 - 3 & 23721.336 & nh@xmath9 & 2(2)-2(2 ) f1=3 - 2 + 13441.4173 & oh & 23/2 j=7/2 f=4 - 4 & 23722.6323 & nh@xmath9 & 2(2)-2(2 ) f1=2 - 2 + 13535.998 & hc@xmath16n & 12 - 11 & 23722.6336 & nh@xmath9 & 2(2)-2(2 ) f1=3 - 3 + 13778.804 & h@xmath11co & 2(1,1)-2(1,2 ) & 23722.6344 & nh@xmath9 & 2(2)-2(2 ) f1=1 - 1 + 13880.54 & unidentified & & 23723.929 & nh@xmath9 & 2(2)-2(2 ) f1=2 - 3 + 13944.832 & hc@xmath10n & 24 - 23 & 23724.691 & nh@xmath9 & 2(2)-2(2 ) f1=2 - 1 + 14488.4589 & h@xmath15co & 2(1,1)-2(1,2 ) f=1 - 1 & 23727.162 & hcccc13cn & 9 - 8 + 14488.4712 & h@xmath15co & 2(1,1)-2(1,2 ) f=1 - 2 & 23804.5 & unidentified & + 14488.4801 & h@xmath15co & 2(1,1)-2(1,2 ) f=3 - 3 & 23811.0 & unidentified & + 14488.4899 & h@xmath15co & 2(1,1)-2(1,2 ) f=2 - 2 & 23817.6153 & oh & 23/2 j=9/2 f=4 - 4 + 14525.862 & hc@xmath10n & 25 - 24 & 23822.265 & hc@xmath10n & 41 - 40 + 14663.993 & hc@xmath16n & 13 - 12 & 23826.6211 & oh & 23/2 j=9/2 f=5 - 5 + 14782.212 & 13ch@xmath9oh & 2(0,2)-3(-1,3 ) e & 23867.805 & nh@xmath9 & 3(3)-3(3 ) f1=2 - 3 + 14812.002 & c - c@xmath9h & 1(1,0)-1(1,1 ) j=3/2 - 1/2 f=2 - 1 & 23868.450 & nh@xmath9 & 3(3)-3(3 ) f1=4 - 3 + 14877.671 & c - c@xmath9h & 1(1,0)-1(1,1 ) j=3/2 - 3/2 f=2 - 1 & 23870.1279 & nh@xmath9 & 3(3)-3(3 ) f1=3 - 3 + 14893.050 & c - c@xmath9h & 1(1,0)-1(1,1 ) j=3/2 - 3/2 f=2 - 2 & 23870.1296 & nh@xmath9 & 3(3)-3(3 ) f1=4 - 4 + 14895.243 & c - c@xmath9h & 1(1,0)-1(1,1 ) j=3/2 - 3/2 f=1 - 1 & 23870.1302 & nh@xmath9 & 3(3)-3(3 ) f1=2 - 2 + 15106.892 & hc@xmath10n & 26 - 25 & 23871.807 & nh@xmath9 & 3(3)-3(3 ) f1=3 - 4 + 15248.225 & c@xmath18h & 23/2 j=11/2 - 9/2 f=6 - 5 f & 23872.453 & nh@xmath9 & 3(3)-3(3 ) f1=3 - 2 + 15248.359 & c@xmath18h & 23/2 j=11/2 - 9/2 f=5 - 4 f & 23922.3132 & 15nh@xmath9 & 6(6)-6(6 ) + 15249.064 & c@xmath18h & 23/2 j=11/2 - 9/2 f=6 - 5 e & 23939.089 & hcc13cccn & 9 - 8 + 15249.198 & c@xmath18h & 23/2 j=11/2 - 9/2 f=5 - 4 e & 23941.99 & hccc13ccn & 9 - 8 + 15687.921 & hc@xmath10n & 27 - 26 & 23959.5 & unidentified & + 15791.986 & hc@xmath16n & 14 - 13 & 23963.901 & hc@xmath14n & 9 - 8 + 15975.966 & hc@xmath14n & 6 - 5 & 23987.5 & unidentified & + 16268.950 & hc@xmath10n & 28 - 27 & 23990.2 & unidentified & + 16849.979 & hc@xmath10n & 29 - 28 & 23996.7 & unidentified & + 16886.312 & dcccn & 2 - 1 f=2 - 1 & 24004.5 & unidentified & + 16886.405 & dc@xmath9n & 2 - 1 f=3 - 2 & 24023.2 & unidentified & + 16919.979 & hc@xmath16n & 15 - 14 & 24037.1 & unidentified & + 17091.742 & ch@xmath9cch & 1(0)-0(0 ) & 24048.5 & unidentified & + 17342.256 & cccs & 3 - 2 & 24139.4169 & nh@xmath9 & 4(4)-4(4 ) + 17431.006 & hc@xmath10n & 30 - 59 & 24205.287 & nh@xmath9 & 10(9)-10(9 ) + 17632.685 & h@xmath19cccn & 2 - 1 f=2 - 2 & 24296.491 & ch@xmath9ocho & 2(0,2)-1(0,1 ) e + 17633.844 & h@xmath19cccn & 2 - 1 f=3 - 2 & 24298.481 & ch@xmath9ocho & 2(0,2)-1(0,1 ) a + 17647.479 & c@xmath8d & 5/2 - 3/2 f=5/2 - 3/2 & 24325.927 & ocs & 2 - 1 + 17647.526 & c@xmath8d & 5/2 - 3/2 f=3/2 - 1/2 & 24375.2 & unidentified & + 17647.716 & c@xmath8d & 5/2 - 3/2 f=7/2 - 5/2 & 24428.652 & ch3c@xmath8h & 6(1)-5(1 ) + 17666.995 & hccc15n & 2 - 1 & 24428.886 & ch3c@xmath8h & 6(0)-5(0 ) + 17683.961 & c@xmath8d & 3/2 - 1/2 f=5/2 - 3/2 & 24532.9887 & nh@xmath9 & 5(5)-5(5 ) + 17684.662 & c@xmath8d & 3/2 - 1/2 f=3/2 - 1/2 & 24788.541 & ch3cccn & 6(1)-5(1 ) + 17736.75 & unidentified & & 24788.780 & ch3cccn & 6(0)-5(0 ) + 17788.570 & h@xmath4cccc & 2(1,2)-1(1,1 ) & 24815.878 & hc@xmath16n & 22 - 21 + 17863.803 & h@xmath4cccc & 2(0,2)-1(0,1 ) & 24928.715 & ch@xmath9oh & 3(2,1)-3(1,2 ) e + 17937.956 & h@xmath4cccc & 2(1,1)-1(1,0 ) & 24933.468 & ch@xmath9oh & 4(2,2)-4(1,3 ) e + 17945.85 & unidentified & & 24934.382 & ch@xmath9oh & 2(2,0)-2(1,1 ) e + 17951.95 & unidentified & & 24959.079 & ch@xmath9oh & 5(2,3)-5(1,4 ) e + 17965.09 & unidentified & & 24984.302 & hc@xmath10n & 43 - 42 + 17974.01 & unidentified & & 24991.19 & sic2 & 8(2,6)-8(2,7 ) + 18012.033 & hc@xmath10n & 31 - 30 & 25018.123 & ch@xmath9oh & 6(2,4)-6(1,5 ) e + 18012.46 & unidentified & & 25023.792 & nh@xmath4d & 4(1,4)-4(0,4 ) + 18017.337 & nh@xmath9 & 7(3)-7(3 ) & 25056.025 & nh@xmath9 & 6(6)-6(6 ) + 18020.574 & c@xmath18h & 23/2 j=6.5 - 5.5 f=7 - 6 e & 25124.872 & ch@xmath9oh & 7(2,5)-7(1,6 ) e + 18020.644 & c@xmath18h & 23/2 j=6.5 - 5.5 f=6 - 5 e & 25249.938 & c5n & 21/2 n=9 - 8 j=9.5 - 8.5 + 18021.752 & c@xmath18h & 23/2 j=6.5 - 5.5 f=7 - 6 f & 25260.649 & c5n & 21/2 n=9 - 8 j=8.5 - 7.5 + 18021.818 & c@xmath18h & 23/2 j=6.5 - 5.5 f=6 - 5 f & 25294.417 & ch@xmath9oh & 8(2,6)-8(1,7 ) e + 18021.86 & unidentified & & 25329.441 & dc@xmath9n & 3 - 2 + 18047.969 & hc@xmath16n & 16 - 15 & 25421.036 & dc5n & 10 - 9 + 18119.029 & hc@xmath19ccn & 2 - 1 f=2 - 1 & 25541.398 & ch@xmath9oh & 9(2,7)-9(1,8 ) e + 18120.773 & hcc@xmath19cn & 2 - 1 f=2 - 1 & 25715.182 & nh@xmath9 & 7(7)-7(7 ) + 18120.865 & hcc@xmath19cn & 2 - 1 f=3 - 2 & 25878.266 & ch@xmath9oh & 10(2,8)-10(1,9 ) e + 18154.884 & sis & 1 - 0 & 25911.017 & ccs & 2,2 - 1,1 + 18186.652 & c@xmath17h & 23/2 15.5 - 15.5 e & 25943.855 & hc@xmath16n & 23 - 22 + 18186.782 & c@xmath17h & 23/2 15.5 - 15.5 f & 26337.414 & c@xmath18h & 23/2 j=19/2 - 17/2 f=10 - 9 f + 18194.9206 & hcccn & 2 - 1 f=2 - 2 & 26337.463 & c@xmath18h & 23/2 j=19/2 - 17/2 f=9 - 8 f + 18195.3176 & hcccn & 2 - 1 f=1 - 0 & 26339.924 & c@xmath18h & 23/2 j=19/2 - 17/2 f=10 - 9 e + 18196.2183 & hcccn & 2 - 1 f=2 - 1 & 26339.973 & c@xmath18h & 23/2 j=19/2 - 17/2 f=9 - 8 e + 18196.3119 & hcccn & 2 - 1 f=3 - 2 & 26363.491 & hcccc@xmath19cn & 10 - 9 + 18197.078 & hcccn & 2 - 1 f=1 - 2 & 26450.598 & h@xmath19cccn & 3 - 2 + 18198.3756 & hcccn & 2 - 1 f=1 - 1 & 26500.462 & hccc@xmath21n & 3 - 2 + 18222.65 & unidentified & & 26518.981 & nh@xmath9 & 8(8)-8(8 ) + 18285.434 & nh@xmath9 & 10(7)-10(7 ) & 26602.181 & hccc@xmath19ccn & 10 - 9 + 18294.20 & unidentified & & 26626.533 & hc@xmath14n & 10 - 9 + 18299.5 & unidentified & & 26682.814 & h@xmath4cccc & 3(1,3)-2(1,2 ) + 18306.3 & unidentified & & 26795.635 & h@xmath4cccc & 3(0,3)-2(0,2 ) + 18320.7 & unidentified & & 26847.205 & ch@xmath9oh & 12(2,10)-12(1,11 ) e + 18343.144 & c - c@xmath9h@xmath4 & 1(1,0)-1(0,1 ) & 26906.891 & h@xmath4cccc & 3(1,2)-2(1,1 ) + 18360.50 & unidentified & & 27071.824 & hc@xmath16n & 24 - 23 + 18363.045 & unidentified & & 27084.348 & c - c@xmath9h@xmath4 & 3(3,0)-3(2,1 ) + 18363.142 & unidentified & & 27178.511 & hc@xmath19ccn & 3 - 2 + 18363.306 & unidentified & & 27181.127 & hcc@xmath19cn & 3 - 2 + 18363.406 & unidentified & & 27292.903 & hcccn & 3 - 2 f=3 - 3 + 18368.0 & unidentified & & 27294.078 & hcccn & 3 - 2 f=2 - 1 + 18379.6 & unidentified & & 27294.295 & hcccn & 3 - 2 f=3 - 2 + 18383.3 & unidentified & & 27294.347 & hcccn & 3 - 2 f=4 - 3 + 18391.562 & nh@xmath9 & 6(1)-6(1 ) & 27296.235 & hcccn & 3 - 2 f=2 - 2 + 18396.7252 & ch@xmath9cn & 1(0)-0(0 ) f=1 - 1 & 27472.501 & ch@xmath9oh & 13(2,11)-13(1,12 ) e + 18397.9965 & ch@xmath9cn & 1(0)-0(0 ) f=2 - 1 & 27477.943 & nh@xmath9 & 9(9)-9(9 ) + 18399.8924 & ch@xmath9cn & 1(0)-0(0 ) f=0 - 1 & 28009.975 & hnccc & 3 - 2 + 18413.822 & c - h13ccch & 1(1,0)-1(0,1 ) & 28169.437 & ch@xmath9oh & 14(2,12)-14(1,13 ) e + 18422.00 & unidentified & & 28199.804 & hc@xmath16n & 25 - 24 + 18485.07 & unidentified & & 28199.805 & hc@xmath16n & 25 - 24 + 18494.1 & ch3sh & 18(2)-17(3 ) a+ & 28316.031 & ch@xmath9oh & 4(0,4)-3(1,2 ) e + 18499.390 & nh@xmath9 & 9(6)-9(6 ) & 28440.980 & ch@xmath4chcn & 3(0,3)-2(0,2 ) + 18513.316 & ch@xmath4chcn & 2(1,2)-1(1,1 ) f=3 - 2 & 28470.391 & hc@xmath10n & 49 - 48 + 18586.06 & unidentified & & 28532.31 & c@xmath8h & 7/2 - 5/2 f=3 - 2 + 18593.060 & hc@xmath10n & 32 - 31 & 28532.46 & c@xmath8h & 7/2 - 5/2 f=4 - 3 + 18638.616 & hc@xmath14n & 7 - 6 & 28542.284 & c@xmath8h & j=5/2 - 5/2 f=3 - 3 + 18650.308 & hcccho & 2(0,2)-1(0,1 ) & 28571.37 & c@xmath8h & 5/2 - 3/2 f=3 - 2 + 18673.312 & hnccc & 2 - 1 & 28571.53 & c@xmath8h & 5/2 - 3/2 f=2 - 1 + 18698.16 & unidentified & & 28604.737 & nh@xmath9 & 10(10)-10(10 ) + 18729.12 & unidentified & & 28903.688 & cccs & 5 - 4 + 18793.92 & unidentified & & 28905.787 & ch@xmath9oh & 15(2,13)-12(1,14 ) e + 18802.235 & h@xmath4cccccc & 7(1,7)-6(1,6 ) & 28919.931 & ch@xmath9cccn & 7(1)-6(1 ) + 18807.888 & nh@xmath4d & 3(1,3)-3(0,3 ) & 28920.209 & ch@xmath9cccn & 7(0)-6(0 ) + 18808.507 & nh@xmath9 & 8(5)-8(5 ) & 28969.954 & ch@xmath9oh & 8(2,7)-9(1,8 ) a + 18817.66 & unidentified & & 28974.781 & h@xmath15co & 3(1,2)-3(1,3 ) f=2 - 2 + 18864.65 & unidentified & & 28974.804 & h@xmath15co & 3(1,2)-3(1,3 ) f=4 - 4 + 18884.695 & nh@xmath9 & 6(2)-6(2 ) & 28974.814 & h@xmath15co & 3(1,2)-3(1,3 ) f=3 - 3 + 18907.54 & unidentified & & 28999.814 & hcccc@xmath19cn & 11 - 10 + 18918.50 & unidentified & & 29051.403 & hc@xmath10n & 50 - 49 + 18961.79 & unidentified & & 29109.644 & c@xmath18h & 23/2 j=21/2 - 19/2 f=11 - 10 f + 18965.588 & ch@xmath4chcn & 2(0,2)-1(0,1 ) f=1 - 0 & 29109.66 & c@xmath18h & 23/2 j=21/2 - 19/2 f + 18966.535 & ch@xmath4chcn & 2(0,2)-1(0,1 ) f=2 - 1 & 29109.686 & c@xmath18h & 23/2 j=21/2 - 19/2 f=10 - 9 f + 18966.616 & ch@xmath4chcn & 2(0,2)-1(0,1 ) f=3 - 2 & 29112.709 & c@xmath18h & 23/2 j=21/2 - 19/2 f=11 - 10 f + 18968.48 & unidentified & & 29112.73 & c@xmath18h & 23/2 j=21/2 - 19/2 e + 18986.20 & unidentified & & 29112.750 & c@xmath18h & 23/2 j=21/2 - 19/2 f=10 - 9 f + 19014.7204 & c@xmath8h & 5/2 - 3/2 f=2 - 1 & 29138.877 & ch@xmath4chcn & 3(1,2)-2(1,1 ) f=3 - 2 + 19015.1435 & c@xmath8h & 5/2 - 3/2 f=3 - 2 & 29139.215 & ch@xmath4chcn & 3(1,2)-2(1,1 ) f=4 - 3 , 2 - 1 + 19025.107 & c@xmath8h & 5/2 - 3/2 f=2 - 2 & 29258.834 & hcc@xmath19cccn & 11 - 10 + 19039.50 & unidentified & & 29289.159 & hc@xmath14n & 11 - 10 + 19043.0 & unidentified & & 29304.09 & c@xmath18h & 21/2 j=21/2 - 19/2 e + 19044.760 & c@xmath8h & 3/2 - 1/2 f=1 - 1 & 29310.5 & unidentified & + 19054.4762 & c@xmath8h & 3/2 - 1/2 f=2 - 1 & 29327.776 & hc@xmath16n & 26 - 25 + 19055.9468 & c@xmath8h & 3/2 - 1/2 f=1 - 0 & 29332.45 & c@xmath18h & 21/2 j=21/2 - 19/2 f + 19099.656 & c@xmath8h & 3/2 - 3/2 f=1 - 1 & 29333.3 & unidentified & + 19119.764 & c@xmath8h & j=3/2 - 3/2 f=2 - 2 & 29337.57 & hc@xmath14n & 11 - 10 v11=1 = 1c + 19174.086 & hc@xmath10n & 33 - 32 & 29342.0 & unidentified & + 19175.958 & hc@xmath16n & 17 - 16 & 29353.8 & unidentified & + 19218.465 & nh@xmath9 & 7(4)-7(4 ) & 29363.15 & hc@xmath14n & 11 - 10 v11=1 = 1d + 19243.521 & ccco & 2 - 1 & 29365.0 & unidentified & + 19262.140 & ch@xmath9cho & 1(0,1 ) - 0(0,0 ) e & 29477.704 & ccs & 2,3 - 1,2 + 19265.137 & ch@xmath9cho & 1(0,1 ) - 0(0,0 ) a++ & 29632.406 & hc@xmath10n & 51 - 50 + 19316.70 & unidentified & & 29632.413 & hc@xmath10n & 51 - 50 + 19325.20 & unidentified & & 29636.920 & ch@xmath9oh & 16(2,14)-12(1,15 ) e + 19336.10 & unidentified & & 29676.14 & cccn & 3 - 2 j=7/2 - 5/2 f=7/2 - 5/2 + 19361.50 & unidentified & & 29676.28 & cccn & 3 - 2 j=7/2 - 5/2 f=9/2 - 7/2 + 19418.661 & c - c@xmath9hd & 1(1,0)-1(0,1 ) f=1 - 1 & 29678.882 & @xmath22 so & 1(0)-0(1 ) + 19418.686 & c - c@xmath9hd & 1(1,0)-1(0,1 ) f=2 - 1 & 29694.99 & cccn & 3 - 2 j=5/2 - 3/2 f=3/2 - 1/2 + 19418.712 & c - c@xmath9hd & 1(1,0)-1(0,1 ) f=1 - 2 & 29695.14 & cccn & 3 - 2 j=5/2 - 3/2 f=7/2 - 5/2 + 19418.724 & c - c@xmath9hd & 1(1,0)-1(0,1 ) f=0 - 1 & 29806.963 & hccnc & 3 - 2 + 19418.740 & c - c@xmath9hd & 1(1,0)-1(0,1 ) f=2 - 2 & 29914.486 & nh@xmath9 & 11(11)-11(11 ) +	the square kilometer array will have the sensitivity , spatial resolution , and frequency resolution to provide new scientific knowledge of evolved stars . four basic areas of scientific exploration are enhanced by the construction of the ska : 1 ) detection and imaging of photospheric radio continuum emission and position correlation with maser distributions , 2 ) imaging of thermal dust emission around evolved stars and the detailed structures of their circumstellar winds ( again , including comparison with maser distributions ) , 3 ) study of cm - wavelength molecular line transitions and the circumstellar chemistry around both o - rich and c - rich evolved stars and 4 ) the possible observation of polarized emission due to the influence of the magnetic fields of agb stars . since this short chapter is not meant to be a review article , a comprehensive reference list has not been generated . i have selected just one or perhaps two references for citations where appropriate .
the central parsec of the milky way ( mw ) is a very special region of our galaxy . it not only hosts the central supermassive black hole , associated with sgr a * , with well determined mass @xcite , but also contains a significant number of early - type stars , including some in the wolf - rayet stage . the central cavity has a mini - spiral structure of ionized gas streamers , encircled by a circum - nuclear disk ( cnd ) of molecular gas . most of the ionized gas emission appears in hydrogen and helium recombination lines , very likely originating from ionization produced by the many young and hot ob stars ; the equivalent to some @xmath5 o9 stars would be required to account for the emission @xcite . an anomalous structure , called mini - cavity , was identified in the mini - spiral , with a diameter of @xmath6 , about @xmath7 to the southwest of sgr a * @xcite , associated with strong [ fe iii ] emission @xcite . for a recent review of the literature on the galactic centre , see @xcite . x - ray binary transients are overabundant in the central parsec of the mw . @xcite have found that the overabundance is at least a factor of 20 , per mass , in the central parsec when compared to the abundance at a distance between 1 and 23 parsecs . it is likely that this overabundance is due to three - body interactions between binary star systems and either black holes or neutron stars that have concentrated in the central parsec through dynamical friction . this is also supposed to be the case with the formation of similar systems in the cores of globular clusters @xcite . the known x - ray transient closest to sgr a * , at only @xmath8 south of the centre of the mw , is cxogx j174540.0 - 290031 , discovered in 2004 @xcite . it was found to be an x - ray binary with an orbital period of 7.9 hr @xcite . from k band observations , an upper limit for any counterpart was set as @xmath9 , suggesting that it is a low - mass x - ray binary ( lmxb ) . this x - ray transient also revealed a radio outburst @xcite . this large radio outburst , as well as its x - ray emission , suggests that this source more likely contains a black hole than a neutron star primary @xcite . in this paper we report the discovery of a high velocity compact nebular filament extending @xmath10 with a strong velocity gradient that could possibly be associated with a colliding shock between the streamers of the mini - spiral . this paper analyzes archive data obtained with the spectrograph for integral field observations in the near infrared ( sinfoni ) on the vlt , available in a public access data bank . the observations were made in k band on april 22 , 2007 . a fore - optics of @xmath11 pixel@xmath0 was used , providing a field of view ( fov ) of @xmath12 . a total of four exposures of the galactic nucleus were retrieved from the public data bank to be used in this work . standard calibration images of linearity lamp , distortion fibre ( used in the data reduction to compute spatial distortions and perform a spatial rectification ) , flat lamp , arc lamp and sky field were obtained from the public data bank . calibration images of an a0v standard star were also retrieved from the data bank . the data reduction was made with gasgano software and included the following steps : determination of the trim , sky subtraction , correction of bad pixels , flat - field correction , spatial rectification ( including correction for spatial distortions ) , wavelength calibration and data cube construction . at the end of the process , four data cubes were obtained , with spaxels ( spatial pixels ) of @xmath13 . afterwards , with a script developed by us and using data from the a0v standard star , we applied , in an iraf environment , telluric absorption removal and flux calibration to all the data cubes . details of the data treatment are given in appendix a. after the data treatment , we applied pca tomography ( see appendix a ) again . tomogram 1 , shown in green in figure 1-a , displays the young stellar population . tomogram 2 , shown in blue in figure 1-a , reveals stars with strong infrared excess , presumably very young stars , like irs 21 @xcite and irs 1 . the red component in figure 1-a corresponds to an image of stars obtained by subtracting the edges of the co bands from the reconstructed data cube , after suppressing eingenvectors 1 and 2 ( see steiner et al . 2009 for a description of `` feature suppression '' ) . the stellar component revealed by figure 1-a basically shows the known young as well as the old stellar populations . irs 16 , the closest irs source to the galactic centre , is resolved in the individual stars , as is irs 33 ( sw of the fov ) . we performed spectral synthesis on the spectrum of each spaxel of the datacube analyzed here , using the starlight software @xcite , which matches the stellar spectrum of a given object with a model corresponding to a combination of template stellar spectra from a pre - established base . in this work , we used the base of stellar spectra miles ( medium resolution int library of empirical spectra ) @xcite , containing the spectra of 150 stellar populations with ages between @xmath14 years and @xmath15 years and with metallicities between 0.0001 and 0.05 ( with @xmath16 ) . the spectral synthesis resulted in a synthetic stellar spectrum for each spaxel of the data cube . before performing spectral synthesis on the data cube , however , we prepared the spectra in two steps : correcting the interstellar extinction due to the galaxy , using @xmath17 and the reddening law of @xcite ; and carrying out a spectral re - sampling with @xmath18 per pixel . the first step used a script written in idl language , while the second used the task `` dispcor '' , of the `` noao '' package , in iraf environment . in order to analyze the emission line spectrum in more detail , we subtracted the synthetic stellar spectra obtained with the spectral synthesis from the observed ones . this procedure resulted in a data cube with emission lines only . we then constructed rgb images of the emission lines br@xmath19 and he i @xmath20 of this emission line data cube . we can see that the recombination lines br@xmath19 and he i @xmath21 are detected as stellar winds in early - type stars and , also , as ionized interstellar lines basically delineating the northern arm . a number of other lines are also seen in the southwest of the fov : @xmath22 , @xmath23 , @xmath24 , and @xmath25 are the strongest among them and are identified as [ fe iii ] lines ( lutz et al . we then constructed rgb images of the emission lines br@xmath19 and [ fe iii ] @xmath26 , shown in figure 1-c and figure 1-d , respectively . we identified a high velocity compact nebular filament , @xmath27 south of sgr a. the structure extends over @xmath10 and presents a velocity gradient of @xmath28 km s@xmath0 arcsec@xmath0 for he i , br@xmath19 and [ fe iii ] @xmath29 lines ( figure 1-e , and figures [ fig2 ] and [ fig4 ] ) . the peak of maximum emission seen in the [ fe iii ] and he i lines is located at @xmath30 and @xmath31 with respect to sgr a. this position is near the star irs 33n , at @xmath32 ; @xmath33 from sgr a. as can be seen in figure [ fig3 ] , the velocity at the peak , as estimated from the br@xmath19 , he i @xmath34 and [ fe iii ] @xmath35 is @xmath36 km s@xmath0 . the filament has a position angle of pa @xmath4 ( see figures [ fig2 ] to [ fig4 ] ) . the observed line ratios at the emission peak of the filament are ( he i @xmath34)/br@xmath19 = @xmath37 and ( [ feiii ] @xmath29)/br@xmath19 = @xmath38 . the filament is positioned in one of the mini - spiral regions nearest to the galactic centre , in the field where the eastern and the northern arms of the mini - spiral cross . it is also at the eastern edge of the mini - cavity . this field shows significant blue - shifted he i emission line ( paumard et al . 2004 ) and also in the h92@xmath39 and h30@xmath39 lines @xcite . the filament that we see may be described as a `` jet - like '' structure , being brighter at the centre and fading towards the borders , at least as it is seen in the he i and the [ fe iii ] lines ( see figure [ fig3 ] ) . this is in the part of the mini - spiral closest to sgr a. the challenge is to explain the compactness of the structure as well as its ionization ( he i ) and excitation ( [ fe iii ] ) . the filament overlaps spatially with the star irs33n ( see figure 1-b and figure [ fig4 ] ) . could this star be the origin of a jet ? irs 33n is a @xmath40 star , presenting a spectral type of b0.5 - 1 i and has a radial velocity of @xmath41 km s@xmath0 @xcite while the centre of the filament is at @xmath3 km s@xmath0 . although close in terms of spatial projected distance , the nebular filament is kinematically decoupled from the star irs 33n . therefore this association can be discarded . km s@xmath0 ) ; 2 ( @xmath42 km s@xmath0 ) ; 3 ( @xmath43 km s@xmath0 ) ; 4 ( @xmath44 km s@xmath0 ) and 5 ( @xmath45 km s@xmath0 ) . the northern edges of the northern and eastern arm streamers , as well as of the mini - cavity , are shown . three stars of irs 16 as well as of irs 33 are also shown . the position of sgr a * is marked as a @xmath46 sign . the error circle of the lmxb ( green ) and the associated radio transients ( red ) are also shown.[fig4 ] ] the orbits of the mini - spiral have been described as containing a one - armed spiral in a keplerian disk ( lacy et al . 1991 ) that properly describes the [ ne ii ] @xmath4712.8 @xmath48 m . an alternative description is given by @xcite , in which the whole mini - spiral is essentially described by three streamers in elliptical orbits around sgr a * , on the basis of h92@xmath39 observations . the northern and eastern arm streamers have high eccentricities ( @xmath49 and @xmath50 , respectively ) and cross near the periastron . the formal solution provides a projected crossing between the two orbits about @xmath51 to the south of the galactic centre . this solution has been disputed by @xcite who find that the one - armed spiral pattern may be a better description of the [ ne ii ] nebular emission . the observations of both h92@xmath39 and the [ ne ii ] lines have good spectral resolution but poorer spatial resolution , of @xmath52 to @xmath6 , than the data reported here . they could not have identified such a strong velocity gradient in a filament so compact . perhaps the best explanation for the existence of this filament is that it is associated to a shock between streamers in the mini - spiral . in the one - arm pattern , the position of the filament is where the one - armed spiral ends ; that could also be seen as the crossing point between the northern arm and the bar . in the three ellipses model , the filament is near the position where the projected orbits of the northern and the eastern arm streamers cross . it is perhaps not a coincidence that the position angle of the filament ( pa @xmath4 ) is similar to that of the eastern arm and also of the bar . perhaps the most challenging aspect of this interpretation is how to explain the large velocity gradient of @xmath28 km s@xmath0 arcsec@xmath0 . it probably has to do with the proximity of the supermassive black hole and its steep gravitational potential gradient . the blue arm of the filament points toward the mini - cavity . this structure was modeled as a bubble of ionized hot gas generated by a fast ( @xmath53 km s@xmath0 ) wind originated from one or several sources , within a few arcseconds from the mini - cavity , and blown into the partially ionized neutral gas streamers orbiting the centre of the galaxy ( eckart et al . 1992 ; wardle & yusef - zadeh 1992 ; lutz et al . 1993 ; melia , coker & yusef - zadeh 1996 ) . in fact , after computing the elliptical keplerian orbits of the streamers , @xcite suggested that the northern and eastern arms may collide in the `` bar '' region , a few arcesconds south of sgr a. this is the region with the highest specific kinetic energy of both northern and eastern arms ( see figure 21 in zhao et al . according to @xcite , the high specific kinetic energy density emphasizes the possibility that the mini - cavity was created by strong shocks resulting from the collisions of the two streamers . the peak of the emission of the filament is located @xmath54 north of the binary x - ray transient cxogx j174540.0 - 290031 , a lmxb with an orbital period of 7.9 hr @xcite . this source is associated to one transient and one persistent albeit variable , radio sources @xcite . the x - ray transient source cxogx j174540.0 - 290031 was detected on july , 5 - 7 , 2004 and is located @xmath8 south of sgr a , with an uncertainty of about @xmath55 @xcite . observations made at 43 ghz a few months earlier , on march 28 , 2004 , detected two sources , named ne ( northeast , the strongest one , reaching 93 mjy at 43 ghz ) and sw ( southwest ) with a peak flux of 48 mjy . the coincidental variability of both radio and x - ray sources strongly suggests that they are associated . @xcite also report radio observations at 1.3 cm and found that the sw source is a persistent one , being detected before the transient phenomenon at the @xmath56 mjy level . the ne component , not detected in previous observations , was detected after the transient . these authors interpret the ne component as a hot spot associated with the head of the jet ejected from the persistent sw radio - variable source . if so , then the stellar counterpart of the x - ray binary is probably near the star irs 33sw , close to the x - ray error box ( figure [ fig4 ] ) . the position angle of the radio jet is pa @xmath57 , quite distinct from the position angle of the nebular filament . as the filament is at @xmath58 from the x - ray source and has a distinct position angle , we may rule out any direct association between the x - ray binary and the origin of the nebular filament . the sinfoni observations ( april 22 , 2007 ) were made a few years after the radio detection ( march 28 , 2004 ) and also after the x - ray transients detections ( july 5 - 7 , 2004 ) . the nebular filament is characterized by a spectrum that is quite distinct from other gas emitting regions in the fov . br@xmath19 and he i @xmath34 are the strongest lines . other gas emitting regions in the fov are probably photoionized by the young stellar population ( such as in irs 16 ) that produces uv photons capable of ionizing h and he ( genzel et al . 2010 ) . such uv photons , because of their low energy , do not heat the nebulae and some 250 o9 equivalent stars are necessary to account for the emission @xcite . the vicinity of the x - ray source is the only region in the fov that presents such strong forbidden lines . but shock waves are also efficient in heating gas and enhancing forbidden line emissions . the field of irs 33 is the hottest in the mini - spiral , as measured from the h30@xmath39 and h92@xmath39 @xcite , with a temperature of @xmath59 k and an electron density of @xmath60 @xmath61 . the temperature of 13000 k is enough to excite the [ fe iii ] emission . it is not enough to ionize the gas to the fe@xmath62 phase . this requires a temperature of @xmath63 k @xcite . but the beam size of the radio observations is @xmath64 ; thus a compact structure with @xmath63 k could certainly exist . for collisional ionization of helium , the temperature must be even higher : @xmath65 k. we conclude that shock heating could both produce and excite the fe@xmath62 ion , but not ionize he . therefore , a combination of non - shock ionization mechanism plus shock heating seems to be required . could the x - ray transient be , at least partially , responsible for photoionizing the filament ? hard x - rays are efficient in photoionizing gas and heating it at the same time @xcite , enhancing the emission from low ionization nebular species . however , the ionization parameter @xmath66 must be @xmath67 in order to produce high ionization species and @xmath68 in order to produce low ionization species @xcite . in the present case , the transient x - ray luminosity was @xmath69 erg s@xmath0 . however , at the time of the transient , diffuse x - ray emission was observed @xmath70 southwest of the source . this suggests that the real x - ray luminosity was @xmath71 erg s@xmath0 @xcite . the discrepant x - ray luminosity seen directly is probably due to the fact that the x - ray binary is seen nearly edge - on . assuming that the x - ray luminosity is @xmath72 erg s@xmath0 we obtain an ionization parameter of @xmath73 at the filament ; this is too low to produce a significant ionization effect . we conclude that the x - ray binary has no ionization or heating effect on the filament unless the source radiates soft x - rays near the eddington limit . this could not have been directly observed due to the strong interstellar absorption and would produce @xmath74 at the filament and could be responsible for the strength of the [ fe iii ] emission and , perhaps , of he i. further studies are required to elucidate the nature of this filament and the role of the lmxb in its neighborhood ; it is important to observe the fov with better spatial resolution . we have analyzed a data cube centered @xmath64 southeast of srg a * , obtained with the sinfoni / vlt ifu spectrograph and studied the kinematics of the gaseous ionized emission in the fov . our main findings are : * we identified a compact nebular filament , @xmath27 south of sgr a. the structure extends over @xmath10 and presents a velocity gradient of @xmath28 km s@xmath0 arcsec@xmath0 . there is a peak of maximum emission seen in [ fe iii ] and he i lines , located at @xmath75 and @xmath76 with respect to sgr a. this position is located at @xmath77 east of the star irs 33n . this emission has the most negative velocity in the fov we have analyzed . * the velocity at the emission peak , as estimated from the br@xmath19 , he i @xmath34 and [ fe iii ] @xmath35 is @xmath36 km s@xmath0 . the nebular filament has a position angle of pa @xmath4 , similar to that of the eastern arm at that position and to the position angle of the mini - spiral bar . although close , the nebular filament is , kinematically decoupled from the star irs 33n , which has a velocity of @xmath41 km s@xmath0 . * the peak position is also located @xmath54 north of the binary x - ray transient cxogx j174540.0 - 290031 , a lmxb with an orbital period of 7.9 hr . this source is associated with one transient plus one persistent and variable radio sources . * the [ fe iii ] line emission is strong in the filament and its vicinity . these lines are probably excited by shock heating . ionization / heating by soft x - ray photons from the nearby lmxb is possible , although unlikely , because of the high luminosity required . * although we can not rule out the possibility that we are dealing with a jet - like structure , we propose that the filament may represent a shock from the collision between the northern and the eastern arm streamers around the centre , or perhaps , between the northern arm and the bar . this research has been supported by the agencies cnpq and fapesp . the data presented herein were obtained with the sinfoni spectrograph on the vlt / eso available on the eso public data bank . bower g.c . , roberts d.a . , yusef - 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zadeh f. , 1992 , nat , 357 , 308 yusef - zadeh f. , morris m. , ekers r. , 1989 , in morris m. , ed . 136 , the center of the galaxy . kluwer , dordrecht , p. 443 yusef - zadeh f. , morris m. , ekers r. , 1990 , nat , 348 , 45 zhao jun - hui . , goss w.m . , lo k.y . , ekers r.d . , 1991 , nat , 354 , 46 zhao jun - hui . , morris m.r . , goss w.m . , an t. , 2009 , apj , 699 , 186 zhao jun - hui . , blundell r. , moran j.m . , downwes d. , schuster k.f . , marrone d.p . , 2010 , apj , 723 , 1097 after the data reduction , a correction of the differential atmospheric refraction was applied to all data cubes , using an algorithm developed by our group . it is important to mention , however , that the relative shifts observed in the images of all data cubes are not compatible with what is expected for the differential atmospheric refraction effect . therefore , we believe that these relative shifts are , at least partially , a consequence of some instrumental effect and not entirely due to the differential atmospheric refraction . in order to combine into one the four data cubes obtained after the correction of the atmospheric differential refraction , we divided the data cubes in two groups with three data cubes each . naming the data cubes 1 , 2 , 3 and 4 ( according to the order of the observations ) , the division in two groups was the following : + + group 1 : 1 , 2 and 3 + group 2 : 2 , 3 and 4 + + we then calculated a median of the data cubes of each group , which resulted in two data cubes at the end of the process . finally , we calculated the average of the two data cubes and obtained the final , combined data cube . by performing a spatial re - sampling to all the images of the data cube , we obtained spaxels of @xmath78 . this procedure does not change the spatial resolution of the observation , but sharpens the contours of the structures in the images . the disadvantage of such spatial re - sampling is that it introduces high spatial frequency components in the images ( usually in the form of small vertical and horizontal stripes ) . the high spatial frequency components , however , are not a serious problem because , after the spatial re - sampling , a butterworth spatial filtering was applied to all the images of the data cube , to remove high spatial frequency noises , including the high frequency components introduced by the spatial re - sampling . the steps of this process were as follows : calculation of the fourier transforms ( @xmath79 ) of all the images of the data cube ; multiplication of the fourier transforms @xmath79 by the image corresponding to the butterworth filter ( @xmath80 ) ; calculation of the inverse fourier transforms of all the products @xmath81 ; extraction of the real part of the calculated inverse fourier transforms . the butterworth filter used in this case corresponds to the product of two identical circular filters , with orders @xmath82 and cutoff frequencies along the horizontal and vertical axis equal to 0.27 ny ( nyquist frequency ) . we detected the presence of a probable instrumental `` fingerprint '' in the data cube , after the butterworth spatial filtering , which appeared in the data cube as a large horizontal strip in the images and also had a low frequency spectral signature . in order to remove it , we applied the pca tomography technique @xcite to the data cube . pca transforms data expressed originally in correlated coordinates into a new system of uncorrelated coordinates ( eigenvectors ) ordered by principal components of decreasing variance . pca tomography consists in applying pca to data cubes . in this case , the variables correspond to the spectral pixels of the data cube and the observables correspond to the spaxels of the data cube . since the eigenvectors are obtained as a function of the wavelength , their shape is similar to spectra and , therefore , we call them eigenspectra . on the other hand , since the observables are spaxels , their projections on the eigenvectors are images , which we call tomograms . the simultaneous analysis of eigenspectra and tomograms allows one to obtain information that , otherwise , would possibly be harder to detect . using pca tomography , we were able to identify and remove the instrumental fingerprint of the data cube analyzed here ( steiner et al . , in preparation ) . finally , a richardson - lucy deconvolution @xcite was applied to all the images of the data cube , using a synthetic gaussian psf . the use of a gaussian psf was possible in this case because we verified that the original psf of this observation had an approximate gaussian shape . the final data cube obtained after the full data treatment has a point - spread function ( psf ) with fwhm @xmath83 and a spectral resolution of @xmath84 .	the central parsec of the milky way is a very special region of our galaxy ; it contains the supermassive black hole associated with sgr a * as well as a significant number of early - type stars and a complex structure of streamers of neutral and ionized gas , within two parsecs from the centre , representing a unique laboratory . we report the identification of a high velocity compact nebular filament 2.2 arcsec south of sgr a. the structure extends over 1 arcsec and presents a strong velocity gradient of 200 km s@xmath0 arcsec@xmath0 . the peak of maximum emission , seen in [ fe iii ] and he i lines , is located at @xmath1 arcsec and @xmath2 arcsec with respect to sgr a. this position is near the star irs 33n . the velocity at the emission peak is @xmath3 km s@xmath0 . the filament has a position angle of pa @xmath4 , similar to that of the bar and of the eastern arm at that position . the peak position is located 0.7 arcsec north of the binary x - ray and radio transient cxogx j174540.0 - 290031 , a low - mass x - ray binary with an orbital period of 7.9 hr . the [ fe iii ] line emission is strong in the filament and its vicinity . these lines are probably produced by shock heating but we can not exclude some x - ray photoionization from the lmxb . although we can not rule out the idea of a compact nebular jet , we interpret this filament as a possible shock between the northern and the eastern arm or between the northern arm and the mini - spiral `` bar '' . [ firstpage ] techniques : imaging spectroscopy ism : structure galaxy : centre x - rays : binaries
bars are ubiquitous in local disk galaxies ( e.g. , rc3 , eskridge et al . 2000 , menendez - delmestre et al . they play an important role in evolving galaxies by transporting vast amounts of gas to the center ( sakamoto et al . 1997 ; sheth et al . 2004 ) , igniting circumnuclear starburst activity ( ho et al . 1997 and references therein ) , reducing the chemical abundance gradient ( martin & roy 1997 ) , and perhaps feeding black holes ( shlosman et al . 1988 , see knapen and laurikainen in this volume ) . bars also form bulges , and perhaps evolve galaxies along the hubble sequence ( e.g. , norman , sellwood & hasan 1996 ; see also kormendy , bournaud , and combes in these proceedings ) . at high redshifts bars were expected to be fairly common because of dynamically colder disks and increased merging activity . so it was a surprise when the first studies of high redshift galaxy morphology found an apparent paucity of barred spirals beyond [email protected] ( van den bergh et al . 1996 ; abraham et al . 1999 ; van den bergh et al . if true these results set strict constraints on cosmological simulations of galaxy formation by requiring that disks were not sufficiently massive , or were dynamically too hot until only @xmath06 gyr ago . these constraints would be in stark contrast to conclusions from studies of the cosmic star formation history ( e.g. , steidel et al . 1999 and references therein ) and thinness of local disks ( e.g. , toth & ostriker 1992 ) which argue that massive disks were already in place by z@xmath01 . bunker ( 1999 ) used a near - infrared 1.6@xmath1 m nicmos gto image to show that in at least one galaxy , a bar was missed by the early studies because they observed the high redshift galaxies in the rest - frame blue / ultraviolet light where bars are difficult to identify . the increasingly better visibility of bars at longer wavelengths is a well - known effect ( see for example figure 1 , and menendez - delmestre et al and references therein in these proceedings ) . we decided to investigate whether the apparent decline in the bar fraction at high redshifts could be due to such a selection effect ? this question was the focal point of sheth et al . ( 2003 ) and whose results we briefly summarize here . image . note how the bar is completely invisible . middle panel : blue - band image . the bar is faint and difficult to identify . right panel : i - band image . only now does the bar become visible.,width=384 ] we analyzed the bar fraction and bar properties in the wfpc2 ( v , and i - band ) and nicmos ( h - band data ) for the northern hubble deep field ( williams et al . 1996 ; dickinson et al . 2000 ) . the nicmos data are ideal for studying galaxies in the redshift range 0.7@xmath2z@xmath31 because these data observe the galaxies in rest - frame v through i - band light . in contrast to previous studies which identified bars by eye , by ellipticity breaks , or a change in isophote position angles between an inner and outermost isophote ( e.g. , van den bergh 1996 ; abraham et al . 1999 ; ) , we applied a more stringent two signature criteria to identify bars . we demanded that a ) the ellipticity increase monotonically and then drop with a sharp change of at least @xmath4 > 0.1 , and b ) the position angle remain constant over the bar region and change by @xmath5 pa > 10@xmath6 after the bar region . we require that the bar be symmetric by fixing the center for the galaxy fitting algorithm . our algorithm misses bars if the galaxy is highly inclined , if the bar position angle is the same as the galactic disk , if the underlying galactic disk is too faint to be adequately imaged , or if the data have inadequate resolution to resolve bars . our method , though perhaps overly strict , has the advantage of ensuring a firm lower limit to the bar fraction . as discussed in sheth et al . ( 2003 ) , at z @xmath3 0.7 , we identify five barred spirals , and two candidate barred spirals , consistent with the prior analysis of the hdfn by abraham et al . ( 1999 ) . at z @xmath7 0.7 we identify four barred spirals and five candidate barred spirals , including two possible candidates at z=1.66 and z=2.37 . the four barred spirals are shown in figure 2 . for comparison , in the previous wfpc2 hdfn studies , van den bergh et al . ( 1996 ) found no barred spirals , and abraham et al . ( 1999 ) found two barred galaxies beyond [email protected] . if we use the abraham et al . ( 1999 ) magnitude cutoff of i(ab)=23.7 , the total number of disk - like galaxies drops to 31 ; amongst these we identify three barred spirals . though we detect a few more bars , the total number of barred spirals is still small . does this reflect a true decline in the bar fraction at z @xmath7 0.7 ? we argue that when the spatial resolution of the observations are considered in the context of bar visibility , there is no evidence of a decline in bars at [email protected] . 0.7 , arranged by redshift . the top row shows the optical ( f606w , v - band ) wfpc2 images and the second row shows the near - infrared ( 1.6@xmath1 m , h - band ) nicmos images . 0.5 scale is shown with a horizontal segment in the lower right of each panel . the rest - frame wavelength for each galaxy is listed inside the top of each panel.,height=384 ] in figure 3 , we show the apparent angular size of various galactic structures as a function of redshift . overlaid are detection limits for various telescopes adopting a five psf detection threshold ; five psfs is an appropriate choice ( menendez - delmestre et al . 2004 , also in this volume ) . the figure shows that at 0.8@xmath1 m wfpc2 data is only marginally capable of detecting a 5 kpc structure beyond [email protected] . the nicmos data have even coarser resolution ( longer @xmath8 and larger pixels ) , and even though these data are not affected by bandshifting until z @xmath7 23 , they only detect structures with sizes @xmath9 10 kpc . the average size of the four bars identified at z @xmath7 0.7 is 12 kpc . m , even the wfpc2 data is only marginally capable of detecting a typical 5 kpc bar at z @xmath7 0.7 ; acs z - band is only slightly better . the nicmos data can only detect the large bars at z @xmath7 0.5 . also shown are capabilities of two new millimeter arrays , carma and alma which will be ideal for probing the gas kinematics in high redshift systems.,height=288 ] the most important point of note from figure 3 is that a measurement of the bar fraction must take into account the size of bars and the available spatial resolution of the data . although we only detect three or four bars in the nicmos data one must note that these data are biased towards detecting only the largest bars . therefore , when we compare the bar fraction at different redshifts , we must compare the fraction for bars of equal sizes . for a representative sample of local galaxies ( song , regan et al . 2001 , sheth et al . 2004 ) , we find that bars with sizes @xmath7 12 kpc are rare ; only one out of 44 galaxies in song has a bar larger than 12 kpc . thus the fraction of bars we detected in the nicmos deep field ( 4/95 galaxies for the entire sample , or 3/31 galaxies using an i - magnitude cutoff ) is similar , and perhaps even , larger than the bar fraction seen in song . thus we concluded that there was no evidence for a decline in the bar fraction at z @xmath7 0.7 , as previously claimed . our results are hampered by small number statistics ( we discovered only a handful of bars at z > 0.7 and our local comparison sample contained forty four galaxies ) , and difficulty in defining comparable samples at high and low redshifts ( see sheth et al . 2003 ) . from the existing analysis , it is difficult to conclusively state how the bar fraction varies as a function of redshift , and how the bar properties vary . in the next two sections we outline new results from two separate on - going studies aimed at overcoming these two main caveats of our nicmos study how well do we know the fraction of nearby bars ? what are their properties ( size , strength ) ? how do these properties depend on the host galaxy properties ? answers to these questions are of fundamental importance before any comparison is made to the high redshift universe . since bars are best studied in the infrared , we decided to address all of these questions using the 2mass large galaxy sample . results of this study were presented in a poster at this conference by menendez - delmestre . we chose all spirals of type sa - sd , with @xmath1065@xmath11 for a sample of 134 galaxies in j+h+k . we ran the same ellipse fitting algorithm on these galaxies as we did previously on the high redshift sample and identified bars using the two signature criteria described above . as shown in figure 2 of menendez - delmestre et al . ( this volume ) , the fraction of barred spirals in the 2mass sample is 58% ( see figure 1 in menendez - delmestre et al . ) . another 21% of the galaxies are identified as candidate bars where usually there is an ellipticity signature but no corresponding position angle change . we examined each of the candidate bars by eye and found that about two - thirds of the candidates were in fact barred . so the total fraction of barred spirals in the infrared @xmath072% , similar to the fraction found by eskridge et al . the bar fraction changes slightly with the t - type with the highest fraction ( 80% ) in t=3 but there is no significant trend in the fraction with the hubble type . unfortunately the 2mass sample has low signal to noise and we were unable to quantify the bar fraction in galaxies of type later than sd . note that the overall bar fraction in rc3 is lower , but not significantly different than the 2mass fraction . this suggests that overall the change in the bar fraction between b - band ( rc3 ) and the near - infrared is not large but also note that the effect may become severe at wavelengths shorter than the b - band , as shown by figure 1 . for the 78 barred spirals in 2mass , we derive a median deprojected bar semi - major axis of 5.1 kpc . this result is different from the song sample we considered in sheth et al . the reason for the difference is that 2mass surveys a larger volume but is shallower . as a result we observe larger and brighter galaxies which tend to have longer bars ( see below ) . we measure a mean ellipticity of 0.45 but the most notable result from the distribution of ellipticities is the scarcity of thin bars ( @xmath12 ) . previously similar results have been interpreted as evidence for a second or later generation of bars ( e.g. , block et al . 2002 , see also bournaud , combes in this volume ) because in numerical simulations the first generation of bars is expected to be strong ( thin ) whereas subsequent generations are expected to be weaker ( fatter ) . however note that the measurement of ellipticity can be affected by bulge light . ellipse fitting does not automatically account for bulge light and as a consequence gives a higher ellipticity than the actual value . when we compare the bar ellipticity and bar length we find no correlation indicating that larger bars are preferentially fatter or thinner . we find that bars come in all shapes and sizes . how do the bar properties depend on the host galaxy ? from the 2mass data , we find a trend of larger bars in larger and brighter galaxies ( figures 4 ) . this is expected given that the bar instability is correlated with the mass of the disk . correlations of the bar strength ( we use ellipticity as a proxy for strength ) and the host galaxy parameters are very weak ( figures 5 ) . unlike the bar length , there is no correlation between the bar ellipticity and galaxy size or brightness . note that these correlations are not immune to biases in the analysis . for instance the ellipse fitting technique always underestimates the bar ellipticity ( sheth et al . 2000 , 2004 ) because the bulge light affects the galaxy isophotes to make the fitted ellipses fatter / thicker than the real underlying bar . hence the lack of `` strong '' bars in early - hubble type galaxies is not necessarily a real effect . another bias is related to resolution . small bars are difficult to identify especially in galaxies with bright bulges and nuclei . so any trend of larger bars in earlier type galaxies should also be viewed with caution . nevertheless the trends of larger bars in larger galaxies and brighter galaxies are relatively robust . the 2mass analysis by menendez - delmestre et al . ( 2004 ) offers an excellent analysis of the local sample of bars and their host galaxies . we can now extend these results to higher redshifts to understand how the bar fraction and bars evolved over time . until recently , the hubble deep fields were the standard windows into the high redshift universe . these data already indicated that defining a comparative sample at high redshifts is difficult . for example the fraction of irregular objects increases to nearly 30% to z@xmath01 ( e.g. , griffiths et al . 1994 , abraham et al . 1996 ) . lilly et al . ( 1996 ) concluded that disks evolve passively in luminosity whereas shude et al . ( 1998 ) argue in favor of significant evolution . certainly some of these discrepancies are a result of cosmic variance and scarcity of high quality data . these gaps , however , are now being filled with surveys like goods , udf , deep2 , and gems . the most ambitious and revolutionary new survey is cosmos ( http://www.astro.caltech.edu/cosmos ) which is observing a contiguous 2-square degree equatorial field using the advanced camera for survey on the hubble space telescope , and multi - wavelength data from x - rays to radio . the size of the field is important because it overcomes cosmic variance and provides a truly representative view of the high redshift universe . cosmos is an ideal treasury program with a vast range of applications . in particular , for the study of galaxy morphology cosmos is a gold mine ( or more appropriately for this conference , a diamond mine ) of opportunity . it will provide spectroscopic redshifts for over 100,000 galaxies with 10,000 galaxies in each of five bins of from z=0.5 to z=2 . the spatial resolution of acs data is exquisite . as shown in figure 3 , one can easily identify structures with sizes as small as 3 kpc in diameter . if figure 6 we show some typical examples of disk galaxies at three photometrically determined redshifts . in order to image such a large field with adequate sensitivity ( cosmos is within half a magnitude of the goods data ) we chose to use the broader i - band filter over the z - band filter for cosmos . hence the analysis of the bar fraction can be done with full confidence to [email protected] and perhaps to even higher redshifts . to date approximately 200 fields out of 600 have been observed . for these fields we first identified a sample of spiral galaxies from a photometric catalog derived from bvriz data from observations on the subaru teelscope for the entire cosmos field . we only chose those galaxies that had photometric redshifts derived with the highest confidence ( @xmath795% ) . we ran the same ellipse fitting algorithm on these galaxies . an example of such a fit is shown in figure 7 . we have done a preliminary analysis for the redshift bin 0.6 @xmath13 z @xmath140.8 for @xmath0 700 galaxies . we find that the fraction of barred spirals is @xmath050% , consistent with the local fraction . although further analysis is necessary with better redshifts and the complete sample , the results already indicate that the bar fraction is not declining significantly , at least out to [email protected] .	an analysis of the nicmos deep field shows that there is no evidence of a decline in the bar fraction beyond [email protected] , as previously claimed ; both bandshifting and spatial resolution must be taken into account when evaluating the evolution of the bar fraction . two main caveats of this study were a lack of a proper comparison sample at low redshifts and a larger number of galaxies at high redshifts . we address these caveats using two new studies . for a proper local sample , we have analyzed 134 spirals in the near - infrared using 2mass ( main results presented by menendez - delmestre in this volume ) which serves as an ideal anchor for the low - redshift universe . in addition to measuring the mean bar properties , we find that bar size is correlated with galaxy size and brightness , but the bar ellipticity is not correlated with these galaxy properties . the bar length is not correlated with the bar ellipticity . for larger high redshift samples we analyze the bar fraction from the 2-square degree cosmos acs survey . we find that the bar fraction at [email protected] is @xmath050% , consistent with our earlier finding of no decline in bar fraction at high redshifts .
several , if not most , mainstream languages include features to support object - oriented programming , yet most of these ( c++ , c # , java , python , etc . ) lack any native language support for the specification and runtime checking of class invariants . while it is usually easy enough to implement the invariant predicates themselves , manual addition imposes further requirements in order to implement the operational requirements of invariant checking and to handle the interplay of invariant specification and inheritance . class invariants are further troublesome in that they involve direct access to an object s attributes . this makes manual addition particularly unappealing , as the available choices are invasive with respect to the original interface and implementation ( to which we may not have access ) , compromise encapsulation , and are error - prone if done manually . this paper presents a lightweight , non - invasive technique for automatically extending a collection of class definitions with a corresponding collection of structural invariant checks . the invariants are given as a stand- alone specification , which is woven together with the original source files to produce a new collection of drop- in replacement classes that are behaviorally indistinguishable from the originals in the absence of invariant- related faults but will expose such faults in a way that the original classes do not . each replacement is defined to be a subclass ( indeed , a _ subtype _ @xcite ) of the original class whose functionality it extends , and it can thus be substituted in any context in which the original occurs . the generation is itself completely automatic , and the incorporation into a test harness or other program is nearly seamless . we focus here on the java language , a choice that complicates the overall strategy in some ways while simplifying it in others . a _ class invariant _ is a conjunction of predicates defined on the values of an object s individual attributes and on the relationships between them . it characterizes an object s `` legal '' states , giving the predicates that must hold if the object is to represent an instance of that abstraction . usually , a class invariant is given in conjunction with the _ contracts _ for each publicly - visible method of a class , _ i.e. _ , the preconditions that must hold on arguments to each method call and the consequent guarantees that are made as postconditions upon the method s return . unlike the contracts , however , a class invariant is a property concerning only an object s _ data values _ , even ( especially ) when those values are not publicly visible . an invariant must hold at every point between the object s observable actions , _ i.e. _ upon creation of any object that is an instance of this class and both before and after every publicly - visible method call @xcite . at other points , including non - visible method calls , it need not hold , and runtime checks are disabled in this case . further , since runtime invariant checks can impose a non - trivial performance penalty on a system , in general , it is desirable to have a mechanism for leaving the checks in place during testing , while removing them from a final , production system . finally , there is an important interplay between the subtype relation ( which determines when one object can safely be substituted in a context calling for another @xcite ) and class invariants : if @xmath0 is a subtype of @xmath1 ( as well as a subclass ) then the invariant for @xmath0 must include all of the constraints in @xmath1 s invariant @xcite . some languages offer native support for invariant checking , but for java and other languages that lack this , including such checks is challenging . a common approach is to make use of the language s assertion mechanism , by including assertions of the invariant at the end of each constructor body and at the beginning and end of the body of each public method @xcite . if the language s assertions mechanism is used , disabling the checking functionality after testing is usually quite easy . however , this approach carries the disadvantage of requiring the class designer to code not only the predicates themselves but also an explicit handling of the inheritance requirements and the full execution model , discussed above . both of these tasks must be implemented for each invariant definition , in each class . to avoid the implementation burden of the assertions approach , we can use a tool that generates the invariant checks from either specialized annotations of the source code @xcite or reserved method signatures @xcite . essentially , such tools offer language extensions to resemble native support for invariant definitions . in comparison to assertion - based approaches , they eliminate the requirement of implementing the execution model , a clear advantage . as with the assertions approach , annotation approaches are invasive , in that they require modification of the original source code . more substantially , the approach generally requires the use of a specialized , nonstandard compiler , whose development may not keep up with that of the language . instead , we can view the addition of runtime invariant checking across a class _ hierarchy _ as a kind of cross - cutting concern , _ i.e. _ code that is defined across several classes and hence resists encapsulation . under this view , it is natural to approach this problem as one of aspect - oriented programming ( aop ) @xcite , in which we can use a tool such as aspectj @xcite to define the checks separately as aspects . the entry and exit points of each method become the join points , the point cuts are inferred from a class s method signatures , and the invariant check itself becomes the advice @xcite . unlike annotation - based approaches , aspect weaving can be done without the need for a non - standard compiler , either through source code transformation or byte code instrumentation @xcite . however , the aop approach also presents several difficulties . for example , balzer _ et al . _ note that mainstream tools such as aspectj lack a mechanism to enforce the requirement that the definition of a class s invariant include the invariant of its parent class @xcite . it is possible to write invariant checking `` advice '' so that it correctly calls the parent class s invariant check , but this must be done manually ( _ e.g. _ @xcite ) . a similar problem occurs in implementing the correct disabling of checks on non - public calls . lastly , because aspects can not in general be prevented from changing an object s state , the weaving of additional aspects may compose poorly with the aspect that provides the invariant check @xcite . it is possible that another aspect could break the class invariant , and since interleaving of multiple aspects is difficult to control , it is possible the two aspects could interleave in such a way as to make the invariant failure go undetected . the work closest in spirit to our own is the design pattern approach of gibbs , malloy , and power ( @xcite . targeting development in the c++ language , they present a choice of two patterns for weaving a separate specification of invariant checks into a class hierarchy , based on the well known _ decorator _ and _ visitor _ patterns @xcite . however , the decorator approach involves a fairly substantial refactoring of the original source code . moreover , the authors note that this technique interacts poorly with the need to structure invariant checks across a full class hierarchy . the refactoring in this case is complex , and it requires the use of multiple inheritance to relate the decorated classes appropriately , making it unsuitable for languages such as java , which support only single inheritance . their alternative is an application of the visitor pattern , in which the invariant checks are implemented as the _ visit _ methods in a single _ visitor _ class . this pattern usually requires that the classes on the `` data side '' implement an _ accept _ method , which is used to dispatch the appropriate _ visit _ method , but in their use of it , only the top of the class hierarchy is modified to be a subclass of an `` invariant facilitator '' , which handles all _ accept _ implementations . however , successful implementation of the _ visit _ methods rests on the assumption that all fields are either publicly visible or have their values readily available through the existence of accessor ( `` getter '' ) methods . unless the language simply lacks a mechanism to hide this representation ( _ e.g. _ python ) , such exposure is unlikely to be the case , as it violates encapsulation , permitting uncontrolled manipulation of an object s parts , either directly or through aliasing @xcite . the central thesis of our work is that , under assumptions common to java and other statically - typed oo languages , these limitations source code modification , multiple inheritance , and public accessibility of fields are unnecessary for a design - pattern approach . the remainder of the present paper shows how to relax them . our approach draws from the gibbs / malloy / power design pattern efforts and from ideas in aop in the treatment of invariant specifications as a cross - cutting concern . we begin with an assumption that the class invariants are given in a single specification file , separate from classes that they document . each constraint is a boolean- valued java expression , with the invariant taken to be the conjunction of these expressions . we assume ( though do not hope to enforce ) that these expressions are free of side effects , and that the invariant given for a child class does not contradict any predicates in inherited invariants . otherwise , the particulars of the specification format are unimportant . the current version of our tool uses json @xcite , but any format for semi - structured data will do . we focus on the java programming language , which means that we assume a statically - typed , object - oriented language , with introspective reflection capabilities , support for type parameters in class definitions , single inheritance ( though implementation of multiple interfaces is possible ) , and a uniform model of virtual method dispatch . we make some simplifications of the full problem . specifically , we work only with synchronization- free , single - threaded , non-_final _ class definitions , and we consider only instance methods of a class that admit overriding , _ i.e. _ , non-_static _ , non-_final _ method definitions . we do not consider anonymous inner class constructs nor the _ lambda expressions _ planned for java 8 @xcite . finally , we assume a class s field visibility grants at least access through inheritance ( _ i.e. _ _ protected _ accessbility or higher ) . this last is made purely for the sake of simplifying the technical presentation , since , as discussed in section [ concl ] , introspection makes it easy to handle variables of any accessibility . as a first effort , we will try an approach that leverages the mechanism of inheritance and the redefinition of inherited method signatures through subtyping polymorphism . the idea is to derive from a class and its invariant a subclass , in which we wrap the invariant in a new , non - public method ( perhaps with additional error reporting features ) , similar to the `` _ repok _ '' approach advocated by liskov and guttag @xcite . to this new subclass , we also add methods @xmath2 and @xmath3 to handle the checking tasks at ( respectively ) method entry and exit points , and we use these to define constructors and overridden versions of every public method . let @xmath1 be a class , with parametric type expression @xmath4 defined on type parameters @xmath5 , field declarations @xmath6 , invariant @xmath7 , constructor definition @xmath8 and public method @xmath9 . we extend @xmath1 with runtime checking of @xmath7 by generating the subclass in fig . [ fig : naive - inh ] , where @xmath10 and @xmath11 are identical to @xmath4 and @xmath12 ( respectively ) , except perhaps for renaming of type parameters ( _ i.e. _ , they are @xmath13-equivalent ) . l `` ` public ` ` class ` @xmath14`<`@xmath10 ` > ` ` extends ` @xmath1`<`@xmath11 ` > ` \ { + ` private int \delta = 0 ; ` + `` public @xmath1`(`@xmath15 ` ) ` \ { + ` super(\overrightarrow{x } ) ; ` + + + ` } ` + `` public @xmath16`(`@xmath17 ` ) ` \ { + + ` return \chi ; ` + ` } ` + ` private boolean inv ( ) { return \rho_a ; } ` + `` private void @xmath2 ( ) \ { + ` if ( \delta = = 0 & & ! inv ( ) ) ` + + + ` } ` + `` private void @xmath3 ( ) \ { + + ` if ( \delta = = 0 & & ! inv ( ) ) ` + + ` } ` + ` } ` + for each constructor in @xmath14 , the body executes the `` real '' statements of the corresponding superclass constructor , followed by a check of @xmath7 , whose execution is itself controlled by the @xmath3 method . likewise , the body of each public method @xmath18 wraps a call to the superclass s version between checks of @xmath7 , with execution controlled by the @xmath2 and @xmath3 methods . if @xmath18 returns a value , then this value is captured in the overridden version in a `` result '' variable , @xmath19 . a method or constructor call is publicly - visible precisely when the call stack depth on a given @xmath14 object is 0 , and this value is tracked by the additional integer - valued field @xmath20 . the @xmath2 and @xmath3 methods increment / decrement @xmath20 as appropriate , evaluating @xmath7 only if @xmath21 . the inheritance - based approach suggests an easy mechanism for reusing code while adding the necessary invariant checks and capturing the distinction between publicly - visible and inner method calls . for the user , the burden consists of replacing constructor calls to @xmath1 with the corresponding calls for @xmath14 . this may be an excessive requirement when @xmath1 objects are used in production - level code , but in many settings where invariant checking is desirable , such constructor calls are limited to only a handful of sites . in the junit framework , for example , integration of @xmath14 objects into unit tests for @xmath1 is likely quite simple , as object construction occurs mainly in the body of a single method , _ setup_. note the assumptions of uniform polymorphic dispatch and non-_final _ declarations here . if a class can not be extended ( _ e.g. _ _ string _ and other objects in the _ java.lang _ package ) , then construction of a subclass that implements the invariant checks is obviously impossible . similarly , a method whose dispatch is statically determined can not be transparently overridden , and if declared _ final _ , it can not be overridden at all . in many languages ( notably , c # and c++ ) the default convention is _ static _ dispatch , with dynamic binding requiring an explicit _ virtual _ designation ; in such cases , the inheritance construction is far less convenient and may be impossible without some refactoring of the original source code . unfortunately , our first attempt fails in two critical ways , which becomes apparent when we attempt to construct the invariant - checking extension across a hierarchy of class definitions . first of all , the inheritance hierarchy of a collection of objects requires a corresponding structure in the composition of invariant checks . this problem is very similar to the one encountered in the `` decorator '' approach of @xcite , but the multiple - inheritance solution given there is unavailable in a single - inheritance language such as java . consider a class @xmath0 that is a subtype of @xmath1 ( written @xmath22 ) : figure [ fig : naive - limit ] depicts the problem to denote the substitution of type expression @xmath23 for the type parameter @xmath24 in expression @xmath25 , and use the shorthand @xmath26 to denote the composition of type expressions @xmath27 . ] . the invariant for a @xmath0 object , @xmath28 , must include the @xmath1 invariant_i.e . _ , @xmath29 . however , a @xmath30 object can not access the fields of its associated @xmath0 object through inheritance and also reuse the functionality of the @xmath31 method . we might choose to have @xmath30 descend from @xmath14 instead , but this only works if all fields in @xmath0 are publicly accessible . as discussed above , this is unlikely to be the case . the second , related failure is that inheritance does not facilitate a correct binding of the type parameters . again , this is clear from fig . [ fig : naive - limit ] . an instantiation of @xmath0 supplies a type @xmath23 to the parameters @xmath32 , which is used in turn to bind the parameters @xmath33 with argument @xmath34 . when we instantiate @xmath30 instead , this same @xmath23 binds the parameters @xmath35 , with the resulting chain of arguments binding @xmath1 s parameters @xmath12 as @xmath36 . for correct use of the @xmath14 invariant check in this @xmath37 object , we would need to bind the type parameter of @xmath14 , @xmath11 , in the same way we do @xmath1 s parameter , @xmath5 ; _ i.e. _ with argument @xmath38 , a binding that can not be ensured , unless @xmath30 is a subclass of @xmath14 . though unsuccessful on its own , we can use the inheritance approach of section [ invcheck : inh - approach ] as the basis for an auxiliary pattern , which we call an _ exposure pattern_. the idea is to construct from the original hierarchy a corresponding set of classes that offers the interface of the original collection and in addition , a controlled exposure of each object s representation . the machinery for checking the invariants is factored into separate classes , as discussed in section [ invcheck ] , below . consider a class definition we derive the _ exposure _ interface and _ exposed _ class where @xmath10 , @xmath39 and @xmath11 , @xmath40 are @xmath13-equivalent to @xmath41 and @xmath33 , as above . note that the fields @xmath42 include all of the original @xmath43 and perhaps others , as discussed on page , below . the constructors and public methods in @xmath44 are overridden in exactly the same manner as in the @xmath14 class of section [ invcheck : inh - approach ] , and likewise the implementation of the @xmath45 and @xmath46 methods . the representation exposure happens through the @xmath47 , a set of raw `` getter '' methods that expose each of the object s fields . in the presence of inheritance , the corresponding structure is realized not in the derived class but in the derived _ interfaces_. thus , for example , gives rise to the interface and class definitions the construction is illustrated in fig . [ fig : exposure - pattern ] . since the type expressions in a class definition are copied to its exposed class and interface ( perhaps with @xmath13-renaming of the parameters ) , it is easy to see that [ prop : type - correctness ] for any type expression @xmath23 , an instance of a class @xmath1 has type @xmath48 if and only if @xmath44 and @xmath49 have types @xmath50 and @xmath51 , respectively . the construction of the accessor methods is less obvious . while we construct @xmath47 for each of the fields @xmath52 , we may need to construct others , as well , in case the invariant @xmath7 makes reference to any inherited fields for which we have not already constructed an interface . this can happen in the case of an incomplete specification of the class hierarchy and invariants . the simplest way to handle this is to include in the interface a @xmath47 for each declared field in the corresponding @xmath1 classes and also for each variable that occurs without explicit declaration in the the predicate @xmath7 . however , we can leverage the inheritance of interfaces to eliminate redundant declarations ( though not implementations , as discussed below ) . to make the construction precise , we denote the _ free variables _ of the predicate @xmath7 by @xmath53 , _ i.e. _ those variables that occur in @xmath7 without being explicitly declared in @xmath7 . conversely , the _ bound _ variables in a class @xmath1 , @xmath54 , are the instance fields declared in @xmath1 . the following definition captures the notion of variables that are `` free '' in @xmath1 through inheritance : [ defn : inh ] let @xmath55 be a specification of a collection of classes and their associated invariants . for a class @xmath1 , _ the set of fields exposed through inheritance in @xmath1 _ , @xmath56 , is defined by @xmath57 we use this to define the necessary method signatures in each exposure interface . [ defn : exp - interface ] given class @xmath1 and invariant @xmath7 , the body of @xmath49 consists of the the signatures @xmath58 where each @xmath59 is the declared type of @xmath60 . [ defn : succ - exposure ] for a field , @xmath61 , either declared in or inherited by a class @xmath1 , we say that @xmath60 is _ successfully exposed for a _ if either * there is an interface @xmath49 and subclass + [ cols= " < " , ] + and for every @xmath44 object @xmath62 , @xmath63 * @xmath1 is a subclass of @xmath64 , and @xmath60 is successfully exposed for @xmath64 . given @xmath1 and @xmath7 , the construction for @xmath49 in definition [ defn : exp - interface ] and the accompanying implementation @xmath44 combine to give us the representation exposure we need for @xmath7 . in particular , [ prop : exposure - correctness ] if @xmath65 , then @xmath66 is successfully exposed for @xmath1 . the primary difference between the exposure pattern construction and the inheritance - based effort of section [ invcheck : inh - approach ] lies in the construction of the exposure interfaces , whose inheritance structure is congruent to that of the original collection of classes . like the earlier attempt , however , the collection of exposed _ classes _ does not share this same relation , and as a consequence , both approaches are subject to some unfortunate redundancy consequences . in particular , we can not reuse code between distinct exposed classes , even when the classes they expose are related by inheritance . for example , if a class @xmath1 contains fields @xmath67 and @xmath68 and public method @xmath69 then the exposed class @xmath44 must override @xmath69 , and it must include exposure methods @xmath70 and @xmath71 , according to the interface @xmath49 . if @xmath22 contains fields @xmath72 , @xmath73 , and method @xmath74 , then it must override not only @xmath74 but also @xmath69 , with the body of the overridden @xmath69 identical to that in @xmath44 . likewise , it must implement not only the @xmath75 and @xmath76 methods from the @xmath77 interface , but also @xmath70 and @xmath71 . happily , all of this is easily automated , and it is reasonable to suppose the space overhead manageable . note first that , with the exception of classes at the top of a specified hierarchy , the size of the interface generated for a class is proportional to the number of fields in that class . recalling definitions [ defn : inh ] and [ defn : exp - interface ] , we can see that this is so because [ prop : inh - trivial ] let @xmath64 be a class included in a specification @xmath55 . for every class @xmath78 , @xmath79 . in other words , only for classes specified at the top of an inheritance hierarchy will we ever need to generate additional @xmath80 declarations in the corresponding interfaces . in all other cases , the accessor interfaces for inherited fields are inherited from the corresponding parent interfaces . hence , the space required to extend a collection of classes depends only on the size of each class and the depth of the inheritance relationship in the collection . specifically , if we assume a bound of @xmath81 new field and method definitions on each class and an inheritance depth of @xmath82 , then the overall space growth is given by @xmath83 it is difficult to give a general characterization of either @xmath81 or @xmath82 , but there is reason to suspect that both are manageable values in practice . mcconnell recommends a limit of 7 new method definitions in a class @xcite . study @xcite finds no significant threshold value for @xmath82 . classes in the jdk s _ java . * _ and _ javax . * _ libraries implement anywhere from less than 10 to over 100 new methods , while the largest depth of any inheritance tree is 8 . as in gibbs / malloy / power @xcite , we implement the runtime invariant checks themselves through an application of the _ visitor pattern _ @xcite , in which the methods implementing the invariant checks are aggregated into a single class ( the `` visitor '' ) , with the appropriate method called from within the class being checked ( the `` acceptor '' ) . unlike their approach , however , our exposure pattern allows us to do this without modification of any part of the original source files , not even at the top of the inheritance hierarchy . suppose we have a class @xmath84 , with invariant @xmath7 . from these , we generate the exposed class @xmath85 and the exposure interface @xmath86 , as in section [ ssec : exposure ] . the specification of @xmath7 and the access methods defined for @xmath49 are used to generate an invariant checking `` visitor '' class : where @xmath87 and @xmath88 are equivalent to @xmath4 and its parameters @xmath33 , as above . runtime checking of @xmath7 is invoked in the @xmath44 methods through calls to that class s @xmath89 method , which serves as the `` accept '' method , handling dispatch of the appropriate invariant check : note that each @xmath90 method in @xmath91 takes an argument of type @xmath49 and not @xmath44 . this is necessary , because of the need to compose an invariant check with that of the object s superclass in each invariant method . for example , if we have @xmath22 , we define @xmath92 as since @xmath44 and @xmath93 are not related by inheritance , it would not be possible to directly cast ` obj ` to its superclass s exposed version . fortunately , the interface is all we need . finally , although we structure our solution here according to the traditional visitor pattern conventions , we do not really need the full generality of that pattern . in particular , it is unnecessary to support full double dispatch , as we only need one instance of @xmath91 , and no @xmath94 method will ever invoke a call back to the @xmath95 method of an object ( not even indirectly , since the @xmath2 and @xmath3 methods in a class prevent a call to @xmath95 if one is already running ) . our implementation of this approach as an eclipse plugin instead drops the @xmath91 parameter from every @xmath89 method , relying instead on a single , static instance of the invariant visitor : method contracts and class invariants are particularly useful in testing . in combination with test oracles , the use of runtime invariant and pre / post - conditions checks improves the exposure of faults as well as the diagnosability of faults when they are detected @xcite . our implementation as an eclipse plug has proven useful in diagnosing invariant - related faults . for example , a simple list interface provides an abstraction for the list data type . a standard way to implement this is with an underlying doubly - linked list , in which we keep a pair of `` sentinel '' head and tail nodes , with the `` real '' nodes in the list linked in between : .... public abstract class abstractlist < t > implements list < t > { protected int size ; ... } public class dlinkedlist < t > extends abstractlist < t > implements list < t > { // inherited from abstractlist : int size protected dnode < t > head , tail ; ... } .... among other predicates , the invariant for _ dlinkedlist _ requires that @xmath96 , @xmath97 . this was given as part of a project for the first author s data structures course , and among the student submissions received was this implementation of _ remove ( ) _ , in which the _ cur.prev _ pointer is not correctly updated : .... public boolean remove(t v ) { dnode < t > cur = head.next ; while ( cur ! = tail ) { if ( cur.data.equals(v ) ) { dnode < t > prev = cur.prev ; cur = cur.next ; prev.next = cur ; size-- ; return true ; } else cur = cur.next ; } return false ; } .... a junit test suite failed to uncover this fault , passing this and the tests for 12 other methods : .... public void testremove ( ) { ls.add("a " ) ; ls.add("b " ) ; ls.add("c " ) ; ls.add("d " ) ; ls.add("a " ) ; ls.add("d " ) ; int sz = ls.size ( ) ; asserttrue(ls.remove("a " ) ) ; asserttrue(ls.size ( ) = = sz - 1 ) ; sz = ls.size ( ) ; asserttrue(!ls.remove ( " * * " ) ) ; asserttrue(ls.size ( ) = = sz ) ; } .... from the original source code and a specification of invariants our tool generates the classes and interfaces .... public interface iexposedabstractlist < t > { int _ getsize ( ) ; } public interface iexposeddlinkedlist < t > extends iexposedabstractlist < t > { dnode < t > _ gethead ( ) ; dnode < t > _ gettail ( ) ; } public abstract class exposedabstractlist < t > extends abstractlist < t > implements iexposedabstractlist < t > { ... } public class exposeddlinkedlist < t > extends dlinkedlist < t > implements iexposeddlinkedlist < t > { ... } public class repokvisitor { ... public < t > void visit(iexposedabstractlist < t > _ inst ) { ... } public < t > void visit(iexposeddlinkedlist < t > _ inst ) { ... } ... } .... objects in a junit test suite are constructed in the _ setup ( ) _ method , and a simple modification was all that was needed to cause _ testremove ( ) _ to fail appropriately : .... protected void setup ( ) { // ls = new dlinkedlist < string > ( ) ; ls = new exposeddlinkedlist < string > ( ) ; } .... the design pattern given here provides a fairly seamless approach for adding correct runtime invariant checking to a class hierarchy , through the construction of drop - in replacements that can be removed as easily as inserted . in addition to the core material presented here , there are a number of extensions possible . for example , the presentation in this paper relies on the assumption above that all fields in a class are accessible through inheritance . happily , this is an easy if tedious limitation to overcome . if instead the field is declared with only intra - object or intra - class access ( _ e.g. _ java s `` private '' ) , we can use the introspective capabilities of the language to manufacture a locally - visible _ get _ method . to access a ` private ` field @xmath66 , for example , our implementation generates a @xmath98 that handles the unwieldy details of java introspection : finally , the work described here incorporates only the invariant checks , rather than full contracts , and it would clearly be useful to extend our design pattern to support this . while we conjecture that our technique is easily extendable to this purpose , the invariant checks present the most interesting problems , owing to their need for attribute access and hierarchical definition . philosophically , ordinary unit testing already performs at least the behavioral components of contract checking , _ i.e. _ the checks of pre and post - conditions . what unit testing can not do is determine whether the invariant continues to hold , as it is often impossible to access an object s fields . the difference lies in the fact that both pre and post conditions are inherently extensional specifications . they impose requirements on method arguments and return values , but on the object itself , all constraints are made upon the abstraction of the object , not the concrete implementation . that implementation whose consistency with the abstraction is the core assertion of a class invariant is by definition opaque to an object s user . the ideas in this paper began with an assignment in the first author s spring 2010 software construction class , and the students there provided valuable feedback . our thanks also to prof . peter boothe of manhattan college , for help in analyzing the inheritance and method complexity of the jdk . klaeren , h. , pulvermller , e. , rashid , a. , speck , a. : aspect composition applying the design by contract principle . in butler , g. , jarzabek , s. , eds . : gcse 00 . volume 2177 of lncs . , springer ( 2000 ) 5769	we present a technique for automatically weaving structural invariant checks into an existing collection of classes . using variations on existing design patterns , we use a concise specification to generate from this collection a new set of classes that implement the interfaces of the originals , but with the addition of user - specified class invariant checks . our work is notable in the scarcity of assumptions made . unlike previous design pattern approaches to this problem , our technique requires no modification of the original source code , relies only on single inheritance , and does not require that the attributes used in the checks be publicly visible . we are able to instrument a wide variety of class hierarchies , including those with pure interfaces , abstract classes and classes with type parameters . we have implemented the construction as an eclipse plug - in for java development .
superclusters are the most extensive density enhancements in the universe of common origin . investigation of large systems of galaxies was pioneered by the study of the _ local supercluster _ by de vaucouleurs @xcite , and by abell @xcite using rich clusters of galaxies by abell @xcite and abell et al . superclusters consist of galaxy systems of different richness : single galaxies , galaxy groups and clusters , aligned in chains ( jeveer , einasto & tago @xcite , gregory & thompson @xcite , zeldovich , einasto & shandarin @xcite ) . new deep galaxy surveys , such as the las campanas galaxy redshift survey , the 2 degree field galaxy redshift survey ( 2dfgrs , colless et al . @xcite , @xcite ) and the sloan digital sky survey ( sdss , adelman - mccarthy et al . @xcite ) cover large areas in the sky and are almost complete up to fairly faint apparent magnitudes . thus these surveys are convenient to detect superclusters using both galaxy and cluster data . this possibility has been used by basilakos @xcite , basilakos et al . @xcite , erdogdu et al . @xcite , porter and raychaudhury @xcite , and einasto et al . @xcite , @xcite , @xcite , @xcite . the goal of the present review is to analyse properties of superclusters based on the 2df galaxy redshift survey and the sloan digital sky survey data release 4 by einasto et al . @xcite , @xcite and @xcite , and to compare properties of real superclusters with theoretical models . for comparison we use the superclusters found for the millennium run mock galaxy catalogue by croton et al . @xcite , that itself based on the millennium simulation of the evolution of the universe by springel et al . @xcite . both real and model superclusters were found using the luminosity density fields calculated using epanechnikov smoothing with a radius of 8 . superclusters were defined as connected non - percolating systems with densities above a certain threshold density . these density fields were normalized to identical mean levels , and all regions above a threshold density 6 ( in units of the mean density ) were considered as superclusters . the density fields were calculated for a grid step of 1 , which allows to investigate the detailed spatial structure of superclusters . the 2dfgrs superclusters were found separately for the northern and southern regions of the 2df survey , and sdss superclusters for the high - declination region of the sdss dr4 survey in the northern hemisphere . the 2df northern and southern regions together contain 544 superclusters , the sdss northern survey has 911 superclusters . the comparison model samples have 1733 and 1068 superclusters ( the full model sample and the simulated 2df sample , respectively ) . + + in figs . [ fig:1 ] and [ fig:2 ] we show high - resolution density fields on the 2dfgrs and sdss surveys . all wedges are about 10 degrees thick , thus near to the observer they are thin . these figures show the cosmic web a continuous network of galaxy systems of various luminosity densities , and voids between them . all luminous regions seen in these figures are superclusters . we see that they have very different richness , some are very small and resemble the local supercluster around the virgo cluster , some are large and very rich . for all superclusters their geometric and physical properties were found . among the geometrical properties are the position ( ra , dec and distance ) , the size , and the offset of the geometrical center from the dynamical one , defined as the center of the main ( most luminous ) cluster . the physical properties are the mean and maximum luminosity densities , the total luminosity and the luminosity of the main cluster and of the main galaxy ( the brightest galaxy of the main cluster ) . comparison of properties of model superclusters with properties of real superclusters shows that they are very similar . superclusters consist of several chains ( filaments ) of galaxies , groups and clusters . these chains have various length , thus superclusters are asymmetrical in shape . the degree of asymmetry is higher in rich superclusters . rich superclusters are also denser and contain luminous knots high - density nuclei . one important property of superclusters is different in real and model samples the supercluster richness . to characterise the richness we used two independent characteristics : the total luminosity and the number of rich clusters , i.e. the multiplicity . the multiplicity was derived from the number of high - density knots of the density field . we call these knots df - clusters . the spatial density of df - clusters is about two times higher than the spatial density of abell clusters in the same volume , thus the expected number of df - clusters in superclusters is about two times higher than the number of abell clusters . both functions were determined separately for the 2dfgrs and sdss superclusters , and for the total observational sample . the total luminosity was calculated by summing the luminosities of galaxies and clusters of galaxies inside the density contour , which defines the boundary of the supercluster . in our case the threshold density was chosen to be 6 ( in units of the mean density ) . in calculating total luminosities we used weights for galaxies which take into account the galaxies outside the observational luminosity window of a survey . to avoid complications due to the use of different color systems and mean luminosities , we used _ relative _ luminosities , normalized by the mean luminosity of poor superclusters , i.e. the superclusters that contain only one df - cluster . for model samples we calculated these functions for two cases . one sample uses all model galaxies and can be considered as the true model sample . the second model sample simulates the 2df sample , where an observer was put into one corner of the sample , and only these galaxies were included , which satisfy the same selection criteria as used in the real 2df sample . the luminosity and multiplicity functions of real and model samples are compared in fig . [ fig:3 ] . we see that both functions show for real samples much more rich superclusters than for model samples . this difference is the major cosmological result of the analysis of our supercluster survey . the presence of very rich superclusters in our vicinity is well known ; good examples are the shapley supercluster and the horologium - reticulum supercluster ( see fleenor et al . @xcite , proust et al . @xcite , nichol et al . @xcite and ragone et al . @xcite and references therein ) . but until recently the number of such extremely massive superclusters was too small to make definite conclusions about their abundance . when comparing models with observations we have to use the simulated 2df sample , which is formed using the same selection criteria as used for the observational sample . the most luminous simulated superclusters of this sample have a relative luminosity of about 15 in terms of the mean luminosity of richness class 1 superclusters , whereas the most luminous superclusters of real samples have a relative luminosity about 100 , i.e. they are about 6 times more luminous . the richest model superclusters have a multiplicity of 10 , whereas the multiplicity of the richest real superclusters is over 70 . the number of abell clusters in the richest abell supercluster is 34 ( einasto et al . @xcite ) . figures [ fig:1 ] and [ fig:2 ] show that very luminous superclusters are located in _ all subsamples _ ( the northern and southern regions of the 2dfgrs , and in subregions of the sdss dr4 sample , if divided into 3 wedges of equal width ) . these subsamples have characteristic volumes of about 10 million cubic , whereas model samples of 10 times larger volume have no extremely rich superclusters . to check these results we used a number of independent numerical simulations , carried out for simulation boxes of size of 500 and 768 , using @xmath0 dark matter particles . we found dm - halos in simulations , and used them to calculate the smoothed density field as for real and millennium simulation . for all simulations we then found simulated superclusters as previously , and found the distribution of dense knots ( simulated rich clusters ) . these calculations confirmed our previous result : the number of dense knots in simulated superclusters is much lower than in real superclusters . this striking conflict between model and reality needs explanation . in order to understand the formation of rich superclusters we used wavelet analysis to investigate the role of density waves of different scales . the trous wavelet technique we used allows to divide the density field into components of various wavelength bands , so that the field is restored by summing all components . the wavelet analysis was carried out both for real and model samples . our results show that in all cases superclusters form only in regions where _ large density waves combine in similar local phases to generate high density peaks_. very rich superclusters are objects where density waves of all large scales ( up to a wavelength @xmath1 ) have similar phases . the smaller is the maximum wavelength of such phase synchronization , the lower is the richness of superclusters . similarly , large voids are caused by large - scale density perturbations of wavelength @xmath2 , here large - wavelength modes combine _ in similar local phases to generate under - densities_. superclusters of galaxies are formed by density perturbations of large scales . these perturbations evolve very slowly . as shown by kofman & shandarin @xcite , the present structure on large scales is built - in already in the initial field of linear gravitational potential fluctuations . actually they are remnants of the very early evolution and stem from the inflationary stage of the universe ( see kofman et al . @xcite . the distribution of luminosities of superclusters allows us to probe processes acting at these very early phases of the evolution of the universe . there are two possible explanations for the large difference between the distribution of luminosities of real and simulated samples . one possibility is that in present simulations the role of very large density perturbations , responsible for the formation of these very luminous superclusters , is underestimated . the other feasible explanation of the differences between models and reality may be the presence of some unknown processes in the very early universe which give rise to the formation of extremely luminous and massive superclusters . 1 . geometric properties of superclusters are well explained by current models . there are much more very rich superclusters than models predict . large perturbations evolve very slowly and represent the fluctuation field at the epoch of inflation . the difference between observations and models can be explained in two ways : + large - scale perturbations are not incorporated in the models , i.e. models need improvement ; + there ocurred presently unknown processes during inflation . the present review is based on talks held in budapest on april 20 , 2006 in detre centenarium , in uppsala university on april 27 , 2006 and in aspen workshop on cosmic voids on june 6 , 2006 . i thank my collaborators maret einasto , enn saar , erik tago and volker mller for permission to use results of our common work in this review . we are pleased to thank the 2dfgrs and sdss teams for the publicly available data releases . the present study was supported by estonian science foundation grants no . 4695 , 5347 and 6104 , and estonian ministry for education and science support by grant to 0060058s98 . i thank astrophysikalisches institut potsdam ( using dfg - grant 436 est 17/2/05 ) and uppsala university for hospitality where part of this study was performed . 2dfgrs supercluster catalogues are available at http://www.aai.ee/ maret/2dfscl.html , sloan dr4 supercluster catalogues at http://www.aai.ee/ maret / sdssdr4scl.html .	a review of the study of superclusters based on the 2dfgrs and sdss is given . real superclusters are compared with models using simulated galaxies of the millennium run . we show that the fraction of very luminous superclusters in real samples is about five times greater than in simulated samples . superclusters are generated by large - scale density perturbations which evolve very slowly . the absence of very luminous superclusters in simulations can be explained either by non - proper treatment of large - scale perturbations , or by some yet unknown processes in the very early universe .
in this paper , we will work in either the smooth or the piecewise linear category . for basic terminologies of knot theory , see @xcite . the notion of _ tangles _ was introduced by j. conway @xcite as the basic building blocks of links in the 3-dimensional sphere @xmath0 . slightly abusing the notation , a tangle @xmath12 is a pair @xmath13 , where @xmath14 is a 3-dimensional ball and @xmath12 is a proper 1-dimensional submanifold of @xmath14 with 2 non - circular components . the points in @xmath15 will be fixed once and for all . recall that a link @xmath16 is a submanifold of @xmath0 homeomorphic to a disjoint union of several copies of the circle @xmath17 . a tangle @xmath18 can be embedded in a link @xmath16 in @xmath0 if there is an embedding @xmath19 such that @xmath20 . using the kauffman bracket at @xmath2 , a necessary condition that one can embed a tangle @xmath12 in a link @xmath16 is given by d. krebes in @xcite . one of the purposes of this paper is to give a generalization to krebes theorem . suppose that @xmath21 tangles @xmath22 , @xmath23 , are given . they can be embedded disjointly in a link @xmath16 if there are embeddings @xmath24 such that @xmath25 for all @xmath26 and @xmath27 for all @xmath28 with @xmath29 . a necessary condition similar to that of krebes will be given for the existence of such a disjoint embedding of tangles in a link ( see theorem 3.7 ) . in order to prove this generalization of krebes theorem , we will study a class of topological objects in @xmath0 called _ @xmath1-punctured ball tangles_. this class of topological objects has rich contents in the theory of _ operads _ @xcite , which is beyond the scope of this paper . our main interest lies in a special type of @xmath1-punctured ball tangles , which in the case of @xmath8 , corresponds exactly to conway s notion of tangles in the 3-ball @xmath14 . using the kauffman bracket at @xmath2 , we will define an invariant for this special type of @xmath1-punctured ball tangles . for an @xmath1-punctured ball tangle @xmath12 , this invariant @xmath30 is an element in @xmath4 , that is the set of @xmath5 matrices over @xmath6 modulo the scalar multiplication of @xmath7 . when @xmath8 , @xmath30 is krebes invariant . suppose now that we have @xmath21 tangles @xmath31 , embedded disjointly in a link @xmath16 . let @xmath32 , \,i=1,2,\dots , k,\ ] ] and let @xmath33 be the kauffman bracket of @xmath16 at @xmath2 . then theorem 3.7 says that @xmath34 divides @xmath35 . when @xmath36 , this is exactly krebes theorem . the proof of theorem 3.7 is based on the fact that the invariant @xmath3 behaves well under the operadic composition of @xmath1-punctured ball tangles . in the second part of this paper , we study the invariant @xmath3 in some more details when @xmath9 . in this case , @xmath1-punctured ball tangles are called _ spherical tangles_. for a given spherical tangle @xmath11 , @xmath37 is a well - defined integer . using a theorem of s. matveev , h. murakami and y. nakanishi in @xcite , we will show that @xmath37 is either 0 or 1 modulo 4 ( theorem 4.29 ) . thus , not every element in @xmath38 can be realized as @xmath39 for some spherical tangle @xmath11 . this is in contrary with the case of @xmath8 , where the invariant @xmath3 is onto . we organize the paper as follows : in section 2 , we formally define the notion of @xmath1-punctured ball tangles . we also recall the kauffman bracket at @xmath2 and krebes theorem in this section . in section 3 , we define our invariant @xmath3 for a special class of @xmath1-punctured ball tangles . a key result is about the behave of the invariant @xmath3 under operadic composition of @xmath1-punctured ball tangles ( theorems 3.6 ) . our generalization of krebes theorem ( theorem 3.7 ) will follow easily from this result . finally , in section 4 , we study the surjectivity of the invariant @xmath3 in the case of @xmath40 . as mentioned before , we will show that @xmath3 is surjective when @xmath8 but not surjective when @xmath9 . in the final section , we pose some questions related with this work which we do not know how to answer at this moment . notice that d. ruberman has given a topological interpretation of krebes theorem @xcite . we do nt know if our generalization of krebes theorem could have a similar topological interpretation . in particular , it will be very nice if there is a topological interpretation of the restriction on @xmath37 for spherical tangles @xmath11 ( theorem 4.29 ) . we define a topological object in the 3-dimensional sphere @xmath0 called an _ @xmath1-punctured ball tangle _ or , simply , an _ @xmath1-tangle_. to study this object , we consider a model for a class of objects and an equivalence relation on it . let @xmath1 be a nonnegative integer , and let @xmath41 be a 3-dimensional closed ball , and let @xmath42 be pairwise disjoint 3-dimensional closed balls contained in the interior @xmath43 of @xmath41 . for each @xmath44 , take @xmath45 distinct points @xmath46 of @xmath47 for some positive integer @xmath48 . then a 1-dimensional proper submanifold @xmath12 of @xmath49 is called an @xmath1-punctured ball tangle with respect to @xmath50 , @xmath51 , and @xmath52 if @xmath53 . hence , @xmath54 for each @xmath55 . note that an @xmath1-tangle @xmath12 with respect to @xmath56 , @xmath51 , and @xmath52 can be regarded as a @xmath57-tuple @xmath58 . let @xmath59 , and let @xmath60 be the class of all @xmath1-punctured ball tangles with respect to @xmath56 , @xmath51 , and @xmath52 , and let @xmath61 . define @xmath62 on @xmath60 by @xmath63 if and only if there is a homeomorphism @xmath64 such that @xmath65 , @xmath66 , and @xmath67 is isotopic to @xmath68 relative to the boundary @xmath69 for all @xmath70 . then @xmath62 is an equivalence relation on @xmath60 , where @xmath68 is the identity map from @xmath71 to @xmath71 . note that @xmath63 if and only if there are a homeomorphism @xmath64 with @xmath65 and @xmath66 and a continuous function @xmath72 such that @xmath73 is a homeomorphism with @xmath74 for each @xmath75 and @xmath76 and @xmath77 , where @xmath78 $ ] . let us denote @xmath79 by @xmath80 for all @xmath81 and @xmath75 , so @xmath82 for each @xmath75 . for every @xmath83 , @xmath84 since @xmath68 and the 1st projection @xmath85 satisfy the condition . suppose that @xmath63 and @xmath86 is a homeomorphism with @xmath87 such that @xmath66 and @xmath72 is a continuous function such that @xmath88 is a homeomorphism with @xmath89 for each @xmath75 and @xmath90 and @xmath91 . define @xmath92 by @xmath93 for all @xmath81 and @xmath94 . then @xmath95 and @xmath96 make @xmath97 . to show the transitivity of @xmath62 , suppose that @xmath63 and @xmath98 is a homeomorphism with @xmath65 such that @xmath66 and @xmath99 is a continuous function such that @xmath100 is a homeomorphism with @xmath101 for each @xmath75 and @xmath90 and @xmath91 and @xmath102 and @xmath103 is a homeomorphism with @xmath104 such that @xmath105 and @xmath106 is a continuous function such that @xmath107 is a homeomorphism with @xmath108 for each @xmath75 and @xmath109 and @xmath110 . define @xmath111 by @xmath112 for all @xmath81 and @xmath75 . then @xmath113 and @xmath114 make @xmath115 . therefore , @xmath62 is an equivalence relation on @xmath60 . let @xmath116 and @xmath117 be @xmath1-punctured ball tangles in @xmath60 . then @xmath116 and @xmath117 are said to be equivalent or of the same isotopy type if @xmath63 . also , for each @xmath1-punctured ball tangle @xmath12 in @xmath60 , the equivalence class of @xmath12 with respect to @xmath62 is denoted by @xmath118 $ ] . by the context , without any confusion , we will also use @xmath12 for @xmath118 $ ] . there are many models for a class of @xmath1-punctured ball tangles . it is convenient to use normalized ones . one model for a class of @xmath1-punctured ball tangles is as follows : \(1 ) @xmath119 and @xmath120 for each @xmath121 . here @xmath122 is the 3-dimensional ball in @xmath123 with center @xmath124 and radius @xmath125 . \(2 ) @xmath46 are @xmath45 distinct points of @xmath47 in the @xmath126-plane for each @xmath44 . \(3 ) @xmath12 is a 1-dimensional proper submanifold of @xmath127 such that @xmath128 . in order to study an @xmath1-punctured ball tangle @xmath12 through its diagram @xmath129 , we consider the @xmath126-projection @xmath130 defined by @xmath131 for all @xmath132 . a point @xmath133 of the image @xmath134 is called a multiple point of @xmath12 if the cardinality of @xmath135 is greater than 1 . in particular , @xmath133 is called a double point of @xmath12 if the cardinality of @xmath136 is 2 . if @xmath133 is a double point of @xmath12 , then @xmath135 is called the crossing of @xmath12 corresponding to @xmath133 and the point in the crossing whose @xmath137-coordinate is greater is called the overcrossing of @xmath12 corresponding to @xmath133 and the other is called the undercrossing . an @xmath1-punctured ball tangle @xmath12 is said to be in regular position if the only multiple points of @xmath12 are double points and each double point of @xmath12 is a transversal intersection of the images of two arcs of @xmath12 and @xmath138 . note that for each @xmath83 , there is @xmath139 such that @xmath140 is in regular position and @xmath141 . furthermore , @xmath140 has a finite number of crossings . consider the image @xmath134 of an @xmath1-punctured ball tangle @xmath12 in regular position . for each double point of @xmath12 , take a sufficiently small closed ball centered at the double point such that the intersection of @xmath134 and the closed ball is an x - shape on the @xmath126-plane . we may assume that the closed balls are pairwise disjoint . now , modify the interiors of the closed balls keeping the image @xmath134 to assign crossings corresponding to the crossings of @xmath12 . as a result , we have a representative @xmath129 of @xmath12 which is ` almost planar ' and @xmath142 . @xmath129 is called a diagram of @xmath12 and we usually use this representative . to deal with diagrams of @xmath1-punctured ball tangles in the same isotopy type , we need reidemeister moves among them . for link diagrams or ball tangle diagrams , we have 3 kinds of reidemeister moves . however , we need one and only one more kind of moves which are called the reidemeister moves of type iv . the reidemeister moves for diagrams of @xmath1-punctured ball tangles are illustrated in the following figure . = 5.8 in figure 1 . tangle reidemeister moves . like link diagrams , tangle diagrams also have reidemeister theorem involving the reidemeister moves of type iv . let us call reidemeister moves including type iv tangle reidemeister moves . let @xmath1 be a nonnegative integer , and let @xmath143 and @xmath144 be diagrams of @xmath1-punctured ball tangles . then @xmath145 if and only if @xmath144 can be obtained from @xmath143 by a finite sequence of tangle reidemeister moves . remark that , even though we may have different models for @xmath1-punctured ball tangles , we may regard them as the same @xmath1-punctured ball tangle if there are suitable model equivalences among them . our invariant is based on the kauffman bracket at @xmath2 . in this section , we recall the kauffman bracket which is a regular isotopy invariant of link diagrams . that is , it will not be changed under reidemeister moves of type ii and iii . assume that @xmath16 is a link diagram with @xmath1 crossings and @xmath146 is a crossing of @xmath16 . take a sufficiently small disk at the projection of @xmath146 to get an x - shape on the projection plane of @xmath16 . now , we have 4 regions in the disk . rotate counterclockwise the projection of the over - strand in the disk which is an arc of @xmath16 containing the overcrossing for @xmath146 to pass over 2 regions . these 2 regions and the other 2 regions are called the @xmath147-regions and the @xmath148-regions of @xmath146 , respectively . we consider 2 ways of splitting the double point in the disk . @xmath147-type splitting is to open a channel between the @xmath147-regions so that we have 1 @xmath147-region and 2 @xmath148-regions in the disk and @xmath148-type splitting is to open a channel between the @xmath148-regions so that we have 2 @xmath147-regions and 1 @xmath148-region in the disk . a choice of how to destroy all of @xmath1 double points in the projection of @xmath16 by @xmath147-type or @xmath148-type splitting is called a state of @xmath16 . notice that we regard a state @xmath149 of the link diagram @xmath16 with @xmath1 crossings as a function @xmath150 , where @xmath151 is the set of all crossings of @xmath16 and @xmath152 is the set of @xmath147-type and @xmath148-type splitting functions , respectively . therefore , a link diagram @xmath16 with @xmath1 crossings has exactly @xmath153 states of it . apply a state @xmath149 to @xmath16 in order to change @xmath16 to a diagram @xmath154 without any crossing . let @xmath16 be a link diagram . then the kauffman bracket @xmath155 , or simply , @xmath156 , is defined by @xmath157 where @xmath11 is the set of all states of @xmath16 , @xmath158 , @xmath159 , and @xmath160 is the number of circles in @xmath154 . we have the following _ skein relation of the kauffman bracket_. let @xmath16 be a link diagram , and let @xmath146 be a crossing of @xmath16 . then if @xmath161 and @xmath162 are link diagrams obtained from @xmath16 by @xmath147-type splitting and @xmath148-type splitting only at @xmath146 , respectively , then @xmath163 . the following lemma is useful in our discussion of the kauffman bracket . let @xmath16 be a link diagram . then states @xmath149 and @xmath164 of @xmath16 are of the same parity , i.e. , @xmath165 mod 2 , if and only if @xmath149 and @xmath164 differ at an even number of crossings , where @xmath160 and @xmath166 are the numbers of circles in @xmath154 and @xmath167 , respectively . let @xmath149 be a state of a link diagram @xmath16 with @xmath1 crossings @xmath168 . change the value of @xmath149 at only one crossing @xmath169 to get another state @xmath170 and observe what happens to @xmath171 , where @xmath172 . we claim that @xmath149 and @xmath170 have different parities , more precisely , @xmath173 . hence , we will have @xmath174 mod 2 . now , to consider @xmath175 and @xmath176 , take a sufficiently small neighborhood @xmath177 at the projection of @xmath169 so that the intersection of @xmath178 and the set of all double points of @xmath16 is the projection of @xmath169 and the intersection of @xmath179 and the projection of @xmath16 has exactly 4 points on the projection plane of @xmath16 which are not double points of @xmath16 . _ case 1_. if these 4 points are on a circle in @xmath154 , then @xmath180 _ case 2_. if two of 4 points are on a circle and the other points are on another circle in @xmath154 , then @xmath181 now , it is easy to show the lemma . suppose that @xmath149 and @xmath164 are states of @xmath16 which differ at @xmath21 crossings of @xmath16 for some @xmath44 . then @xmath182 mod 2 . if @xmath183 mod 2 , then @xmath21 is even . conversely , if @xmath184 mod 2 , then @xmath185 is even , that is , @xmath21 is odd . this proves the lemma . following @xcite , a state @xmath149 of a link diagram @xmath16 is called a monocyclic state of @xmath16 if @xmath186 . that is , we have only one circle when we remove all crossings of @xmath16 by @xmath149 . from now on , we consider only the kauffman brackets at @xmath187 . since @xmath188 , the determinant @xmath189 of @xmath16 is an isotopy invariant . it is easy to show that reidemeister move of type i dose not change @xmath189 by the skein relation of kauffman bracket and @xmath188 . notice that @xmath190 if @xmath2 . therefore , @xmath191 where @xmath192 is the set of all monocyclic states of @xmath16 . as a corollary of lemma 2.7 , monocyclic states @xmath149 and @xmath164 of @xmath16 differ at an even number of crossings . if @xmath16 is a link diagram , then there are @xmath193 and @xmath194 such that @xmath195 and @xmath196 . suppose that @xmath149 and @xmath164 are states of a link diagram @xmath16 such that @xmath149 and @xmath164 differ at only one crossing . then either @xmath197 or @xmath198 . hence , either @xmath199 or @xmath200 . that is , @xmath201 . if @xmath202 and @xmath203 is a state of @xmath16 such that @xmath149 and @xmath203 differ at @xmath21 crossings , where @xmath204 is the number of crossings of @xmath16 , then @xmath205 because there are exactly @xmath206 sequences with @xmath21 terms consisting of @xmath207 and @xmath208 and the product of all terms of each of the sequences is either @xmath209 or @xmath210 . hence , @xmath211 if @xmath21 is odd and @xmath212 if @xmath21 is even . now , let us take a monocyclic state @xmath213 of @xmath16 , and let @xmath214 . then @xmath195 and @xmath196 for some @xmath193 by the corollary above . this proves the lemma . in this subsection , we introduce some notations and krebes theorem @xcite . a ball tangle @xmath148 , which is a 0-punctured ball tangle with @xmath215 , is said to be embedded in a link @xmath16 if there are a diagram @xmath216 of @xmath148 , a diagram @xmath217 of @xmath16 , and a 3-dimensional closed ball @xmath218 such that @xmath219 and @xmath216 are of the same isotopy type . given a ball tangle diagram @xmath148 , we consider 3 kinds of closures as in figure 2 a ) . = 5 in figure 2 . a ) closures , b ) diagrams by a numerator state and a denominator state . the link diagrams @xmath220 and @xmath221 are called the numerator closure and the denominator closure of @xmath148 , respectively . a monocyclic state of @xmath220 is called a numerator state of @xmath148 and that of @xmath221 is a denominator state of @xmath148 . notice that a numerator state @xmath149 and a denominator state @xmath164 of a ball tangle diagram @xmath148 differ at an odd number of crossings . to see this , we think of a diagram of another closure @xmath16 of @xmath148 which has only one more crossing @xmath146 at the outside of ball containing @xmath148 ( see @xmath16 in figure 2 ) . we have two monocyclic states of @xmath16 from the numerator state @xmath149 and the denominator state @xmath164 , respectively , which differ at @xmath146 . hence , @xmath149 and @xmath164 differ at an odd number of crossings . the following notations throughout the rest of the paper : @xmath222 @xmath223 i.e. @xmath224 is the set of 8-th roots of unity ; and @xmath225 . @xmath222 @xmath226 is the set of all @xmath227 matrices over @xmath228 , and @xmath229 is the quotient of @xmath226 under the scalar multiplication by @xmath7 . @xmath222 @xmath230 is the class of diagrams of 0-punctured ball tangles with @xmath215 ( i.e. ball tangles ) . @xmath222 @xmath231 is the class of diagrams of 1-punctured ball tangles with @xmath232 ( they will be called spherical tangles ) . if @xmath233 , then @xmath234 if and only if @xmath235 or @xmath236 mod 4 . suppose that @xmath237 and @xmath238 mod 4 for some @xmath239 . then @xmath240 for some @xmath241 , hence , @xmath242 . therefore , @xmath243 is not in @xmath244 . conversely , if @xmath235 or @xmath236 mod 4 , then @xmath245 . we have @xmath246 for the links @xmath16 , @xmath220 , and @xmath221 in figure 2 . if @xmath247 and @xmath248 , by proposition 2.10 , we have @xmath249 mod 4 . so there is a unique @xmath250 such that @xmath251 \in pm_{2\times 1}(\mathbb{z}).\ ] ] define @xmath252 by @xmath253 for each @xmath254 . this is krebes tangle invariant . notice that reidemeister move of type i dose not change @xmath255 . so @xmath255 is a ball tangle invariant . the following lemmas about the ball tangle invariant @xmath256 are proved in @xcite . if @xmath257 and @xmath258 are diagrams of ball tangles with @xmath259 $ ] and @xmath260 $ ] , then @xmath261 $ ] , where @xmath262 stands for the horizontal addition of ball tangles ( see figure 8 a ) ) . if @xmath148 is a diagram of ball tangle with @xmath263 $ ] , then we have @xmath264\qquad\text{and}\qquad f(b^r)=\left[\begin{matrix } q \\ -p \end{matrix}\right],\ ] ] where @xmath265 is the mirror image of @xmath148 and @xmath266 is the @xmath267 counterclockwise rotation of @xmath148 on the projection plane . ( krebes @xcite ) if @xmath16 is a link and @xmath148 is a ball tangle embedded in @xmath16 with @xmath268 $ ] , then @xmath269 divides @xmath189 . let @xmath1 be a positive integer . then an @xmath1-punctured ball tangle @xmath270 with @xmath56 and @xmath271 can be regarded as an @xmath1-variable function @xmath272 defined as @xmath273 is a tangle filled up in the @xmath26-th hole @xmath274 of @xmath270 by @xmath275 for each @xmath121 , where @xmath276 is a class of @xmath277-punctured ball tangles with @xmath278 such that @xmath279 for each @xmath121 and @xmath280 is a class of tangles . however , this representation of @xmath1-punctured ball tangles as @xmath1-variable functions is not perfect in the sense that @xmath1-punctured ball tangles are equivalent only if they induce the same function . @xmath1-punctured ball tangles which induce the same function need not be equivalent . that is , we can say that tangles are stronger than functions . roughly speaking , the class of @xmath1-punctured ball tangles as only @xmath1-variable functions gives us an _ operad _ , a mathematical device which describes algebraic structure of many varieties and in various categories . see @xcite . from now on , we consider only @xmath1-punctured ball tangles with @xmath283 . to construct the invariant @xmath282 of @xmath1-punctured ball tangle @xmath270 , let us regard @xmath270 as a ` hole - filling function ' , in sense described as above @xmath284 , where @xmath285 . to construct our invariant of @xmath1-punctured ball tangles with @xmath283 , we need to use some quite complicated notations . let us start with a gentle introduction to our notations : \(1 ) for a diagram of 0-punctured ball tangle @xmath286 ( a ball tangle ) , we can produce 2 links @xmath287 and @xmath288 , which are the numerator closure and the denominator closure of @xmath286 , respectively . \(2 ) for a diagram of 1-punctured ball tangle @xmath289 ( a spherical tangle ) , we can produce @xmath290 links @xmath291 , @xmath292 ; @xmath293 , @xmath294 , where the subscript 1(1 ) means to take the numerator closure of @xmath12 with its hole filled by the fundamental tangle 1 . \(3 ) for a diagram of 2-punctured ball tangle @xmath295 , we can produce @xmath296 links @xmath297 , @xmath298 , @xmath299 , @xmath300 ; @xmath301 , @xmath302 , @xmath303 , @xmath304 . = 1.7 in figure 3 . an @xmath1-punctured ball tangle with @xmath283 . = 3.5 in figure 4 . a hole - filling function @xmath270 , a ) @xmath305 , b ) @xmath306 , c ) @xmath307 . if @xmath1 is a positive integer , @xmath308 , and @xmath309 , then @xmath310 is linearly ordered by a dictionary order , or lexicographic order , consisting of @xmath153 ordered @xmath1-tuples each of whose components is either 1 or 2 . that is , if @xmath311 and @xmath312 , @xmath313 , then @xmath314 if and only if @xmath315 or there is @xmath316 such that @xmath317 . \(4 ) @xmath318 and @xmath319 , where @xmath320 is the dictionary order on @xmath310 . hence , @xmath321 is the least element @xmath322 and @xmath323 is the greatest element @xmath324 of @xmath310 . let us denote @xmath325 for each @xmath326 . \(5 ) for a diagram of @xmath1-punctured ball tangle @xmath270 , we can produce @xmath327 links @xmath328 ; @xmath329 . \(6 ) the sequence @xmath330 is defined recursively as follows : \1 ) @xmath331 ; \2 ) if @xmath332 , then @xmath333 for each @xmath334 . note that @xmath335 for each @xmath336 . now , we define our invariant of @xmath1-punctured ball tangles with @xmath283 inductively . for each @xmath337 , define @xmath338 by @xmath339 for each @xmath340 . then @xmath282 is an isotopy invariant of @xmath1-punctured ball tangle diagrams . in particular , @xmath341 is krebes ball tangle invariant @xmath256 . let @xmath342 . then tangle reidemeister moves of type ii , iii , and iv do not change @xmath343 because kauffman bracket is a regular invariant of link diagrams . also , it is easy to show that tangle reidemeister move of type i does not change @xmath344 by the skein relation of kauffman bracket . hence , it is enough to show that @xmath345 consists of two elements differ by a scalar multiplication of @xmath346 . by lemma 2.8 , for each @xmath347 , there are @xmath348 and @xmath349 such that @xmath350 and @xmath351 . notice that , for each @xmath347 , @xmath321 and @xmath352 differ at only @xmath353 coordinates . hence , @xmath305 and @xmath354 differ at only @xmath353 holes . if @xmath355 and @xmath356 and @xmath357 differ at only 1 hole , then @xmath358 by proposition 2.6 , lemma 2.8 , and proposition 2.10 . thus , @xmath359 . since @xmath360 for each @xmath361 , @xmath362 . this shows that @xmath363 . therefore , @xmath364 if @xmath116 and @xmath117 are isotopic @xmath1-punctured ball tangle diagrams . = 5 in figure 5 . the skein relation of kauffman bracket at the @xmath26-th hole . for each nonnegative integer @xmath1 , @xmath282 is called the @xmath1-punctured ball tangle invariant , simply , the @xmath1-tangle invariant . now , in order to think of an @xmath1-punctured ball tangle @xmath270 as a ` hole - filling function ' , we define a function which makes a dictionary order on complex numbers . let @xmath1 be a positive integer , and let @xmath365 be an @xmath1-tuple of positive integers , and let @xmath366 . then @xmath367 is linearly ordered by a dictionary order , where @xmath368 for each @xmath369 . ( 4@xmath370 ) @xmath371 and @xmath372 , where @xmath320 is the dictionary order on @xmath367 . hence , @xmath373 is the least element @xmath322 and @xmath374 is the greatest element @xmath375 of @xmath367 . let us denote @xmath376 for each @xmath377 . for each @xmath378 and @xmath1-tuple @xmath365 of positive integers , define @xmath379 by @xmath380 for all @xmath381 . then @xmath382 is well - defined and called the dictionary order function on @xmath383 with respect to @xmath384 . also , the @xmath26-th projection of @xmath382 is denoted by @xmath385 for each @xmath386 . in particular , we simply denote @xmath382 by @xmath387 when @xmath388 . denote by @xmath389 the @xmath21-dimensional column vector space over @xmath390 , so the map @xmath391 is to transpose row vectors to column vectors . let @xmath392 . if @xmath393 , then we denote by @xmath394^\dag=\{(v_1,\dots , v_k)^\dag,(-v_1,\dots ,- v_k)^\dag\}\ ] ] the corresponding element in @xmath395 . remark that , we may extend the above notation to matrices modulo @xmath7 . under this extension , matrix multiplication is well - defined . that is , if @xmath147 and @xmath148 are matrices and @xmath396 is defined , then @xmath397[b]=[a][-b]=[-a][b]=[-a][-b]=[-ab]=[ab]$ ] . for each @xmath398 and @xmath1-tuple @xmath365 of positive integers , define @xmath399:p\mathbb{c}^{k_1\dag}\times\cdots\times p\mathbb{c}^{k_n\dag } \longrightarrow p\mathbb{c}^{k_1\cdots k_n\dag}\ ] ] by @xmath399(\left[\begin{matrix } v_1 ^ 1 \\ \cdot \\ \cdot \\ \cdot \\ v_{k_1}^1 \end{matrix}\right ] , \dots , \left[\begin{matrix } v_1^n \\ \cdot \\ \cdot \\ \cdot \\ v_{k_n}^n \end{matrix}\right])=\left[\begin{matrix } \prod_{j=1}^n v_{\alpha_{1j}^{n , k_1,\dots , k_n}}^j \\ \\ \cdot \\ \cdot \\ \prod_{j=1}^n v_{\alpha_{k_1\cdots k_nj}^{n , k_1,\dots , k_n}}^j \end{matrix}\right]\ ] ] for all @xmath381 . then @xmath400 $ ] is well - defined and called the dictionary order function induced by @xmath382 . suppose that @xmath401 and @xmath402 are in @xmath403 such that @xmath404^\dag,\dots,[x_n]^\dag)=([y_1]^\dag,\dots,[y_n]^\dag ) \in p\mathbb{c}^{k_1\dag}\times\cdots\times p\mathbb{c}^{k_n\dag}$ ] . then @xmath405 and @xmath406 . hence , @xmath407^\dag= [ \xi^{n , k_1,\dots , k_n}(y_1,\dots , y_n)]^\dag.\ ] ] therefore , @xmath400([x_1]^\dag,\dots,[x_n]^\dag)= [ \xi^{n , k_1,\dots , k_n}]([y_1]^\dag,\dots,[y_n]^\dag)$ ] . this shows that @xmath400 $ ] is well - defined . as another notation , if @xmath16 is a link diagram and @xmath270 is a diagram of @xmath1-punctured ball tangle for some @xmath408 , then the sets of all crossings of @xmath16 and @xmath270 are denoted by @xmath204 and @xmath409 , respectively . if @xmath378 and @xmath270 is an @xmath1-punctured ball tangle diagram and @xmath410 are ball tangle diagrams , then @xmath411 we denote the set of all monocyclic states of a link diagram @xmath16 by @xmath412 . let @xmath413 . then @xmath149 is a monocyclic state of @xmath220 if and only if there is a unique @xmath326 such that @xmath414 , @xmath415 . note that @xmath416 . let us denote a state @xmath149 of @xmath220 by @xmath417 , where @xmath418 . then @xmath419 and @xmath420 similarly , @xmath421 . this proves the lemma . for each @xmath378 , @xmath282 is an @xmath1-punctured ball tangle invariant such that @xmath422(f^0(b^{(1 ) } ) , \dots , f^0(b^{(n)}))$ ] for all @xmath423 . suppose that @xmath270 is an @xmath1-punctured ball tangle such that @xmath424 $ ] for some @xmath425 and @xmath410 are ball tangles such that @xmath426,\dots , f^0(b^{(n)})=\left[\begin{matrix } z_n\langle b_1^{(n ) } \rangle \\ iz_n\langle b_2^{(n ) } \rangle \end{matrix}\right]\ ] ] for some @xmath427 , where @xmath428 and @xmath429 are the numerator closure and the denominator closure of @xmath430 , respectively , for each @xmath121 . then @xmath431(f^0(b^{(1)}),\dots , f^0(b^{(n)}))\\ & = \left[\begin{matrix } \begin{pmatrix } ( -i)^{t_1}z\langle t_{1\alpha_1^n}^n \rangle & \cdots & ( -i)^{t_{2^n}}z\langle t_{1\alpha_{2^n}^n}^n \rangle \\ ( -i)^{t_1}iz\langle t_{2\alpha_1^n}^n \rangle & \cdots & ( -i)^{t_{2^n}}iz\langle t_{2\alpha_{2^n}^n}^n \rangle \end{pmatrix } \begin{pmatrix } i^{t_1}z_1\cdots z_n\langle b_{\alpha_{11}^n}^{(1 ) } \rangle \cdots \langle b_{\alpha_{1n}^n}^{(n ) } \rangle \\ \cdot \\ \cdot \\ \cdot \\ i^{t_{2^n}}z_1\cdots z_n\langle b_{\alpha_{2^n1}^n}^{(1 ) } \rangle \cdots \langle b_{\alpha_{2^nn}^n}^{(n ) } \rangle \end{pmatrix } \end{matrix}\right ] \end{aligned}\ ] ] @xmath432\\ & = \left[\begin{matrix } zz_1\cdots z_n \sum_{i=1}^{2^n } \langle t_{1\alpha_i^n}^n \rangle \langle b_{\alpha_{i1}^n}^{(1 ) } \rangle \cdots \langle b_{\alpha_{in}^n}^{(n ) } \rangle \\ izz_1\cdots z_n \sum_{i=1}^{2^n } \langle t_{2\alpha_i^n}^n \rangle \langle b_{\alpha_{i1}^n}^{(1 ) } \rangle \cdots \langle b_{\alpha_{in}^n}^{(n ) } \rangle \end{matrix}\right]=\left[\begin{matrix } zz_1\cdots z_n \langle t^n(b^{(1)},\dots , b^{(n)})_1 \rangle \\ izz_1\cdots z_n\langle t^n(b^{(1)},\dots , b^{(n)})_2 \rangle \end{matrix}\right]\\ & = f^0(t^n(b^{(1)},\dots , b^{(n ) } ) ) \end{aligned}\ ] ] by lemma 3.5 . suppose that a ball tangle diagram @xmath148 is embedded in a link diagram @xmath16 . since the complement of a ball in @xmath0 is still a ball , we may think of @xmath433 as another ball tangle diagram embedded in @xmath16 and @xmath434 , that is , @xmath16 is the numerator closure of horizontal addition of @xmath148 and @xmath435 . if @xmath436 $ ] and @xmath437 $ ] , then @xmath438 $ ] and @xmath439 . hence , @xmath440 divides @xmath189 . this is krebes theorem ( see theorem 2.14 ) . we have the following generalization of krebes theorem . let @xmath16 be a link , and let @xmath410 be ball tangles with the invariants @xmath441 , \dots , \left[\begin{matrix } p_n \\ q_n \end{matrix}\right]$ ] , respectively . if @xmath410 are embedded in @xmath16 disjointly , then @xmath442 divides @xmath189 . denote by @xmath443 for each @xmath444 . let @xmath445 , and let @xmath446 $ ] . then @xmath447 and @xmath448 $ ] , hence , @xmath449 . notice that we can regard @xmath450 as an @xmath451-punctured ball tangle with its holes filled up by @xmath452 . hence , @xmath453 for some @xmath451-punctured ball tangle @xmath454 . let @xmath455 $ ] . then , by lemma 3.4 and theorem 3.6 , we have @xmath456(f^0(b^{(1 ) } ) , \dots , f^0(b^{(n-1)}))\\ & = \left[\begin{matrix } \begin{pmatrix } a_{11 } & a_{12 } & \cdots & a_{12^{n-1 } } \\ a_{21 } & a_{22 } & \cdots & a_{22^{n-1 } } \end{pmatrix } \begin{pmatrix } p_1p_2\cdots p_{n-1 } \\ p_1p_2\cdots q_{n-1 } \\ \cdot \\ \cdot \\ \cdot \\ q_1q_2\cdots q_{n-1 } \end{pmatrix } \end{matrix}\right ] \end{aligned}\ ] ] @xmath457.\ ] ] let @xmath458 . then @xmath459 and there are @xmath460 such that @xmath461 . since @xmath462 divides each term of the above linear combination , @xmath462 divides @xmath463 . hence , @xmath464 divides @xmath465 and @xmath465 divides @xmath466 . therefore , @xmath467 divides @xmath189 . we use the following notation throughout this section : \(1 ) the subscripts 1,2 of ball tangles will no longer used to denote different kinds of closures . they will be used simply to distinguish different ball tangles . \(2 ) the ball tangle invariant @xmath341 will be denoted by @xmath256 with values in @xmath468 and the spherical tangle invariant @xmath469 will be denoted by @xmath3 with values in @xmath470 . let @xmath471 be 4 points in @xmath472 , and let @xmath473 and @xmath474 for each @xmath475 . then a 1-dimensional proper submanifold @xmath11 of @xmath476 , @xmath78 $ ] , is called a spherical tangle about @xmath477 ( or simply , spherical tangle ) if @xmath478 and @xmath479 . note that @xmath480 is homeomorphic to @xmath476 . define @xmath62 on the class of all spherical tangles about @xmath471 by @xmath481 if and only if there is a homeomorphism @xmath482 such that @xmath483 and @xmath484 rel @xmath485 for spherical tangles @xmath486 and @xmath487 . then @xmath62 is an equivalence relation on it . @xmath486 and @xmath487 are said to be isotopic , or of the same isotopy type , if @xmath481 and , for each spherical tangle @xmath11 , the equivalence class @xmath488 $ ] is called the isotopy type of @xmath11 . remark that we usually use @xmath11 for @xmath488 $ ] and consider only diagrams for spherical tangles and ball tangles . now , let us define the product of spherical tangle diagrams as follows : @xmath489 \circ [ s_1]=[s_2(s_1)],\ ] ] or simply , @xmath490 for all spherical tangle diagrams @xmath486 and @xmath487 , where , roughly speaking , @xmath491 means to put @xmath486 inside of @xmath487 , using the identification @xmath492)_1\coprod ( s^2\times[0,1])_2 } { ( s^2\times 1)_1=(s^2\times 0)_2}=s^2\times [ 0,1].\ ] ] it is clear that @xmath493 is associative and @xmath494 is the identity spherical tangle for @xmath493 . thus , the class @xmath231 of spherical tangle diagrams with @xmath493 forms a monoid . recall lemma 2.12 , and lemma 2.13 : \(1 ) if @xmath495 and @xmath496 , f(b_2)=\left[\begin{matrix } r \\ s \end{matrix}\right]$ ] , then @xmath497 $ ] . so if we denote @xmath498 $ ] by @xmath499 + _ h \left[\begin{matrix } r \\ s \end{matrix}\right]$ ] , then we have @xmath500 . \(2 ) if @xmath254 and @xmath501 $ ] , then @xmath502 $ ] and @xmath503 $ ] , where @xmath265 is the mirror image of @xmath148 and @xmath266 is the @xmath267 rotation of @xmath148 counterclockwise on the projection plane . so if we denote @xmath504 $ ] by @xmath499^$ ] and @xmath505 $ ] by @xmath499^r$ ] , then we have @xmath506 and @xmath507 . to avoid complication , we use the same notations for @xmath508 , @xmath370 , and @xmath509 applied to @xmath230 and @xmath510 . we shall be able to understand the meaning of different operations by their contexts . \(3 ) if @xmath254 , then @xmath511 but @xmath512 need not be the same as @xmath148 . \(4 ) if @xmath513 , then @xmath514 and @xmath515 . notice that , if an element @xmath147 in @xmath510 can be obtained by applying @xmath508 , @xmath370 , and @xmath509 to finitely many invariants of ball tangles , then @xmath147 itself is the invariant of a ball tangle . let us calculate @xmath256 for ball tangles in figure 6 . the ball tangles _ b _ * and * _ c _ * have invariants @xmath516 $ ] and @xmath517 $ ] , respectively . the ball tangle * _ a _ * has invariant @xmath518 $ ] because @xmath519 and @xmath518= \left[\begin{matrix } 1 \\ 0 \end{matrix}\right ] + _ h \left[\begin{matrix } 1 \\ 0 \end{matrix}\right]$ ] . the ball tangles * _ d _ * and * _ e _ * have invariants @xmath520 $ ] and @xmath521 $ ] , respectively . they are the mirror images each other . \4 . the ball tangle * _ f _ * has invariant @xmath522 + _ h \left[\begin{matrix } 1 \\ 1 \end{matrix}\right]=\left[\begin{matrix } 2 \\ 1 \end{matrix}\right]$ ] , and the ball tangle * _ g _ * has invariant @xmath523 $ ] , where @xmath133 is the number of horizontal twists in * _ g_. \5 . ball tangles _ j _ * and * _ k _ * have invariants @xmath524 $ ] and @xmath525 $ ] , respectively , where @xmath526 is the number of vertical twists in * _ k_. \6 . the ball tangle _ h _ * has invariant @xmath527 $ ] because @xmath527=\left[\begin{matrix } 1 \\ 3 \end{matrix}\right ] + _ h \left[\begin{matrix } 1 \\ 0 \end{matrix}\right]$ ] . the ball tangle * _ l _ * is @xmath528 and has invariant @xmath529=\left[\begin{matrix } 0 \\ 3 \end{matrix}\right]$ ] . \7 . the ball tangle * _ m _ * has invariant @xmath530 $ ] because @xmath531 and @xmath530= \left[\begin{matrix } 1 \\ 1 \end{matrix}\right ] + _ h \left[\begin{matrix } 0 \\ 3 \end{matrix}\right]$ ] . @xmath532 . hence , the ball tangles * _ i _ * and * _ m _ * have the same invariant but they are apparently not isotopic . = 4 in figure 6 . ball tangle diagrams . to prove the surjectivity of @xmath256 , we use euclidean algorithm . ( euclidean algorithm ) if @xmath533 and @xmath534 , then there are uniquely @xmath369 and @xmath535 and @xmath536 such that @xmath537 and @xmath538 and @xmath539 and @xmath540 and @xmath541 for each @xmath542 . the ball tangle invariant @xmath543 is onto . it is enough to show that there is @xmath254 such that @xmath544 $ ] if @xmath545 and @xmath534 . suppose that @xmath545 and @xmath534 . then , by euclidean algorithm , there are uniquely @xmath369 and @xmath546 and @xmath547 such that @xmath537 and @xmath548 = \left[\begin{matrix } q_1 \\ 1 \end{matrix}\right ] + _ h \left[\begin{matrix } r_1 \\ a \end{matrix}\right ] , 0<r_1<a,\ ] ] @xmath549 = \left[\begin{matrix } q_2 \\ 1 \end{matrix}\right ] + _ h \left[\begin{matrix } r_2 \\ r_1 \end{matrix}\right ] , 0<r_2<r_1,\ ] ] @xmath550 = \left[\begin{matrix } q_2 \\ 1 \end{matrix}\right ] + _ h \left[\begin{matrix } r_3 \\ r_2 \end{matrix}\right ] , 0<r_3<r_2,\ ] ] @xmath551 @xmath552 = \left[\begin{matrix } q_k \\ 1 \end{matrix}\right ] + _ h \left[\begin{matrix } r_k \\ r_{k-1 } \end{matrix}\right ] , 0<r_{k-1}<r_{k-2},\ ] ] @xmath553 = \left[\begin{matrix } q_{k+1 } \\ 1 \end{matrix}\right ] + _ h \left[\begin{matrix } 0 \\ r_k \end{matrix}\right ] , r_{k+1}=0.\ ] ] since @xmath554,\dots,\left[\begin{matrix } q_{k+1 } \\ 1 \end{matrix}\right]$ ] , and @xmath555 $ ] are realizable by ball tangles and @xmath556 = \left[\begin{matrix } r_{i-1 } \\ r_i \end{matrix}\right]^{r}$ ] for each @xmath557 , @xmath558 $ ] corresponds a ball tangle . therefore , there is @xmath254 such that @xmath559=f(b)$ ] . this proves the theorem . we can define vertical connect sum of ball tangle diagrams by horizontal connect sum and rotations . define @xmath560 on @xmath230 by @xmath561 for all @xmath562 . then @xmath560 is called the vertical connect sum on @xmath230 . ( 2 ) if @xmath495 and @xmath496 , f(b_2)=\left[\begin{matrix } r \\ s \end{matrix}\right]$ ] , then @xmath563 $ ] . so if we denote @xmath564 $ ] by @xmath499 + _ v \left[\begin{matrix } r \\ s \end{matrix}\right]$ ] , we have @xmath565 . note that @xmath566 and @xmath567 are noncommutative monoids with identities * _ c _ * and * _ b _ * in figure 6 , respectively . on the other hand , @xmath568 and @xmath569 are commutative monoids with identities @xmath570 $ ] and @xmath571 $ ] , respectively . the ball tangle invariant @xmath256 is a monoid epimorphism from @xmath566 and @xmath567 to @xmath568 and @xmath572 , respectively . the following lemma tells us a unique commutative square . for each @xmath573 , there is a unique function @xmath574 such that @xmath575 . furthermore , @xmath576 is the function from @xmath510 to @xmath510 defined by @xmath577 for each @xmath513 . let @xmath573 . then @xmath578 for each @xmath254 ( theorem 3.6 ) . hence , @xmath579 . the uniqueness of @xmath576 follows from the surjectivity of @xmath256 . to show the uniqueness of @xmath576 , suppose that @xmath580 and @xmath581 are functions from @xmath510 to @xmath510 such that @xmath582 and @xmath583 , respectively , and @xmath513 . then there is @xmath584 such that @xmath585 by the surjectivity of @xmath256 ( theorem 4.4 ) and @xmath586 . hence , @xmath587 , that is , @xmath576 is unique . this proves the lemma . if @xmath588 , then @xmath589 . suppose that @xmath588 . then @xmath590 and @xmath591 . hence , @xmath592 . therefore , by the uniqueness of @xmath593 , @xmath594 . let us identify @xmath576 with @xmath39 for each @xmath595 . since @xmath596 is the composition of @xmath486 and @xmath487 , we have the following lemma immediately . if @xmath588 , then @xmath597 . notice that lemma 4.8 does not depend on the surjectivity of @xmath256 . we can prove lemma 4.8 by theorem 3.6 and the following lemma without using the surjectivity of @xmath256 . if @xmath598 and @xmath599= b\left[\begin{matrix } 1 \\ 0 \end{matrix}\right ] , a\left[\begin{matrix } 0 \\ 1 \end{matrix}\right]= b\left[\begin{matrix } 0 \\ 1 \end{matrix}\right ] , a\left[\begin{matrix } 1 \\ 1 \end{matrix}\right]= b\left[\begin{matrix } 1 \\ 1 \end{matrix}\right]$ ] , then @xmath600 . by theorem 3.6 , we have that @xmath601 and @xmath602 are matrices in @xmath603 such that @xmath604=f(s_2)f(s_1)\left[\begin{matrix } 1 \\ 0 \end{matrix}\right]$ ] , @xmath605=f(s_2)f(s_1)\left[\begin{matrix } 0 \\ 1 \end{matrix}\right]$ ] , and @xmath606=f(s_2)f(s_1)\left[\begin{matrix } 1 \\ 1 \end{matrix}\right]$ ] . hence , @xmath597 by lemma 4.9 . let us introduce the elementary operations on @xmath231 . let @xmath11 be a spherical tangle diagram . then \(1 ) @xmath607 is the mirror image of @xmath11 , \(2 ) @xmath608 is the spherical tangle diagram obtained by interchanging the inside hole with the outside hole of @xmath11 , \(3 ) @xmath609 is the spherical tangle diagram obtained by only rotating inside hole of @xmath11 @xmath267 counterclockwise on the projection plane , \(4 ) @xmath610 is the spherical tangle diagram obtained by only rotating outside hole of @xmath11 @xmath267 counterclockwise on the projection plane , \(5 ) @xmath611 is the spherical tangle diagram obtained by the @xmath267 rotation of @xmath11 itself counterclockwise on the projection plane . = 4.5 in figure 7 . elementary operations on @xmath231 . note that @xmath612 , @xmath613 , and @xmath614 for each @xmath573 . if @xmath573 with the invariant @xmath615 $ ] , then \(1 ) @xmath616 $ ] , ( 2 ) @xmath617 $ ] , ( 3 ) @xmath618 $ ] , \(4 ) @xmath619 $ ] , ( 5 ) @xmath620 $ ] . let @xmath573 with @xmath615 $ ] . then there is @xmath621 such that @xmath622 , @xmath623 , @xmath624 , @xmath625 . here the link @xmath626 , @xmath627 , is obtained by taking the numerator closure ( @xmath628 ) or the denominator closure ( @xmath629 ) of @xmath11 with its hole filled by the fundamental tangle @xmath630 . therefore , @xmath631=\left[\begin{matrix } u^{-1}\alpha u & u^{-1}(-i)\gamma iu \\ u^{-1}i\beta ( -i)u & u^{-1}\delta u \end{matrix}\right].\ ] ] now we have \(1 ) @xmath632 , \(2 ) @xmath633 , \(3 ) @xmath634 . hence , @xmath616 $ ] , @xmath617 $ ] , @xmath635= \left[\begin{matrix } -\gamma & \alpha \\ -\delta & \beta \end{matrix}\right]$ ] . since @xmath612 and @xmath636 , ( 4 ) and ( 5 ) are easily proved by ( 2 ) and ( 3 ) . like the case of ball tangle operations and invariants , it is convenient to use the following notations . notation : let @xmath637 \in pm_{2\times2}$ ] . then \(1 ) @xmath637^=\left[\begin{matrix } \alpha & -\gamma \\ -\beta & \delta \end{matrix}\right]$ ] , ( 2 ) @xmath637 ^ -=\left[\begin{matrix } \delta & \gamma \\ \beta & \alpha \end{matrix}\right]$ ] , ( 3 ) @xmath637^{r_1}=\left[\begin{matrix } -\gamma & \alpha \\ -\delta & \beta \end{matrix}\right]$ ] , \(4 ) @xmath637^{r_2}=\left[\begin{matrix } -\beta & -\delta \\ \alpha & \gamma \end{matrix}\right]$ ] , ( 5 ) @xmath638^r=\left[\begin{matrix } \delta & -\beta \\ -\gamma & \alpha \end{matrix}\right]$ ] . with these notations , we can write : @xmath639 if @xmath640 . the determinant function det is well - defined on @xmath603 since @xmath641 for each @xmath642 . notice that the 5 elementary operations on @xmath231 do not change the determinant of invariants of spherical tangles . if @xmath588 , then \(1 ) @xmath643 , ( 2 ) @xmath644 , ( 3 ) @xmath645 , \(4 ) @xmath646 , ( 5 ) @xmath647 . notice that a spherical tangle has exactly 2 holes which are inside and outside . let @xmath254 , and let @xmath573 . then \(1 ) the 1st and the 2nd outer horizontal connect sums of @xmath148 and @xmath11 are the spherical tangle diagrams denoted by @xmath648 and @xmath649 , respectively , \(2 ) the 1st and the 2nd outer vertical connect sums of @xmath148 and @xmath11 are the spherical tangle diagrams denoted by @xmath650 and @xmath651 , respectively ( see figure 8) . we also define the connect sums at the inside hole by @xmath652 as follows . let @xmath254 , and let @xmath573 . then \(1 ) the 1st and the 2nd inner horizontal connect sums of @xmath148 and @xmath11 are the spherical tangle diagrams @xmath653 and @xmath654 defined by @xmath655 and @xmath656 , respectively , \(2 ) the 1st and the 2nd inner vertical connect sums of @xmath148 and @xmath11 are the spherical tangle diagrams @xmath657 and @xmath658 defined by @xmath659 and @xmath660 , respectively , where @xmath661 and @xmath662 are the @xmath663 rotation of @xmath148 with respect to the vertical axis and the horizontal axis of the projection plane , respectively ( see figure 9 ) . = 4.5 in figure 8 . connect sums of ball tangles and outer connect sums . = 4.7 in figure 9 . inner connect sums and rotations of ball tangles about axes . let us give definitions of monoid actions . this is just a generalization of group actions . let @xmath192 be a monoid with the identity @xmath664 , and let @xmath71 be a nonempty set . then \(1 ) a function @xmath665 is called a left monoid action of @xmath192 on @xmath71 if @xmath666 and @xmath667 for all @xmath668 and @xmath81 , \(2 ) a function @xmath669 is called a right monoid action of @xmath192 on @xmath71 if @xmath670 and @xmath671 for all @xmath668 and @xmath81 . in this sense , the connect sums of diagrams of ball tangles and spherical tangles are monoid actions on @xmath231 . hence , we have 8 monoid actions on @xmath231 by @xmath230 which are similar . also , the composition on @xmath231 induces a left monoid action and a right monoid action on @xmath230 . in particular , a monoid action is onto like a group action because of identity . let @xmath254 , and let @xmath573 . then \(1 ) @xmath672 , ( 2 ) @xmath673 , \(3 ) @xmath674 , ( 4 ) @xmath675 . note that @xmath676 for each @xmath254 and @xmath677 for each @xmath573 . the following lemma tells us that the other outer horizontal sum and two outer vertical sums can be expressed in terms of the 1st horizontal sum and @xmath509 which is the rotation for ball tangle diagrams or spherical tangle diagrams . let @xmath254 , and let @xmath573 . then \(1 ) @xmath678 , \(2 ) @xmath679 , \(3 ) @xmath680 . let @xmath254 , and let @xmath573 . then \(1 ) @xmath681 for some @xmath682 . hence , @xmath683 . take @xmath684 and @xmath685 . then @xmath686 , \(2 ) @xmath687 for some @xmath682 . hence , @xmath688 . take @xmath689 and @xmath690 . then @xmath679 , \(3 ) @xmath691 for some @xmath692 . hence , @xmath693 . take @xmath694 and @xmath695 . then @xmath696 . now , we consider the invariants of spherical tangles obtained from the various connect sums with ball tangles and their determinants . note that @xmath697 for each @xmath584 and @xmath698 for each @xmath595 . also , @xmath699 for each @xmath595 . if @xmath254 with @xmath501 $ ] and @xmath573 with @xmath615 $ ] , then \(1 ) @xmath700 $ ] , @xmath701 , \(2 ) @xmath702 $ ] , @xmath703 , \(3 ) @xmath704 $ ] , @xmath705 , \(4 ) @xmath706 $ ] , @xmath707 . \(1 ) let @xmath708 \in pm_2 $ ] . then there is @xmath709 such that @xmath710 \in pm_2 $ ] . we have @xmath711 + _ h \left[\begin{matrix } \alpha & \gamma \\ \beta & \delta \end{matrix}\right ] \left[\begin{matrix } x \\ y \end{matrix}\right]\\ & = \left[\begin{matrix } p \\ q \end{matrix}\right ] + _ h \left[\begin{matrix } \alpha x + \gamma y \\ \beta x + \delta y \end{matrix}\right ] = \left[\begin{matrix } p\beta x + p\delta y + q\alpha x + q\gamma y \\ q\beta x + q\delta y \end{matrix}\right]\\ & = \left[\begin{matrix } p\beta + q\alpha & p\delta + q\gamma \\ q\beta & q\delta \end{matrix}\right ] \left[\begin{matrix } x \\ y \end{matrix}\right ] = \left[\begin{matrix } p\beta + q\alpha & p\delta + q\gamma \\ q\beta & q\delta \end{matrix}\right]f(x ) . \end{aligned}\ ] ] by lemma 4.9 , @xmath712 $ ] . hence , @xmath713 . also , @xmath714 . therefore , @xmath715 . this proves ( 1 ) . \(3 ) since @xmath716 , @xmath717 $ ] , and @xmath718 , we have @xmath719 since @xmath720 $ ] , we have @xmath721\ ] ] and @xmath722 . this proves ( 3 ) . similarly , ( 2 ) and ( 4 ) can be proved . a spherical tangle diagram @xmath11 is said to be @xmath723-reducible if there are @xmath398 , @xmath724 , @xmath725 such that @xmath726 and there is only one of @xmath727 equal to @xmath723 , where @xmath728 , @xmath729 , @xmath730 , and @xmath731 . in general , a spherical tangle @xmath11 is @xmath732-reducible if , in above definition , we replace @xmath723 by another spherical tangle @xmath732 . in other words , a spherical tangle diagram @xmath11 is @xmath723-reducible if @xmath11 can be decomposed by finitely many ball tangle diagrams and only one identity spherical tangle diagram with respect to the inner and the outer connect sums and their opposite operations . note that , in definition 4.19 , the expression of @xmath11 can be written as @xmath733 . in this case , the order of operations in the expression is important . if a spherical tangle @xmath11 is @xmath723-reducible , then @xmath734 for some @xmath735 . furthermore , if @xmath11 is @xmath732-reducible , then @xmath736 for some @xmath737 . it follows from lemma 4.18 immediately . by lemma 4.18 , we know that a ball tangle connected to a spherical tangle in the sense of definition 4.13 and definition 4.14 contributes a square of integer to the determinant of the invariant of connect sum . some examples of spherical tangles are given in figure 10 . = 4.3 in figure 10 . spherical tangle diagrams . \1 . the spherical tangle _ a _ * is @xmath723 and has invariant @xmath738 $ ] . \2 . the spherical tangle * _ b _ * has invariant @xmath739 $ ] . \3 . the spherical tangle * _ c _ * is @xmath740 and has invariant @xmath739 \left[\begin{matrix } 1 & 0 \\ 1 & 1 \end{matrix}\right]= \left[\begin{matrix } 1 & 0 \\ 2 & 1 \end{matrix}\right]$ ] . the spherical tangle * _ d _ * has invariant @xmath741 $ ] . . the spherical tangle * _ e _ * has invariant @xmath571 + _ { h_1 } \left[\begin{matrix } 1 & 0 \\ 1 & 1 \end{matrix}\right ] = \left[\begin{matrix } 1 & 1 \\ 0 & 0 \end{matrix}\right]$ ] . ( see lemma 4.18 ) \6 . the spherical tangle * _ f _ * has invariant @xmath742 $ ] . the spherical tangle @xmath743 has invariant @xmath744 \left[\begin{matrix } 1 & 1 \\ 0 & 0 \end{matrix}\right ] = \left[\begin{matrix } 1 & 1 \\ 1 & 1 \end{matrix}\right]$ ] and @xmath745 has invariant @xmath746 $ ] . recall the @xmath747-move on knot diagrams introduced in @xcite . it is illustrated in figure 11 ( a ) . if we apply the kauffman states to the diagrams involved in the @xmath747-move , we get the 5 basis diagrams without closed components as shown in figure 11 ( b ) . = 3.8 in figure 11 . ( a ) a @xmath747-move on a diagram ; ( b ) the 5 basis diagrams . link diagrams @xmath143 and @xmath144 are said to be @xmath747-equivalent if @xmath144 can be obtained from @xmath143 by a finite sequence of @xmath747-moves and reidemeister moves . for a spherical tangle @xmath11 , we can get four links @xmath748 by taking closures of @xmath11 with its hole filled by fundamental tangles . we say that two spherical tangles @xmath11 and @xmath749 are @xmath747-equivalent if each @xmath750 can be obtained from @xmath626 by a finite sequence of @xmath747-moves and reidemeister moves . let @xmath11 and @xmath749 be spherical tangles such that @xmath11 and @xmath749 are @xmath747-equivalent . if @xmath751 $ ] and @xmath752 $ ] , then @xmath753 mod 4 , @xmath754 mod 4 , @xmath755 mod 4 , @xmath756 mod 4 , for some @xmath757 . suppose that @xmath16 is a link diagram and @xmath758 is a link diagram obtained from @xmath16 by a single @xmath747-move . then @xmath759 and @xmath760 hence , @xmath761 suppose that @xmath762 , @xmath763 , @xmath764 , @xmath765 for some @xmath766 . then @xmath767 . _ case 1_. if @xmath768 , then @xmath769 . so we must have @xmath770 mod 4 . _ case 2_. if @xmath771 , then @xmath772 . so we must have @xmath773 mod 4 . _ case 3_. if @xmath774 , then @xmath775 implies @xmath776 mod 4 and @xmath777 mod 4 . therefore , we always have @xmath778 mod 4 with @xmath779 . in general , suppose that a link @xmath780 can be obtained from @xmath16 by a finite sequence of @xmath747-moves and reidemeister moves of type ii and iii . if @xmath781 and @xmath782 , then @xmath783 mod 4 and @xmath779 depends only on the powers @xmath21 and @xmath784 . for the spherical tangle @xmath11 , we need to consider 4 links @xmath748 . if @xmath785 $ ] , then @xmath786 , @xmath787 , @xmath788 , and @xmath789 . also , if @xmath790 $ ] , then @xmath791 , @xmath792 , @xmath793 , and @xmath794 . notice that since a @xmath747-move will not change the writhe and we can postpone all reidemeister moves of type i in any finite sequence of diagram moves to the end of that sequence , we do not need to worry that the kauffman bracket is only a regular isotopy invariant . thus , for the four corresponding links we obtained from @xmath749 , the sign @xmath795 is a constant . thus , we have @xmath796 mod 4 , @xmath754 mod 4 , @xmath755 mod 4 , @xmath756 mod 4 . we will make use of the following theorem . ( matveev @xcite and murakami - nakanishi @xcite ) : oriented links @xmath797 and @xmath798 are @xmath747-equivalent if and only if @xmath799 for all @xmath28 with @xmath800 . suppose that a spherical tangle diagram @xmath11 has no circle components . it has 4 components @xmath801 . we will look at a diagram of @xmath11 and orient each @xmath802 arbitrarily . we use @xmath803 to mean to reverse the orientation of @xmath802 . we define @xmath804 to be a half of the sum of the signs of crossings between @xmath802 and @xmath805 . we have @xmath806 . note : for each @xmath748 , we get a link whose components are unions of some of @xmath801 . if components of @xmath807 etc . are oriented , they are unions of some of @xmath808 , @xmath809 , @xmath810 , @xmath811 , where @xmath812 . let @xmath749 be another spherical tangle with components @xmath813 . suppose the end points of @xmath814 are the same as the end points of @xmath802 . i.e. , @xmath815 , @xmath816 , @xmath817 , @xmath818 . so we can orient each @xmath802 and @xmath814 consistently . in this case , we can orient the links @xmath626 and @xmath750 consistently in the sense that the corresponding components of @xmath626 and @xmath750 are the same union of @xmath819 and @xmath820 , respectively . the linking numbers of @xmath626 and @xmath750 are equal for all @xmath821 if @xmath822 for all @xmath823 . this is obvious from the definition of consistent orientations of @xmath626 and @xmath750 and @xmath824 =3 in figure 12 . the spherical tangle @xmath825 . let @xmath825 be the spherical tangle shown in figure 12 . let @xmath826 be the number of half twists inside of the balls marked by 1,2,3,4 , respectively . then @xmath827 this is by a direct calculation . we have @xmath828.\ ] ] we calculate the determinant of @xmath829 and get @xmath830 . let @xmath11 be a spherical tangle without closed components . then either there is a spherical tangle @xmath749 which is either @xmath723-reducible or @xmath825-reducible , with @xmath825 as shown in figure 12 or that @xmath825 after some possible operations of @xmath831 and/or @xmath832 , such that @xmath833 and @xmath834 for all @xmath823 . if @xmath11 has one component whose end points lie on different boundary components of @xmath835 , then we can find such a spherical tangle @xmath749 that is @xmath723-reducible . suppose now @xmath11 has no such components . the linking number of two components whose end points lie on on the same boundary component of @xmath835 can be realized by adding ball tangles . so we may assume that there is no linking between such components . then , after some possible operations of of @xmath831 and/or @xmath832 , we can take @xmath749 as @xmath825 with the number of full twistings equal to the linking numbers of @xmath11 . if @xmath11 is a spherical tangle diagram without closed components , then @xmath836 mod 4 for some integer @xmath1 . let @xmath749 be the spherical tangle in lemma 4.26 . then @xmath626 and @xmath750 have the same linking numbers , for each @xmath837 . by theorem 4.23 , @xmath11 and @xmath749 are @xmath747-equivalent . the theorem then follows from lemma 4.22 , theorem 4.20 , and lemma 4.25 . suppose now that a spherical tangle @xmath11 has closed components , we have the following theorem . if @xmath573 with @xmath615 $ ] and @xmath11 has closed components , then @xmath838 mod 2 , @xmath839 mod 2 , @xmath840 mod 2 , @xmath841 mod 2 . see figure 13 ( top left ) , where a closed component of @xmath11 is hooked with another component of @xmath11 as shown . applying the kauffman skein relation to the local picture there , we get two diagrams with coefficients @xmath842 and 2 , respectively . the diagram with coefficient 2 has one less closed component than @xmath11 and the diagram with coefficient @xmath842 is obtained from @xmath11 by unhook the closed component at that place . we keep performing this unhooking process until the closed component is hooked with another component only once , as illustrated in figure 13 ( bottom left ) . applying the kauffman skein relation again , we can unhook this closed component entirely and we end up with a diagram having one less closed component and a coefficient 2 . note that this closed component may itself being knotted . but this is not important since a knotted closed component separating from other components will make no contribution to the kauffman bracket when @xmath187 . what we have shown is the fact that when @xmath11 has a closed component , then 2 divides @xmath843 for all @xmath837 . this proves the theorem . = 4.7 in figure 13 . the case that @xmath11 has closed components . note that @xmath844 or 1 mod 4 . so combine theorem 4.27 and theorem 4.28 , we get the following theorem . for every @xmath573 , either @xmath845 mod 4 or @xmath846 mod 4 . from this theorem , we can conclude that there is no spherical tangle @xmath11 such that @xmath847,\ ] ] since the determinant of the matrix above is not equal to 0 or 1 mod 4 . here are some open questions that we are unable to answer at this moment . s. v. matveev , _ generalized surgeries of three - dimensional manifolds and representations of homology spheres _ , matematicheskie zametki 42 ( 1987 ) , no . 2 , 268278 . ( english translation in mathematical notes 42 ( 1987 ) 651656 . )	we consider a class of topological objects in the 3-sphere @xmath0 which will be called _ @xmath1-punctured ball tangles_. using the kauffman bracket at @xmath2 , an invariant for a special type of @xmath1-punctured ball tangles is defined . the invariant @xmath3 takes values in @xmath4 , that is the set of @xmath5 matrices over @xmath6 modulo the scalar multiplication of @xmath7 . this invariant leads to a generalization of a theorem of d. krebes which gives a necessary condition for a given collection of tangles to be embedded in a link in @xmath0 disjointly . we also address the question of whether the invariant @xmath3 is surjective onto @xmath4 . we will show that the invariant @xmath3 is surjective when @xmath8 . when @xmath9 , @xmath1-punctured ball tangles will also be called _ spherical tangles_. we show that @xmath10 or 1 mod 4 for every spherical tangle @xmath11 . thus , @xmath3 is not surjective when @xmath9 .
let @xmath3 be the space of degree @xmath4 polynomials in @xmath5 variables over an algebraically closed field @xmath1 of characteristic @xmath6 . let @xmath7 be the inverse limit of the @xmath3 , equipped with the zariski topology and its natural @xmath2 action ( see [ ss : note ] ) . this paper is concerned with the following question : [ mainques ] is the space @xmath7 noetherian with respect to the @xmath2 action ? that is , can every zariski - closed @xmath2-stable subspace be defined by finitely many orbits of equations ? this question may seem somewhat esoteric , but it is motivated by recent work in the field of representation stability , in particular the theory of twisted commutative algebras ; see [ ss : tca ] . it is also connected to certain uniformity questions in commutative algebra in the spirit of ( the now resolved ) stillman s conjecture ; see [ ss : stillman ] . for @xmath8 the question is easy since one can explicitly determine the @xmath2 orbits on @xmath7 . for @xmath9 this is not possible , and the problem is much harder . the purpose of this paper is to settle the @xmath10 case : [ mainthm ] question [ mainques ] has an affirmative answer for @xmath10 . in fact , we prove a quantitative result in finitely many variables that implies the theorem in the limit . this may be of independent interest ; see [ ss : overview ] for details . the key concept in the proof , and the focus of most of this paper , is the following notion of rank for cubic forms . let @xmath11 with @xmath12 . we define the * q - rank. ] of @xmath13 , denoted @xmath14 , to be the minimal non - negative integer @xmath15 for which there is an expression @xmath16 with @xmath17 and @xmath18 , or @xmath19 if no such @xmath15 exists ( which can only happen if @xmath20 ) . [ ex1 ] for @xmath12 , the cubic @xmath21 has q - rank @xmath5 . this is proved in [ sec : example ] . in particular , infinite q - rank is possible when @xmath20 . the cubic @xmath22 has q - rank 1 , as follows from the identity @xmath23 the cubic @xmath24 therefore has q - rank at most @xmath5 , and we expect it is exactly @xmath5 . the notion of q - rank is similar to some other invariants in the literature : 1 . hochster @xcite define a homogeneous polynomial to have strength * @xmath25 if it does not belong to an ideal generated by @xmath26 forms of strictly lower degree . for cubics , q - rank is equal to strength plus one . the paper @xcite ( inspired by tao s blog post @xcite ) introduced the notion of `` slice rank '' for tensors . q - rank is basically a symmetric version of this . let @xmath27 be the locus of forms @xmath13 with @xmath28 . this is the image of the map @xmath29 the main theorem of @xcite implies that the domain of the above map is @xmath2-noetherian , and so , by standard facts ( see @xcite ) , its image @xmath27 is as well . it follows that any @xmath2-stable closed subset of @xmath0 of bounded q - rank is cut out by finitely many orbits of equations . theorem [ mainthm ] then follows from the following result : [ mainthm2 ] any @xmath2-stable subset of @xmath0 containing forms of arbitrarily high q - rank is zariski dense . to prove this theorem , one must show that if @xmath30 is a sequence in @xmath0 of unbounded q - rank then for any @xmath4 there is a @xmath26 such that the orbit - closure of @xmath31 projects surjectively onto @xmath32 . we prove a quantitative version of this statement : [ mainthm3 ] let @xmath11 have q - rank @xmath33 ( in fact , @xmath34 suffices ) , and suppose @xmath35 . then the orbit closure of @xmath13 surjects onto @xmath32 . the proof of this theorem is really the heart of the paper . the idea is as follows . suppose that @xmath36 has large q - rank . we establish two key facts . first , after possibly degenerating @xmath13 ( i.e. , passing to a form in the orbit - closure ) one can assume that the @xmath37 s and @xmath38 s are in separate sets of variables , while maintaining the assumption that @xmath13 has large q - rank . this is useful when studying the orbit closoure , as it allows us to move the @xmath39 s and @xmath40 s independently . second , we show that @xmath40 s have large rank in a very stong sense : namely , that within the linear span of the @xmath40 s there is a large - dimensional subspace such that every non - zero element of it has large rank . the results of @xcite then imply that the orbit closure of @xmath41 in @xmath42 surjects onto @xmath43 , and this yields the theorem . we now explain one source of motivation for question [ mainques ] . an * ideal invariant * is a rule that assigns to each homogeneous ideal @xmath44 in each standard - graded polynomial @xmath1-algebra @xmath45 ( in finitely many variables ) a quantity @xmath46 , such that @xmath47 only depends on the pair @xmath48 up to isomorphism . we say that @xmath49 is * cone - stable * if @xmath50}(i[x])=\nu_a(i)$ ] , that is , adjoining a new variable does not affect @xmath49 . the main theorem of @xcite is ( in part ) : [ thm : ess ] the following are equivalent : 1 . let @xmath49 be a cone - stable ideal invariant that is upper semi - continuous in flat families , and let @xmath51 be a tuple of non - negative integers . then there exists an integer @xmath52 such that @xmath47 is either infinite or at most @xmath52 whenever @xmath44 is an ideal generated by @xmath15 elements of degrees @xmath53 . ( crucially , @xmath52 does not depend on @xmath45 . ) 2 . for every @xmath54 as above , the space @xmath55 is @xmath56-noetherian . define an ideal invariant @xmath49 by taking @xmath47 to be the projective dimension of @xmath44 as an @xmath45-module . this is cone - stable and upper semi - continuous in flat families . the boundedness in theorem [ thm : ess](a ) for this @xmath49 is exactly stillman s conjecture , proved in @xcite . theorem [ thm : ess ] shows that question [ mainques ] is intimately connected to uniformity questions in commutative algbera in the style of stillman s conjecture . the results of @xcite are actually more precise : if part ( b ) holds for a single @xmath54 then part ( a ) holds for the corresponding @xmath54 . thus , combined with theorem [ mainthm ] , we obtain : [ thm : cubicinv ] let @xmath49 be a cone - stable ideal invariant that is upper semi - continuous in flat families . then there exists an integer @xmath52 such that @xmath57 is either infinite or at most @xmath52 , whenever @xmath44 is generated by a single cubic form . the following two consequences of theorem [ thm : cubicinv ] are taken from @xcite . given a positive integer @xmath58 there is an integer @xmath52 such that the following holds : if @xmath59 is a cubic hypersurface containing finitely many codimension @xmath58 linear subspaces then it contains at most @xmath52 such subspaces . given a positive integer @xmath58 there is an integer @xmath52 such that the following holds : if @xmath59 is a cubic hypersurface whose singular locus has codimension @xmath58 then its singular locus has degree at most @xmath52 . it would be interesting if these results could be proved by means of classical algebraic geometry . it would also be interesting to determine the bound @xmath52 for some small values of @xmath58 . in this section we put @xmath60 . our original motivation for considering question [ mainques ] came from the theory of twisted commutative algebras . recall that a * twisted commutative algebra * ( tca ) over the complex numbers is a commutative unital associative @xmath61-algebra @xmath45 equipped with a polynomial action of @xmath2 ; see @xcite for background . the easiest examples of tca s come by taking the symmetric algebra on a polynomial representation of @xmath2 : for example , @xmath62 or @xmath63 . tca s have appeared in several applications in recent years , for instance : * modules over the tca @xmath62 are equivalent to @xmath64-modules , as studied in @xcite . the structure of the module category was worked out in great detail in @xcite . * finite length modules over the tca @xmath63 are equivalent to algebraic representations of the infinite orthogonal group @xcite . * modules over tca s generated in degree 1 were used to study @xmath65-modules in @xcite , with applications to syzygies of segre embeddings . a tca @xmath45 is * noetherian * if its module category is locally noetherian ; explicitly , this means that any submodule of a finitely generated @xmath45-module is finitely generated . a major open question in the theory , first raised in @xcite , is : is every finitely generated tca noetherian ? so far , our knowledge on this question is extremely limited . for tca s generated in degrees @xmath66 ( or more generally , `` bounded '' tca s ) , noetherianity was proved in @xcite . ( it was later reproved in the special case of @xmath64-modules in @xcite . ) for the tca s @xmath63 and @xmath67 , noetherianity was proved in @xcite . no other cases are known . we remark that these known cases of noetherianity , limited though they are , have been crucial in applications . since noetherianity is such a difficult property to study , it is useful to consider a weaker notion . a tca @xmath45 is * topologically noetherian * if every radical ideal is the radical of a finitely generated ideal . the results of @xcite show that tca s generated in degrees @xmath68 are topologically noetherian . topological noetherianity of the tca @xmath69 is equivalent to the noetherianity of the space @xmath70 appearing in question [ mainques ] . thus theorem [ mainthm ] can be restated as : the tca @xmath71 is topologically noetherian . this is the first noetherianity result for an unbounded tca generated in degrees @xmath72 . using similar methods , we can prove the following result : the space @xmath73 is noetherian with respect to the action of the group @xmath74 , where @xmath75 denotes the completed tensor product . we plan to write a short note containing the proof . in [ sec : qrank ] we establish a number of basic facts about q - rank . in [ sec : proof ] we use these facts to prove the main theorem . finally , in [ sec : example ] , we compute the q - rank of the cubic in example [ ex1 ] . this example is not used in the proof of the main theorem , but we thought it worthwhile to include one non - trivial computation of our fundamental invariant . throughout we let @xmath1 be an algebraically closed field of characteristic @xmath6 . the symbols @xmath76 , @xmath77 , and @xmath78 will always denote @xmath1-vector spaces , perhaps infinite dimensional . we write @xmath79 for the space of degree @xmath4 polynomials on @xmath77 equipped with the zariski topology . precisely , we identify @xmath80 with the spectrum of the ring @xmath81 . when @xmath77 is infinite dimensional the elements of @xmath80 are certain infinite series and the functions on @xmath80 are polynomials in coefficients . whenever we speak of the orbit of an element of @xmath80 , we mean its @xmath82 orbit . we thank bhargav bhatt , jan draisma , daniel erman , mircea mustata , and steven sam for helpful discussions . in this section , we establish a number of basic facts about q - rank . throughout @xmath77 will denote a vector space and @xmath13 a cubic in @xmath83 . initially we allow @xmath77 to be infinite dimensional , but following proposition [ prop : inf ] it will be finite dimensional ( though this is often not necessary ) . our first result is immediate , but worthwhile to write out explicitly . [ prop : subadd ] suppose @xmath84 . then @xmath85 we defined q - rank from an algebraic point of view ( number of terms in a certain sum ) . we now give a geometric characterization of q - rank that can , at times , be more useful . [ prop : geom ] we have @xmath28 if and only if there exists a linear subspace @xmath78 of @xmath77 of codimension at most @xmath15 such that @xmath86 . first suppose @xmath28 , and write @xmath87 . then we can take @xmath88 . this clearly has the requisite properties . now suppose @xmath78 of codimension @xmath15 is given . let @xmath89 be a basis for @xmath78 , and complete it to a basis of @xmath77 be adding vectors @xmath90 . let @xmath91 be dual to @xmath92 . we can then write @xmath93 , where every term in @xmath94 uses one of the variables @xmath95 , and these variables do not appear in @xmath96 . since @xmath97 by assumption and @xmath98 by its definition , we find @xmath99 . but @xmath96 only uses the variables @xmath100 , and these are coordinates on @xmath78 , so we must have @xmath101 . thus every term of @xmath13 has one of the variables @xmath102 in it , and so we can write @xmath103 for appropriate @xmath104 , which shows @xmath28 . in the above proposition , @xmath97 means that the image of @xmath13 in @xmath105 is 0 . it is equivalent to ask @xmath106 for all @xmath107 . the next result shows that one does not lose too much q - rank when passing to subspaces . [ prop : qsubsp ] suppose @xmath108 has codimension @xmath4 . then for @xmath109 we have @xmath110 if @xmath87 then we obtain a similar expression for @xmath111 , which shows that @xmath112 . suppose now that @xmath113 , and let @xmath114 be a codimension @xmath15 subspace such that @xmath115 ( proposition [ prop : geom ] ) . then @xmath116 has codimension @xmath117 in @xmath77 , and so @xmath118 ( proposition [ prop : geom ] again ) . our next result shows that if @xmath77 is infinite dimensional then the q - rank of @xmath109 can be approximated by the q - rank of @xmath111 for a large finite dimensional subspace @xmath78 of @xmath77 . this will be used at a key juncture to move from an infinite dimensional space down to a finite dimensional one . [ prop : inf ] suppose @xmath119 ( directed union ) . then @xmath120 . we first give two lemmas . in what follows , for a finite dimensional vector space @xmath78 we write @xmath121 for the grassmannian of codimension @xmath15 subspaces of @xmath78 . for a @xmath1-point @xmath122 of @xmath121 , we write @xmath123 for the corresponding subspace of @xmath78 . by `` variety '' we mean a reduced scheme of finite type over @xmath1 . [ lem : grmap ] let @xmath108 be finite dimensional vector spaces , and let @xmath124 be a closed subvariety . suppose that for every @xmath1-point @xmath125 of @xmath126 the space @xmath127 has codimension @xmath15 in @xmath78 . then there is a unique map of varieties @xmath128 that on @xmath1-points is given by the formula @xmath129 . let @xmath130 be the scheme of all linear maps @xmath131 , and let @xmath132 be the open subscheme of surjective linear maps . we identify @xmath133 with the quotient of @xmath132 by the group @xmath134 . the quotient map @xmath135 sends a surjection to its kernel . let @xmath136 be the inverse image of @xmath126 . there is a natural map @xmath137 given by restricting . by assumption , every closed point of @xmath138 maps into @xmath139 under this map . since @xmath139 is open , it follows that the map @xmath140 factors through a unique map of schemes @xmath141 . since this map is @xmath134-equivariant , it descends to the desired map @xmath128 . if @xmath125 is a @xmath1-point of @xmath126 then it lifts to a @xmath1-point @xmath142 of @xmath138 , and the corresponding map @xmath143 has @xmath144 . the image of @xmath125 in @xmath121 is @xmath145 , which establishes the stated formula for our map . [ lem : invsys ] let @xmath146 be an inverse system of non - empty proper varieties over @xmath1 . then @xmath147 is non - empty . if @xmath60 then @xmath148 is a non - empty compact hausdorff space , and the result follows from the well - known ( and easy ) fact that an inverse limit of non - empty compact hausdorff spaces is non - empty . for a general field @xmath1 , we argure as follows . ( we thank bhargav bhatt for this argument . ) let @xmath149 be the zariski topological space underlying the scheme @xmath150 , and let @xmath126 be the inverse limit of the @xmath149 . since each @xmath149 is a non - empty spectral space and the transition maps @xmath151 are spectral ( being induced from a map of varieties ) , @xmath126 is also a non - empty spectral space ( * ? ? ? * lemma 5.24.2 , 5.24.5 ) . it therefore has some closed point @xmath125 . let @xmath152 be the image of @xmath125 in @xmath149 . we claim that @xmath152 is closed for all @xmath153 . suppose not , and let @xmath154 be such that @xmath155 is not closed . passing to a cofinal set in @xmath44 , we may as well assume @xmath156 is the unique minimal element . let @xmath157 be the residue field of @xmath152 , and let @xmath158 be the direct limit of the @xmath157 . the point @xmath152 is then the image of a canonical map of schemes @xmath159 . since @xmath155 is not closed , it admits some specialization , so we may choose a valuation ring @xmath160 in @xmath158 and a non - constant map of schemes @xmath161 extending @xmath162 . since @xmath150 is proper , the map @xmath163 extends uniquely to a map @xmath164 . by uniqueness , the @xmath165 s are compatible with the transition maps , and so we get an induced map @xmath166 extending the map @xmath167 . since @xmath168 is induced from @xmath165 , it follows that @xmath165 is non - constant . the image of the closed point in @xmath169 under @xmath165 is then a specialization of @xmath125 , contradicting the fact that @xmath125 is closed . this completes the claim that @xmath152 is closed . since @xmath152 is closed , it is the image of a unique map @xmath170 of @xmath1-schemes . by uniqueness , these maps are compatible , and so give an element of @xmath147 . first suppose that @xmath171 is finite dimensional for all @xmath153 . for @xmath172 we have @xmath173 by proposition [ prop : qsubsp ] , and so either @xmath174 or @xmath175 stabilizes . if @xmath174 then @xmath176 by proposition [ prop : qsubsp ] and we are done . thus suppose @xmath175 stabilizes . replacing @xmath44 with a cofinal subset , we may as well assume @xmath175 is constant , say equal to @xmath15 , for all @xmath153 . we must show @xmath177 . proposition [ prop : qsubsp ] shows that @xmath178 , so it suffices to show @xmath28 . let @xmath179 be the closed subvariety consisting of all codimension @xmath15 subspaces @xmath180 such that @xmath181 . this is non - empty by proposition [ prop : geom ] since @xmath182 has q - rank @xmath15 . suppose @xmath172 and @xmath125 is a @xmath1-point of @xmath183 , that is , @xmath184 is a codimension @xmath15 subspace of @xmath185 on which @xmath13 vanishes . of course , @xmath13 then vanishes on @xmath186 , which has codimension at most @xmath15 in @xmath171 . since @xmath182 has q - rank exactly @xmath15 , it can not vanish on a subspace of codimension less than @xmath15 ( proposition [ prop : geom ] ) , and so @xmath186 must have codimension exactly @xmath15 . thus by lemma [ lem : grmap ] , intersecting with @xmath171 defines a map of varieties @xmath187 . this maps into @xmath150 , and so for @xmath172 we have a map @xmath188 . these maps clearly define an inverse system . appealing to lemma [ lem : invsys ] we see that @xmath147 is non - empty . let @xmath189 be a point in this inverse limit , and put @xmath190 . thus @xmath191 is a codimension @xmath15 subspace of @xmath171 on which @xmath13 vanishes , and for @xmath172 we have @xmath192 . it follows that @xmath193 is a codimension @xmath15 subspace of @xmath77 on which @xmath13 vanishes , which shows @xmath28 ( proposition [ prop : geom ] ) . we now treat the general case , where the @xmath171 may not be finite dimensional . write @xmath194 with @xmath195 finite dimensional . then @xmath196 , so @xmath197 this completes the proof . for the remainder of this section we assume that @xmath77 is finite dimensional . if @xmath77 is @xmath4-dimensional then the q - rank of any cubic in @xmath83 is obviously bounded above by @xmath4 . the next result gives an improved bound , and will be crucial in what follows . [ prop : qbd ] suppose @xmath198 . then @xmath199 , where @xmath200 note that @xmath201 . let @xmath26 be the largest integer such that @xmath202 . then the hypersurface @xmath203 contains a linear subspace of dimension at least @xmath26 by ( * ? ? ? * lemma 3.9 ) . it follows from proposition [ prop : geom ] that @xmath204 . some simple algebra shows that @xmath205 . suppose that @xmath206 is a cubic . eventually , we want to show that if @xmath13 has large q - rank then its orbit under @xmath82 is large . for studying the orbit , it would be convenient if the @xmath37 s and the @xmath38 s were in separate sets of variables , as then they could be moved independently under the group . this motivates the following definition : we say that a cubic @xmath109 is * separable * if there is a direct sum decomposition @xmath207 and an expression @xmath206 with @xmath208 and @xmath209 . now , if we have a cubic @xmath13 of high q - rank we can not conclude , simply based on its high q - rank , that it is separable . fortunately , the following result shows that if we are willing to degenerate @xmath13 a bit ( which is fine for our ultimate applications ) , then we can make it separable , while retaining high q - rank . [ prop : srk ] suppose that @xmath109 has q - rank @xmath15 . then the orbit - closure of @xmath13 contains a separable cubic @xmath94 satisfying @xmath210 . let @xmath211 be a basis for @xmath212 . after possibly making a linear change of variables , we can write @xmath213 . write @xmath214 , where @xmath215 is homogeneous of degree @xmath153 in the variables @xmath216 . since @xmath217 has degree 3 in the variables @xmath216 , it can contain no other variables , and can thus be regarded as an element of @xmath218 . therefore , by proposition [ prop : qbd ] , we have @xmath219 . after possibly making a linear change of variables in @xmath102 , we can write @xmath220 for some @xmath221 . let @xmath222 ( resp . @xmath223 ) be the result of setting @xmath224 in @xmath13 ( resp . @xmath225 ) , for @xmath226 . we have @xmath227 by proposition [ prop : qsubsp ] . of course , @xmath228 , so @xmath229 . by subadditivity ( proposition [ prop : subadd ] ) , at least one of @xmath230 or @xmath231 has q - rank @xmath232 . we have @xmath233 where @xmath234 is a quadratic form in the variables @xmath235 with @xmath236 . thus @xmath237 , and @xmath238 , is separable . we have @xmath239 where @xmath240 is a linear form in the variables @xmath235 with @xmath236 . thus @xmath241 , and @xmath242 , is separable . to complete the proof , it suffices to show that @xmath230 and @xmath231 belong to the orbit - closure of @xmath13 , as we can then take @xmath243 or @xmath244 . it is clear that @xmath222 is in the orbit - closure of @xmath13 , so it suffices to show that @xmath230 and @xmath231 are in the orbit - closure of @xmath222 . consider the element @xmath245 of @xmath246 defined by @xmath247 then @xmath248 and @xmath249 . thus @xmath250 . a similar construction shows that @xmath231 is in the orbit - closure of @xmath222 . suppose that @xmath206 is a cubic of high q - rank . one would like to be able to conclude that the @xmath38 then have high ranks as well . we now prove two results along this line . for a linear subspace @xmath251 , we let @xmath252 be the maximum of the ranks of elements of @xmath253 , and we let @xmath254 be the minimum of the ranks of the non - zero elements of @xmath253 ( or 0 if @xmath255 ) . [ prop : maxrank ] suppose @xmath206 has q - rank @xmath15 , and let @xmath251 be the span of the @xmath38 . then for every subspace @xmath256 of @xmath253 we have @xmath257 we may as well assume that @xmath37 and @xmath38 are linearly independent . thus @xmath258 . let @xmath256 be a subspace of dimension @xmath259 . after making a linear change of variables in the @xmath40 s and @xmath39 s , we may as well assume that @xmath256 is the span of @xmath260 . let @xmath261 . we must show that @xmath262 . let @xmath263 have rank @xmath264 . choose a basis @xmath211 of @xmath212 so that @xmath265 . if some @xmath38 for @xmath266 had a term of the form @xmath267 with @xmath268 then some linear combination of @xmath38 and @xmath269 would have rank @xmath270 , a contradiction . thus every term of @xmath38 , for @xmath266 , has a variable of index @xmath271 , and so we can write @xmath272 where @xmath273 . but now @xmath274 where @xmath275 . this shows @xmath276 , which completes the proof . in our eventual application , it is actually @xmath277 that is more important than @xmath278 . fortuantely , the above result on @xmath278 automatically gives a result for @xmath277 , thanks to the following general proposition . [ prop : minrank ] let @xmath251 be a linear subspace and let @xmath15 be a positive integer . suppose that @xmath279 holds for all linear subspaces @xmath280 . let @xmath26 and @xmath281 be positive integers satisfying @xmath282 then there exists a @xmath26-dimensional linear subspace @xmath280 with @xmath283 . [ lem : minrank ] let @xmath284 be quadratic forms of rank @xmath285 . suppose there is a linear combination of the @xmath40 s that has rank at least @xmath264 . then there is a linear combination @xmath269 of the @xmath40 s satisfying @xmath286 . let @xmath287 be a linear combination of the @xmath40 s with @xmath288 and @xmath26 minimal . since @xmath289 , it follows that @xmath290 . thus if @xmath291 then @xmath292 would have rank @xmath293 , contradicting the minimality of @xmath26 . therefore @xmath294 . let @xmath295 be a basis for @xmath253 so that @xmath296 is lexicographically minimal . in particular , this implies that @xmath297 . if @xmath298 then lexicographic minimality ensures that any non - trivial linear combination of @xmath299 has rank at least @xmath281 , and so we can take @xmath256 to be the span of these forms . thus suppose that @xmath300 . in what follows , we put @xmath301 . note that @xmath302 . in fact , @xmath303 , and so @xmath304 . for @xmath305 , consider the following statement : * there exist linearly independent @xmath306 such that : ( i ) @xmath307 is a linear combination of @xmath308 ; ( ii ) @xmath309 ; and ( iii ) the span of @xmath306 has minrank at least @xmath281 . we will prove @xmath310 by induction on @xmath39 . of course , @xmath311 implies the proposition . first consider the case @xmath312 . the statement @xmath313 asserts that there exists a non - zero linear combination @xmath314 of @xmath315 such that @xmath316 . since the span of @xmath315 has codimension @xmath317 in @xmath253 , our assumption guarantees that some linear combination @xmath314 of these forms has rank at least @xmath281 . since each form has rank @xmath285 , lemma [ lem : minrank ] ensures we can find @xmath314 with @xmath318 . we now prove @xmath310 assuming @xmath319 . let @xmath320 be the tuple given by @xmath319 . the span of @xmath321 has codimension @xmath322 in @xmath253 , and so our assumption guarantees that some linear combination @xmath323 has rank at least @xmath324 . by lemma [ lem : minrank ] , we can ensure that this @xmath323 has rank at most @xmath325 . thus ( i ) and ( ii ) in @xmath310 are established . we now show that any non - trivial linear combination @xmath326 has rank at least @xmath281 , which will show that the @xmath314 s are linearly independent and establish ( iii ) in @xmath310 . if @xmath327 then the rank is at least @xmath281 by the assumption on @xmath320 . thus assume @xmath328 . we have @xmath329 since @xmath330 , we thus see that @xmath326 has rank at least @xmath281 , which completes the proof . proposition [ prop : minrank ] is not specific to ranks of quadratic forms : it applies to any subadditive invariant on a vector space . combining the propositions [ prop : maxrank ] and [ prop : minrank ] , we obtain : [ cor : minrank ] suppose @xmath206 has q - rank @xmath15 , let @xmath253 be the span of the @xmath38 s , and let @xmath26 and @xmath281 be positive integers such that holds . then there exists a @xmath26-dimensional linear subspace @xmath280 with @xmath283 . we now prove the main theorems of the paper . we require the following result ( see ( * ? ? ? * proposition 3.3 ) and its proof ) : [ thm : deg2 ] let @xmath122 be a point in @xmath331 , with @xmath77 finite dimensional . write @xmath122 as @xmath332 , and let @xmath251 be the span of the @xmath38 . let @xmath78 be a @xmath4-dimensional subspace of @xmath77 . suppose that @xmath333 are linearly independent and that @xmath334 . then the orbit - closure of @xmath122 surjects onto @xmath335 . we begin by proving an analog of the above theorem for @xmath83 : [ thm : surj ] suppose @xmath77 is finite dimensional . let @xmath109 have q - rank @xmath15 and let @xmath78 be a @xmath4-dimensional subspace of @xmath77 with @xmath336 then the orbit - closure of @xmath13 surjects onto @xmath105 . applying proposition [ prop : srk ] , let @xmath94 be a separable cubic in the orbit - closure of @xmath13 satisfying @xmath210 . write @xmath337 where @xmath208 and @xmath209 and @xmath207 and the @xmath39 s and @xmath40 s are linearly independent . let @xmath253 be the span of the @xmath40 s . put @xmath338 and @xmath339 . note that @xmath340 by corollary [ cor : minrank ] we can therefore find a @xmath339 dimensional subspace @xmath256 of @xmath253 with @xmath283 . making a linear change of variables , we can assume @xmath256 is the span of @xmath341 . let @xmath342 . this is in the orbit - closure of @xmath94 ( and thus @xmath13 ) since it is obtained by setting @xmath343 for @xmath344 . it is crucial here that the @xmath40 s and @xmath39 s are in different sets of variables , so that setting some @xmath39 s to 0 does not change the @xmath40 s . by theorem [ thm : deg2 ] , the orbit closure of @xmath345 in @xmath346 surjects onto @xmath347 . now let @xmath348 . since @xmath349 we can write @xmath350 with @xmath351 and @xmath352 . pick @xmath353 such that @xmath354 in the image of @xmath355 . then @xmath96 is the image of @xmath356 , which completes the proof . suppose that @xmath109 has q - rank @xmath34 and let @xmath78 be a subspace of @xmath77 of dimension @xmath4 with @xmath357 then the orbit - closure of @xmath13 surjects onto @xmath105 . by definition of @xmath358 , we have @xmath359 ( for an integer @xmath360 ) if and only if @xmath361 . thus the condition in the theorem is equivalent to @xmath362 , where @xmath363 is the left side of the inequality in the theorem . this expression is equal to @xmath364 plus lower order terms , and is therefore less than @xmath365 for @xmath366 ; in fact , @xmath367 is sufficient . thus for @xmath367 it is enough that @xmath368 ; since @xmath369 , it is enough that @xmath370 . thus for @xmath371 , the orbit closure of @xmath13 surjects onto @xmath105 . but it obviously then surjects onto smaller subspaces as well , so we only need to assume @xmath372 . let @xmath77 be infinite dimensional . suppose @xmath373 is zariski closed , @xmath82-stable , and contains elements of arbitrarily high q - rank . then @xmath374 . it suffices to show that @xmath126 surjects onto @xmath105 for all finite dimensional @xmath108 . thus let @xmath78 of dimension @xmath4 be given . let @xmath15 be sufficiently large so that the inequality in theorem [ thm : surj ] is satisfied and let @xmath375 have q - rank at least @xmath15 . by proposition [ prop : inf ] , there exists a finite dimensional subspace @xmath376 of @xmath77 containing @xmath78 such that @xmath377 has q - rank at least @xmath15 . theorem [ thm : surj ] implies that the orbit - closure of @xmath377 surjects onto @xmath105 . since @xmath126 surjects onto the orbit closure of @xmath377 , the result follows . it was explained in the introduction how this implies theorem [ mainthm ] , so the proof is now complete . fix a positive integer @xmath5 , and consider the cubic @xmath378 in the polynomial ring @xmath379_{1 \le i \le n}$ ] introduced in example [ ex1 ] . we now show : it is clear that @xmath380 . to prove equality , it suffices by proposition [ prop : geom ] to show that @xmath381 if @xmath77 is a codimension @xmath382 subspace of @xmath383 . this is exactly the content of the following proposition : arrange the given elements in a matrix as follows : @xmath387 note that we are free to permute the rows and apply permutations within a row without changing the value of @xmath13 , e.g. , we can switch the values of @xmath388 and @xmath389 , or switch @xmath390 with @xmath391 , without changing @xmath13 . we now proceed to find a basis for @xmath77 among the elements in the matrix according to the following three - phase procedure . _ phase 1 . _ find a non - zero element of the matrix , and move it ( using the permutations mentioned above ) to the @xmath388 position . now in rows @xmath392 find an element that is not in the span of @xmath388 ( if one exists ) and move it to the @xmath393 position . now in rows @xmath394 find an element that is not in the span of @xmath388 and @xmath393 ( if one exists ) and move it to the @xmath395 position . continue in this manner until it is no longer possible ; suppose we go @xmath15 steps . at this point , @xmath95 are linearly independent , and @xmath235 , @xmath396 , and @xmath152 , for @xmath397 all belong to their span . _ phase 2 . _ from rows @xmath398 find an element in the second or third column not in the span of @xmath95 and move it ( using permutations that fix the first column ) to the @xmath389 position . next from rows @xmath399 find an element in the second or third column not in the span of @xmath400 and move it to the @xmath401 position . continue in this manner until it is no longer possible ; suppose we go @xmath281 steps . at this point , @xmath402 form a linearly independent set , and the elements @xmath403 for @xmath404 belong to their span . the conclusion from phase 1 still holds as well . _ phase 3 . _ now carry out the same procedure in the third column . that is , from rows @xmath405 find an element in the third column not in the span of @xmath402 and move it ( by permuting rows ) to the @xmath406 position . then from rows @xmath407 find an element in the third column not in the span of @xmath408 and move it to the @xmath409 position . continue in this manner until it is no longer possible ; suppose we go @xmath264 steps . at this point , @xmath410 forms a basis of @xmath77 . the conclusions from phases 1 and 2 still hold . for clarity , we write @xmath411 for our basis . we note that because @xmath412 we must have @xmath413 . the ring @xmath414 is identified with the polynomial ring in the @xmath415 , @xmath416 , @xmath126 variables . we now determine the coefficient of @xmath417 in @xmath418 . if @xmath236 then @xmath419 has degree 3 in the @xmath415 variables , and so the coefficient is 0 . if @xmath420 then @xmath419 has degree 0 in the @xmath126 variables , and so again the coefficient is 0 . finally , suppose that @xmath421 . then @xmath422 . the only way this can contain @xmath423 is if @xmath424 . we thus see that the coefficient of @xmath423 in @xmath419 is 0 except for @xmath424 , in which case it is 1 , and so @xmath425 is non - zero .	let @xmath0 be the space of complex cubic polynomials in infinitely many variables over the field @xmath1 . we show that this space is @xmath2-noetherian , meaning that any @xmath2-stable zariski closed subset is cut out by finitely many orbits of equations . our method relies on a careful analysis of an invariant of cubics we introduce called q - rank . this result is motivated by recent work in representation stability , especially the theory of twisted commutative algebras . it is also connected to uniformity problems in commutative algebra in the vein of stillman s conjecture .
the hadronic particle - antiparticle correlation was already pointed out in the beginning of the nineties . however , the final formulation of these hadronic squeezed or back - to - back correlations was proposed only at the end of that decade@xcite , predicting that such correlations were expected if the masses of the mesons were modified in the hot and dense medium formed in high energy nucleus - nucleus collisions . soon after that , it was shown that analogous correlations would exist in the case of baryons as well . an interesting theoretical finding was that both the fermionic ( fbbc ) and the bosonic ( bbbc ) back - to - back correlations were very similar , both being positive and treated by analogous formalisms . in what follows , we will focus our discussion to the bosonic case , illustrating the effect by considering @xmath0 and @xmath1 pairs , considered to be produced at rhic energies@xcite . let us discuss the case of @xmath3-mesons first , which are their own antiparticles , and suppose that their masses are modified in hot and dense medium . naturally , they recover their asymptotic masses after the system freezes - out . therefore , the joint probability for observing two such particles , i.e. , the two - particle distribution , @xmath4 , can be factorized as @xmath5 $ ] , after applying a generalization of wick s theorem for locally equilibrated systems@xcite . the first term corresponds to the product of the spectra of the two @xmath3 s , @xmath6 , being @xmath7 and @xmath8 the free - particle creation and annihilation operators of scalar quanta , and @xmath9 means thermal averages . the second term contains the identical particle contribution and is represented by the square modulus of the chaotic amplitude , @xmath10 . together with the first term , it gives rise to the femtoscopic or hanbury - brown & twiss ( hbt ) effect . the third term , the square modulus of the squeezed amplitude , @xmath11 , is identically zero in the absence of in - medium mass - shift . however , if the particle s mass is modified , together with the first term it leads to the squeezing correlation function . the annihilation ( creation ) operator of the asymptotic , observed bosons with momentum @xmath12 , @xmath13 ( @xmath14 ) , is related to the in - medium annihilation ( creation ) operator @xmath15 ( @xmath16 ) , corresponding to thermalized quasi - particles , by the bogoliubov - valatin transformation , @xmath17 , where @xmath18 , @xmath19 . the argument , @xmath20 $ ] , is the _ squeezing parameter_. in terms of the above amplitudes , the complete @xmath0 correlation function can be written as c_2(k_1,k_2 ) = 1 + + , [ fullcorr ] where the first two terms correspond to the identical particle ( hbt ) correlation , whereas the first and the last terms represent the correlation function between the particle and its antiparticle , i.e. , the squeezed part . the in - medium modified mass , @xmath21 , is related to the asymptotic mass , @xmath22 , by @xmath23 , here assumed to be a constant mass - shift . the formulation for both bosons and fermions was initially derived for a static , infinite medium @xcite . more recently , it was shown@xcite in the bosonic case that , for finite - size systems expanding with moderate flow , the squeezed correlations may survive with sizable strength to be observed experimentally . similar behavior is expected in the fermionic case . in that analysis , a non - relativistic treatment with flow - independent squeezing parameter was adopted for the sake of simplicity , allowing to obtain analytical results . the detailed discussion is in ref . @xcite , where the maximum value of @xmath24 , was studied as a function of the modified mass , @xmath21 , considering pairs with exact back - to - back momentum , @xmath25 ( in the identical particle case , this procedure would be analogous to study the behavior of the intercept of the hbt correlation function ) . although illustrating many points of theoretical interest , this study in terms of the unobserved shifted mass and exactly back - to - back momenta was not helpful for motivating the experimental search of the bbc s . a more realistic analysis would involve combinations of the momenta of the individual particles , @xmath26 , into the average momentum of the pair , @xmath27 . since the maximum of the bbc effect is reached when @xmath28 , this would correspond to investigate the squeezed correlation function , @xmath29 , close to @xmath30 . for a hydrodynamical ensemble , both the chaotic and the squeezed amplitudes , @xmath31 and @xmath32 , respectively , can be written in a special form derived in @xcite and developed in @xcite . therefore , within a non - relativistic treatment with flow - independent squeezing parameter , the squeezed amplitude is written as in @xcite , i.e. , @xmath33 + 2 n^_0 r_^3 \exp\bigl[-\frac{(\mathbf{k}_1-\mathbf{k}_2)^2}{8 m _ * t}\bigr ] \exp \bigl[-\frac{im\langle u\rangle r(\mathbf{k_1 } + \mathbf{k_2})^2}{2 m _ * t_}\bigr ] \exp\bigl[- \bigl ( \frac{1}{8 m _ t _ * } + \frac{r_^2}{2 } \bigr ) ( \mathbf{k_1 } + \mathbf{k_2})^2\bigr ] \bigl\ } $ ] , and the spectrum , as @xmath34 , where @xmath35 and @xmath36 @xcite . we adopt here @xmath37 . inserting these expressions into eq . ( [ fullcorr ] ) and considering the region where the hbt correlation is not relevant , we obtain the results shown in figure 1 . part ( a ) shows the squeezed correlation as a function of @xmath38 , for several values of @xmath39 . the top plot shows results expected in the case of a instant emission of the @xmath0 correlated pair . if , however , the emission happens in a finite interval , the second term in eq . ( [ fullcorr ] ) is multiplied by a reduction factor , in this case expressed by a lorentzian ( @xmath40 ^ -1 $ ] ) , i.e. , the fourier transform of an exponential emission . the result is shown in the plot in the middle of figure 1(a ) . we see that this represents a dramatic reduction in the signal , even though its strength is sizable for being observed experimentally . if the system expands with radial flow ( @xmath41 ) , the result is shown in the plot at the bottom of figure 1(a ) , again considering that the @xmath3 s are emitted during a finite period of time , @xmath42 fm / c . we see that , in the absence of flow , the squeezed correlation signal grows faster for higher values @xmath43 than the corresponding case in the presence of flow . however , this last one is stronger in all the investigated @xmath43 region , showing that the presence of radial flow enhances the signal . the sensitivity of the squeezed - pair correlation to the size of the region where the mass - shift occurs is shown in figure1(b ) for two values of radii , @xmath44 fm and @xmath45 fm , keeping @xmath46 gev / c fixed . the differences are reflected in the inverse width of the curves , plotted as a function of @xmath47 . in case of no in - medium mass modification , the squeezed correlation functions would be unity for all values of @xmath48 in both plots . fm ( top ) and @xmath45 fm ( bottom).,title="fig : " ] fm ( top ) and @xmath45 fm ( bottom).,title="fig : " ] in the case of the squeezed correlations of @xmath1 pairs , we show in figure 2(a ) results for the generated momenta of the pairs within the narrow interval @xmath49 mev / c , by plotting the squeezed correlation , @xmath50 versus @xmath21 and @xmath51 . for the kaons , we can fixe the value of the shifted mass to be @xmath52 mev , corresponding to one of the maxima in figure 2(a ) , and then procedure similarly to what was done in the @xmath0 case . the result is shown in figure 3 of ref.@xcite . also in this case the intensity of the squeezed correlation would be large enough to be searched for experimentally . next , we investigate how the behavior of the identical particle correlations could be affected in case of in - medium mass modification , since the femtoscopic correlation function also depends on the squeezing factor , @xmath53 . the hbt correlation function is obtained by inserting the chaotic amplitude , @xmath54 + n^_0 r_^3 ( \|c_{_0}\|^2+\|s_{_0}\|^2 ) \exp\bigl[-\frac{(\mathbf{k_1}+\mathbf{k}_2)^2}{8 m _ t_}\bigr ] \exp\bigl[-\bigl(\frac{im\langle u\rangle r}{2 m _ t_}\bigr)(\mathbf{k}_1 ^ 2-\mathbf{k}_2 ^ 2)\bigr ] \exp\bigl[-\bigl ( \frac{1}{8 m _ t } + \frac{r^2_}{2}\bigr)(\mathbf{k_1}-\mathbf{k_2})^2\bigr ] \bigr\ } , $ ] together with the expression for the spectrum , into eq.([fullcorr ] ) . we use the case of identical @xmath55 pairs as illustration , as seen in figure 2(b ) . the investigation is extended to both the cases of instant emission ( @xmath56 ) and finite emission ( @xmath57=2fm / c ) . in this figure , we can see the well - known result corresponding to the narrowing of the femtoscopic correlation function with increasing emission times , as well as the broadening the curve with flow in the absence of squeezing , as expected . however , if the squeezing originated in the mass - shift is present , its effects tend to oppose to those of flow ( for large @xmath58 , it practically cancels the broadening of the correlation function due to flow ) , another striking indication of mass - modification , even in hbt ! in the present work we suggest an effective way to search for the back - to - back squeezed correlations in heavy ion collisions at rhic , and later at lhc energies , by investigating the squeezed correlation function , @xmath29 , in terms of @xmath59 , for different values of @xmath60 . we showed that , in the presence of flow , the signal is stronger over the momentum regions analyzed in the plots , suggesting that flow may help to effectively discover the bbc signal experimentally . another important point that we find , within this simplified model and in the non - relativistic limit considered here , is that the squeezing would distort significantly the hbt correlation function as well , tending to oppose to the flow effects on those curves , practically neutralizing it for large values of @xmath58 . the analysis in terms of the variable @xmath61 would not be suited for a genuine relativistic treatment . in this case , however , a momentum variable could be constructed , as @xmath62 . in fact , it would be preferable to redefine this variable as @xmath63 , whose non - relativistic limit is @xmath64 , as discussed in ref.@xcite . finally , it is important to emphasize that all the effects and signals discussed here would exist only if the particles analyzed had their masses modified by interactions in the hot and dense medium . m. asakawa m , t. csrg and m. gyulassy _ phys . lett . _ 83 * , 4013 ( 1999 ) . p. k. panda , t. csrg , y. hama , g. krein and sandra s. padula , _ phys b * 512 * , 49 ( 2001 ) . sandra s , padula , y. hama , g. krein , p. k. panda and t. csrg , _ phys . _ c * 73 * , 044906 ( 2006 ) . m. gyulassy , s. k. kaufmann , and l. w. wilson , phys . c * 20 * , 2267 ( 1979 ) . a. makhlin and yu . sinyukov , _ sov . phys . _ * 46 * , 354 ( 1987 ) ; yu . sinyukov , _ nucl . phys . _ a * 566 * , 589c ( 1994 ) . sandra s. padula , o. socolowski jr . , t. csrg , m. i. nagy , proc . quark matter 2008 , _ j. phys . g : nucl . part . phys . _ * 35 * , 104141 ( 2008 ) . sandra s. padula , o. socolowski jr . and danuce m. dudek , arxiv:0812.1784v1[nucl - th ] , to be published in the proc . of the xxxviii international symposium on multiparticle dynamics ( ismd 2008 )	a novel type of correlation involving particle - antiparticle pairs was found out in the 1990 s . currently known as squeezed or back - to - back correlations ( bbc ) , they should be present if the hadronic masses are modified in the hot and dense medium formed in high energy heavy ion collisions . although well - established theoretically , such hadronic correlations have not yet been observed experimentally . in this phenomenological study we suggest a promising way to search for the bbc signal , by looking into the squeezed correlation function of @xmath0 and @xmath1 pairs at rhic energies , as function of the pair average momentum , @xmath2 . the effects of in - medium mass - shift on the identical particle correlations ( hanbury - brown & twiss effect ) are also discussed .
the formation and evolution of galaxies remains one of the big questions in astronomy . in the currently favoured @xmath2cdm model , galaxies are built up over time via the accretion of smaller systems ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the picture is not wholly satisfactory and some parts of the milky way may have formed in an initial collapse of baryonic material , somewhat akin to the model of galaxy formation proposed by @xcite . one firm prediction of the @xmath2cdm model is that this accretion of smaller systems should still be ongoing and that the milky way halo should contain a large number of satellite systems . it has been suggested that , given the model , there are too few satellites actually within the milky way halo @xcite , although the extent of this discrepancy has been a matter of some debate . many attempts have been made to resolve this issue ( e.g. * ? ? ? * ) by correcting the measured velocity dispersion of the local satellites to a velocity dispersion of the dark matter components , while more recently @xcite used numerical methods to highlight a scenario whereby these low mass systems are evaporated at the epoch of re - ionization . in the meantime , more and more surveys are probing the halo of the milky way revealing structures which may point to the formation history of our galaxy . the tidal dismemberment of a dwarf galaxy as it falls through the milky way halo is a slow process , with extensive streams of tidal debris existing for long periods of time @xcite . while ancient remnants have been identified , via phase - space analysis , in our own galactic neighbourhood @xcite , more extensive surveys of the galactic halo , such as the spaghetti survey @xcite and utilizing 2mass @xcite have concluded that there is only a single , major on - going accretion event , that of the sagittarius dwarf galaxy @xcite . while this accretion event is adding mass to the galactic halo and provides an important probe of the shape of the dark matter potential ( e.g. * ? ? ? * ; * ? ? ? * ) , the lack of other other major accretion events is somewhat disconcerting given the predictions from @xmath2cdm . as will be discussed in detail in @xmath3[ring ] , the recently discovered monoceros ring ( mri ) can be interpreted as an additional on - going accretion event within the milky way . investigating the density and extent of this structure is important when trying to fully understand the impact this type of event is having on the evolution of our galaxy both in the past and into the future . if the mri is instead the outermost edge of the milky way , mapping the outer reaches of the disk will provide insight into the milky way s past . to this end , we have used the isaac newton telescope wide field camera to continue a campaign to detect this stellar population around the galactic plane mapping out the extent of the mri . this paper presents the results of a wide - field camera survey of the extensive stellar population thought to represent a continuation of the previous work in this field . the layout of the paper is as follows ; @xmath3[ring ] summarises the extant knowledge of the mri and the associated population of stars while @xmath3[obs ] describes the observational procedure and data reduction . @xmath3[analysis ] describes the analysis procedure , and the conclusions of this study are presented in @xmath3[conclusions ] . the first sign of a new structure in the galaxy , the monoceros ring , came from a study of colour selected f - stars drawn from commissioning data from the sloan digital sky survey [ sdss , @xcite ] . obtained in a narrow strip around the celestial equator these revealed two major stellar overdensities in the galactic halo , consistent with an intersection of the streams of tidal debris torn from the sagittarius dwarf galaxy @xcite . accompanying these , however , was an additional significant overdensity in the direction of the galactic anti - centre , interpreted as being another tidal stream about the galaxy , just past the edge of the stellar disk , at a galactocentric distance of @xmath4kpc and with a thickness of @xmath5kpc , covering over @xmath6 of sky within @xmath7 of the galactic equator . given the results of the study of the sdss , @xcite searched for the signature of this stellar population in fields obtained for the isaac newton telescope wide field camera ( int / wfc ) survey ( see * ? ? ? identifying the stream as a distinct population in colour - magnitude diagrams ( cmds ) , this study found the stars to be extensively distributed ; over 100@xmath0 of the sky within @xmath8 of the galactic equator , suggesting that the stream completely rings the galaxy . main sequence fitting reveals that the galactocentric distance to the stream varies between @xmath9kpc and @xmath10kpc , with an apparent scale - height of @xmath11kpc . @xcite suggested that this stream represents debris of an accreting dwarf galaxy , although pointed out that the extant data did not rule out alternatives such as an outer spiral arm or unknown warp / flare of the galactic disk . in fact , they favoured the interpretation of it being a perturbation of the disc , possibly the result of ancient warps . simultaneously , @xcite presented an analysis of a larger sdss catalog of halo stars obtaining a number of radial velocities in several fields over the stream , finding a velocity dispersion of @xmath12km / s , inconsistent with any known galactic component . furthermore , these kinematics indicate that the stream possesses a prograde orbit about the milky way with a circular velocity of @xmath13km / s [ note : this revised value was presented in @xcite ] . metallicity estimates from these spectra indicate the stars are relatively metal poor @xmath14\sim-1.6)$ ] . @xcite also concluded the stream represents a cannibalized dwarf galaxy undulating about the edge of the galactic disk . several other studies have focused upon this stream population ; using the 2-micron all sky survey ( 2mass ) @xcite identified m - giant stars beyond the disk of the milky way , consistent with the population detected in the optical . again , at a galactocentric distance ( r@xmath15 ) of @xmath16kpc , this arc of stars covers @xmath17 , with a higher metallicity than previously determined @xmath14=-0.4\pm0.3)$ ] . @xcite extended this study , obtaining stellar velocities of m - giants selected from 2mass . confirming a velocity dispersion of @xmath10km / s , this study concluded the stream orbits the milky way in a prograde fashion on an orbit with very little eccentricity . while this may seem problematic for accretion models preferring elliptical orbits , numerical simulations by @xcite suggest that tidal streams in the plane of the milky way can possess quite circular orbits ; these numerical studies , however , also suggest that the accretion event must be young , less than @xmath18gyr since its first perigalactic passage , or any coherent structure would have dissolved . finally , @xcite noted five globular clusters which are apparently associated with the stellar stream , as well as @xmath9 outer , old stellar clusters , bolstering the argument that it represents an accreting dwarf galaxy . @xcite also employed the 2mass catalogue to identify m - giants beyond the disk of the milky way . by considering the projected density of these stars , this study uncovered north - south anisotropies in the density of m - stars , with arcs above and below the plane of the galaxy . significantly , @xcite identified a strong overdensity of these stars , @xmath19 of them in roughly 20@xmath0 of the sky at @xmath20 , the constellation of canis major ( cma ) . this is a similar number of m - stars to that seen in the sagittarius dwarf galaxy and @xcite similarly interpret this population of stars as a dwarf galaxy . it is probably the progenitor , with an original mass of @xmath21 , of the extensive stream of stars around the edge of the disk of the milky way . given its mass , if cma does represent an equatorial accretion event , it will , when fully dissolved , increase the mass of the thick disk by @xmath22 . additional evidence for this accretion interpretation of the cma dwarf comes from @xcite who found the signature of the main body of the dwarf in the background to several galactic open clusters , with the analysis of this population suggesting that it is somewhat metal rich with an age of @xmath23 gyrs , although a blue plume indicates a younger population . this study also identified several globular and open clusters associated with the dwarf . recently , @xcite have shown that the globular clusters associated with cma possess their own age - metallicity relationship which is distinct from that of the galactic population . furthermore , the clusters are smaller than expected , if drawn from the galactic population , strengthening the interpretation that they are of non - galactic origin . , by re - analysing the 2mass data claim the overabundance of m - giants in cma is simply a signature of the warp in the milky way . in response , @xcite used 2df data of the canis major region to highlight the velocity disparities between the milky way disk stars and the m - giant overdensity stars . this has been repeated more recently by @xcite suggesting the dwarf galaxy cma can now be tentatively accepted as a real entity despite the current debate . connecting cma to the mri is more problematic due to gaps in the detection of the ring and poor kinematic knowledge . a recently completed 2df kinematic survey @xcite of the canis major region may yield results in this area although currently any general conclusions linking it with the mri are speculative . more information about halo substructure has been recently discovered as @xcite have identified another structure in triangulum - andromedae ( triand ) which extends much further out than the mri . currently it is not known whether the two are related , although interestingly the detection of the mri by @xcite resides at the edge of this new structure in triand suggesting a connection . the latest evidence supporting the interpretation of the canis major dwarf galaxy has been presented by @xcite , showing the colour - magnitude diagram of a region ( 0.5@xmath0 x 0.5@xmath0 ) centred on ( _ l_,_b _ ) = ( 240,-8)@xmath0 , the overdensity proposed by @xcite . an upper limit to the heliocentric distance of the galaxy is found to be [email protected] to [email protected] kpc ; this is comparable with previous estimates . by measuring the surface brightness and total luminosity of the dwarf galaxy an estimate of the mass range is found to be 1.0@xmath25 m@xmath26 @xmath27 5.5@xmath2810@xmath29 . the tightness of the main sequence they found contradicts the claims that the canis major overdensity is the signature of the galactic warp . @xcite have completed extensive modelling of the mri and while not conclusively establishing a connection between canis major and the mri , their results are highly suggestive of such a link . @xcite using all of the current information known about the mri , undertook thousands of simulations , prograde and retrograde , in an attempt to find a model which best fitted the data . while some retrograde models were marginally acceptable , their preferred model was a prograde orbit which includes multiple wraps of the milky way . this allows for the scenario that the tidal stream has both near and far components . @xmath3[123m19 ] discusses a field in which this phenomenon appears . if the dwarf galaxy has completed more than one orbit then the stream must be a much older structure than previously assumed [ cf . @xcite ] , raising the possibility that the newly discovered triand structure @xcite could be the distant arm of a multiply wrapped tidal stream . the model of @xcite and the continuing work of those studying this new galactic feature are slowly piecing together this structure , although currently there is no real coverage around the entire galactic plane , these observations hopefully extend the knowledge of the mri and it s potential progenitor the canis major dwarf . the data was obtained on the isaac newton telescope wide field camera ( int / wfc ) at roque de los muchachos in la palma , canary islands . mounted at the telescope prime focus , this covers 0.29 square degrees field per pointing , imaging onto four 4k@xmath282k ccds . these possess 13.5@xmath30 m pixels , corresponding to a pixel scale of 0.33 arcsec per pixel . nine survey regions were chosen , roughly equally spaced between _ _ = @xmath31 . to aid in determining the location of the fields , the model of @xcite ( see figure [ figaitoff ] ) was used to predict where we might expect the spread of debris from an equatorial accretion . for each galactic longitude , two regions symmetrically placed above and below the galactic plane , were imaged . each target region is a composite of overlapping fields , the number of which depended on time available to observe in each region ( details of the observations are summarised in table [ obstable ] and presented graphically in figure [ figaitoff ] ) , but typically with a total area of @xmath5 square degrees . with the representative integration times , the limiting magnitudes are on average for these observations @xmath32 23.3 , @xmath33 22.2 , @xmath34 23.8 and @xmath35 22.8 . archival data of ( 123,-19)@xmath0 , taken from the m31 survey @xcite , is used as the opposing field to ( 118,+16)@xmath0 . de - biassing and trimming , vignetting correction , astrometry and photometry were all undertaken with the casu data reduction pipeline @xcite . the flat fielding employed a master twilight flat generated over the entire observing run . each star was individually extinction corrected using the dust_getval.c program supplied by schlegelschlegel / dust / data / data.html ] which interpolates the extinction using the maps presented by @xcite . observing several standard fields per night allows the calibration of the photometry to be determined , as described by @xcite , deriving the ccd zero - points . these are consistent to within a few percent on photometric nights . the data reduction pipeline produces a catalogue of all images in each colour - band . rejecting non - stellar images , the catalogues can be cross - correlated and the colour for individual stars can be determined . near the limiting magnitude , however , galaxies can appear stellar and so galactic contamination is expected for the faintest sources . [ cols=">,^,^,^,^,^,^,^ , < " , ] [ disttable ] bcc would like to thank his wife , lll , for kindly supplementing his scholarship income , the university of sydney for the university postgraduate award and the cambridge astronomical survey unit at cambridge university and mike irwin for their hospitality during my week there . bcc would also like to thank jorge pearrubia , for access to his monoceros ring model and the anonymous referee for their many helpful suggestions . gfl acknowledges the support of the discovery project grant dp0343508 . the research of amnf has been supported by a marie curie fellowship of the european community under contract number hpmf - ct-2002 - 01758 . gfl would also like to thank triple j for their chillout session on sunday mornings which drowns out his fighting children and also their three hours of power in case they choose to fight between the hours of 11 and 1 at night .	we present the results of a wide - field camera survey of the stars in the monoceros ring , thought to be an additional structure in the milky way of unknown origin . lying roughly in the plane of the milky way , this may represent a unique equatorial accretion event which is contributing to the thick disk of the galaxy . alternatively , the monoceros ring may be a natural part of the disk formation process . with ten pointings in symmetric pairs above and below the plane of the galaxy , this survey spans 90 degrees about the milky way s equator . signatures of the stream of stars were detected in three fields , ( _ l_,_b _ ) = ( 118,+16)@xmath0 and ( 150,+15)@xmath0 plus a more tentative detection at ( 150,-15)@xmath0 . galactocentric distance estimates to these structures gave @xmath117 , @xmath117 , and @xmath113 kpc respectively . the monoceros ring seems to be present on both sides of the galactic plane , in a form different to that of the galactic warp , suggestive of a tidal origin with streams multiply wrapping the galaxy . a new model of the stream has shown a strong coincidence with our results and has also provided the opportunity to make several more detections in fields in which the stream is less significant . the confirmed detection at ( _ l_,_b _ ) = ( 123,-19)@xmath0 at @xmath114 , kpc from the galactic centre allows a re - examination revealing a tentative new detection with a galactocentric distance of @xmath121 kpc . these detections also lie very close to the newly discovered structure in triangulum - andromedae hinting of a link between the two . the remaining six fields are apparently non - detections although in light of these new models , closer inspection reveals tentative structure . with the overdensity of m - giant stars in canis major being claimed both as a progenitor to the monoceros ring and alternatively a manifestation of the milky way warp , much is still unknown about this structure and its connection to the monoceros ring . further constraints are needed for the numerical simulations to adequately resolve the increasingly complex view of this structure . latexl-.36em.3ex-.15em t-.1667em.7ex-.125emx [ section ] [ firstpage ] galaxy : formation galaxy : structure galaxies : interactions
the phenomenon of localization manifests itself in many quantum mechanical systems ranging from the localization of light in a random medium @xcite , to anderson localization of an electronic wave @xcite and to atoms in time - dependent laser fields @xcite . in all these cases the underlying classical system is chaotic and shows diffusion as a function of time . in contrast , the quantum mechanical counterpart has a localized wave function whose width is governed by the classical diffusion and planck s constant @xcite . in the present paper we show that there exists an additional quantum parameter that controls the localization length . in the system of a two - level ion stored in a paul trap @xcite and interacting with a standing wave it is the detuning between the transition frequency of the ion and the laser frequency . a recent paper @xcite has shown that a stored ion moving along a far - detuned standing wave shows localization in position and momentum variables . in this work we have neglected the internal structure of the ion . we have therefore only considered the quantum dynamics of a particle moving in a one - dimensional time - dependent potential . in the present paper we extend this analysis and consider the motion of an ion taking into account its internal dynamics . our paper is organized as follows : in sec . [ 2 ] we first summarize the essential ingredients of the problem . in particular , we introduce the relevant equations and describe the methods of solution for the classical and quantum mechanical equations of motion . we also define the quantities which we calculate such as the position and momentum distributions and their moments . in sec . [ 3 ] we discuss the dependence of the classical and quantum mechanical position and momentum distributions on the detuning . we find characteristic oscillations in the widths of these distributions as a function of the detuning . these oscillations are absent in the corresponding classical curves . we explain these structures by transforming the hamiltonian into an interaction picture . we emphasize that this is different to the case of atoms moving in a phase modulated standing wave . there the corresponding oscillations appear in the widths of the quantum as well as of the classical momentum distributions @xcite . hence , the detuning is an additional parameter controlling the width and the shape of the quantum distributions . moreover , in the quantum systems discussed so far in the context of dynamical localization the quantum diffusion is always slower than the classical one . however , in the present model the new control parameter detuning can create situations in which the quantum diffusion temporarily exceeds the classical one . quantum interference effects such as dynamical localization are extremely sensitive to decoherence arising for example from noise or spontaneous emission @xcite . we are therefore forced to investigate in sec . [ 4 ] the influence of spontaneous emission . we find that the drastic difference between quantum and classical behavior is preserved in the limit of a far off resonance situation . we conclude in sec . [ 5 ] by summarizing our main results . we consider the standard paul trap set - up realized experimentally in many labs @xcite : a standing electromagnetic wave of frequency @xmath0 and wave number @xmath1 aligned along the @xmath2-axis couples the internal states @xmath3 and @xmath4 of a single two - level ion of mass @xmath5 to the center - of - mass motion . the resulting dynamics of the ion follows from the time - dependent schrdinger equation with the hamiltonian & = & + ^2 + _ 0 _ z + & + & _ 0 _ x ( k ) ( _ l ) . [ schr ] here the parameters @xmath6 and @xmath7 are proportional @xcite to the dc and ac voltages applied to the trap and @xmath8 are the standard pauli matrices . moreover , we denote the frequencies of the ac field , the atomic transition , the rabi frequency by @xmath9 , and @xmath10 , respectively . we introduce the dimensionless position @xmath11 , time @xmath12 and momentum @xmath13 . when we transform into an interaction picture with the unitary transformation @xmath14 the dimensionless hamiltonian in rotating wave approximation reads @xmath15 \hat{x}^2 - { k\hspace{-0.45em}\raisebox{0.7ex}{-}\hspace{0.2em}}\delta \hat{\sigma}_z \nonumber \\ & + & { k\hspace{-0.45em}\raisebox{0.7ex}{-}\hspace{0.2em}}\omega_0 \hat{\sigma}_x \cos \hat{x}. \label{ham } \end{aligned}\ ] ] here we have defined the dimensionless rabi frequency @xmath16 and detuning @xmath17 . the dynamics of the ion follows from the time dependent schrdinger equation @xmath18 for the state vector @xmath19 \|x\rangle . \label{t_ev } \end{aligned}\ ] ] here the effective planck constant @xmath20 is consistent with the commutation relation @xmath21 = 2 k^2/(m \omega)[\hat{\tilde{x}},\hat{\tilde{p } } ] = 2 k^2/(m \omega ) i \hbar \equiv i { k\hspace{-0.45em}\raisebox{0.7ex}{-}\hspace{0.2em}}$ ] . the state vector @xmath22 provides the probabilities p_g(x , t ) \|_g(x , t)\|^2 or p_e(x , t ) \|_e(x , t)\|^2 to find the ion at the time @xmath23 at position @xmath2 given it is in the internal state @xmath24 or @xmath25 , respectively . when we are not interested in the internal states the position probability distribution @xmath26 reads p(x , t ) \|_g(x , t)\|^2 + we simulate the effect of spontaneous emission by the quantum monte carlo method @xcite using the effective non hermitian hamiltonian @xmath27 here we introduced the atomic operators @xmath28 and @xmath29 , and @xmath30 is the spontaneous decay rate scaled by @xmath31 . the moments of time when a spontaneous emission event takes place are chosen at random . then the wave function is projected onto the ground state and renormalized , that is @xmath32 . the recoil @xmath33 $ ] is chosen randomly according to the probability distribution @xcite @xmath34 of dipole radiation . when the results of single runs are averaged , we obtain the same result as predicted by a master equation @xcite . as an initial condition for the internal states we use the superposition @xmath35 and the the center - of - mass wave function is a gaussian of width @xmath36 . in order to investigate localization we start with this wave packet at the origin where the classical phase space is a stochastic sea @xcite . we calculate the time evolution using the split - operator method @xcite with a grid of 8192 points . we control numerical errors using an adaptive time step size algorithm @xcite . this numerical integration allows us to find the time dependence of the position probability distributions @xmath37 and @xmath26 . moreover , we calculate the corresponding momentum distributions where the wave functions @xmath38 and @xmath39 in position space are replaced by the wave functions in momentum space . we compare and contrast these quantum results to the dynamics @xcite resulting from the classical equations of motion @xcite & = & p + & = & -(a+2q(2 t ) ) x + k_0 ( x ) r_1 [ bl1 ] for the center - of - mass motion . these equations are driven by the bloch equations @xcite _ 1 & = & - 2 r_2 + _ 2 & = & 2 r_1 + 2 _ 0 ( x ) r_3 [ bl2 ] + _ 3 & = & - 2 _ 0 ( x ) r_2 describing the internal dynamics . here @xmath40 , @xmath41 , and @xmath42 denote the in and out of phase quadratures of the dipole moment , and the atomic inversion , respectively . for the comparison with the quantum mechanical results we calculate 4096 trajectories starting from a classical gaussian ensemble centered initially at the origin and having the same widths in position and momentum as the quantum wave packet . in this way we obtain the classical phase space distribution from which we find by integration over @xmath2 or @xmath43 the classical momentum or position distributions @xmath44 or @xmath45 . in the present section we study the influence of the detuning of the laser field with respect to the atomic transition on the dynamics of the system . here we consider the following quantities of interest : the classical and quantum distributions , and their width as a function of time and detuning . in fig . 1 we show the influence of the internal structure of the ion on the classical and on the quantum diffusion . here we analyze for various detunings @xmath46 the time dependence of the widths @xmath47 ( left column ) of the classical ( thin lines ) and quantum mechanical ( thick lines ) position distributions @xmath48 and @xmath26 ( right column ) . whereas the left column emphasizes the time dependence of the widths , the right column shows position distributions averaged over the time interval @xmath49 $ ] . in order to bring out most clearly the influence of the detuning we have neglected spontaneous emission in this figure . for a very small detuning the quantum width @xmath47 lies well below the classical one . the latter increases as a function of time . in contrast , the quantum result displays oscillations around a steady state value . this suppression of the classical diffusion is due to dynamical localization as discussed in ref.@xcite . however , for slightly larger detunings the quantum curve is still oscillating but reaches partially above the classical one . for even larger detunings the quantum curve falls again below the classical one . eventually for larger detunings it again goes partially above . hence , for a fixed detuning , there exist time regimes in which the quantum diffusion is stronger than the classical one . this detuning dependence of the quantum diffusion manifests itself in the shape of the quantum mechanical position distribution . the latter consists of a sharp peak and a broad background . the detuning essentially controls the shape of the background : depending on @xmath46 we find distributions either of negative or positive curvature . in contrast , the detuning hardly influences the classical distributions shown on the right column by thin curves . we emphasize that the momentum distributions not shown here display almost identical behavior . to discuss the influence of the detuning on the localization length we show in fig . 2 the widths @xmath47 of the classical and the quantum distributions averaged from @xmath50 to @xmath51 . we note characteristic oscillations in the quantum mechanical curve as a function of the detuning @xmath46 . these oscillations do not appear in the classical curve which decreases monotonously for increasing @xmath46 . this feature becomes clear when we recall that the width of the classical distribution at a given time is determined by the classical diffusion constant . the latter is proportional to the perturbing potential caused by the laser field . in the limiting case of large detuning , the effective potential becomes @xcite proportional to the effective coupling constant @xmath52 . as @xmath46 increases , the perturbation decreases and hence the diffusion is slower . an interesting domain is a small region around @xmath53 . here the classical diffusion rate sharply decreases . the explanation of this feature follows from the set of eqs . ( [ bl1 ] ) and ( [ bl2 ] ) : the effective potential provided by the laser field in eq . ( [ bl1 ] ) is proportional to the factor @xmath54 . this quantity can take the maximum value of unity . for the resonant case , @xmath55 , we have @xmath56 and hence @xmath40 is independent of time and determined by the initial condition . the bloch vector coordinates corresponding to the initial state @xmath57 are @xmath58 and @xmath59 . hence @xmath60 results in the maximum driving strength of the center - of - mass motion . the curve for the quantum widths shows a distinctly different behaviour with a lot of structure . in particular , for @xmath61 we note a well developed resonance structure and two sharp minima and maxima . the minimum around @xmath62 is related to the separable quantum dynamics in the two diabatic potentials @xmath63 x^2 \pm \hbar \omega_0 \cos x$ ] . to bring this out most clearly we transform from the @xmath64 basis states to their superpositions @xmath65 , and @xmath66 . in this basis the hamiltonian eq . ( [ ham ] ) reads & = & ^2+^2 + & & + k_0 _ z + k_x , [ ham_tr ] and the coupling is now proportional to the detuning @xmath46 . hence we indeed find the potentials @xmath67 . when we now increase the detuning , transitions between the states @xmath68 and @xmath69 contribute to the quantum diffusion . therefore , the quantum diffusion rate quickly rises to the maximal classical one ; this corresponds to the formation of the first maximum . the next minimum fits the condition @xmath70 , where @xmath71 is the secular frequency of the trap . the origin of this resonance becomes clear , when we write the hamiltonian , eq . ( [ ham ] ) , in the interaction picture where the dynamics of the trap as well as the internal state energy are transformed away by unitary transformations @xcite . in this picture the hamiltonian reads h_(t ) = _ n=0^ _ k=-[n/2]^ _ l=-^ k & & _ l^(n , n+2k ) e^2i ( l- k_s+)t + & & _ + + [ hamflo ] with the lamb - dicke parameter @xmath72^{1/2 } } $ ] and & & _ l^(n , n+2k ) _ 0 [ i]^2k + & & _ -/2^/2 dt [ ^(t)]^2k e^-^2\|(t)\|^2 l^2k_n(^2\|(t)\|^2 ) e^-2ilt . here @xmath73 denotes the @xmath74th energy eigenstate of the time independent reference oscillator @xcite with frequency @xmath75 . moreover , we have made use of the floquet solution @xmath76 , where @xmath77 is a periodic function . the solution @xmath78 obeys @xcite the differential equation + & = & 0 [ mathier ] with @xmath79 and @xmath80 . from the term @xmath81 in the hamiltonian ( [ hamflo ] ) we expect resonance effects , when @xmath46 satisfies the condition @xmath82 . for @xmath83 we have @xmath84 , which corresponds to two phonon transitions . this resonance suggests an enhancement of the quantum diffusion rate on resonance . instead , we have a deep minimum . in order to understand this counter - intuitive behavior we calculate the characteristic frequencies @xmath85 for @xmath86 and @xmath87 at different vibrational quantum numbers @xmath74 . from the inset of fig . 2 we recognize that @xmath88 has a deep minimum around @xmath89 , which explains suppression of the diffusion over the vibrational states . as shown by the curve in the inset for @xmath90 there is no minimum just a decay . this causes the quantum diffusion to take on the classical value , which explains the formation of the second maximum @xcite . we conclude this section by noting that the phenomenon of oscillations in the width of distributions also appears in another quantum system showing classical chaos and quantum localization : an atom moving in a phase - modulated standing wave @xcite . however in this case the oscillations are of classical origin and appear both in the classical and quantum widths . we now consider the effect of spontaneous emission using the quantum monte carlo technique @xcite . the purpose of our investigations is twofold . first , we want to show that indeed the phenomena discussed so far are quantum interference effects . they are therefore sensitive to decoherence such as spontaneous emission . the latter predominantely occures when the laser is tuned close to resonance with the atomic transition . second , we want to show that in the far - detuned limit the effect of decoherence is negligible and the phenomenon of dynamical localization survives . in fig . 3 we show the results of our simulations with a realistic rate of spontaneous emission corresponding to the decay rate of the @xmath91 transition at @xmath92 mhz in @xmath93 @xcite . we compare them to the classical and to the quantum result without noise . the loss of the coherence causes destruction of localization . in this case the widths of the quantum mechanical position and momentum distributions are larger than the classical ones . this additional diffusion is caused by the random recoil kicks following each spontaneous emission event . when we neglect the recoil which of course is not realistic the quantum curve follows the classical one . the position and momentum distributions are of classical type , that is on the logarithmic scale they are polynomial curves . note that the pattern of the standing wave appears on top of the position distributions . in fig . 4 we show the results for @xmath94 , corresponding to a detuning of @xmath95 ghz , a value which was mentioned in ref . the small oscillations in the quantum widths @xmath47 and @xmath96 are destroyed , but the main phenomenon , the substantial quantum suppression of classical diffusion is still visible . it is interesting to note that in a single realization of the dynamics the coherence of the two level superposition is completely destroyed by a single spontaneous emission . however , it affects only slightly the motional coherence because the population of the excited state is very low . furthermore , the small difference between the results with and without spontaneous emission in fig . 4 is of the same order of magnitude that was found for the problem of an atom in a phase modulated standing wave @xcite . in the position distribution of the ground state , there is more probability in the classical - like background than in the case of no spontaneous emission . we have found that this background is very slowly growing , and no real steady state can be reached . however , if we increase the detuning , the real steady state is approached asymptotically . since dynamical localization appears for a large range of parameters @xcite , there is a lot of room for optimizing the parameters . thus the phenomenon of dynamical localization in a paul trap could be observed experimentally . in order to observe not only dynamical localization but also the oscillations in the localization length discussed in the previous section , one should consider a configuration where decoherence is weaker even for small detunings , for example a different set of parameters in the present system , or a different system such as a raman transition between two ground states . there are basically two ways of controlling dynamical localization : ( i ) through the classical diffusion and ( ii ) through the quantized nature of the variable showing localization . this stands out most clearly in the estimate @xmath97 for the localization length . here @xmath98 is the classical diffusion coefficient determined by the perturbing potential . the coupling to this potential is called the control parameter , since it is a direct way to control the localization length . the scaled planck constant @xmath99 describes how important the quantization of the system is with respect to the perturbation . in this paper we have shown that , when the perturbing potential has a quantum character as well , the relation @xmath97 is not exactly true anymore ; we observe oscillations in the localization length which do not appear in the classical diffusion rate . the parameter determining the oscillations could be called the quantum control parameter . we thank b. kneer and m. el ghafar for many fruitful discussions . t. and v. s. acknowledge the support of the deutsche forschungsgemeinschaft . we thank the rechenzentrum ulm and the rechenzentrum karlsruhe for their technical support . a. lagendijk and b. a. van tiggelen , phys . rep . 270 * , 143 ( 1996 ) . p. w. anderson , phys . 109 * , 1492 ( 1958 ) . for a review see , for example p. m. koch and k. a. h. van leeuwen , phys . rep . * 255 * , 289 ( 1995 ) ; g. casati , phys . a * 45 * , 7670 ( 1992 ) . f. l. moore , j. c. robinson , c. bharucha , p. e. williams , and m. g. raizen , phys . * 73 * , 2974 ( 1994 ) ; theoretical proposal in r. graham , m. schlautmann , and p. zoller , phys . a * 45 * , r19 ( 1992 ) . f. haake , _ quantum signatures of chaos _ ( springer - verlag , berlin , 1992 ) . , eds . g. casati and b. chirikov ( cambridge university press , 1995 ) . w. paul , rev . phys . * 62 * , 531 ( 1990 ) . m. el ghafar , p. trm , v. savichev , e. mayr , a. zeiler , and w. p. schleich , phys . lett . * 78 * , 4181 ( 1997 ) . j. c. robinson , c. bharucha , f. l. moore , r. jahnke , g. a. georgaki , m. g. raizen , and b. sundaram , phys . rev . lett . * 74 * , 3963 ( 1995 ) . j. bardroff , i. bialynicki - birula , d. s. krhmer , g. kurizki , e. mayr , p. stifter , and w. p. schleich , phys . lett . * 74 * , 3959 ( 1995 ) . in the work of r. graham and s. miyazaki , phys . a * 53 * , 2683 ( 1996 ) the problem of atoms in a phase modulated standing wave was considered without the adiabatic elimination of the upper state . however , the emphasis of that paper was on spontaneous emission , not on the effect of an additional quantum degree of freedom . for experimental studies of the effect of noise and dissipation on dynamical localization see m. arndt , a. buchleitner , r. n. mantegna , and h. walther , phys . lett . * 67 * , 2435 ( 1991 ) ; f. l. moore , j. c. robinson , c. bharucha , b. sundaram , and m. g. raizen , phys . lett . * 75 * , 4598 ( 1995 ) ; r. blmel , a. buchleitner , r. graham , l. sirko , u. smilansky , and h. walther , phys . a * 44 * , 4521 ( 1991 ) , h. ammann , r. gray , i. shvarchuck , and n. christensen , phys . lett . * 80 * , 4111 ( 1998 ) ; b. g. klappauf , w. h. oskay , d. a. steck , and m. g. raizen , phys . rev . lett . * 81 * , 1203 ( 1998 ) . d. m. meekhof , c. monroe , b. e. king , w. m. itano , and d. j. wineland , phys . 76 * , 1796 ( 1996 ) ; c. monroe , d. m. meekhof , b. e. king , and d. j. wineland , science * 272 * , 1131 ( 1996 ) ; d. leibfried , d. m. meekhof , b. e. king , c. monroe , w. m.itano , and d. j. wineland , phys . * 77 * , 4281 ( 1996 ) . g. birkl , j. a. yeazell , r. rckerl , and h. walther , europhys . * 27 * , 197 ( 1994 ) ; h. katori , s. schlipf , and h. walther , phys . * 79 * , 2221 ( 1997 ) . b. appasamy , y. stalgies , and p. e. toschek , phys . * 80 * , 2805 ( 1998 ) . r. dum , a. s. parkins , p. zoller , and c. w. gardiner , phys . a * 46 * , 4382 ( 1992 ) ; k. mlmer , y. castin , and j. dalibard , j. opt . b * 10 * , 524 ( 1993 ) . h. carmichael , _ an open system approach to quantum optics _ ( springer verlag , berlin , 1991 ) j. javanainen and s. stenholm , appl . * 21 * , 35 ( 1980 ) . for more information about the classical dynamics of this system see r. chacn and j. i. cirac , phys . a * 51 * , 4900 ( 1994 ) ; m. el ghafar , e. mayr , v. savichev , p. trm , a. zeiler , and w. p. schleich , j. mod . opt . * 44 * , 1985 ( 1997 ) . m. d. feit , j. a. fleck , and a. steiger , j. of comput . * 47 , * 412 ( 1982 ) . k. riedel , _ quanteneffekte in paul fallen , dynamische lokalisierung und dekohrenz _ , diplomarbeit , universitt ulm , 1997 , unpublished . g. m. zaslavsky , _ chaos in dynamic systems _ ( harwood academic publishers , chur , 1985 ) . in deriving these equations , we have assumed that the quantum mechanical position distribution is very narrow and localized near the classical trajectory @xmath100 . this property allows us to take the slowly varying cosine potential out of the integrals . l. mandel and e. wolf , _ optical coherence and quantum optics _ ( cambridge university press , 1995 ) a. p. kazantsev , g. i. surdutovich , and v. p. yakovlev , _ mechanical action of light on atoms _ ( world scientific , singapore , 1990 ) . j. bardroff , c. leichtle , g. schrade , and w. p. , phys . lett . * 77 * , 2198 ( 1996 ) ; ibid acta physica slovaka * 46 * , 231 ( 1996 ) . r. j. glauber , _ laser manipulation of atoms and ions _ school of physics enrico fermi course 118 , edited by e. arimondo , w. d. phillips , and f. strumia ( north holland , amsterdam , 1992 ) ; see also g. schrade , p. j. bardroff , r. j. glauber , c. leichtle , v. yakovlev , and w. p. schleich , appl . b * 64 * , 181 ( 1997 ) . there is a close similarity between such type behavior and the so called fano resonance : u. fano , phys . rev . * 124 * , 1866 ( 1961 ) . s. r. jefferts , c. monroe , e. w. bell , and d. j. wineland , phys . a * 51 * , 3112 ( 1995 ) . p. goetsch and r. graham , phys . a * 54 * , 5345 ( 1996 ) .	we identify a new parameter that controls the localization length in a driven quantum system . this parameter results from an additional quantum degree of freedom . the center - of - mass motion of a two - level ion stored in a paul trap and interacting with a standing wave laser field exhibits this phenomenon . we also discuss the influence of spontaneous emission . # 1 # 1^ # 1 # 1^ # 1\| # 1 # 1#1 \|
the two - dimensional electron gas ( 2deg ) in a perpendicular magnetic field has a very rich phase diagram that includes several phases such as the laughlin liquids that give rise to the integer and quantum hall effectsgirvin1 , the wigner crystal at small filling factor in each landau level@xcite , the bubble crystals and the stripe phase in higher landau levels@xcite and the skyrme crystal@xcite near filling factor @xmath1 in the lowest landau level . the phase diagram is even more complex when system with extra degrees of freedom such as double quantum wells ( dqws ) are considered@xcite . in dqws , the orientation of the pseudospin vector associated with the layer degree of freedom can be modified by changing the tunneling and electrical bias between the layers . another way to modify the properties of the 2deg is by the addition of a lateral two - dimensional superlattice patterned on top of the gaas / algaas heterojunction hosting the 2deg that creates a spatially modulated potential at the position of the 2deg@xcite . the effect of a one - dimensional periodic potential on the landau levels is particularly interesting@xcite since it leads to commensurability problems due to the presence of different lengths scales : the lattice constant of the external potential @xmath11 the magnetic length @xmath12 ( @xmath13 is the magnetic field ) and the fermi wavelength . novel magnetoresistance oscillations with period different than that of the well - known shubnikov - de haas oscillations have been detected in such systems . even more interesting is the effect of a periodic two - dimensional potential on the band structure of the 2deg@xcite . the intricate pattern of eigenvalues that results from such potential has been studied by many authors and is known as the hofstadter butterfly spectrum@xcite . its observation in gaas / algaas heterojunction is very difficult due to screening and disorder effects but experimental signature in magnetotransport experiments in 2degs with a lateral surface superlattice potential with period of the order of @xmath14 nm and less have been reported@xcite . interest in this problem has been revived recently by the experimental observations of the hofstadter s butterfly spectrum that use the moir superlattices that arise from graphene or bilayer graphene placed on top of hexagonal boron nitride@xcite . another interest of superlattice potentials is their use to create artificial lattices . for example , a lateral superlattice with a honeycomb crystal structure has recently been proposed to create an artificial graphenelike system@xcite in a gaas / algaas heterojunction . in this work , we study theoretically the effect of a square lattice lateral potential with a period @xmath15 on the ground state of the 2deg in gaas / algaas heterojunction at filling factor @xmath1 and in landau level @xmath2 we include the spin degree of freedom and use the hartree - fock approximation to study the combined effects of the external potential and the coulomb interaction . we assume that the potential is sufficiently weak so that landau level mixing can be neglected . we also ignore disorder and work at zero temperature . we vary the potential strength @xmath3 and calculate the ground state for different rational values of @xmath16 $ ] ( where @xmath17 is the flux quantum and @xmath18 are integers with no common factors ) which is the inverse of the number of flux quanta per unit cell of the surface potential . our formalism allows for the formation of uniform as well as spatially modulated ground states with or without spin texture . our calculation indicates that , at a critical value , @xmath4 of the external potential , there is a transition from a uniform fully spin polarized state to a charge density wave ( cdw ) with an intricate spin texture . each unit cell of this cdw contains two positive and two negative amplitude modulations and the vortex spin texture at each maximum(minimum)resembles that of a positively(negatively ) charged meron . the two positively(negatively ) charged merons in each unit cell have the same vorticity but a global phase that differs by @xmath19 these meronlike textures , however , are not quantized since the amplitude of the cdw varies continuously with @xmath3 . in the vortex - cdw , as we call it , the average spin polarization @xmath5 varies with @xmath3 in a continuous or discontinuous manner depending on whether @xmath6 $ ] or @xmath20 $ ] and saturates at a finite , positive , value of @xmath5 that depends only on @xmath21 in most cases . in the special case @xmath22 the vortex - cdw phase is absent and the transition is directly from a fully spin polarized and uniform 2deg to an unpolarized cdw . the phase diagram for @xmath20 $ ] is richer than that of @xmath6 $ ] as it involves the transition between the vortex - cdw and its conjugate phase , the anti - vortex cdw , obtained by reversing @xmath23 and inverting the vorticity of all merons . this transition between the two cdws is accompanied by a discontinuous change of @xmath5 that becomes continuous when the zeeman coupling goes to zero . we study the properties of the vortex - cdw at different values of @xmath21 and with a particular emphasis on its collective excitations which we derive using the generalized random - phase approximation ( grpa ) . the vortex - cdw has collective modes that have much in common with the collective excitations of the skyrme crystal@xcite that is expected to be the ground state near filling factor @xmath1 in @xmath2 namely , the broken u(1 ) symmetry in the vortex - cdw phase leads to a new gapless mode that can provide a fast channel for the relaxation of nuclear spins@xcite . this mode and the meronlike spin texture disappear at larger values of the external potential leaving a ground state that is either unpolarized if @xmath24 or partially polarized if @xmath25 our paper is organized as follows . in sec . ii , we introduce the hamiltonian of the 2deg in the presence of the lateral square lattice potential and briefly review the hartree - fock and generalized random - phase approximation that we use to compute the density of states , the density and spin profiles and the collective excitations of the various phases . in sec . iii , we present our numerical results for the phase diagram of the 2deg as a function of the potential strength @xmath3 and the inverse magnetic flux per unit cell @xmath21 . we conclude in sec . iv with a discussion on the experimental detection of the new vortex - cdw state . the system we consider is a 2deg in a gaas / algaas heterojunction or quantum well submitted to a perpendicular magnetic field @xmath26 and to a lateral superlattice potential @xmath27 the coupling of the electrons to this external potential is given by @xmath28 , where @xmath29 is the total density operator including both spin states @xmath30 ( we take @xmath31)@xmath32 we assume that only the landau level @xmath33 is occupied but our calculation can easily be generalized to any landau level by changing the effective interactions @xmath34 and @xmath35 and the form factor @xmath36 the hamiltonian of the interacting 2deg is given , in the hartree - fock approximation , by @xmath37where @xmath38 is the 2deg area , @xmath39 is the landau level degeneracy , and the form factor for the @xmath33 landau level is@xmath40where @xmath12 is the magnetic length . the averages are over the hartree - fock ground state of the 2deg . the non - interacting single - particle energies , measured with respect to the kinetic energy @xmath41 are given by@xmath42where the zeeman energy @xmath43 with @xmath44 the effective @xmath45factor of bulk gaas and @xmath46 the bohr magneton . in some experiments on skyrmions , the effective @xmath45factor was tuned in the range @xmath47 to @xmath48 by applying hydrostatic pressure to a sample of gaas / algaas modulation doped quantum wellpotemski . in our study , we will thus consider that @xmath49 is not determined by the magnetic field , but is instead a parameter than can be adjusted . the hartree and fock interactions in @xmath33 are given by@xmath50where @xmath51 is the dielectric constant of gaas . finally , the operators @xmath52 are defined by@xmath53where @xmath54 is the operator that creates an electron with guiding - center index @xmath55 ( in the landau gauge ) and spin @xmath56 the four operators @xmath52 are related to the averaged electronic and spin densities in the @xmath57plane by@xmath58 , \\ s_{y}\left ( \mathbf{r}\right ) & = & \frac{1}{2\pi \ell ^{2}}\sum_{\mathbf{q}}{\operatorname{im}}\left [ \left\langle \rho _ { + , -}\left ( \mathbf{q}\right ) \right\rangle e^{-q^{2}\ell ^{2}/4}e^{i\mathbf{q}\cdot \mathbf{r}}\right ] , \\ s_{z}\left ( \mathbf{r}\right ) & = & \frac{\hslash } { 2}\left [ n_{+}\left ( \mathbf{r}\right ) -n_{-}\left ( \mathbf{r}\right ) \right ] .\end{aligned}\]]the @xmath59 can be considered as the order parameters of an ordered phase of the 2deg . the averaged hartree - fock ground - state energy per electron at @xmath1 is given by@xmath60 the order parameters @xmath61 are computed by solving the hartree - fock equation for the single - particle green s function @xmath62 which is defined by@xmath63where@xmath64they are obtained with the relation @xmath65 the hartree - fock equation of motion for the green s function @xmath66 is given@xcite by @xmath67 g_{\alpha , \beta } \left ( \mathbf{q},i\omega _ { n}\right ) \label{motion } \\ & = & \delta _ { \mathbf{q},0}\delta _ { \alpha , \beta } \notag \\ & & -\frac{e}{\hslash s}\sum_{\mathbf{q}^{\prime } } v_{e}\left ( \mathbf{q}-\mathbf{q}^{\prime } \right ) f\left ( \left\vert \mathbf{q}-\mathbf{q}^{\prime } \right\vert \right ) \notag \\ & & \times \gamma _ { \mathbf{q},\mathbf{q}^{\prime } } g_{\alpha , \beta } \left ( \mathbf{q}^{\prime } , i\omega _ { n}\right ) \notag \\ & & + \frac{1}{\hslash } \sum_{\mathbf{q}^{\prime } \neq \mathbf{q}}u^{h}\left ( \mathbf{q - q}^{\prime } \right ) \gamma _ { \mathbf{q},\mathbf{q}^{\prime } } g_{\alpha , \beta } \left ( \mathbf{q}^{\prime } , i\omega _ { n}\right ) \notag \\ & & -\frac{1}{\hslash } \sum_{\mathbf{q}^{\prime } } \sum_{\gamma } u_{\alpha , \gamma } ^{f}\left ( \mathbf{q - q}^{\prime } \right ) \gamma _ { \mathbf{q},\mathbf{q}^{\prime } } g_{\gamma , \beta } \left ( \mathbf{q}^{\prime } , i\omega _ { n}\right ) , \notag\end{aligned}\]]where@xmath68and @xmath69 are fermionic matsubara frequencies , @xmath70 is the chemical potential and we have defined the potentials @xmath71 these potentials depend on the order parameters @xmath72 that are unknown . the equation of motion for @xmath73 must thus be solved numerically@xcite by using a seed for the order parameters and then iterate eq . ( [ motion ] ) until a convergent solution is found . in case several solutions are found ( corresponding to different choice for the initial seed ) , we choose the one with the lowest energy and compute the dispersion relation of its collective modes to make sure that it is a stable solution . we remark that this method does not guarantee that the true ground state is the solution that we keep . the density of states @xmath74 is obtained from the single - particle green s function by using the relation @xmath75 .\ ] ] to find the dispersion relation of the collective modes , we derive the equation of motion in the generalized random - phase approximation for the two - particle green s function@xmath76this equation is@xmath77where @xmath78 is a bosonic matsubara frequency . equation ( [ grpa ] ) represents the summation of bubble ( polarization effects ) and ladder ( excitonic corrections ) diagrams . the hartree - fock two - particle green s function ( the single - bubble feynman diagram with hartree - fock propagators ) that enters this equation is obtained from the hartree - fock equation of motion for @xmath79 and is given by @xmath80 \chi _ { \alpha , \beta , \gamma , \delta } ^{\left ( 0\right ) } \left ( \mathbf{q},\mathbf{q}^{\prime } , i\omega _ { n}\right ) \\ & = & \hslash \left [ \gamma _ { \mathbf{q},\mathbf{q}^{\prime } } ^{\ast } \left\langle \rho _ { \alpha , \delta } \left ( \mathbf{q - q}^{\prime } \right ) \right\rangle \delta _ { \beta , \gamma } -\gamma _ { \mathbf{q},\mathbf{q}^{\prime } } \left\langle \rho _ { \gamma , \beta } \left ( \mathbf{q - q}^{\prime } \right ) \right\rangle \delta _ { \alpha , \delta } \right ] \notag \\ & & -\frac{e}{s}\sum_{\mathbf{q}^{\prime \prime } } v_{e}\left ( \mathbf{q}-\mathbf{q}^{\prime \prime } \right ) f\left ( \left\vert \mathbf{q}-\mathbf{q}^{\prime \prime } \right\vert \right ) \left [ \gamma _ { \mathbf{q},\mathbf{q}^{\prime \prime } } ^{\ast } -\gamma _ { \mathbf{q},\mathbf{q}^{\prime \prime } } \right ] \notag \\ & & \times \chi _ { \alpha , \beta , \gamma , \delta } ^{\left ( 0\right ) } \left ( \mathbf{q}^{\prime \prime } , \mathbf{q}^{\prime } , i\omega _ { n}\right ) \notag \\ & & -\sum_{\mathbf{q}^{\prime \prime } \neq 0}u^{h}\left ( \mathbf{q - q}^{\prime \prime } \right ) \left [ \gamma _ { \mathbf{q},\mathbf{q}^{\prime \prime } } ^{\ast } -\gamma _ { \mathbf{q},\mathbf{q}^{\prime \prime } } \right ] \notag \\ & & \times \chi _ { \alpha , \beta , \gamma , \delta } ^{\left ( 0\right ) } \left ( \mathbf{q}^{\prime \prime } , \mathbf{q}^{\prime } , i\omega _ { n}\right ) \notag \\ & & + \sum_{\alpha ^{\prime } } \sum_{\mathbf{q}^{\prime \prime } } u_{\alpha , \alpha ^{\prime } } ^{f}\left ( \mathbf{q - q}^{\prime \prime } \right ) \gamma _ { \mathbf{q},\mathbf{q}^{\prime \prime } } ^{\ast } \chi _ { \alpha ^{\prime } , \beta , \gamma , \delta } ^{\left ( 0\right ) } \left ( \mathbf{q}^{\prime \prime } , \mathbf{q}^{\prime } , i\omega _ { n}\right ) \notag \\ & & -\sum_{\beta ^{\prime } } \sum_{\mathbf{q}^{\prime \prime } } u_{\beta ^{\prime } , \beta } ^{f}\left ( \mathbf{q - q}^{\prime \prime } \right ) \gamma _ { \mathbf{q},\mathbf{q}^{\prime \prime } } \chi _ { \alpha , \beta ^{\prime } , \gamma , \delta } ^{\left ( 0\right ) } \left ( \mathbf{q}^{\prime \prime } , \mathbf{q}^{\prime } , i\omega _ { n}\right ) . \notag\end{aligned}\ ] ] by defining the super - indices @xmath81 and @xmath82 eq . ( [ grpa ] ) can be rewritten as a @xmath83 matrix equation for the matrix of green s functions @xmath84 this equation has the form @xmath85 \chi = b.$ ] the matrix @xmath86 , which depends only on the @xmath87 is then diagonalized numerically to find @xmath88 . the retarded response functions are obtained with the analytic continuation @xmath89 we compute the following density and spin responses:@xmath90 \right\rangle \theta \left ( t - t^{\prime } \right ) \right ] _ { \omega } , \label{response}\]]where @xmath91 and the operators @xmath92 , \\ \rho _ { x } & = & \frac{1}{2i}\left [ \rho _ { + , -}-\rho _ { -,+}\right ] , \\ \rho _ { z } & = & \frac{1}{2}\left [ \rho _ { + , + } -\rho _ { -,-}\right ] , \\ \rho _ { n } & = & \rho _ { + , + } + \rho _ { -,-}.\end{aligned}\ ] ] in a uniform phase , the order parameters @xmath93 are finite only when @xmath94 while in a two - dimensional cdw , they can be non zero each time @xmath95 , where @xmath96 is a reciprocal lattice vector of the cdw . for the response function , we have to compute @xmath97 in the uniform phase and @xmath98 in the cdw where @xmath99 is , by definition , a vector in the first brillouin zone of the cdw . in the cdw , the grpa matrices @xmath86 and @xmath13 have dimensions @xmath100 , where @xmath101 is the number of reciprocal lattice vectors considered in the numerical calculation . we typically take @xmath102 the formalism developed in this section can also be applied to the 2deg in graphene if the electrons are assumed to occupy only one of the two valleys . in landau level @xmath33 of graphene ( and only in this level ) , the form factor @xmath103 and the hartree and fock interactions @xmath104 and @xmath35 are the same as those given in eqs . ( [ form],[inter ] ) . we study the phase diagram of the 2deg at filling factor @xmath1 in landau level @xmath33 and at temperature @xmath106 k. for the external potential , we choose the simple square lattice form@xmath107 , \]]so that @xmath108 in eq . ( [ hamiltonien ] ) with the vectors @xmath109 this external potential tries to impose a two - dimensional density modulation of the 2deg with a square lattice constant @xmath110 we allow the spin texture ( if any ) to have the bigger lattice constant @xmath111by considering the order parameters @xmath112 with reciprocal lattice vectors given by @xmath113 where @xmath114 the _ density _ unit cell has a lattice constant @xmath15 while the _ magnetic _ unit cell has a lattice constant @xmath115 for the potential strength , we use @xmath116 [ see eq . ( motion ) ] , where @xmath117 . the critical values of @xmath3 ( not @xmath118 ) for the transition between the uniform and the modulated phases at different value of @xmath21 are similar . hereafter , we give all energies in units of @xmath119 . we make the important assumption that landau level mixing by both the coulomb interaction and the external potential can be neglected i.e. we work in the limit of a weak superlattice potential . we also neglect disorder effect and work at zero temperature . the ratio @xmath120 that enters the hartree - fock energy and equation of motion for the single - particle green s function is given by@xmath121where @xmath17 is the flux quantum . the important parameter @xmath122 is the number of flux quanta piercing a _ density _ unit cell area . with this definition , the factor @xmath123 we limit our analysis to @xmath124 where @xmath125 and @xmath126 are integers with no common factors . in landau level @xmath127 a wigner crystal@xcite with a triangular lattice can form at sufficiently small filling factor @xmath128@xcite . at @xmath129 however , the ground state of the 2deg is a uniform electron liquid with full spin polarization i.e. a quantum hall ferromagnet@xcite ( qhf ) whose energy per electron is given by@xmath130(neglecting the kinetic energy that is a constant in @xmath33 ) . the qhf remains the ground state even when the zeeman coupling goes to zero because a perfect alignment of the spins minimizes the coulomb exchange energy [ the second term on the right - hand side of eq . ( [ fock ] ) ] . in a uniform state , the coulomb hartree energy is cancelled by the neutralizing uniform positive background . we first consider the case @xmath6 .$ ] figure 1 shows the ground state energy and spin polarization @xmath131 as a function of the potential @xmath3 for @xmath132 and for a zeeman coupling @xmath133 . the ground state is spatially uniform and has an energy @xmath134 and a spin polarization @xmath5 that remain constant until a critical field @xmath135 this uniform state is described by only one order parameter , i.e. @xmath136 and is fully spin polarized i.e. the spin per electron is @xmath137 the corresponding change in the density of states ( dos ) with @xmath3 is shown in fig . 2 . in the absence of external potential and coulomb interaction , the dos has two peaks at energies @xmath138 corresponding to the two spin states . with coulomb interaction , the zeeman gap @xmath49 is strongly renormalized [ see fig . 2 ( a ) ] as is well known . when the external potential is present , the dos for each spin orientation , has @xmath126 peaks corresponding to the number of subbands expected when an electron is submitted to both a magnetic field and a weak periodic potentialhofstadter . this is clearly visible in fig . 2 ( a),(d),(e ) for @xmath139 . the external potential increases the width of the peaks in the dos and decreases the renormalized zeeman gap ( which is also the transport gap ) . we remark that the rapid oscillations in some of the graphs at @xmath140 are a numerical artefact . they depend strongly on the number of reciprocal lattice vectors kept in the calculation . as a function of the applied external field @xmath3 for different values of @xmath21 at filling factor @xmath105 in landau level @xmath33 and for the zeeman coupling @xmath141 if we enforce the uniform solution beyond the critical value @xmath142 , we find that the transport gap closes at @xmath143 for @xmath144 where the system becomes metallic . our code no longer converges in this case . but , this transition to a metallic state does not occur because the uniform state becomes unstable at @xmath145 the stability of a state is evaluated by computing the dispersion relation of its collective modes . for the uniform state , the collective excitations reduce to a spin - wave mode@xmath32 when @xmath146 the spin - wave dispersion is given by@xcite in landau level @xmath33 for zeeman coupling @xmath147 ( a)-(c ) @xmath144 and @xmath148 respectively ; ( d)-(e ) @xmath149 and @xmath150 respectively ; ( f ) @xmath151 and @xmath152 . the rapid oscillations in some of the graphs are a numerical artefact . all energies are in units of @xmath153 @xmath154 . \label{spinwave}\ ] ] this mode is gapped at the bare zeeman energy and saturates at @xmath155 figure 3 shows its dispersion for @xmath144 and @xmath156 the wave vector @xmath99 runs along the path @xmath157 i.e. along the edges of the irreducible _ density _ brillouin zone ( with @xmath158 in units of @xmath159 ) . the spin - wave mode softens at a finite wave vector @xmath99 as @xmath160 increases so that the uniform state becomes unstable at @xmath161 which is also the value at which the 2deg is seen to enter a new phase in fig . when plotted in the reduced zone scheme as in fig . 3 , the spin - wave mode is split into several branches that accumulate into a very dense manifold near @xmath162 . only some of these branches are shown in fig . 3 since we are interested only in the low - energy sector . the spin - wave dispersion is obtained by following the pole of the response functions @xmath163 with @xmath164 for different values of @xmath165 we remark that the softening of the spin - wave mode by a _ one - dimensional _ external potential was reported previously by bychkov et al.@xcite . these authors suggested that the resulting condensation of the spin excitons at the softening wave vector would create a new spin density wave ground state . this is precisely what we find , but this time , for a _ two - dimensional _ surface potential . for @xmath166 @xmath133 and different values of the external potentiel @xmath3 . the wave vector @xmath99 follows the path : @xmath167 along the irreducible ( density ) brillouin zone of the square lattice . all energies are in units of @xmath168 the ground state in a small region of @xmath3 after @xmath169 is a charge density wave with a vortex spin texture . hereafter , we refer to this state as the vortex - cdw . the range of @xmath3 where the vortex - cdw is the ground state depends on @xmath21 and @xmath170the electronic density and spin texture of the vortex - cdw are shown in fig . 4 for the parameters @xmath144 and @xmath171 the value @xmath172 is close to @xmath173 so that the amplitude of the cdw in this figure is small . the amplitude increases with @xmath3 however . minima and maxima of the cdw have the same amplitude and there is no net induced charge in a unit cell as expected . the spin density @xmath23 ( not shown in the figure ) varies only slightly . the spin texture of the vortex - cdw is interesting . there is a @xmath174 spin vortex at each positive and negative modulation of the density . since the @xmath175 component of the spin is everywhere positive and because of the spin - charge coupling inherent to a qhf@xcite , the positive and negative modulations have opposite vorticity we could , loosely speaking , refer to the positive and negative modulations as merons and antimerons . a meron is an excitation of a unit vector field @xmath176 that has @xmath177 at its center and @xmath178 far away from the center where the vectors lie in the @xmath57plane and form a vortex configuration with vorticity @xmath179 as @xmath180 increases from the meron core , the spins smoothly rotate up ( if @xmath181 ) or down ( if @xmath182 ) towards the @xmath57plane . there are four flavors of meron@xcite with a topological charge given by @xmath183 n_{v}.$ ] in a qhf , merons carry half an electron charge . in our vortex - cdw however , we are dealing with a spin density @xmath184 that does not have @xmath185 everywhere in space ( the vectors do not just rotate ) so that our merons do not have a quantized charge . but , for a given sign of @xmath186 the meron and antimeron have opposite vorticity and so opposite electrical charge . moreover , our merons are closely packed in a square lattice and have a large core so that the spin vector tilts towards the @xmath57plane but @xmath186 does not go to zero between two adjacent merons . in each magnetic unit cell of the vortex - cdw , there are two merons and two antimerons with the same vorticity but opposite global phase for two merons or antimerons . this bipartite meron lattice is similar to the square lattice antiferromagnetic state ( sla ) of the skyrme crystal that was predicted to occur in a 2deg near ( but not at ) filling factor @xmath1 in the absence of an external potential@xcite . in the skyrme crystal , the electrons ( or holes ) added to the qhf state at @xmath1 crystallize in the form of skyrmions for @xmath187 or antiskyrmions for @xmath188 . the vortex - cdw that we find here occurs _ at _ precisely @xmath189 figure 1(b ) shows that the average spin @xmath5 decreases with @xmath3 in the vortex - cdw and saturates at the precise value @xmath190 for @xmath191 . in the saturation region for @xmath5 , the spin texture has disappeared and the cdw has very little modulation in @xmath192 we will call this phase , the normal - cdw . there is no saturation for the two cases @xmath193 we assume that this is due to the fact that there is another phase very close in energy that wins over the vortex - cdw for @xmath194 in these two cases ( and probably at a larger value of @xmath3 for the other cases ) but we have not been able to stabilize this other phase . we thus limit our analysis to the range @xmath195 $ ] for most values of @xmath21 in this work . the change in @xmath5 induced by the external potential should be detectable experimentally . in particular , the vortex - cdw is absent for @xmath24 and thus the 2deg makes a transition from a fully polarized to an unpolarized cdw with period @xmath15 instead of @xmath196 . in landau level @xmath33 for @xmath172 and zeeman coupling @xmath197 the electronic density @xmath198 is in units of @xmath199 the the vector field shows the vortex structure in the @xmath57plane for the parallel component of the spin vector . all energies are in units of @xmath168 the density of states for the vortex - cdw is shown in fig . 5 for @xmath149 and @xmath200 the subband structure gets more and more different from that of the uniform phase as @xmath3 increases [ compare with fig . the electron - hole gap decreases slowly with @xmath3 in the vortex - cdw and normal - cdw phases . in landau level @xmath33 for @xmath149 , zeeman coupling @xmath133 and different values of @xmath3 ( all energies are in units of @xmath201 ) . ] figure 6 shows the hartree , fock ( or exchange ) , external potential and zeeman contributions to the total energy of the vortex - cdw and uniform state for @xmath144 and zeeman coupling @xmath197 the hartree energy is zero in the uniform phase and small in the vortex - cdw phase . the zeeman energy is also very small in both phases . the competition is between the fock and the external potential energies . the former is minimal in the uniform state and increases with @xmath3 in the vortex - cdw phase . the latter is zero in the uniform phase but decreases with @xmath3 in the vortex - cdw state . figure 6 shows that the increase in exchange energy is more than compensated by the decrease in the external potential energy when the vortex - cdw is formed . of different contributions to the total energy of the uniform and vortex - cdw states for @xmath144 and the zeeman coupling @xmath197 ] the energy of the vortex - cdw state does not depend on the global phase of its vortices . this @xmath203 symmetry , which is broken in a particular realization of the vortex - cdw state , leads to a gapless phase mode ( a goldstone mode ) . this is clearly seen in fig . 7(b ) where the two modes for @xmath204 are the spin wave mode which is gapped at @xmath49 and the gapless phase mode . in fig . 7 , @xmath149 and the wave vector @xmath99 now follows the path @xmath205 along the edges of the irreducible _ magnetic _ brillouin zone of the square lattice [ with @xmath206 in units of @xmath207 . to obtain the dispersions in the cdw phases , we have computed the response functions @xmath208 with @xmath209 keeping the first @xmath210 reciprocal lattice vectors in the summation . the summation allows the capture of modes that originate from a folding of the full dispersion into the first brillouin zone . it also captures the electron - hole continuum@xcite that starts at the hartree - fock gap . in fig . 7 , we have cut the dispersions at a frequency corresponding to the onset of this continuum . figure 7 shows the dispersion for : ( a ) the uniform phase , ( b ) the vortex - cdw , and ( c ) the normal - cdw . in landau level @xmath33 for @xmath149 and zeeman coupling @xmath197 ( a ) @xmath211 ( b ) @xmath212 and ( c ) @xmath213 all energies are in units of @xmath168 the vortex and normal cdws have a phonon mode gapped by the external potential . the branch we indicate as the gapped phonon mode in fig . 7(b ) has the strongest peak in the response function @xmath214 ( no summation over @xmath96 ) as @xmath215 while the spin wave and phase modes are stronger in @xmath216 @xmath217 and @xmath218 respectively . at @xmath219 for @xmath220 the ground state has transited to the normal - cdw and the phase mode is gapped as shown in fig . 7(c ) . the phase diagram for @xmath20 $ ] is different from that of @xmath6 .$ ] for @xmath20 , $ ] we find a transition between two types of vortex - cdw phases . the first vortex - cdw is the one described in the previous section , the second one , the antivortex - cdw has the sign of all vortices and @xmath5 inverted ( but different amplitude for the charge and spin modulations ) . this antivortex - cdw evolves from a uniform state that has all spin down as shown in fig . 8 . at @xmath221 these two cdw are degenerate in energy . at finite zeeman coupling , there is a crossing between the energy curves of these two phases . the ground state thus evolves from the uniform state with all spins up , to the vortex - cdw , then to the antivortex - cdw and finally into the normal cdw . figure 8 shows these transitions for the special cases of @xmath222 and @xmath223 the corresponding behavior of @xmath5 is also shown . the region where @xmath5 varies in each graph is where the vortex-(or antivortex)-cdw is the ground state . as the zeeman coupling gets smaller , this region increases . the value of @xmath3 for the crossing between the two vortex - cdw states is shown by the dashed vertical line in the @xmath5 vs @xmath3 curves . the average spin @xmath5 changes discontinuously at this point but this discontinuity goes to zero as @xmath224 the value of @xmath5 is always positive , however . for @xmath6 , $ ] the energy curve for the anti - vortex cdw is above that of the vortex - cdw for all values of @xmath3 . the two curves merge at @xmath225 but there is no crossing between the two solutions . if we take advantage of the possibility of changing the value of the @xmath45factor independently of the magnetic field in gaas / algaas heterojunctions , then it is possible to reduce the zeeman coupling and , as fig . 9 clearly shows , to increase the transition region where the vortex - cdw is expected . per electron ( right ) of the vortex - cdw ( squares ) and antivortex - cdw ( triangles ) for @xmath222 and zeeman couplings : ( a),(b ) @xmath226(c),(d ) @xmath227(e),(f ) @xmath228 in units of @xmath229 the vertical dashed lines indicate the value of the potential @xmath3 at which the transition between the two vortex cdws takes place . ] figure 9 shows the behavior of @xmath5 in the ground state for @xmath230 and a very small zeeman coupling @xmath231 the vertical dashed lines indicate where the transition between the vortex and antivortex - cdw phases occurs for each value of @xmath232 the spin starts at @xmath233 in the uniform state then decreases in the vortex - cdw ( open symbols in fig . 9 ) . when the antivortex - cdw replaces the vortex - cdw as the ground state of the system , the value of @xmath5 changes discontinuously . this jump is more apparent for @xmath234 in fig . 9 . after this discontinuity , @xmath5 increases ( filled symbols in fig . 9 ) until it reaches the finite value @xmath235 at large @xmath8 a value that is independent of the zeeman coupling @xmath170 the behaviour of @xmath5 is not monotonous . it the limit @xmath236 the @xmath5 curves for the vortex- and antivortex - cdws would cross at @xmath237 and there would be no discontinuity . in the special case @xmath22 the transition is directly from the uniform and fully polarized state with @xmath233 to the normal cdw where @xmath238 there is thus an important discontinuity in @xmath5 in this case . as a function of the applied external potential @xmath3 for several values of @xmath239 and zeeman coupling @xmath240 the vertical dashed lines indicate the potential strength @xmath3 for each value of @xmath21 where the transition from the vortex ( filled symbols ) to the anti - vortex cdw ( open symbols ) takes place . for @xmath22 the transition is from the uniform ( with @xmath241 to the normal cdw with @xmath238 ] as we mentioned above , our formalism can equally well be used to discuss the energy of the electron gas in landau level @xmath33 in graphene if the electrons are assumed to occupy only one valley@xmath32 an exact diagonalization study by ghazaryan and chakraborty@xcite for a 2deg in graphene finds transition between unpolarized and partially polarized ground states induced by the external potential when @xmath144 ( their @xmath243 ) . the equation ( [ motion ] ) was also used@xcite to study the effect of coulomb interaction on the density of states for graphene in a modulated potential but the vortex - cdw state that we found was not considered in that work . we have computed the phase diagram of the 2deg at @xmath1 in landau level @xmath33 in the presence of an applied external potential with a square lattice periodicity for several rational values of @xmath244 @xmath245 $ ] . we restricted our analysis to @xmath195 .$ ] in this range , the 2deg evolves first from a uniform state with full spin polarization then to a vortex - cdw and ( if @xmath246 ) antivortex - cdw state and finally into a normal cdw with no spin texture but with a finite spin polarization @xmath5 if @xmath25 the change in the spin polarization @xmath5 with the applied field ( smooth for @xmath247 and abrupt for @xmath248 ) is one feature of the phase transition described in this work that should be measurable experimentally . another one is the gapless spin mode due to the broken u(1 ) symmetry in the vortex - cdw phase . the same mode occurs in a skyrme crystal . in that system , it was shown that such mode could provide a fast channel for the relaxation of the nuclear spin in nuclear magnetic resonance experiments@xcite . indeed , the bare zeeman gap in the dispersion of the spin - wave mode is orders of magnitude larger than the nuclear spin splitting , impeding the creation of spin waves by nuclear spins . the softening of the spin wave mode in the uniform phase may also lead to an increase in nuclear spin - lattice relaxation time as suggested by bychkov@xcite . we have used @xmath3 for the external potential because the transition from the uniform to the vortex - cdw takes place at roughly the same value of @xmath249 when the potential is expressed in terms of @xmath3 . the actual external potential however is @xmath250 this means that the critical field @xmath251 translates into different real critical fields for different values of @xmath21 i.e. from @xmath252 for @xmath234 to @xmath253 for @xmath254 it is not clear , then if our assumption of neglecting landau - level mixing can be justified for @xmath21 near unity . we assumed that the external lattice parameter @xmath15 is fixed experimentally . when @xmath21 is also given , all other parameters are determined : the magnetic field , the electronic density @xmath255 ( at @xmath1 ) and the ratio , @xmath256 of the coulomb interaction to the cyclotron energy : @xmath257where @xmath258 is the effective bohr radius and @xmath259 is the lateral superlattice constant in nm . we used , for gaas : @xmath51 and @xmath260 ( where @xmath261 is the electron mass ) . for @xmath262so that with a very small ( but physically feasible@xcite ) superlattice period of @xmath263 nm , we get @xmath264 t , @xmath265 @xmath266 while for @xmath267 we get @xmath268 t , @xmath269@xmath270 the magnetic field and density pose no problem , but @xmath271 is not small , especially when @xmath272 clearly , a more sophisticated calculation including a certain amount of landau - level mixing and screening is required to confirm that the vortex - cdw phase is effectively the ground state in this system . we leave this to further work . for reviews , see physics of the electron solid , edited by s.t . chui ( international , boston , 1994 ) ; h. fertig and h. shayegan , in _ perspectives in quantum hall effects _ , edited by s. das sarma and a. pinczuk ( wiley , new york , 1997 ) , chaps . 3 and 9 , respectively . for a review , see m. m. fogler , in high magnetic fields : applications in condensed matter physics and spectroscopy , edited by c. berthier , l .- p . levy , and g. martinez ( springer- verlag , berlin , 2002 ) , chap . 4 , pp . 98138 . r. r. gerhardts , d. weiss , and k. v. klitzing , phys . . lett . * 62 * , 1173 ( 1989 ) ; r. w. winkler , j. p .. kotthaus and k. ploog , phys . lett . * 62 * , 1177 ( 1989 ) ; vidar gudmundsson and rolf r. gerhardts , phys . ref . b * 52 * , 16744 ( 1995 ) . d. j. thouless , in _ the quantum hall effect _ , edited by r. e. prange and s. m. girvin , graduate texts in contemporary physics ( springer - verlag , new york , 1987 ) , p. 101 ; m. c. geisler , j. h. smet , v. umansky , k. von klitzing , b. naundorf , r. ketzmerick , and h. schweizer , physica e * 25 * , 227 ( 2004 ) . s. melinte , mona berciu , chenggang zhou , e. tutuc , s. j. papadakis , c. harrison , e. p. de poortere , mingshaw wu , p.m . chaikin , m . shayegan , r.n . bhatt , and r.a . register , phys . lett . * 92 * , 036802 ( 2004 ) .	the combined effect of a lateral square superlattice potential and the coulomb interaction on the ground state of a two - dimensional electron gas in a perpendicular magnetic field is studied for different rational values of @xmath0 the inverse of the number of flux quanta per unit cell of the external potential , at filling factor @xmath1 in landau level @xmath2 when landau level mixing and disorder effects are neglected , increasing the strength @xmath3 of the potential induces a transition , at a critical strength @xmath4 from a uniform and fully spin polarized state to a two - dimensional charge density wave ( cdw ) with a meronlike spin texture at each maximum and minimum of the cdw . the collective excitations of this vortex - cdw are similar to those of the skyrme crystal that is expected to be the ground state _ near _ filling factor @xmath1 . in particular , a broken u(1 ) symmetry in the vortex - cdw results in an extra gapless phase mode that could provide a fast channel for the relaxation of nuclear spins . the average spin polarization @xmath5 changes in a continuous or discontinuous manner as @xmath3 is increased depending on whether @xmath6 $ ] or @xmath7 .$ ] the phase mode and the meronlike spin texture disappear at large value of @xmath8 leaving as the ground state partially spin - polarized cdw if @xmath9 or a spin - unpolarized cdw if @xmath10
pressure is a fundamental thermodynamic variable in many industrial processes . the study , understanding and prediction of how changes in pressure affect different properties such as solubility parameters , activity coefficients , flory - huggins parameters and interfacial tension , amongst others , is fundamental in the design and application of multi - component products . in particular , the understanding of hydrocarbon - water mixtures at different thermodynamic conditions is essential in the process of oil recovery and other industrial applications . in these systems the effect of high pressure and temperature in different properties is particularly relevant in order , for example , to improve the displacement kinetics involved during the extraction of oil . complex capillary responses due to changes in interfacial tension and in cohesion parameters originated by variable thermodynamic conditions in the oil reservoir could have important economic consequences . the evaluation of these effects is not easy to perform in the laboratory due to the fact that experiments in such extreme conditions are expensive and difficult to handle . for this reason there are many measurements under atmospheric conditions , but only a few studies at high pressure and temperature are available@xcite . interfacial tension @xmath0 and cohesion parameters such as the cohesive energy density @xmath1 , the solubility @xmath2 and the flory - huggins @xmath3 parameters are important quantities widely used in different industrial areas such as paints and coatings , pharmaceutics , bio - polymers , membranes , smart materials , etc . a paramount goal in this area is to establish an accurate and accessible methodology to obtain these parameters for complex mixtures as a function of pressure and temperature . usually , the solubility parameters are obtained experimentally by means of the heat of vaporization at atmospheric conditions , but in a complex fluid system the components are not totally volatile at different thermodynamic conditions @xcite . other alternatives such as the use of equations of state could be employed but in this case it is necessary to have a good description of the volume and the density behavior at different temperatures and pressures which is not an easy task . another common solution has been to consider the internal pressure as a substitute for the solubility parameters @xcite but these two concepts do not describe the same phenomena especially under different thermodynamic conditions @xcite . as an alternative for the study of this kind of complex systems , multiscale numerical simulation has shown to be a very promissing option @xcite . as different time and length scales and a big number of components are involved in complex fluids , multiscale modeling involving atomistic and mesoscopic approaches has been considered as an attractive combination for their study . the use of these two techniques permits to simulate large complex systems taking advantage of the collective properties at a mesoscopic level with relatively cheap computational requirements , and also to calculate properties at the atomistic level when necessary . as it is well known , the collective and cooperative behavior emerge in systems with many particles making possible the use of coarse grained simulations and scaling concepts for their study . one of these numerical techniques is the dissipative particle dynamic ( dpd ) methodology @xcite which is specially attractive to simulate correctly the hydrodynamics of complex liquids ( for an extensive review of this methodology see for example @xcite ) . even though the use of classical dpd has been applied to simulate different kinds of systems , its use to model and reproduce real behavior in a dpd fluid at different thermodynamic conditions ( different pressures or temperatures ) , not only in a qualitative but also in a quantitatively way , remains a challenge due to the restricted thermodynamic behavior given by the functional expressions for the conservative force employed . some efforts have been done in order to describe the thermodynamic properties of real systems with dpd using modifications of the traditional technique through , for example , the so called multibody - dpd ( mdpd)@xcite . in mdpd the conservative force depends not only on the inter - particle separation but also on the instantaneous local particle density which depends on the positions of all other neighboring particles . for this reason , the conservative force in mdpd is a many - body force . using an improved mdpd model some authors @xcite , have performed simulations for single and multicomponent systems at constant pressure , including a modified version of the andersen barostat to suppress the unphysical volume oscillations due to pressure changes , simplifying the equilibration of the system . nevertheless , the effect of pressure using the classical dpd technique has remained unexplored . in traditional dpd the thermodynamic quantities are obtained using only a single parameter @xmath4 in the conservative force . recently , one of us has presented the methodology to model the effect of temperature in this kind of simulations via a temperature - dependent repulsive @xmath5 parameter @xcite . this parametrization allows one to consider the thermodynamic conditions using information at an atomistic level . in this contribution we present the technique to study the effect of pressure with the dpd approach , following a similar idea . to achieve this , we obtain and analyze first the effect of pressure and temperature on the cohesive energy density @xmath6 for the pure components via atomistic simulations , and then calculate the solubility parameters @xmath7 and the flory - huggins @xmath8 parameters in binary liquid - liquid mixtures at different pressures and temperatures . following the standard dpd methodology , we assume that the temperature and pressure dependence of the dpd @xmath9 parameters could be associated directly with @xmath8 . we probe this direct dependence by modeling the interfacial tension between binary mixtures performing dpd simulations . we use our model to predict the interfacial tension of benzene - water and n - decane - water mixtures at t = 298 , 323 and 373 k and p = 200 , 400 , 600 , 700 , 800 , 1000 and 1200 atmospheres , and compare with reported experimental data @xcite obtaining an excellent agreement . to our knowledge , this is the first time that dpd simulations at different pressures and temperatures are reported with such exactitude . in section ii we describe the general characteristics of dpd methodology and the procedure followed to obtain the pressure dependence of the dpd interaction parameters via the solubility and flory - huggins parameters . in section iii we discuss the effect of pressure and temperature in the solubility parameters and interfacial tension . section iv presents the simulation details and the results obtained for the cohesive density energy @xmath1 and the solubility parameters @xmath2 by atomistic simulations at several pressures and temperatures , as well as for the interfacial tension @xmath0 of benzene - water and n - decane - water mixtures performing dpd coarse grained simulations . finally , some conclusions are discussed in section v. the core structure of the dissipative particle dynamics ( dpd ) simulation method @xcite is fundamentally the same as a classic molecular dynamics ( md ) algorithm but , in dpd , the particles correspond to coarse grained structures representing molecular or atomic clusters instead of individual atoms . the momentum and position of each dpd particle are calculated by solving newton s second law of motion using the total force acting on it at finite time steps . the main difference involving md and dpd methodology is that , in dpd , the functional structure of the interacting force linking any two particles @xmath10 and @xmath11 is constituted by the sum of three components : a conservative ( @xmath12 ) , a dissipative ( @xmath13 ) , and a random ( @xmath14 ) force . the total force felt by particle @xmath10 due to the presence of all other particles is thus @xmath15\ ] ] the time evolution of velocities and positions are calculated from @xmath16 and @xmath17 , where @xmath18 is the velocity and @xmath19 the position of particle @xmath10 . a soft , linearly decaying repulsive interaction is used for the conservative force between each particle pair : @xmath20 where @xmath21 and @xmath22 . in this expression @xmath4 is known as the repulsive dpd parameter acting between the pair of particles and @xmath23 represents a cutoff distance . the dissipative and the random forces are defined as : @xmath24\hat { \bm{r}}_{ij } \,,\\ \bm{f}_{ij}^r & = & \sigma\omega^r(r_{ij})\xi_{ij}\hat{\bm{r}}_{ij}\,,\end{aligned}\ ] ] respectively . here , @xmath0 is the noise amplitude and @xmath25 is the friction coefficient . to guarantee that a boltzmann distribution is achieved at equilibrium , @xmath0 and @xmath25 are related by @xmath26 as a consequence of the fluctuation - dissipation theorem @xcite , keeping the temperature internally fixed . here @xmath27 is boltzmann s constant , @xmath28 is a random number distributed between @xmath29 and @xmath30 with gaussian distribution and unit variance , and @xmath31 is the relative velocity between the particles . the weight functions @xmath32 and @xmath33 depend on distance and vanish for @xmath34 , they are commonly established as : @xmath35 ^ 2 = \begin{cases } ( 1-r_{ij}/r_c)^2 , & \text{$(r_{ij } \leq r_c)$ } \\ 0 , & \text{$(r_{ij } > r_c)$}. \end{cases}\ ] ] the cutoff radius @xmath23 is commonly chosen as the reduced unit of length , @xmath36 , and corresponds to the intrinsic length scale of the dpd model . the masses of all particles are chosen to be equal to the reduced unit of mass @xmath37 . the cut - off radius @xmath23 could be written as : @xmath38 where @xmath39 is the bead volume , @xmath40 is the coarse graining factor ( it is the number of water molecules per dpd particle ) and can be considered as a real - space renormalization factor , and @xmath41 is the number of dpd beads in a lattice box of side @xmath23 and volume @xmath42 . for example , for @xmath43 , @xmath44 , and @xmath45@xmath46 , we have @xmath47 . as we have mentioned , thermodynamic quantities in dpd are calculated using only the conservative force @xcite , and for this reason a correct estimation and scaling for the repulsive @xmath4 parameter is fundamental to simulate realistic systems at different thermodynamic conditions . the effect of changes in the random and dissipative forces has been recently studied @xcite showing that the average contribution of these two forces to the pressure is negligible . for this reason we will focus only on the conservative parameter @xmath4 . the dpd parametrization needs to be appropriately selected to give an accurate estimate of the real system . for sufficiently large number densities the dpd equation of state for a monocomponent system is @xcite @xmath48 where @xmath49 is a numerical constant and @xmath50 are the interaction parameters for identical dpd particles . this equation shows the dependence on @xmath51 of the interaction parameters . to describe the real system , fluctuations in the liquid must be described adequately , and these are determined by the compressibility of the system . the definition for the dimensionless isothermal compressibility is @xmath52 where @xmath53 is the number density of molecules in the medium and @xmath54 is the usual isothermal compressibility , @xmath55 . the dimensionless isothermal compressibility for water at standard conditions is @xmath56 , and may be considered to be constant in the pressure range from @xmath57 to @xmath58 mpa to be considered @xcite . using the dpd equation of state the following relationship emerges : @xmath59 then , the conservative force parameter for particles of the same type , @xmath50 , may be obtained as @xmath60 k_b t. \label{aii}\ ] ] these equations give the relationship between the mesoscopic model parameter and the real compressibility of the system . the free energy density for a dpd monocomponent system is @xmath61 and for a two - component system @xmath62 where the indices @xmath63 refer to species @xmath10 and @xmath11 respectively . writing @xmath64 , @xmath65 and @xmath66 , we have @xmath67 where @xmath68 has been identified with the well known flory - huggins parameter . groot and warren @xcite found that there is a linear relation between @xmath3 and @xmath69 given by @xmath70 for @xmath71 . we propose that the dependence on temperature and pressure could be then included in the dpd repulsive parameter via the pressure and temperature dependent flory - huggins parameter @xmath8 , and can be generalized as @xmath72 where @xmath73 is a numerical constant @xcite and @xmath8 is given by @xmath74 ^ 2 . \label{chipt}\ ] ] here , @xmath75 is the solubility parameter of the @xmath10-th particle , which we take to be that of the species it represents even if the particle does not cover a full molecule of it ( _ vide infra _ for simulation details ) , and @xmath76 is the partial molar volume of particle @xmath10 at temperature @xmath77 and pressure @xmath51 . interfacial tensions are obtained by the irving - kirkwood @xcite method expressing the surface tension from the local components of the pressure tensor . we use the virial theorem route @xcite and the components of the pressure tensor @xmath78 obtained from the total conservative force , and time averages over the simulation time . for an interface at the x - y plane we have @xmath79\}dz \label{tensionsim}\end{aligned}\ ] ] were @xmath80 is the length of the simulation box in the z - direction , the brackets indicate time average over the integration phase of the simulation , and @xmath81 are the temperature and pressure dependent component of the pressure tensor in the @xmath10-direction ; similarly for other interfaces . it is known that the effect of temperature in the interfacial tension is much greater than that of the pressure . usually , the maximum change in the interfacial tension in binary mixtures as a function of pressure , in the range @xmath57 to @xmath58 mpa , is around @xmath82 to @xmath83 dynes / cm ; for this reason the effect of pressure is difficult to observe in the laboratory . experimental studies using binary mixtures ( benzene / water and n - decane / water ) have suggested the following phenomenological relation over a range of 20 to 150 @xmath84c and for pressures of 200 to 700 atm @xcite : @xmath85 where @xmath51 is the pressure in atmospheres and @xmath86 in @xmath87c . the values of the coefficients @xmath88 , @xmath89 and @xmath90 depend on the kind of liquids in the system . some authors @xcite have reported that the pressure coefficient @xmath89 is very small and positive for the case of benzene / water and n - decane / water systems for pressures lower than 700 atmospheres , but experimental results obtained in @xcite for the benzene / water system at pressures higher than 700 atmospheres show a negative pressure coefficient as a consequence of the increased influence in the solubility of the phases when the pressure is increased . the decrease in the interfacial tension when the pressure is increased has been reported also for binary mixtures of co@xmath91/alkane systems @xcite . the forces involved in the interfacial tension of multicomponent fluids are intimately related to the solubility and flory - huggins parameters . the concept of the solubility parameter @xmath2 introduced by hildebrand and scott in 1950 @xcite has been fundamental in the analysis of mixtures and pure compounds in many industrial areas . it is usually assumed that the solubility parameter consists of a linear combination of contributions from dispersion interactions , polar interactions and hydrogen bonding @xcite : @xmath92 commonly , this parameter is calculated at atmospheric pressure via the heat of vaporization , but in many cases high pressures and temperatures are involved in industrial processes and this procedure could lead to a poor estimate for @xmath2 . a more adequate estimation of @xmath7 under different conditions of pressure and temperature is necessary , and we can express it in terms of the cohesive energy of the liquid @xmath93 and its molar volume @xmath94 as @xmath95 where the cohesive energy density ( @xmath1 ) is a measure of the whole molecular cohesion per unit volume . the effect of pressure and temperature on the solubility parameter has been estimated by null and palmer @xcite for vapor pressure calculations as @xmath96^{1/2 } \label{deltanull}\ ] ] where @xmath97 is the molar volume of the liquid phase , @xmath98 the gas constant and @xmath99 , @xmath100 are the constants of antoine s equation @xcite @xmath101 with @xmath51 and @xmath77 the pressure and temperature of the vapor phase . antoine s equation permits a good estimation of the vapor pressure as a function of temperature , but unfortunately the constants are not available for all systems.those for the species considered in this work are shown in table [ table0 ] . for the dependence of the cohesive energy on pressure and temperature we use barton s expression @xcite @xmath102 where @xmath103 and @xmath104 are the internal energy of the vapor and liquid phases respectively @xcite , and @xmath105 is the residual internal energy . macdonald and hyne @xcite determined the cohesive energy density of alcohol - water mixtures at different temperatures and at atmospheric pressure finding that the cohesive energy density has a monotonous behavior with p over a very large range . in this contribution we present an alternative to obtain all this information via multi - scale simulations using molecular dynamics and dissipative particle dynamics simulations and compare the results with the experimental data . the solubility parameters @xmath7 and cohesive energy densities @xmath6 at temperatures @xmath106 and pressures @xmath107 atm for benzene , n - decane and water were calculated performing atomistic molecular dynamics simulations . we consider periodic cells of amorphous fluid structures , using the amorphous cell module of the materials studio suite @xcite . the dimension of the simulation box was chosen in all cases to be @xmath108 @xmath109 and the compass force field was used to model the interatomic interactions . we developed @xmath110 dynamics simulations in order to equilibrate the density of the system at the temperature and pressure of interest . we then used the discover molecular dynamics engine to evolve the systems at these thermodynamic conditions obtaining statistically independent structures . from this information we obtained the total solubility parameter @xmath2[j/@xmath111 as well as the electrostatic @xmath112[j/@xmath111 and dispersive @xmath113[j/@xmath111 contributions for each component , the cohesive energy density @xmath1[j / m ] , the density @xmath114[gr/@xmath111 , and the molar volume @xmath115[@xmath116 . the results for the solubility parameter @xmath2 and molar volume @xmath115 are presented in table [ table1 ] . figure [ psvsp ] shows our results for water , benzene and n - decane compared to the theoretical prediction using the null - palmer equation and the molar volume obtained by md simulations . the b , c antoine s constants were taken from @xcite and summarized in table [ table0 ] . we can observe , as is expected , that at fixed temperature the solubility parameter increases only slightly with pressure . the agreement is striking , with the benefit that our methodology may be employed for any species . [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] [ table3 ] even though the maximum experimental change in interfacial tension is less than 3 [ dynes / cm ] over the entire pressure range @xcite due to the low mutual miscibility between the liquids , as can be observed in the density profiles shown in figure [ dpdpd ] our simulations can capture this effect because of the direct dependence of the conservative force with the solubility parameters . differences between benzene - water and n - decane / water mixtures indicate that there are weaker attractive forces between n - decane and water than between benzene and water , which is evident from the @xmath117 values obtained . a methodology is presented for modeling the pressure dependence of interfacial tension in binary mixtures using a combination of molecular dynamics and dissipative particle dynamics simulations . the cohesive energy density and the solubility parameters are calculated at different temperatures and pressures using molecular dynamics simulations and the thermodynamic properties of real systems are reproduced with excellent agreement . the pressure and temperature dependence of the flory - huggins and repulsive interaction parameters are also obtained and the experimental behavior of the considered systems is reproduced . the methodology described works in a very large range of parametric conditions , is sensitive to changes of only a few dynes / cm , and is amenable to applications were experimentation may be difficult or even impossible . we thank dgtic - unam for computational support . mayoral , e ; nahmad - achar , e. parametrisation in dissipative particle dynamics : applications in complex fluids , in _ selected topics of computational and experimental fluid mechanics _ ; klapp et al . ( springer international publishing : switzerland , 2015 . gregorowicz , j. ; kiciak , k ; malanowski , s. vapour pressure data for 1-butanol , cumene , n - octane and n - decane and their statistically consistent reduction with the antoine equation . _ fluid phase equilib . _ , * 1987 * , 38 , 97 - 107 gama , a. ; balderas , m.a . ; hernndez , j.d . ; prez e. the role of the dissipative and random forces in the calculation of the pressure of simple fluids with dissipative particle dynamics . _ , * 2015 * , 188 , 76 - 81 mejia , a. ; segura , h. ; vega , l.f . ; wisniak , j. simultaneous prediction of interfacial tension and phase equilibria in binary mixtures : an approach based on cubic equations of state with improved mixing rules . _ fluid phase equilib . _ , * 2005 * , 227 , 225 - 238 georgiadis , a. ; llovell , f. ; bismarck , a. ; blas , f.j . ; galindo , a. ; maitland , g.c . ; martin trusler j.p . ; jackson , g. interfacial tension measurements and modelling of ( carbon dioxide plus n - alkane ) and ( carbon dioxide plus water ) binary mixtures at elevated pressures and temperatures . _ j. supercritical fluids _ , * 2010 * , 55 , 743 - 754	we study and predict the interfacial tension , solubility parameters and flory - huggins parameters of binary mixtures as functions of pressure and temperature , using multiscale numerical simulation . a mesoscopic approach is proposed for simulating the pressure dependence of the interfacial tension for binary mixtures , at different temperatures , using classical dissipative particle dynamics ( dpd ) . the thermodynamic properties of real systems are reproduced via the parametrization of the repulsive interaction parameters as functions of pressure and temperature via molecular dynamics simulations . using this methodology , we calculate and analyze the cohesive density energy and the solubility parameters of different species obtaining excellent agreement with reported experimental behavior . the pressure - and temperature - dependent flory - huggins and repulsive dpd interaction parameters for binary mixtures are also obtained and validated against experimental data . this multiscale methodology offers the benefit of being applicable for any species and under difficult or non - feasible experimental conditions , at a relatively low computational cost .
spatially discretized partial differential equations ( or , equivalently , chains of coupled ordinary differential equations ) have attracted considerable attention recently . one of the issues that has been vigorously debated and that will concern us in this paper , is whether discrete systems can support solitary waves travelling without losing energy to resonant radiation and decelerating as a result . we address this issue for one of the prototype models of nonlinear physics , the @xmath0-theory : @xmath1 the @xmath0-equation ( [ continuous - phi4 ] ) is lorentz - invariant , and so the existence of the travelling kink @xmath2 where @xmath3 and @xmath4 , is an immediate consequence of the existence of the stationary kink for @xmath5 . on the other hand , if we discretize equation ( [ continuous - phi4 ] ) in @xmath6 , @xmath7 the translation and lorentz invariances are lost , and the existence of the travelling kink ( and even of an arbitrarily centred stationary one ) becomes a nontrivial matter . in equation ( [ phi-4 ] ) , @xmath8 , @xmath9 , @xmath10 , @xmath11 is the lattice spacing , and the nonlinearity @xmath12 satisfies the continuity condition @xmath13 we restrict ourselves to symmetric discretizations , i.e. @xmath14 equation ( [ continuous - phi4 ] ) results from ( [ phi-4 ] ) in the continuum limit , where @xmath15 , @xmath16 and @xmath17 . in this limit , the truncation error of the taylor series is @xmath18 . we shall be concerned with monotonic kink solutions of ( [ phi-4 ] ) : @xmath19 for all @xmath20 . as @xmath17 , such monotonic discrete kinks approach the continuous kink ( [ travelling ] ) . the most common , one - site discretization of the nonlinearity function is given by @xmath21 it is a well established fact @xcite , however , that the discrete klein - gordon equation ( [ phi-4])+([nonlinearity - phi4 ] ) admits only a countable set of stationary monotonic kinks with the boundary conditions @xmath22 physically , this fact is related to the presence of the peierls - nabarro barrier , an effective potential periodic with the spacing of the lattice . half of the stationary kinks are centred at the minima ( the on - site kinks ) and the other half ( the off - site kinks ) at the maxima of the peierls - nabarro potential . there are no continuous families of stationary discrete kinks of the form @xmath23 , with @xmath24 a free parameter , which would interpolate between the two solutions . in an abuse of terminology , we will be calling such families `` translationally invariant kinks '' although , in the first place , translation invariance is a property of an equation rather than a solution , and in the second , all lattice equations are of course _ not _ translationally invariant . as for propagating waves , of special importance are kinks moving at constant speed and without the emission of radiation . we will be referring to such kinks , i.e. solutions of the form @xmath25 where @xmath26 is a monotonically growing function satisfying the boundary conditions ( [ bcs ] ) , as _ sliding kinks _ , to emphasise the fact that they do not experience any radiative friction . being an obstacle to the translationally invariant " kinks , the peierls - nabarro barrier is also detrimental to the existence of sliding kinks at least for small @xmath27 ( see reviews in @xcite and @xcite ) . in an attempt to find a discrete model with translationally invariant " and sliding kinks , speight and ward @xcite considered a hamiltonian discretization of the form @xmath28 in the static limit , the corresponding energy admits a topological lower bound which is saturated by a first- ( rather than second- ) order difference equation . this equation is readily shown to have a one - parameter continuous family of stationary kink solutions @xmath23 for @xmath29 ( see proposition 1 in @xcite ) . the parameter @xmath24 of the family defines the position of the kink relative to the lattice . since all members of the family have the same ( lowest attainable ) energy , the stationary kink experiences no peierls - nabarro barrier . as for travelling kinks , speight and ward s numerical simulations revealed that although moving kinks in this model do lose energy to cherenkov radiation and decelerate as a result , this happens at a slower rate than a similar process in equation ( [ nonlinearity - phi4 ] ) ( see figures 4 and 5 in @xcite ) . another line of attack was chosen by bender and tovbis @xcite who proposed a different discretization supporting a continuous family of arbitrarily centred stationary kinks : @xmath30 in this case , the family arises due to the suppression of the stationary kink s resonant radiation . in fact , the family of stationary kinks can be found explicitly as @xmath31,\ ] ] where @xmath32 for all @xmath33 . ( the solution ( [ al - kink ] ) coincides with the stationary dark soliton of the repulsive ablowitz - ladik equation @xcite . ) finally , the nonlinearity @xmath34 was introduced by kevrekidis @xcite , who demonstrated the existence of a two - point invariant and hence a first - order difference equation associated with the stationary equation . consequently , the discretization ( [ nonlinearity - panos ] ) also supports a continuous family of stationary kinks for all @xmath35 $ ] with some @xmath36 . ( for general discussion , see @xcite , @xcite and @xcite . ) a relevant property of the model ( [ nonlinearity - panos ] ) , which is related to the existence of a two - point invariant @xcite and indicates some additional underlying symmetry , is the conservation of momentum . ( see also @xcite . ) since the reasons for the nonexistence of `` translationally invariant '' kinks and of sliding kinks are apparently related ( the breaking of symmetries of the underlying continuum theory or , speaking physically , the presence of the peierls - nabarro barrier ) , the availability of translation - invariant " stationary kinks in the models ( [ nonlinearity - speight ] ) , ( [ nonlinearity - tovbis ] ) and ( [ nonlinearity - panos ] ) suggests that they might have sliding kinks as well . it is the purpose of the present study to find out whether this is indeed the case . we shall analyse the persistence of continuous families of stationary kinks @xmath23 for nonzero velocities ; in other words , examine the existence of solutions of the form @xmath37 where @xmath38 is a monotonically growing function satisfying ( [ bcs ] ) , and @xmath39 . we develop an accurate numerical test in the limit @xmath17 which shows whether or not standing and travelling kinks of the discrete @xmath0 model ( [ phi-4 ] ) bifurcate from the exact kink solutions ( [ travelling ] ) of its continuous counterpart ( [ continuous - phi4 ] ) . the analysis of this bifurcation poses a singular problem in perturbation theory which can be analysed using two ( inner and outer ) matched asymptotic scales on the complex plane @xcite . in particular , the nonvanishing of the stokes constant in the inner asymptotic equation serves as a sufficient condition for the non - existence of continuous solutions of the difference equations @xcite . our test will be based on computing the stokes constant for the differential - difference equation underlying the lattice system . we will examine all four discretizations of the @xmath0 theory mentioned above , i.e. equations ( [ nonlinearity - phi4 ] ) , ( [ nonlinearity - speight ] ) , ( [ nonlinearity - tovbis ] ) and ( [ nonlinearity - panos ] ) . since translationally invariant stationary kinks @xmath40 do exist for the three exceptional nonlinearities ( [ nonlinearity - speight ] ) , ( [ nonlinearity - tovbis ] ) and ( [ nonlinearity - panos ] ) , the stokes constant is _ a priori _ vanishing for @xmath5 in these three cases . however , we will show that in all three cases the stokes constant acquires a nonzero value as soon as @xmath27 deviates from zero . it remains nonzero for all @xmath27 except a few isolated values which define the particular velocities of the sliding kinks in the corresponding model . there is one such isolated zero of the stokes constant for the nonlinearity ( [ nonlinearity - speight ] ) and three _ sliding velocites _ for the discretization ( [ nonlinearity - panos ] ) . consequently , the main conclusion of this work is that the sliding kinks , i.e. kinks travelling at a constant speed without the emission of radiation , can occur only at particular values of the velocity . the sliding velocities are , of course , functions of the discretization spacing @xmath11 , so that sliding kinks arise along continuous curves on the @xmath41-plane . we conclude this introduction with a remark on a convention adopted in the remainder of this paper namely , that the linear part of the function @xmath12 in ( [ phi-4 ] ) can always be fixed to @xmath42 without loss of generality . indeed , the most general function satisfying ( [ nonlinearity - continuous ] ) and ( [ nonlinearity - symmetry ] ) is @xmath43 , where @xmath44 is arbitrary . since @xmath45 in ( [ phi-4 ] ) is also a free parameter , we can always make a replacement @xmath46 such that @xmath47 . this gives @xmath48 where @xmath49 is a homogeneous polynomial of degree 3 which is independent of the parameter @xmath11 . the plan of this paper is as follows . in the next section ( section [ section2 ] ) we review the construction of the outer and inner asymptotic solutions in the limit @xmath50 . section [ section3 ] contains details of the numerical computation of the stokes constants while the last section ( section [ section4 ] ) summarises the results of our work . we are looking for a sliding - kink solution of the discrete @xmath0 models ( [ phi-4 ] ) in the form @xmath51 where @xmath52 is assumed to be a twice differentiable function of @xmath53 , that satisfies the differential advance - delay equation @xmath54 with the boundary conditions @xmath55 as @xmath56 . the velocity @xmath27 is assumed to be smaller than 1 in modulus . if a solution to this boundary - value problem ( i.e. a heteroclinic orbit ) exists , then the parameter @xmath24 is arbitrary due to the translation invariance of the advance - delay equation ( [ advance - delay ] ) . the scaling parameter @xmath11 ( which stands for the lattice step - size ) can be used to reduce equation ( [ advance - delay ] ) to a singularly perturbed differential equation as @xmath17 @xcite . formal asymptotic solutions of the problem ( [ advance - delay ] ) can be constructed at the inner and outer asymptotic scales . the formal series represent convergent asymptotic solutions of the singular perturbation problem only if the stokes constants are all zero @xcite . asymptotic analysis beyond all orders of perturbation theory was pioneered by kruskal and segur @xcite and has been utilised by many authors . it was extended by pomeau et . @xcite to allow the computation of radiation coefficients from borel summation of series rather than from the numerical solution of differential equations . essentially the same method has been applied to different problems by grimshaw and joshi @xcite and tovbis and collaborators @xcite . in this paper , we shall work with formal inner and outer asymptotic series for the problem ( [ advance - delay ] ) without attempting rigorous analysis of their asymptoticity . assuming that the solution @xmath52 is a real analytic function of @xmath57 , we consider the taylor series for the second difference in a strip @xmath58 since the cubic polynomial @xmath59 satisfies the continuity and symmetry relations ( [ nonlinearity - continuous ] ) and ( [ nonlinearity - symmetry ] ) , the nonlinearity of ( [ advance - delay ] ) can also be expanded in a taylor series in the same strip : @xmath60 where the coefficients @xmath61 depend on even derivatives and even powers of odd derivatives of @xmath52 , and also @xmath62 . the differential advance - delay equation ( [ advance - delay ] ) can thus be written as @xmath63 for @xmath64 , equation ( [ singular - perturbation ] ) becomes the travelling wave reduction of the continuous model ( [ continuous - phi4 ] ) , with the explicit solution @xmath65 we will search for solutions of equation ( [ singular - perturbation ] ) of the form @xmath66 substituting the expansion ( [ perturbation - series ] ) into ( [ singular - perturbation ] ) we get , at order @xmath67 , @xmath68 where the linearised operator @xmath69 is given by @xmath70 and @xmath71 are polynomials in @xmath72 and their derivatives . the kernel of @xmath73 is one - dimensional , and spanned by an even eigenfunction @xmath74 . the rest of the spectrum of @xmath73 is positive . it is not difficult to prove by induction that if @xmath75 are all odd in @xmath57 for @xmath76 , the nonhomogeneous term @xmath71 is also odd in @xmath57 and hence , by the fredholm alternative , there exists a unique odd bounded solution @xmath77 for @xmath53 . moreover , since @xmath71 decays to zero exponentially fast as @xmath78 , the function @xmath77 is also exponentially decaying for any @xmath79 . the perturbation @xmath80 in particular satisfies the nonhomogeneous equation @xmath81 where the numerical coefficients depend on whether the nonlinearity function @xmath82 is given by ( [ nonlinearity - phi4 ] ) , ( [ nonlinearity - speight ] ) , ( [ nonlinearity - tovbis ] ) or ( [ nonlinearity - panos ] ) : @xmath83 the odd bounded solution @xmath80 of the nonhomogeneous equation ( [ first - order ] ) is : @xmath84 where @xmath85 the hat in the series ( [ perturbation - series ] ) indicates that the series is formal , i.e. it may or may not converge @xcite , depending on the choice of @xmath27 and @xmath49 in equation ( [ advance - delay ] ) . we shall be referring to ( [ perturbation - series ] ) as the outer asymptotic expansion . the leading - order term ( [ leading - order ] ) of the outer expansion ( [ singular - perturbation ] ) has poles at @xmath86 , where @xmath20 . we apply the scaling transformation @xmath87 to equation ( [ advance - delay ] ) in order to study the convergence of the formal asymptotic solution ( [ perturbation - series ] ) near the pole @xmath88 ( see @xcite ) . this yields the following differential advance - delay equation for @xmath89 : @xmath90 the following are the cubic functions @xmath49 for each of the four discretizations that we deal with in this paper : @xmath91 ; \\ \fl \quad \textrm{bender - tovbis ( \ref{nonlinearity - tovbis } ) } : & q & = \frac{1}{4 } \psi^2(\zeta ) \left [ \psi(\zeta+1 ) + \psi(\zeta-1 ) \right ] ; \\ \fl \quad \textrm{kevrekidis ( \ref{nonlinearity - panos } ) } : & q & = \frac{1}{8 } \left [ \psi^3(\zeta+1 ) + \psi^2(\zeta+1)\psi(\zeta-1 ) + \psi(\zeta+1)\psi^2(\zeta-1 ) \right . \\ & \phantom{q } & \phantom{= } \left . + \psi^3(\zeta-1 ) \right].\end{aligned}\ ] ] we note that the heteroclinic orbit becomes small as @xmath17 under the normalization ( [ scaling ] ) : if @xmath55 as @xmath92 , then @xmath93 as @xmath94 . the formal asymptotic series ( [ perturbation - series ] ) in the new variables ( [ scaling ] ) becomes a new formal series @xmath95 where each term @xmath96 can be expanded in a formal series in descending powers of @xmath97 . in particular , the leading - order function @xmath98 has the general form @xmath99 by comparing the series ( [ perturbation - series - inner ] ) and ( [ leading - order - inner ] ) with the solutions ( [ leading - order ] ) and ( [ first - order - explicit ] ) in variables ( [ scaling ] ) , we note the correspondence : @xmath100 we shall be referring to ( [ perturbation - series - inner ] ) as the inner asymptotic expansion . the odd powers of @xmath11 in the inner asymptotic expansion ( [ perturbation - series - inner ] ) appear due to the matching conditions with the outer asymptotic expansion ( [ perturbation - series ] ) under the scaling ( [ scaling ] ) , as well as due to the non - zero boundary conditions for the heteroclinic orbits @xmath93 as @xmath101 . convergence of the formal inverse - power series ( [ leading - order - inner ] ) for the leading - order solution @xmath98 depends on the values of the stokes constants @xcite . computation of the stokes constants is based on borel laplace transforms of the inner equation ( [ inner ] ) @xcite . assuming continuity in @xmath11 , we study the leading - order solution @xmath102 of the truncated inner equation @xmath103 by substituting the series ( [ leading - order - inner ] ) into equation ( [ inner - limiting ] ) , one can derive a recurrence relation between the coefficients in the set @xmath104 . the stokes constants can be computed from the asymptotic behavior of the coefficients @xmath105 for large @xmath106 . alternatively , the leading - order solution @xmath107 and the stokes constants can be defined by using the borel laplace transform : @xmath108 the choice of the contour of integration @xmath109 determines the domain of @xmath107 in the complex @xmath97-plane . we define two solutions @xmath110 and @xmath111 , which lie on the stable and unstable manifolds respectively , such that @xmath112 we note that the stable and unstable solutions tend to the stationary point at the origin , since the heteroclinic orbits connect the stationary points at @xmath113 which move to the origin as @xmath17 . the three stationary points coalesce to become a degenerate stationary point at the origin within the truncated inner equation ( [ inner - limiting ] ) . the borel laplace transform ( [ laplace ] ) produces the stable solution @xmath110 when the contour of integration @xmath114 lies in the first quadrant of the complex @xmath115-plane and extends from @xmath116 to @xmath117 . similarly , it produces the unstable solution @xmath111 when the contour of integration @xmath118 lies in the second quadrant . we choose the integration contours in such a way that @xmath119 as @xmath120 , so that the solutions @xmath110 and @xmath111 are defined by ( [ laplace ] ) for all complex @xmath97 with @xmath121 . the borel transform @xmath122 satisfies the following integral equation , which follows from ( [ inner - limiting ] ) and ( [ laplace ] ) : @xmath123.\ ] ] here , @xmath124 $ ] denotes a double convolution of @xmath125 with itself ( in this case , the hat is used to denote an operator ) . we list below the convolutions @xmath124 $ ] for each of the four models under consideration : @xmath126 + 3 \left\ { \cosh p \ ; \left [ v(p ) \ast v(p ) \right ] \right\ } \\ \fl & \ast v&(p ) + 3 \left [ \cosh p \ ; v(p ) \right ] \ast v(p ) \ast v(p ) + 2 v(p ) \ast v(p ) \ast v(p ) ; \\ \fl \quad \textrm{bender - tovbis ( \ref{nonlinearity - tovbis } ) } : & 2 \hat{q } & = \left [ \cosh p \ ; v(p ) \right ] \ast v(p ) \ast v(p ) ; \\ \fl \quad \textrm{kevrekidis ( \ref{nonlinearity - panos } ) } : \quad & 4\hat{q } & = \cosh p \ ; \left [ v(p ) \ast v(p ) \ast v(p ) \right ] \\ \fl & & \phantom{= } + \left [ \cosh p \ ; v(p ) \right ] \ast \left [ \rme^p\ , v(p ) \right ] \ast \left [ \rme^{-p}\,v(p ) \right],\end{aligned}\ ] ] where the asterisk @xmath127 denotes the convolution integral for the borel laplace transform : @xmath128 and the integration is performed from the origin to the point @xmath115 on the complex plane , along the contour @xmath109 . the inverse power series ( [ leading - order - inner ] ) for the limiting solution @xmath107 becomes the following power series for the borel transform @xmath122 : @xmath129 where @xmath130 . the hat denotes a formal series which might only converge for some values of @xmath115 . the virtue of the integral form ( [ integral - limiting ] ) is that the limiting behavior of @xmath131 for large @xmath106 can be related to singularities of @xmath122 , which in turn correspond to the oscillatory tails of @xmath107 . if the sliding kink exists , the inverse - power series for @xmath107 will converge for all @xmath132 such that @xmath121 . this implies that the stable and unstable solutions @xmath110 and @xmath111 coincide , i.e. that the contour @xmath114 in the right half of the complex @xmath115-plane can be continously deformed to the contour @xmath118 in the left half - plane ( see figure [ pathofintegrationfig ] ) . if , however , there are any singularities between the two contours , then a continuous deformation is possible only if the residues are zero . the residues are proportional to the values of the stokes constants . when the stokes constants are nonzero , the formal power series ( [ laplace - power ] ) for the solution @xmath122 of the integral equation ( [ integral - limiting ] ) diverges for some values of @xmath115 in the sector between the contours @xmath114 and @xmath118 . the borel transform @xmath122 is singular near the points in the @xmath115-plane where the coefficient in front of @xmath122 on the left - hand side of the integral equation ( [ integral - limiting ] ) vanishes @xcite , except for the point @xmath133 where the right hand side is also zero . that is , singularities occur when @xmath134 . the location of these singularities is important because the stable and unstable solutions are not , in fact , uniquely defined by ( [ stable - unstable ] ) ; different solutions are generated depending on where the contours lie relative to the singularities of @xmath122 with @xmath135 . exploiting this nonuniqueness , we wish to choose the contours @xmath114 and @xmath118 to lie above all the singularities with nonzero real part ; this will minimise the number of singularities between the stable and unstable solutions . it is not difficult to show that the contour @xmath114 extending from 0 to @xmath136 can be chosen in such a way , i.e. so that that there are no singularities between it and the imaginary axis . indeed , assume , for definiteness , that @xmath137 . let @xmath138 be the number of positive roots of the equation @xmath139 and denote the real and imaginary parts of @xmath140 by @xmath141 and @xmath142 : @xmath143 . in the @xmath144-plane , consider a rectangular region @xmath145 bounded by the horizontal segments @xmath146 and @xmath147 at the top and bottom , and vertical segments @xmath148 and @xmath149 on the left and right . here @xmath106 is any positive integer greater than @xmath150 and @xmath151 is taken to be small . using the argument principle , we can count the number of ( complex ) roots of the equation @xmath152 in the region @xmath145 . we have @xmath153 on the right lateral side , where @xmath154 , this becomes @xmath155 as we move from @xmath147 to @xmath156 , the numerator in ( [ arg2 ] ) will change sign @xmath138 times . in a similar way , moving down along the left side there will be @xmath138 more zero crossings , while no zero crossings will occur along the horizontal segments . this means that the argument can change by no more than @xmath157 and hence there are at most @xmath138 roots in the region @xmath145 , no matter how large @xmath106 is . similarly , we can show that the equation @xmath158 has no more than @xmath159 roots in the region @xmath145 , if @xmath159 is the number of positive roots of @xmath160 . the upshot is that for any finite @xmath27 , there are only a finite number of singularities with small real parts ; the singularities can not accumulate to the imaginary axis . for @xmath161 , the singularities with nonzero real parts lie on the curves @xmath162 accordingly , in order for the integration contours @xmath114 and @xmath118 to lie above these singularities , they must be curvilinear ( and not just rays ) as shown in figure [ pathofintegrationfig ] . having chosen the contours @xmath114 and @xmath118 to lie above the singularities in the first and second quadrants respectively , the only singularities of @xmath122 that determine whether the stable solution @xmath110 can be continuously transformed into the unstable solution @xmath111 are those at non - zero pure imaginary values of @xmath115 . we will be referring to these values as resonances . the set of resonances @xmath163 is defined by the transcendental equation @xmath164 when @xmath165 , the set @xmath166 is infinite - dimensional and can be described explicitly : @xmath167 let @xmath168 be the smallest imaginary root in the set @xmath163 . it is clear from ( [ resonances ] ) that @xmath169 for @xmath170 , so that @xmath171 as @xmath172 and @xmath173 as @xmath174 . the set of resonances @xmath163 is finite - dimensional for @xmath170 and it consists of only one root @xmath168 for @xmath175 , where @xmath176 . due to the resonances , a function @xmath107 that satisfies the truncated inner equation ( [ inner - limiting ] ) may have oscillatory tails as @xmath177 . adding the solutions of equation ( [ inner - limiting ] ) linearised about @xmath98 , the general bounded solution of ( [ inner - limiting ] ) in the limit @xmath178 can be represented as @xcite : @xmath179 here , @xmath98 is given by the power series ( [ leading - order - inner ] ) ; @xmath180 are coefficients which we will be referring to as amplitudes in what follows ; @xmath181 are roots of ( [ resonances ] ) for @xmath182 , and the functions @xmath183 , @xmath184 , satisfy the linearised truncated inner equation ( [ inner - limiting ] ) . in particular , the equation for the leading - order term @xmath185 is @xmath186 where @xmath187 are the partial derivatives of @xmath188 with respect to its first , second and third argument respectively , evaluated at @xmath189 . if the amplitude @xmath180 is nonzero for some @xmath106 , the formal power series ( [ leading - order - inner ] ) does not converge because the solution ( [ wave - decomposition ] ) does not decay as @xmath190 . the amplitudes @xmath180 are proportional to the stokes constants computed for the formal power series ( [ leading - order - inner ] ) . each oscillatory term in the sum ( [ wave - decomposition ] ) becomes exponentially small in @xmath11 when we transform from @xmath97 to @xmath57 using the transformation ( [ scaling ] ) . since @xmath168 is the element of @xmath163 with the smallest imaginary part , it follows that the @xmath191 term dominates the sum in ( [ wave - decomposition ] ) when the transformation ( [ scaling ] ) is made ( unless @xmath192 ) . when @xmath175 , where @xmath176 , it is the _ only _ term in the sum since the resonant set @xmath163 consists of just the one root @xmath168 . we shall , therefore , only be concerned with the leading - order stokes constant , which multiplies the function @xmath193 . if @xmath98 is given by the power series ( [ leading - order - inner ] ) , the solution of the linearized equation ( [ inner - limiting - first - order ] ) can also be represented by a formal power series : @xmath194 where we can set @xmath195 due to the linearity of ( [ inner - limiting - first - order ] ) . substituting ( [ leading - order - inner ] ) and ( [ power - series - phi ] ) into ( [ inner - limiting - first - order ] ) and using ( [ resonances ] ) , the coefficient @xmath196 can be determined from @xmath197 + { \mathcal{o}}(\zeta^{r-3 } ) = 0.\end{aligned}\ ] ] in this equation , the coefficient of each power of @xmath97 should be set to zero . in order to set the coefficent in front of the first term to zero in the situation where @xmath161 , we must choose @xmath198 . the second term then gives @xmath199 after which all the other coefficients @xmath200 , @xmath201 , , can be computed recursively . on the other hand , in the situation with @xmath165 , we have @xmath202 and the coefficient in front of @xmath203 is zero regardless of the value of @xmath204 . setting the coefficient in front of @xmath205 to zero requires that we choose either @xmath206 or @xmath207 , and hence we have two different descending - power series , one starting with @xmath208 and the other one with @xmath209 . we shall focus on the former as it dominates the latter in the limit @xmath210 . again , the succeeding terms in ( [ power - series - phi ] ) are determined recursively . thus , we have established that the leading - order oscillatory term in the expansion ( [ wave - decomposition ] ) behaves as @xmath211\rme^{- \rmi k_1\zeta } & for $ c \neq 0 $ and \\ \alpha_1\left[\zeta^3 + b_1\zeta^2 + { \mathcal{o}}{\left(\zeta^1\right)}\right]\rme^{-2\pi\rmi\zeta } & for $ c = 0$. } \label{tails}\ ] ] for @xmath161 , the two leading order terms in the expression above are generated by , respectively , a simple pole and a logarithmic singularity of the borel transform @xmath122 at @xmath212 . for @xmath165 they are generated by a quadruple pole of @xmath122 at @xmath213 . from the fact that @xmath214 is an even function of @xmath115 , we deduce the structure of this function near the poles : @xmath215 as @xmath216 . here @xmath217 and @xmath218 are the leading - order stokes constants for @xmath161 and @xmath165 , respectively ; @xmath219 and @xmath220 are independent of @xmath115 , and @xmath221 stands for terms with even slower growth as @xmath222 . to show that these singularities do indeed give rise to the oscillatory tails in ( [ tails ] ) , we compare the two integrals @xmath223 and @xmath224 for a given value of @xmath97 . to this end , we deform the paths of integration @xmath114 and @xmath118 to @xmath225 and @xmath226 respectively , without crossing any singularities . this is illustrated in figure [ pathofintegrationfig ] . there are two contributions to the difference @xmath227 . the first comes from integrating around the pole at @xmath212 , and is equal to @xmath228 times the residue of the function @xmath229 at @xmath212 , determined from ( [ poles ] ) . the second contribution ( manifest only in the @xmath161 case ) arises because the _ integrand _ increases as the singularity is encircled , since it is a branch point of the logarithm . since @xmath230 can be written as @xmath231 , where @xmath232 , we see that @xmath122 increases by @xmath233 as the branch point @xmath212 is encircled in the @xmath161 case . therefore , the difference in the integrand of ( [ laplace ] ) along the paths @xmath225 and @xmath226 is @xmath234 , which must be integrated along the path of integration from @xmath212 to infinity , to give @xmath235 . ( we have considered the integration on a riemann surface in order to account for branch points . ) adding together the two contributions discussed above , we have @xmath236 \rme^{-\rmi k_1\zeta } & for $ c \neq 0$\\ -\frac{1}{128}[16 \pi^3 \rmi s_1 \zeta^3 \\ \qquad + ( 192\pi^4s_1+\rho)\zeta^2 + { \mathcal{o}}(\zeta^1 ) ] \rme^{-2\pi\rmi\zeta } & for $ c = 0$. } \label{stable - minus - unstable}\ ] ] if we take the limit @xmath237 , then the unstable solution @xmath111 decays to zero as a power law , according to the expansion ( [ leading - order - inner ] ) . thus , the stable solution @xmath110 has the oscillatory tail given by the representation ( [ wave - decomposition ] ) with the amplitude factor @xmath238 similarly , if we take the limit @xmath239 , then the stable solution @xmath110 decays to zero , while the unstable solution @xmath111 has the representation ( [ wave - decomposition ] ) with the amplitude factor given by the negative of expression ( [ alpha-1-expression ] ) . by comparing the other terms on the right - hand side of ( [ stable - minus - unstable ] ) to the corresponding terms in ( [ tails ] ) , @xmath219 and @xmath220 can be uniquely determined . we now match the leading - order singular behaviour of @xmath122 near @xmath240 , given by ( [ poles ] ) , to the formal power series ( [ laplace - power ] ) . expanding the expressions in ( [ poles ] ) as power series gives us @xmath241 p^{2n } & for $ c = 0 $ , } \label{poles - expanded}\ ] ] as @xmath216 . these series converge for all @xmath242 ; in particular , they are valid for @xmath216 , provided @xmath242 . hence we can replace ( [ poles ] ) with ( [ poles - expanded ] ) in this neighbourhood . in ( [ poles - expanded ] ) , the ellipses stand for coefficients of the expansion of terms with a slower growth as @xmath243 which were dropped in ( [ poles ] ) . the discarded terms would modify the coefficients of the power series ( [ poles - expanded ] ) ; however , there are terms which would not be affected by these modifications , namely terms with large @xmath106 . for example , the coefficients proportional to @xmath219 and @xmath220 in ( [ poles - expanded ] ) are a factor of @xmath106 smaller than those proportional to @xmath217 and @xmath218 ; the discarded coefficients would be even smaller . therefore the leading singular behaviour of @xmath122 as @xmath216 is determined just by the large-@xmath106 coefficients of the power series ( [ poles - expanded ] ) , and hence only the large-@xmath106 coefficients should be matched to the coefficients of the expansion ( [ laplace - power ] ) . this gives the stokes constant as a limit of the coefficients @xmath244 of the series ( [ laplace - power ] ) : @xmath245 this formula is used in the next section for numerical computations of the leading - order stokes constant @xmath217 for @xmath161 . note that , since ( [ laplace - power ] ) matches ( [ poles - expanded ] ) in the limit @xmath246 , the formal power series @xmath247 also has radius of convergence @xmath248 . however , the formal inverse - power series @xmath98 diverges for all @xmath97 unless @xmath247 converges everywhere ( which requires that all the stokes constants be zero ) . next , we note that as @xmath249 , the stokes constant @xmath217 does not tend to @xmath218 , its value at @xmath165 . this discontinuity is due to the fact that , as @xmath249 , pairs of simple roots in the resonant set @xmath163 coalesce . ( e.g. @xmath250 coalesces with @xmath251 at @xmath252 , and so on . ) as a result , all roots are double and the representation of @xmath193 is discontinuous at @xmath165 , with the power degree @xmath204 of the prefactor in ( [ power - series - phi ] ) jumping from @xmath198 for @xmath161 to @xmath206 for @xmath165 . in particular , in exceptional models , i.e. , discrete models with continuous families of stationary kinks ( like ( [ nonlinearity - speight ] ) , ( [ nonlinearity - tovbis ] ) and ( [ nonlinearity - panos ] ) ) the constant @xmath218 is _ a priori _ zero while the limit of @xmath217 as @xmath253 may be nonvanishing . in fact , numerical computations of the top limit in ( [ stokes ] ) indicate that the stokes constant blows up as @xmath253 . renormalisation of @xmath217 for small @xmath27 is , however , a nontrivial asymptotic problem which is beyond the scope of our current investigation . for @xmath175 , where @xmath176 , the resonant set @xmath163 contains only one root @xmath250 and , therefore , there is just one stokes constant @xmath217 , which completely determines the convergence of the formal power series for @xmath98 . if @xmath254 at some point @xmath255 , the stable and unstable solutions @xmath110 and @xmath111 coincide to leading order . arguments based on the implicit function theorem ( see @xcite ) reveal a heteroclinic bifurcation which occurs on crossing a smooth curve @xmath256 on the @xmath41-plane , with @xmath257 . on the other hand , for @xmath258 the resonant set @xmath163 contains more than one root . if @xmath259 for some @xmath260 , this alone is not sufficient for the convergence of the formal power series @xmath261 . the higher - order stokes constants @xmath262 , @xmath263 , , must be introduced and computed from the asymptotic behavior of the power series @xmath247 . as we shall show in the next section , the function @xmath217 does have zeros in the case of the discretizations ( [ nonlinearity - speight ] ) and ( [ nonlinearity - panos ] ) . all these zeroes are `` safe '' ; that is , all @xmath264 values lie in the interval @xmath265 , so that the higher - order stokes constants do not have to be computed . in this section , we report on the numerical computation of the stokes constants @xmath217 for the four different discretizations of the @xmath0 model ( [ phi-4 ] ) under consideration . our numerical method utilises the expression ( [ stokes ] ) of the stokes constant in terms of the coefficients of the formal power series solution ( [ laplace - power ] ) . first , we obtain the recurrence relation for the coefficients in the set @xmath266 by substituting the power series expansion ( [ laplace - power ] ) into the limiting integral equation ( [ integral - limiting ] ) , and using the convolution formula @xmath267 after that , we compute the asymptotic behavior of these coefficients as @xmath246 and evaluate the limit ( [ stokes ] ) numerically for a fixed value of @xmath161 . in order to calculate the stokes constant for the four models in a uniform way , we write a general symmetric homogeneous cubic polynomial @xmath59 as @xmath268 where @xmath269 are numerical coefficients , with @xmath270 and @xmath271 . the symmetry implies that @xmath272 and therefore it is sufficient to specify just six coefficients . the values of these coefficients for the four nonlinearities in question are given in table [ coeffs ] . .[coeffs]the coefficients @xmath273 of the cubic polynomial ( [ cubic - polynomial ] ) for the four models under consideration . [ cols="<,^,^,^,^,^,^",options="header " , ] by applying the borel laplace transform ( [ laplace ] ) to equation ( [ inner - limiting ] ) with @xmath49 as in ( [ cubic - polynomial ] ) , we obtain the corresponding cubic convolution function @xmath274 $ ] on the right hand side of the integral equation ( [ integral - limiting ] ) : @xmath275 = \sum_{\alpha = -1}^{1}\sum_{\beta = \alpha}^{1}\sum_{\gamma = \beta}^{1}a_{\alpha,\beta,\gamma } \ ; \rme^{\alpha p } v(p ) \ast \rme^{\beta p}v(p ) \ast \rme^{\gamma p}v(p).\ ] ] to derive the recurrence formula for the coefficents @xmath244 in ( [ laplace - power ] ) , it will be more convenient to consider the power series expansion which consists of both even and odd powers of @xmath115 : @xmath276 we now substitute the series ( [ general - laplace - power ] ) into ( [ integral - limiting ] ) with @xmath124 $ ] given by ( [ integraleq ] ) and use the convolution formula ( [ convolution - powers ] ) . equating the coefficients of @xmath277 where @xmath278 , in the resulting equation , we find that @xmath279 } \frac{2}{(2i + 2 ) ! } v_{n-2i } - c^2 v_n = \sum_{\alpha = -1}^{1}\sum_{\beta = \alpha}^{1}\sum_{\gamma = \beta}^{1 } \frac{a_{\alpha,\beta,\gamma}}{(n+2)(n+1)}\bigg\ { \sum_{i=0}^n\left(\sum_{k=0}^{n - i}\frac{\alpha^k}{k!}v_{n - i - k}\right ) \nonumber\\ \times \left[\sum_{j=0}^i\left(\sum_{l=0}^j\frac{\beta^l}{l!}v_{j - l}\right ) \left(\sum_{m=0}^{i - j}\frac{\gamma^m}{m!}v_{i - j - m}\right)\frac{j!(i - j)!}{i!}\right]\frac{i!(n - i)!}{n!}\bigg\},\end{aligned}\ ] ] where @xmath280 $ ] is the integer part of @xmath281 and @xmath282 . equation ( [ recurrence ] ) is a recurrence relation between the coefficients @xmath266 . solving equation ( [ recurrence ] ) with @xmath283 , we get @xmath284 . note that this result is independent of the choice of @xmath285 , i.e. independent of the model . letting @xmath286 and making use of the symmetry of @xmath49 , one can show by induction that the coefficients of all odd powers in ( [ general - laplace - power ] ) are zero ( as we concluded previously on the basis that the outer expansion is odd ) . to prevent overflow or underflow when evaluating the recurrence relation numerically , we shall work with the normalised coefficients @xmath287 so that the stokes constant ( [ stokes ] ) for @xmath161 is given by @xmath288 reformulating ( [ recurrence ] ) in terms of @xmath289 , we use the relation ( [ numerical - limit ] ) to compute @xmath289 numerically . we truncate the sums involving @xmath290 , @xmath291 and @xmath292 when these factors become smaller than @xmath293 , and evaluate the sums involving the combinatorial factors in two halves . in the first , the summation index increases from zero to the halfway point , and in the second it decreases from its maximum . this ensures that the combinatorial factors are always decreasing from one step to the next so that they can be accurately determined recursively . we also truncate these sums when the combinatorial factors fall below @xmath293 . these expedients result in a numerical routine fast enough to allow for evaluation of the recurrence relation up to very large @xmath106 ; this is essential given the slow convergence of @xmath289 to a constant . matching ( [ laplace - power ] ) to ( [ poles - expanded ] ) yields @xmath294 \quad \textrm{as $ n \to \infty$};\ ] ] therefore , the rate at which @xmath289 converges to @xmath217 is of order @xmath295 : @xmath296 although the convergence of @xmath289 to @xmath217 is extremely slow , we can accelerate the process by using ( [ convergence - of - w_n ] ) . defining @xmath297 we get @xmath298 the convergence of the sequence @xmath299 is much faster than that of @xmath289 ; see figure [ sequencegraphs ] . the relative error @xmath300 can be written as @xmath301 plus terms of order @xmath302 . this gives an empirical criterion for the termination of the process . we continued our computations until @xmath303 reached a value smaller than @xmath304 , i.e. until the percentage error dropped below 0.1% . for @xmath305 , the value of @xmath106 to which we have to compute in order to achieve this accuracy is less than @xmath306 , increasing for smaller values of @xmath27 to approximately @xmath307 for @xmath308 . consequently , the above numerical algorithm is not suited to the study of the @xmath249 limit , and would have to be modified for that purpose . figure [ stokesgraphs ] displays the stokes constant computed using the above numerical procedure , for the four models of table [ coeffs ] . we see that the stokes constant @xmath217 vanishes almost nowhere in @xmath161 in all four models . there are , however , several isolated zeros : @xmath309 for @xmath310 in the case of the speight - ward nonlinearity ( [ nonlinearity - speight ] ) and for @xmath311 , @xmath312 and @xmath313 in the case of the kevrekidis discretization ( [ nonlinearity - panos ] ) . importantly , all of these lie in the region @xmath314 where the resonance set ( [ resonances ] ) consists of only one value , @xmath168 . ( here @xmath176 . ) therefore , there is a sliding kink in the @xmath17 limit for each of these isolated values of velocity . furthermore , strong parallels between our current setting and that of solitons of the fifth - order kdv equation @xcite suggest that sliding kinks should exist along a curve on the @xmath41 plane emanating from each of the points @xmath315 . in other words , we conjecture that there is a radiationless kink travelling with a certain particular speed @xmath316 for each @xmath11 in the case of the speight - ward nonlinearity , and that there are three such velocities ( for each @xmath11 ) in the case of the kevrekidis model . for small @xmath11 , @xmath316 should be close to the above values @xmath264 . in order to verify the existence of kinks sliding at these isolated velocities by an independent method , we solved the differential advance - delay equation ( [ advance - delay ] ) numerically . the infinite line was approximated by an interval of length @xmath317 , with the antiperiodic boundary conditions @xmath318 . we utilised newton s iteration with an eighth - order finite - difference approximation of the second derivative ; the step size was chosen to be @xmath319 . the continuum solution ( [ travelling ] ) was used as an initial guess . if we find a solution to the advance - delay equation with @xmath52 decaying to a constant for large positive and negative @xmath57 , then we regard this solution as ( a numerical approximation to ) a radiationless travelling kink . we were able to tune @xmath27 for a fixed value of @xmath11 so that the radiation was reduced to the order of @xmath320 , whereupon the finite accuracy of our numerical scheme prevented any further reduction . to make sure that the radiation does vanish rather than reaching a local minimum but remaining nonzero , we plot the average magnitude of the radiation near the ends of the interval as a function of @xmath27 , for fixed @xmath11 . this is defined as the average of @xmath321 ^ 2 $ ] over the last 20 units of the interval , where @xmath322 is the average value of @xmath52 over these last 20 units . the results are shown in figure [ radiationzero ] . note the straight - line behavior of the graphs near the isolated values of @xmath27 ; this indicates that the coefficient of the sinusoid superimposed over the kink s flat asymptote crosses through zero ( rather than attaining a small but nonzero minimum ) . the supression of radiation at the isolated points is thereby verified . finally , the last question that we would like to address here is whether the intensity of the radiation from the moving discrete kink depends on the type of discretization . more specifically , we would like to know whether the choice of one of the exceptional discretizations ( which , by definition , support translationally invariant _ stationary _ kinks ) serves to reduce the radiation from the _ moving _ kinks . speight and ward have already given an affirmative answer for their exceptional discretization ; here we consider the one - parameter nonlinearity @xmath323 which interpolates between the one - site nonlinearity ( [ nonlinearity - phi4 ] ) ( for which @xmath324 ) and the exceptional discretization ( [ nonlinearity - tovbis ] ) of bender and tovbis ( for which @xmath325 ) . figure [ hybridgraph ] shows the stokes constant for the model ( [ hybridmodel ] ) , as @xmath326 changes from 0 to 1 for fixed values of @xmath27 . the stokes constant is indeed seen to be drastically reduced as @xmath326 approaches @xmath327 that is , in the limit of the exceptional discretization . ( it nonetheless remains nonzero , of course , unless @xmath165 . ) in this paper we have investigated the existence of sliding kinks i.e. discrete kinks travelling at a constant velocity over a flat background , without emitting any radiation in four discrete versions of the quartic - coupling theory . one of these models is the most common , one - site , discretization . as the overwhelming majority of discrete @xmath0-equations , it supports travelling kinks , but these kinks do radiate and decelerate as a result . the other three discretizations we considered are all exceptional in the sense that they all support one - parameter continuous families of stationary kinks where the free parameter defines the position of the kink relative to the lattice . this property is clearly nongeneric ; the translation invariance of the continuous @xmath0-theory is broken by the discretization and hence in generic disretizations kinks may only be centred at a site or midway between two sites . since the nonexistence of `` translationally - invariant '' and of sliding kinks in the generic models can be ascribed to similar factors , _ viz . _ , the breaking of the translation and lorentz invariances , it was hoped that the exceptional discretizations might turn out to be equally exceptional from the point of view of sliding kinks . our approach was based on the computation of the stokes constants associated with the putative sliding kink in a given equation . the main conclusion of our work is that the sliding kinks do exist in the discrete @xmath0 theories , but only with special , isolated , velocities ( which of course depend on @xmath11 ) . there is one such velocity in the exceptional model of speight and ward , and three different sliding velocities in the discretization of kevrekidis . it is natural to expect that the sliding kinks should play the role of attractors similarly to the fronts moving with `` stable velocities '' in dissipative systems ; that is , radiating travelling kinks should evolve into kinks travelling with the sliding velocities if there are such velocities in the system . not every discretization supports sliding kinks , of course ; in particular , no sliding velocities arise for the generic , one - site , nonlinearity and even for the exceptional discretization of bender and tovbis . one natural way of trying to construct the sliding kinks is via power series expansions in powers of @xmath328 ; for the exceptional discretizations , this construction can be carried out to any order . this approach was pursued in the recent work of ablowitz and musslimani @xcite . our results indicate , however , that these power series will not converge and exponentially small terms ( terms lying beyond all orders of the power expansion ) emerge because of the singular behaviour of the stokes constant @xmath217 as @xmath249 . detailed studies of this singular limit will be presented elsewhere . the exceptional discretizations have richer underlying symmetries than generic nonlinearities but the `` translation invariance '' of the stationary kink alone does not automatically guarantee the existence of the sliding velocities . the exact relation between the `` translational invariance '' and mobility of kinks is still to be clarified ; at this stage it is worth mentioning that the stokes constant associated with ( and hence the intensity of radiation from ) a moving kink is several orders of magnitude smaller in exceptional models than in generic discretizations . finally , it is instructive to point out some parallels with an earlier work of flach , zolotaryuk and kladko @xcite who also studied the phenomenon of kink sliding in klein - gordon lattices . in the scheme of @xcite , one postulates an analytic expression for the sliding kink , @xmath329 , with some explicit function @xmath52 , and then reconstructs the klein - gordon nonlinearity for which this is an exact solution . our present conclusions are in agreement with the results of these authors who observed that for a given @xmath11 , the kink may only slide at a particular , isolated , velocity . the two approaches , ours and that of @xcite , are reciprocal ; while we examine the existence of sliding kinks for particular discretizations of the @xmath0-theory , with fixed parameters independent of the kink s velocity , in the `` inverse method '' of @xcite one assumes an explicit solution of a particular form but does not have any control over the resulting nonlinearities . consequently , the discrete klein - gordon models generated by the `` inverse method '' are not discretizations of the @xmath0-theory and do include explicit dependence on the velocity of the sliding kink . o.o . was supported by funds provided by the south african government and the university of cape town . d.p . thanks the department of mathematics at uct for hospitality during his visit and the nrf of south africa for financial support which made the visit possible . i.b . is a harry oppenheimer fellow ; also supported by the nrf under grant 2053723 . dky05b j.a . sepulchre , `` energy barriers in coupled oscillators : from discrete kinks to discrete breathers '' , in _ proceedings of the conference on localization and energy transfer in nonlinear systems , june 17 - 21 , 2002 , san lorenzo de el escorial , madrid , spain _ , eds . l. vazquez , r.s . mackay , m.p . zorzano ( world scientific , 2003 ) , pp . 102129 . a. tovbis , m. tsuchiya , and c. jaff , `` exponential asymptotic expansions and approximations of the unstable and stable manifolds of singularly perturbed systems with the henon map as an example '' , chaos * 8 * , 665681 ( 1998 ) .	in recent years , three exceptional discretizations of the @xmath0 theory have been discovered @xcite which support translationally invariant kinks , i.e. families of stationary kinks centred at arbitrary points between the lattice sites . it has been suggested that the translationally invariant stationary kinks may persist as _ sliding _ kinks , i.e. discrete kinks travelling at nonzero velocities without experiencing any radiation damping . the purpose of this study is to check whether this is indeed the case . by computing the stokes constants in beyond - all - order asymptotic expansions , we prove that the three exceptional discretizations do not support sliding kinks for most values of the velocity just like the standard , one - site , discretization . there are , however , isolated values of velocity for which radiationless kink propagation becomes possible . there is one such value for the discretization of @xcite and three _ sliding velocities _ for the model of @xcite .
coherent control utilizes the wave properties of matter to steer a quantum dynamical process to a desired outcome . the source of control is interference , constructive to achieve the goal and destructive to eliminate unwanted consequences @xcite . the agents of control are external fields , in particular electromagnetic fields . the experimental and theoretical challenge lies in identifying these control fields . the present study is aimed at finding control fields which are constrained to limit the damage which the control field may cause to the controlled system . theoretically , the control problem can be formulated as an inversion : finding the field subject to the quantum dynamics which leads to the desired outcome . optimal control theory ( oct ) has been developed as a tool to address this problem @xcite . it can be formulated starting from a variational ansatz @xcite or using krotov s method @xcite . recently the krotov s method has been extended to include a strict limitation on the spectrum of the optimized field @xcite . the most well studied task in oct has been the goal of a state - to - state transition . given an initial state @xmath1 and a closed quantum system , the field needs to be found which drives the system to a specific final state @xmath2 . this task has been shown to be completely controllable @xcite if the fields are not restricted . moreover the control landscape is favorable composed of flat ridges such that the climb in the gradient direction will lead to one of the many possible solutions @xcite . a more involved control task is to optimize the expectation value of an operator at a final time , @xmath3 . this task can be formulated in the framework of open quantum systems . the oct approach yields an iterative solution to the inversion problem which is based on propagating the system density operator @xmath4 forward in time and the target operator @xmath5 backward in time @xcite . the prospect of quantum computing has posed an even more complex control problem : imposing a unitary transformation @xmath6 on a subset of quantum states which act as the quantum register . the unitary transformation carries out a specific computational task . this control task is equivalent to @xmath7 simultaneous state - to - state transformations @xcite . the solution of the iterative set of equations has been shown to become exponentially more difficult with the size @xmath7 of the unitary transformation @xcite . these findings are in accordance with a very complex control landscape @xcite . a further step up in complexity is the task of imposing a unitary transformation under dissipative conditions . this task emerges in the quantum governor @xcite , in quantum information processing and it is a traditional task in nuclear magnetic resonance ( nmr ) spectroscopy @xcite . in any practical implementation the positive task of obtaining the final goal has to be weighted by possible negative consequences . for example control fields of high intensity can damage the system by causing ionization or dissociation . a remedy for this problem consists in restricting the population in certain lossy excited state manifolds . this task has been the motivation for the development of local control theory ( lct ) @xcite . lct has been applied to lock unwanted electronic excitations @xcite and recently to the problem of quantum information processing where avoiding population loss becomes crucial @xcite . however , oct is more powerful than lct and it is therefore desirable to incorporate constraints describing negative consequences of the control process into the algorithm . such constraints depend on the state of the system at intermediate times @xcite . for example , the system can simply be restricted to remain in an `` allowed '' or to avoid a `` forbidden '' subspace during its evolution . more elaborate examples include imposing a predefined path between an initial and a final state @xcite , and maximizing the expectation value of a time - dependent operator throughout the optimization time interval @xcite . previous oct studies which impose state - dependent constraints @xcite were performed for state - to - state optimization and are based on the variational approach . in the present work , an optimization algorithm including state - dependent constraints is obtained using the krotov method for the state - to - state case as well as for unitary transformations . the krotov method offers the advantage that the monotonic convergence of the algorithm can be ensured by the choice of the imposed constraints @xcite . a brief comparison with the algorithms using the variational ansatz @xcite will be given for the state - to - state optimization . the paper is organized as follows : the state - dependent constraints are formulated in sec . [ sec : formalism ] , and the resulting algorithm is presented for optimization of state - to - state transition and of a unitary transformation . a review of the krotov method together with an outline of the derivation of the equations presented in sec . [ sec : formalism ] is given in the appendix . [ sec : results ] introduces a model example and illustrates optimization under state - dependent constraints for a state - to - state transition and for a unitary transformation . our findings are compared to related approaches in sec . [ sec : disc ] . finally , sec . [ sec : concl ] concludes . represents the state of the system at time @xmath9 , and @xmath10\,=\,{\boldsymbol{\mathsf{\hat{h}}}}_0\,-\,{\boldsymbol{\mathsf{\hat{\mu}}}}\epsilon(t)\,,\ ] ] is the system+control hamiltonian . @xmath11 denotes the field - free hamiltonian , @xmath12 the semiclassical control field and @xmath13 is a system operator describing the coupling between system and field . the objective of the optimization is to find a field which drives the system from an initial state at @xmath14 , @xmath15 to a target subspace at time @xmath16 representing the final time objective , such that a minimum ( or maximum ) expectation value of the time - dependent operator @xmath17 is maintained throughout the complete time interval @xmath18 $ ] . the target subspace at time @xmath16 is described by the projector @xmath19 , e.g. , @xmath20 for a single target state . in oct , these requirements are formulated as a functional which depends on the system state and the control , in such a way that an optimal field corresponds to an extremum of the functional . that functional can be expressed as a sum over terms related to the different conditions imposed on the system evolution . the complete functional is obtained as a sum over functionals corresponding to the final time objective , to the state - dependent intermediate - time objective ( or constraint ) , and to the constraint over the field . the term corresponding to the objective at the final time @xmath16 , the actual target , can be expressed as @xmath21\,=\,\lambda_0\ , \langle\varphi(t)\|{\boldsymbol{\mathsf{\hat{d}}}}\|\varphi(t)\rangle\,,\ ] ] where @xmath22 is a real parameter , which can be negative or positive , depending on whether the functional is minimized or maximized during the optimization . @xmath23 $ ] emphasizes the bilinearity of the functional with respect to the system state at time @xmath16 . other possibilities for expressing this term exist @xcite , but the resulting optimization algorithms are very similar . the state - dependent intermediate - time objective or constraint is represented by the functional , @xmath24=\int_0^t\,g_b[\varphi,\varphi^\dagger]\,dt\,,\ ] ] where @xmath25 is taken to be @xmath26=\lambda_b\langle\varphi(t)\|{\boldsymbol{\mathsf{\hat{p}}}}(t)\|\varphi(t)\rangle\,.\ ] ] @xmath27 is a real parameter which can be positive or negative , as discussed later . more complicated dependences of @xmath25 on the operator @xmath17 and on the state @xmath28 are conceivable . to obtain a closed algorithm , the complete functional has to include a term depending on the field @xcite , @xmath29=\int_0^t\,g_a[\epsilon]\,dt\,.\ ] ] generally , @xmath30 can be written as @xmath31=\lambda_a(t)\,[\epsilon(t)-\epsilon_r(t)]^2\,,\ ] ] where @xmath32 denotes a reference field and @xmath33 corresponds to the common choice of minimizing the field energy . @xmath34 $ ] represents an intermediate - time objective , but one which does not depend on the state of the system . the complete functional is given by @xmath35\,=\,j_0[\varphi_t,\varphi_t^\dagger]\,+\,j_a[\epsilon ] \,+\,j_b[\varphi,\varphi^\dagger]\,.\ ] ] for simplicity , we omit the dependence of @xmath28 and @xmath36 on time , except for the final time @xmath16 . the optimization problem is now equivalent to the minimization or maximization of this functional . for that purpose , the krotov method is employed . since the krotov method operates with real functions , a complete presentation of the equations for this problem is somewhat cumbersome @xcite . an outline of the derivation is given in the appendix and only the final result is presented below . a guess field is denoted by @xmath37 and the corresponding state @xmath38 is given by the evolution eq . ( [ eq : schrodinger ] ) with the initial condition @xmath39 . in the krotov method a new field @xmath40 which decreases ( or increases ) the functional value is obtained by the following equations : a new `` state '' @xmath41 is determined using the inhomogeneous equation @xmath42\ , with the `` initial '' condition @xmath43,[eq : chicondition ] ) of the appendix . it corresponds to the common oct result modified by the inhomogeneous term @xmath44 which arises from the state - dependent constraint . the state @xmath41 is used to determine the new control field , @xmath45 cf . ( [ eq : newfield ] ) of the appendix , where @xmath46 was chosen . this is an implicit equation since the state @xmath47 which depends on @xmath48 appears in the right - hand side of eq . ( [ eq : eps1 ] ) . the numerical discretization of this implicit equation has been widely discussed for the homogeneous case ( see for example ref . @xcite ) . the inhomogeneous term in eq . ( [ eq : chistate ] ) requires a modification of the time propagation method . a symmetrical propagation scheme is employed based on the diagonalization of the hamiltonian in the interleaved time grid points , @xmath49 . the inhomogeneous term is evaluated as @xmath50 the iterative algorithm is constructed with @xmath48 as input to the next step of the iteration and the process is repeated until the required convergence is achieved . the monotonic convergence of the algorithm is analyzed defining @xmath51 as the difference between the functional values before and after one iteration , @xmath52\,-\ , j[\varphi^{(1)},\varphi^{\dagger(1)},\epsilon^{(1 ) } ] \nonumber\\ & = & \delta_1+\int_0^t\,\left(\delta_{2a}(t)+\delta_{2b}(t)\right)\,dt\,.\end{aligned}\ ] ] the terms @xmath53 are derived in the appendix and can be evaluated using eqs . ( [ eq : j_0]-[eq : g_a ] ) . this yields @xmath54 cf . ( [ eq : delta_1 ] ) , with the definition @xmath55 furthermore , @xmath56+g_a[\epsilon^{(0)}]\,+\,\nonumber \\ & & \quad\left[\frac{\partial g_a}{\partial\epsilon}\right]_{(1)}\ , \left(\epsilon^{(1)}-\epsilon^{(0)}\right)\,,\end{aligned}\ ] ] cf . ( [ eq : delta_2 ] ) , which yields for our choice @xmath57 @xmath58 and @xmath59 cf . ( [ eq : delta_2 ] ) . the algorithm converges monotonically to a minimum ( maximum ) of the functional if @xmath60 ( @xmath61 ) in each iteration step . a sufficient but not necessary condition consists in all @xmath53 being larger ( smaller ) than zero . let the operators @xmath19 and @xmath17 be positive - semidefinite . sufficient conditions are then given by @xmath62 for minimization , and by @xmath63 for maximization . this result leads to some curious consequences . for example , let the system be described by a discrete number of levels and assume its hilbert space can be split into two subspaces , the `` allowed '' subspace , described by the projector @xmath64 , and the `` forbidden '' subspace , described by @xmath65 ( @xmath66 ) . the objective of the optimization consists in some transition inside the allowed subspace , avoiding any population transfer to the forbidden one . in the case of minimization of the functional @xmath67 , the latter requirement can be expressed by one of the two following choices for @xmath68 , @xmath69 in the case of maximization , the possibilities are @xmath70 in both cases , @xmath71 and @xmath72 have the same physical meaning , remaining in the allowed subspace , or equivalently , avoiding the forbidden subspace . the choice @xmath72 is more appealing in principle , since the inhomogeneous term of eq . ( [ eq : chistate ] ) would decrease and eventually become negligible when approaching an optimal solution . however , only @xmath71 fulfills the sufficient conditions for monotonic convergence . a note of caution must be made at this point . ( [ eq : cmin ] ) and ( [ eq : cmax ] ) are sufficient but not necessary conditions . monotonic convergence can therefore be found for values of @xmath73 not fulfilling eqs . ( [ eq : cmin ] ) and ( [ eq : cmax ] ) . this can happen if the values of @xmath53 compensate each other to give a convergent total @xmath51 . in addition , the analysis assumes an exact solution of the control equations . a limited accuracy of the numerical implementation of the algorithm and a poor accuracy of the propagation method can lead to the breakdown of the monotonic convergence @xcite . the objective consists in implementing a given unitary transformation @xmath74 , up to a global phase , in a given subspace @xmath75 of dimension @xmath76 described by the projector @xmath77 , @xmath78 to this end , the parameter @xmath79 is defined , @xmath80 where @xmath81 the modulus of @xmath79 is equal to @xmath76 when the target unitary transformation is implemented in the subspace @xmath75 by the field @xmath36 @xcite . the optimization problem is again formulated as a functional minimization ( maximization ) . the final time term is now defined by @xmath82 & = & \lambda_0\,\|\tau\|^2 \\ & = & \lambda_0 \sum_{n=1}^{n_r}\,\langle\varphi_{fn}\|\varphi_n(t)\rangle \sum_{n'=1}^{n_r}\,\langle\varphi_{n'}(t)\|\varphi_{fn'}\rangle\,,\nonumber\end{aligned}\ ] ] where @xmath83 denote the set of states @xmath84 ( @xmath85 ) . other choices of @xmath86 are possible @xcite . the intermediate - time state - dependent term takes the form , @xmath87&=&\lambda_b\,{\rm tr } \left\{{\boldsymbol{\mathsf{\hat{u}}}}(t,0;\epsilon)^\dagger{\boldsymbol{\mathsf{\hat{p}}}}(t){\boldsymbol{\mathsf{\hat{u}}}}(t,0;\epsilon){\boldsymbol{\mathsf{\hat{r } } } } \right\ } \nonumber \\ & = & \lambda_b\,\sum_{n=1}^{n_r}\,\langle\varphi_{n}(t)\|{\boldsymbol{\mathsf{\hat{p}}}}(t)\|\varphi_n(t)\rangle\,.\end{aligned}\ ] ] the constraint over the field is taken to be the same as in the state - to - state case , cf . ( [ eq : g_a ] ) . the equations defining the algorithm are obtained using the krotov method as outlined in the appendix . they read as follows : @xmath76 `` states '' @xmath41 are given by the inhomogeneous evolution equation , @xmath88\ , with the `` initial '' condition latexmath:[\ ] ] if the objective is to minimize the functional @xmath67 ( @xmath216 ) , the sufficient but not necessary conditions for monotonic convergence are given by @xmath217 , ( @xmath218 ) . analogously , in order to maximize the functional it is sufficient that @xmath219 , ( @xmath218 ) . whether the @xmath53 are positive or negative is determined by the particular choice of @xmath86 , @xmath30 , and @xmath25 . in sec . [ sec : formalism ] the values of @xmath51 are analyzed for the cases under study . the previous approach can easily be generalized to the case of unitary transformations . the functional @xmath67 depends then on @xmath220 real functions denoted by the set @xmath221 . as a consequence , the scalar functions @xmath165 , @xmath222 and @xmath104 will also depend on all of them . moreover , the new dependence must be taken into account in the derivatives of the previous equations by the substitution @xmath223 the remaining procedure is analogous to the state - to - state case , and the relations ( [ eq : correspondence ] ) can be used to obtain the equations for the optimization algorithm given in sec . [ subsec : form_utrafo ] .	optimal control theory is developed for the task of obtaining a primary objective in a subspace of the hilbert space while avoiding other subspaces of the hilbert space . the primary objective can be a state - to - state transition or a unitary transformation . a new optimization functional is introduced which leads to monotonic convergence of the algorithm . this approach becomes necessary for molecular systems subject to processes implying loss of coherence such as predissociation or ionization . in these subspaces controllability is hampered or even completely lost . avoiding the lossy channels is achieved via a functional constraint which depends on the state of the system at each instant in time . we outline the resulting new algorithm , discuss its convergence properties and demonstrate its functionality for the example of a state - to - state transition and of a unitary transformation for a model of cold rb@xmath0 .
the stellar imf at high redshift is of great importance for a wide range of astrophysical problems , such as the ionization and enrichment of the intergalactic medium , the extragalactic background light , the visibility of galaxies and the rate at which baryons are locked - up into stars and stellar remnants . there are two complementary approaches to the determination of the imf long ago and far away : one is to observe directly high redshift objects , and attempt inferences on the stellar imf from the integrated spectrum and photometry , while the second approach analyses the fossil record in old stars at low redshift . the characterization of the stellar imf in external galaxies , compared to that in the milky way , is a crucial step in deciphering the important physical processes that determine the distribution of stellar masses under a range of different physical conditions . the low mass stellar imf at the high redshifts at which these stars formed is directly accessible through star counts , plus a mass - luminosity relation . the high mass imf at these high redshifts is constrained by the chemical signatures in the low mass stars that were enriched by the supernovae from the high mass stars . i will discuss both ends of the imf at high redshift , in an external galaxy . simulations of galaxy formation within the framework of the ` concordance ' ( @xmath1)cdm cosmology agree that the first stars form within structures that are less massive than a typical l@xmath2 galaxy today ( e.g. kauffmann , white & guiderdoni 1993 ; cole et al . large galaxies form hierarchically , through the merging and assimilation of such smaller systems . satellite galaxies of the milky way are survivors of this merging ( e.g. bullock , kravtsov & weinberg 2000 ) . the stars that formed at early times are found , at the present day , throughout large galaxies , and also in satellite galaxies . environments with little subsequent star formation are the best places to find and study old stars the stellar halo of the milky way , and a few of the dwarf spheroidal satellite galaxies . the dwarf spheroidal galaxy in ursa minor ( umi dsph ) , like all members of its morphological class ( gallagher & wyse 1994 ) , has extremely low surface brightness , with a central value of only @xmath3 v mag / sq . arcsec , or @xmath4/pc@xmath5 . the total luminosity is in the range @xmath6 ( kleyna et al . 1998 ; palma et al . 2003 ) , equal to that of a luminous galactic globular cluster . again similar to a globular cluster , the ursa minor dsph contains little or no gas and has apparently not formed a significant number of stars for @xmath7 gyr ( e.g. hernandez et al . 2000 ; carrera et al . 2002 ) , or since a redshift @xmath8 . the metallicity distribution of the stars is narrow , with a mean of [ fe / h ] @xmath9 dex and a dispersion of @xmath10 dex ( e.g. bellazzini et al . the stellar line - of - sight velocity dispersion is @xmath11 km / s ( e.g. wilkinson et al . 2004 ) , sufficiently large that , unlike globular clusters , equilibrium models have a large mass - to - light ratio , @xmath12 perhaps as high as several hundred in solar units if the mass is @xmath13 ( wilkinson et al . 2004 ) , indicating a non - baryonic dark halo . non - equilibrium models are rather contrived and themselves fail to explain the data ( e.g. wilkinson et al . models of the evolution of dwarf spheroidals are by no means well developed , but very likely the stars formed in an environment rather different than that of globular cluster stars , or of current star - forming regions in the disk of the milky way . the distance of the ursa minor dsph is only @xmath14 kpc , close enough that a determination of the luminosity function of low - mass main sequence stars through star counts is feasible , particularly using the hubble space telescope . the ( unusually ) simple stellar population of this dwarf spheroidal essentially of single age , single metallicity makes the derivation of the luminosity function from star counts a robust procedure . this determines the low - mass stellar imf at redshifts of @xmath15 . high - resolution spectroscopy of the luminous evolved stars is also possible , yielding elemental abundances which constrain the high - mass stellar imf that enriched the low - mass stars we observe . the ( very ) dominant stellar population in the umi dsph is very similar to that of a classical galactic halo globular cluster . the most robust constraint on the low - mass stellar imf is then obtained by a direct comparison between the faint stellar luminosity functions of the umi dsph and of representative globular clusters of the same age and metallicity , such as m15 and m92 , observed in the same bandpasses , same telescope and detector . with the same stellar populations , this is a comparison between _ mass _ functions , and differences may be ascribed to variations in the low - mass imf . we therefore obtained deep images with the hubble space telescope in a field close to the center of the umi dsph , using stis as the primary instrument ( optical long pass filter ) , with wfpc2 ( v@xmath16 and i@xmath17 filters ) and nicmos ( nic2/h - band ) in parallel . the wfpc2 filters matched those of extant data for m15 and m92 ; we obtained our own stis / lp and nic2/h - band data for m15 . similarly exposed data for an off field , at 23 tidal radii from the centre of umi dsph , were also acquired . the detailed paper presenting the results from the full dataset is wyse et al . ( 2002 ) ; preliminary results from a partial wfpc2 dataset were presented in feltzing , wyse & gilmore ( 1999 ) . the images are not crowded and standard photometric techniques were used to derive the luminosity functions . for the wfpc2 data , the luminosity functions were based only upon stars ( unresolved objects ) that lie close to the well - defined umi dsph main sequence locus in the colour - magnitude diagram ( cmd ; see figure 1 ) . the off field cmd confirmed little contamination from galactic stars . the stis luminosity function was based on one band only , and we employed various approaches to background subtraction . the nicmos data served only to exclude a hypothetical population of very red stars and will not be discussed further here . the extant wfpc2 data for the globular clusters ( piotto et al . 1997 ) are from fields at intermediate radii within the clusters , where effects of dynamical evolution on the mass function should be minimal . -0.25 cm -7.cm -1 cm the comparisons with the wfpc2 colour - magnitude based v - band and i - band luminosity functions are shown in figure 2 . we adopted 0.5 mag bins to have reasonable numbers in each bin , and to minimize effects of e.g. reddening and distance moduli uncertainties . the 50% completeness limits for the umi dsph data are equivalent to absolute magnitudes of @xmath18 and @xmath19 , which using the baraffe et al . ( 1997 ) models _ both _ correspond to masses of @xmath20 . the stis / lp data provide an independent check , by both a direct lp - luminosity function comparison between m15 and umi dsph , and a derived stis - based i - band luminosity function . all show that the globular cluster stars and the umi stars have indistinguishable faint luminosity functions , down to an equivalent mass limit of @xmath21 ( see wyse et al . 2002 for details ) . -4.51 cm 2.32 in -0.25truecm -0.5 cm we employed various statistical tests to quantify the agreement of the various datasets e.g. stis - derived i - band _ vs. _ wfpc2 i - band etc : @xmath22 linear , least - square fits to the ( log ) counts as a function of apparent magnitude , using various ranges of magnitude and differing bin choices , consistently found agreement to better than 2@xmath23 . @xmath22 kolmogorov - smirnov tests on the unbinned data for a variety of magnitude ranges ; the results depend on systematics such as the relative distance moduli , but again there is general agreement to better than 5% significance level . @xmath22 @xmath24square tests were carried out on the binned data , using a variety of bin centers ( maintaining 0.5 mag bin widths ) and magnitude ranges and again agreement to better than 5% significance level . the main result is that the underlying mass functions of low - mass stars in galactic halo globular clusters and in the external galaxy the umi dsph are indistinguishable . this is a comparison between two different galaxies , and systems of very different baryonic densities and dark matter content . adopting the baraffe et al . ( 1997 ) models , the 50% completeness limits for the luminosity functions of the stars in the umi dsph correspond to @xmath21 , and the mass function may be fit by a power law , with slope somewhat flatter than the salpeter ( 1955 ) value , over the range we test , of @xmath25 . this is consistent with the solar neighbourhood mass function over this mass range , and indeed the universal mass function that appears to be the conclusion of this meeting . however , the light - to - mass transformation is not robustly defined for k / m dwarfs , especially as a function of age and metallicity . calibration of this is best achieved by analysis of low - mass stars in detached eclipsing binary systems , and we have recently undertaken a photometric survey of open clusters to identify candidate low - mass binary systems to be followed up with spectroscopy for radial velocity curves ; this forms the phd thesis of leslie hebb at johns hopkins university . our sample consists of six open clusters of known age and metallicity ( from the brighter turn - off stars ) , old enough to have low - mass stars on the main sequence , age @xmath26 yr , with the oldest being @xmath27 gyr . we used both the wide field camera on the 2.5 m isaac newton telescope and the mosaic camera on the kitt peak 4 m telescope , each of which provide a field of view of @xmath28 . the observing strategy we adopted was designed to enable the detection of a 0.05 mag amplitude eclipse in a target 0.3 @xmath29 star , monitored on timescales of fraction of an hour , hours and days . the low probability of eclipse means that populous clusters must be observed for many days . we expect our survey to find 35 low - mass eclipsing systems . the details of the survey are presented in hebb , wyse & gilmore ( 2004 ) . the imaging data has all been acquired , differential photometry obtained and we are now analyzing the derived light curves . an example of the light curve of a candidate eclipsing low - mass system is shown in figure 3 ; the candidate was identified by applying a box - fitting algorithm to the photometric time series . our photometry , plus infrared data from 2mass , is consistent with this being an m - dwarf system . we have applied for follow - up spectroscopic data , together with higher time - sampling photometric data , for this system and for our other candidates . the high - mass stellar mass function long ago can be constrained by the elemental abundances in the long - lived low - mass stars that formed from gas that was enriched by the type ii supernovae from the massive stars of a previous generation ( see e.g. review of wyse 1998 ) . interpretation of the pattern of elemental abundances is easiest for low - mass stars that formed early in a star - formation event , and were enriched by _ only _ massive stars . most of the dwarf spheroidal companions to the milky way have had extended star formation , and so are expected to show evidence in the elemental abundances for the incorporation of iron from long - lived type ia supernovae . the ursa minor dsph is the best candidate for having had a sufficiently short duration of star formation that a significant fraction of its low - mass stars formed prior to the onset of type ia supernovae in sufficient numbers to be noticed in the chemical elemental abundances ; this timescale is uncertain , but likely to be of the order of 12 gyr . the elemental mix produced by a generation of massive stars depends on the massive - star mass function , because the yields of a given type ii supernova depends on its mass . in particular , the @xmath30-element yields ( nuclei formed by successive addition of a helium nucleus ) vary more strongly with progenitor mass than does the iron yield ( e.g. figure 1 of gibson 1998 ) . there appears to have been surprisingly good mixing at early times , at least in the stellar halo of the milky way ( see the remarkably low scatter in the ratio of [ @xmath30/fe ] at [ fe / h ] @xmath31 in the sample analysed by cayrel et al . 2004 ) , so that a well - defined value of [ @xmath30/fe ] is produced by a generation of massive stars of given imf . this is seen as the ` type ii plateau ' in [ @xmath30/fe ] for metal - poor galactic stars . the available elemental abundance data for a handful of individual stars in the umi dsph are consistent with the same value for the type ii plateau as seen in stars of the milky way ( shetrone et al . 2001 ) , with some downturn for more metal - rich stars , as expected if there is an age spread of 12 gyr and an age - metallicity relationship . the simplest interpretation is that the high mass imf was the same in the umi dsph as in the galaxy and that imf is a power - law with salpeter ( 1955 ) slope . most of the stars in the other dwarf spheroidals have low values of [ @xmath30/fe ] , consistent with a standard salpeter imf for massive stars and an extended star formation history , as implied by their colour - magnitude diagram ( see e.g. venn et al . the fossil record in low - mass stars at the present time allows the derivation of the stellar imf at high redshift . the low - mass luminosity function is accessible through star counts , most robustly in a system with a simple stellar population . we have found that the low - mass imf is invariant between globular clusters in the halo of the milky way and an external galaxy , the dwarf spheroidal in ursa minor . the underlying mass function is apparently the same as that for present - day star formation in the local disk of the milky way . the low - mass imf is remarkably invariant , over a broad range of metallicities , age , star - formation rate , baryonic density , dark matter content indeed most of the parameters that _ a priori _ one might have expected to be important in determining the masses of stars . the high - mass imf is also apparently independent of these parameters . this invariance is particularly surprising if the jeans mass plays an important role . i thank my colleagues and collaborators sofia feltzing , jay gallagher , gerry gilmore , leslie hebb , mark houdashelt and tammy smecker - hane for their contributions to the results described here . i would also like to thank the tireless organizers of this stimulating meeting for inviting me . carrera , r. , aparicio , a. , martinez - delgado , d. & alonso - garcia , j. 2002 , aj , 123 , 3199 cayrel , r. et al . 2004 , a&a , 416 , 1117 cole , s. , lacey , c. , baugh , c. & frenk , c. 2000 , mnras , 319 , 168 feltzing , s. , wyse , r.f.g . & gilmore , g. 1999 , apjl , 516 , 17 gallagher , j.s . & wyse , r.f.g . 1994 , pasp , 106 , 1225 gibson , b. 1998 , apj , 501 , 675 hebb , l. , wyse , r.f.g . & gilmore , g. 2004 , aj , december issue ( astro - ph/0409289 ) hernandez , x. , gilmore , g. & valls - gabaud , d. 2000 , mnras , 317 , 831 kauffmann , g. , white , s.d.m . & guiderdoni , b. 1993 , mnras , 264 , 201 kleyna , j. , geller , m. , kenyon , s. , kurtz , m. & thorstensen , j. 1998 , aj , 115 , 2359 palma , c. , majewski , s. , et al . 2003 , aj , 125 , 1352 piotto , g. , cool , a. & king , i. 1997 , aj , 113 , 1345 salpeter , e.e . 1955 , apj , 121 , 161 shetrone , m. , ct , p. & sargent , w. 2001 , apj , 548 , 592 venn , k. et al . 2004 , aj , 128 , 1177 wilkinson , m. , kleyna , j. , evans , w. , gilmore , g. , irwin , m. & grebel , e. 2004 , apjl , 611 , 21 wyse , r.f.g . 1998 , in the stellar imf , asp conf series 142 , eds gilmore & howell , p89 wyse , r.f.g . et al . 2002 , new ast , 7 , 395	the dwarf spheroidal galaxy in ursa minor is apparently dark - matter dominated , and is of very low surface brightness , with total luminosity only equal to that of a globular cluster . indeed its dominant stellar population is old and metal - poor , very similar to that of a classical halo globular cluster in the milky way galaxy . however , the environment in which its stars formed was clearly different from that in the globular clusters in the milky way galaxy what was the stellar imf in this external galaxy a long time ago ? the fossil record of long - lived , low - mass stars contains the luminosity function , derivable from simple star counts . this is presented here . the mass function requires a robust mass - luminosity relation , and we describe the initial results to determine this , from our survey for eclipsing low - mass binaries in old open clusters . the massive star imf at early times is constrained by elemental abundances in low - mass stars , and we discuss the available data . all data are consistent with an invariant imf , most probably of salpeter slope at the massive end , with a turnover at lower masses . # 1#1@xmath0 # 1
the radio emission of active galactic nuclei ( agns ) is synchrotron radiation generated in the relativistic jets that emerge from the nucleus of the galaxy , presumably along the rotational axis of a central supermassive black hole . synchrotron radiation can be highly linearly polarized , up to @xmath1 in the case of a uniform magnetic ( * b * ) field ( pacholczyk 1970 ) . linear polarization observations are essential , as they give information about the orientation and degree of order of the * b * field , as well as the distribution of thermal electrons and the * b-field geometry in the vicinity of the agn . many theorists have suggested that the magnetic fields of these sources are closely connected with the collimation of the jets , and could determine whether sources have prominent jets or not ( eg . meier et al . thus , information on the magnetic fields of these sources is essential in helping us better understand various physical processes in agn jets . vlbi polarization observations of bl lac objects have shown a tendency for the polarization e * vectors in the parsec - scale jets to be aligned with the local jet direction , which implies that the corresponding * b * field is transverse to the jet , because the jet is optically thin ( gabuzda , pushkarev & cawthorne 2000 ) . it seems likely that many of these transverse * b * fields represent the ordered toroidal component of the intrinsic * b * fields of the jets , as discussed by gabuzda et al . ( 2008 ) , see also references therein . depending on the observer s viewing angle and the helix s pitch angle , helical jet * b * fields can also give rise to a ` spine - sheath ' polarization structure in the frame of the observer , with a region of longitudinal polarization ( transverse * b-vectors ) along the central ` spine ' of the jet surrounded by regions of transverse polarization ( longitudinal b-vectors ) near the edges of the jet . the presence of transverse polarization near the edges of the jet could be a natural consequence of a helical jet b * field , although it has also been suggested to be due to interaction with the surrounding medium ( laing 1996 ; lyutikov , pariev & gabuzda 2005 ; attridge , roberts & wardle 1999 ; pushkarev et al . 2005 ) . faraday rotation studies can play a key role in determining the intrinsic * b * field geometries associated with the jets . faraday rotation of the plane of linear polarization occurs during the passage of an electromagnetic wave through a region with free electrons and a magnetic field with a non - zero component along the line - of - sight . the amount of rotation is proportional to the integral of the density of free electrons @xmath2 multiplied by the line - of - sight * b * field , the square of the observing wavelength @xmath3 , and various physical constants ; the coefficient of @xmath3 is called the rotation measure ( rm ) : @xmath4 the intrinsic polarization angle can be obtained from the relation : @xmath5 where @xmath6 is the observed polarization angle , @xmath7 is the intrinsic polarization angle in the absence of faraday rotation and @xmath8 is the observing wavelength . simultaneous multifrequency observations thus potentially enable the determination of the rm , as well as identification of the intrinsic polarization angles . systematic gradients in the faraday rotation measure ( rm ) have been reported previously across the parsec - scale jets of several agns , interpreted as reflecting the systematic change in the line - of - sight component of a toroidal or helical jet * b * field across the jet ( blandford 1993 , asada et al . 2002 , gabuzda , murray & cronin 2004 , zavala & taylor 2005 , gabuzda et al . 2008 , asada et al . 2008a , b,2010 , mahmud , gabuzda & bezrukovs 2009 ) . such fields would come about in a natural way as a result of the ` winding up ' of an initial ` seed ' field by the differential rotation of the central accreting objects ( e.g. nakamura et al . 2001 , lovelace et al . 2002 ) . we consider here two objects in which we have detected transverse rm gradients in both the core region and jet : 0716 + 714 and 1749 + 701 . in both cases , there is a reversal of the direction of the rm gradient between these two regions . we discuss a possible explanation of this phenomena based on magnetic - tower - type models for jet launching . throughout we assume @xmath9 = 71 km / s / mpc , @xmath10 = 0.73 and @xmath11 = 0.27 . very long baseline array ( vlba ) polarization observations of the sources included in this paper were carried out as part of two different studies of the same sample of bl lac objects : one at 4.615.4 ghz and one at 1.361.67 ghz . the high - frequency observations of 0716 + 714 were on 22 march 2004 and of 1749 + 701 were on 22 august 2003 ; the low - frequency observations of 1749 + 701 were on 17 january 2004 . in both cases , the distributions of @xmath12@xmath13 points were virtually identical for the different frequencies observed during a single set of observations , with the baseline lengths scaled in accordance with the individual observing frequencies . standard tasks in the nrao aips package were used for the amplitude calibration and preliminary phase calibration . the instrumental polarizations ( ` d - terms ' ) were determined with the task ` lpcal ' , solving simultaneously for the source polarization . in all cases , the reference antenna used was los alamos . the electric vector position angle ( evpa ) calibration was done using integrated polarization observations of bright , compact sources , obtained with the very large array ( vla ) near in time to our vlba observations , by rotating the evpa for the total vlbi polarization of the source to match the evpa for the integrated polarization of that source derived from vla observations . the observations were carried out at six frequencies : 4.612 , 5.092 , 7.916 , 8.883 , 12.939 and 15.383 ghz . each source was observed for about 2530 minutes at each frequency , in a ` snap - shot ' mode with 810 scans spread out over the observing time period . presented in this paper are the results for 0716 + 714 ( observed on 22 march 2004 ) and 1749 + 701 ( observed on 22 august 2003 ) . the instrumental polarizations ( ` d - terms ' ) were determined using observations of 1156 + 295 ( 22 august 2003 ) and 0235 + 164 ( 22 march 2004 ) . the source of the integrated vla polarizations for the evpa calibration was the nrao website ( www.aoc.nrao.edu/ smyers / calibration/ ) . the vla observations were made at frequencies 5 , 8.5 , 22 and 43 ghz . we found these evpa values to be consistent with a linear @xmath3 law ( faraday rotation ) and were thus able to interpolate the corresponding values for our non - standard frequencies ( see mahmud et al . the sources used were 1803 + 784 and 2200 + 420 . we refined our initial evpa calibration by examining the resulting polarization images for several sources with simple structures and checking for consistency . this led to adjustments of @xmath14 for several of the evpa corrections . this procedure improved the overall self consistency of the polarization and rm maps for virtually all of the sources observed . we estimate that our overall evpa calibration is accurate to within @xmath15 . a summary of our final 4.615.4 ghz evpa corrections is given in mahmud et al . ( 2009 ) . [ tab : evpa_red ] .evpa calibrations for 17 january 2004 [ cols="<,<",options="header " , ] taylor & zavala ( 2010 ) have recently proposed four criteria for the reliable detection of transverse faraday rotation gradients , the most stringent of which is that the observed rm gradient span at least three `` resolution elements '' across the jet . this criterion reflects the desire to ensure that it is possible to distinguish properties between regions located on opposite sides of the jets . the criterion of three `` resolution elements '' has been taken to correspond to three beamwidths , and coincides with the general idea that structures separated by less than a beamwidth are not well resolved . to test the validity of this criterion of taylor & zavala ( 2010 ) , we constructed core jet - like sources with various intrinsic widths and with transverse rm gradients present across their structures , and carried out monte carlo simulations based on these model sources . a description of our monte carlo simulations and the results they yielded are presented in the appendix . the transverse source widths for our model sources correspond to intrinsic widths of about 1/2 , 1/3 , 1/5 , 1/10 and 1/20 of the beam full - width at half - maximum ( fwhm ) in the direction across the jet . the simulations show that the transverse rm gradients introduced into the model visibility data remain visible in the rm maps constructed from `` noisy '' data having the same distribution of @xmath16 points as our observations of 0716 + 714 , even when the intrinsic width of the structure is much smaller than the beam width . both uni - directional model rm gradients and model rm structure containing two oppositely directly transverse gradients in the core region and jet are visible for all the jet widths considered . the results of these new monte carlo simulations thus directly demonstrate that the three - beamwidth criterion of taylor & zavala ( 2010 ) is overly restrictive , since the simulations directly show the possibility of detecting transverse rm gradients even when the intrinsic widths of the corresponding source structures are much less than the beamwidth , resulting in rm distributions that span only @xmath17 beamwidths . this demonstrates that the relatively modest widths spanned by the transverse rm gradients in 0716 + 714 and 1749 + 701 that we report here should not be taken by themselves as grounds to question the reliability of these gradients . we note here that our monte carlo simulations are not intended to provide a physical model for our observations , or to reproduce our observed rm distributions in any detail ; instead , they are intended solely to demonstrate the possibility of detecting a transverse rm gradient in real data , even if the intrinsic jet width is much smaller than the beam fwhm . inspection of fig . 30 of hovatta et al . ( 2012 ) indicates that the fraction of `` false positives '' , i.e. , spurious rm gradients , that were obtained in their monte carlo simulations did not exceed @xmath18 when a @xmath19 criterion was imposed for the rm gradient , even when the observed width of the rm gradient was less than 1.5 beamwidths . this suggests that there may be up to a @xmath18 probability that the rm gradients we report here are spurious , due to their relatively limited widths , although we consider this to be unlikely , given that the rm differences involved correspond to as much as @xmath20 . with regard to the other criteria for reliability of transverse rm gradients proposed by taylor & zavala ( 2010 ) , the criterion that the change in the rm across the jet be at least @xmath19 is satisfied by the rm images in figs . [ fig:0716_rm ] and [ fig:1749_rm ] ( see also table 2 ) . the differences in the rms across the core region and jet of 0716 + 714 ( fig . [ fig:0716_transdist ] ) are approximately @xmath21 ; the differences in the rms across the core region and jet of 1749 + 701 ( fig . [ fig:1749_transdist ] ) are approximately @xmath22 . the criterion that the change in the rm be monotonic and smooth within the errors is also satisfied by the gradients in both 0716 + 714 and in 1749 + 701 . although the gradients suggested by the slices displayed in figs . [ fig:0716_rm ] and [ fig:1749_rm ] are not constant ( linear ) , they are nevertheless monotonic . it is interesting to note here that the simulated rm maps of broderick & mckinney ( 2010 ) typically do not show rm gradients with a constant slope all across the rm distribution after convolution , even though the intrinsic predicted gradients are monotonic [ see , for example , their fig . 8 bottom right panels ] . the remaining criterion proposed by taylor & zavala ( 2010 ) is that the spectrum be optically thin at the location of the observed rm gradient . this criterion is motivated by two factors : ( i ) the desire to avoid possible jumps in the observed polarization angles due to optically thick thin transitions with the observed frequency range , and ( ii ) the fact that the fractional polarization can change rapidly with optical depth in the optically thick regime , leading to the possibility of wavelength - dependent polarization effects when regions having different optical depths at different frequencies are superposed , which could in principle lead to the fitting of spurious rm values in optically thick regions when these are inhomogeneous . this criterion is clearly satisfied by the gradients across the jets of 0716 + 714 and 1749 + 701 , which are all optically thin . the core regions of these two objects are also predominantly , but not fully , optically thin . the core - region spectral indices and @xmath23 values provide no evidence for a transition between optically thick and optically thin in the frequency rangees considered , consistent with the fact that the observed faraday rotations in the polarization angles are all no greater than a few tens of degrees . thus , there is no reason to suspect that jumps in the observed polarization angles due to optical - depth transitions are contributing to the observed core - region rms . we can not completely rule out the possibility that the polarization angles in the core - region are subject to wavelength - dependent optical - depth effects , however , we consider this to be unlikely , for two reasons : ( i ) the degrees of polarization in the core regions are @xmath24 for 0716 + 714 and @xmath25 for 1749 + 701 , indicating a substantial contribution from optically thin regions ; and ( ii ) the quality of the @xmath26 fits for the core regions is no worse than for the optically thin jet regions . thus , our detection of the rm gradients across the jets can be considered firm , while the detection of the oppositely directed rm gradients across the core may be somewhat more tentative , due to the small possibility that the observed polarization could be affected by optical depth effects at some of the observed frequencies . this is much less likely to be the case for 1749 + 701 , since the observed frequencies span the relatively narrow range from 1.361.67 ghz . in both 0716 + 714 and 1749 + 701 , the tentative transverse rm gradients detected in the core region are opposite in direction to the rm gradients detected across the jets ( figs . [ fig:0716_rm][fig:1749_rm ] and figs . [ fig:0716_transdist][fig:1749_transdist ] ) . in fact , a similar behaviour is shown by the parsec - scale rm distribution for 3c 120 presented by gmez et al . ( 2011 ) : their rm map for january 1999 shows higher positive values on the southern side of the jet at the distance of components l and k ( about 4 mas from the core ) , but more negative values on the southern side of the jet at the distance of component o ( about 2 mas from the core ) . at first , this seems difficult to understand , since the direction of an rm gradient associated with a helical * b * field is essentially determined by the direction of the rotation of the central accretion disc and the direction of the poloidal field it winds up , both of which we would expect to be constant in time . we can offer several possible explanations for this result . we briefly discuss these below , and explain our reasoning for identifying the one that we think is the most likely ( see also mahmud et al . 2009 ) . * torsional oscillations of the jet . * one possible interpretation of oppositely directed core and jet transverse rm gradients , could be that the direction of the azimuthal * b * field component changed as a result of torsional oscillations of jet ( bisnovatyi kogan 2007 ) . such torsional oscillations , which may help stabilize the jets , could cause a flip of the azimuthal * b * field from time to time , or equivalently with distance from the core , given the jet outflow . in this scenario , we expect that the direction of the observed transverse rm gradients may reverse from time to time when the direction of the torsional oscillation reverses ; this reversal pattern would presumably then propagate outward with the jet . * reversal of the `` pole '' facing the earth . * another possible interpretation could be that the `` pole '' of the black hole facing the earth reversed . one way to retain a transverse rm gradient in a helical magnetic field model but reverse the direction of this gradient , is if the direction of rotation of the central black hole ( i.e. the direction in which the field threading the accretion disc is `` wound up '' ) remains constant , but the `` pole '' of the black hole facing the earth changes from north to south , or vice versa . to our knowledge , it is currently not known whether such polarity reversals are possible for the central black hole of agn , or on what time scale they could occur . * nested - helix b - field structure . * a simpler and more likely explanation is a magnetic - tower - type model , with poloidal magnetic flux and poloidal current concentrated around the central axis ( lynden - bell 1996 ; nakamura et al . fundamental physics dictates that the magnetic - field lines must close ; in this picture , the magnetic field forms meridional loops that are anchored in the inner and outer parts of the accretion disc , which become twisted due to the differential rotation of the disc . this should essentially give rise to an `` inner '' helical b field near the jet axis and an `` outer '' helical field somewhat further from the jet axis . these two regions of helical field will be associated with oppositely directed rm gradients , and the total observed rm gradient will be determined by which region of helical field dominates the observed rms . thus , the presence of a change in the direction of the observed transverse rm gradient between the core / innermost jet and jet regions well resolved from the core could represent a transition from dominance of the inner to dominance of the outer helical * b * fields in the total observed rm . this seems to provide the simplest explanation for the rm - gradient reversals we observe in these two objects . typically , we would expect the direction of the rm gradients in the core and jet ( i.e. , the regions whose net rm is determined by the inner / outer helical fields ) to remain constant in time , since they should be determined by the source geometry and viewing angle . mahmud et al . ( 2009 ) discuss the possibility that this type of `` nested helical field '' structure could also occasionally give rise to changes in the direction of the observed rm gradients with time within a given source . the polarization rotation - measure images for the two bl lac objects presented here provide new evidence in support of helical magnetic fields associated with the jets of these agn , most importantly , the presence of transverse rotation measure gradients across the jets of both objects . there is also a dominance of transverse * b * fields in the jets of 0716 + 714 and 1749 + 701 , and signs of ` spine - sheath ' polarization structures or orthogonal polarization offset toward one side of the jet in both these sources , consistent with the possibility that these jets carry a helical magnetic - field component : this type of structure can also come about naturally in the case of a helical jet * b * field ( e.g. lyutikov et al . 2005 ; pushkarev et al . we interpret the observed transverse rm gradients as being due to the systematic variation of the toroidal component of a helical * b * field across the jet ( blandford 1993 ) . we note in this connection that the transverse rm gradients in both 0716 + 714 and 1749 + 701 have opposite signs on either side of the jet , making it impossible to explain the gradients as an effect of changing thermal - electron density alone ( there must be a change in the direction of the line - of - sight magnetic field ) . we have also detected tentative transverse rm gradients in the region of the observed vlbi core in both bl lac objects , which can be interpreted as being associated with helical * b * fields in the innermost jets of these sources . further , we have found a striking new feature of the rm distributions in these objects : a reversal in the direction of the transverse rm gradients . similar reversals can be seen in the rm images for 3c 120 presented by gmez et al . ( 2011 ) . at first , this seems difficult to understand , since the direction of the rm gradient associated with a helical * b * field is essentially determined by the direction of the rotation of the central accretion disc and the direction of the poloidal field it winds up . we suggest that the most likely explanation for these reversals is that we are dealing with a ` nested - helical - field ' structure such as that present in magnetic - tower models , in which poloidal field lines emerging from the inner accretion disc form meridional loops that close in the outer part of the disc , with both sides of the loops ( which have oppositely directed poloidal field components ) getting ` wound up ' by the disc rotation . further observations and studies of the rm - gradient reversals observed in these objects can potentially provide key information about how the geometry of the magnetic fields in these agn jets evolve , and may provide information on the jet dynamics and jet collimation . we are currently using a variety of multi - frequency polarization vlba data to search for additional candidates for agn jets displaying rm gradients and rm - gradient reversals on both parsec and decaparsec scales . the research for this publication was supported by a research frontiers programme grant from science foundation ireland and the irish research council for science engineering and technology ( ircset ) . the national radio astronomy observatory is operated by associated universities inc . we thank r. zavala for kindly providing the modified version of the aips ` rm ' task used in this work . we are also grateful to the referee for his careful reading of the paper and thoughtful , competent and useful comments . we constructed a model source with a transverse rm gradient present across its jet , and carried out monte carlo simulations based on this model source . the model source is cylindrical , with a fall - off in intensity on either side of the cylinder axis , and along the axis of the cylinder from a specified point located near one end of the cylinder ( see fig . the resulting appearance of the model emission region is broadly speaking `` core jet - like '' . model visibility data were generated for each of the six frequencies listed in section 2.1 ( 4.615.4 ghz ) , including the effect of the transverse rm gradient in the @xmath27 and @xmath28 visibility data , and these model visibility data were sampled at precisely the @xmath16 points at which 0716 + 714 was observed at each of the frequencies . random thermal noise and the effect of uncertainties in the evpa calibration by up to @xmath15 were added to the sampled model visibilities . the amount of thermal noise added was chosen to yield rms values in the simulated images that were comparable to those in our actual observations . stokes @xmath29 , @xmath27 and @xmath28 images were constructed from these visibilities in casa , using the same beam as was used in the observations of 0716 + 714 presented here ( @xmath30 mas in @xmath31 , where the dimensions given correspond to the full width at half maximum of the beam along its major and minor axes ) . the polarization of the model was chosen to yield a degree of polarization in the lower half of the convolved model image ( the `` core '' region ) of about 5% and a degree of polarization in the upper half of the convolved model image of about 10% similar to the observed values for 0716 + 714 . the @xmath27 and @xmath28 images were then used to construct the corresponding polarization angle ( pang ) images at each frequency , which were , in turn , used to construct rm images in the usual way . finally , monte carlo rm maps were constructed , based on 200 independent realizations of the thermal noise and evpa calibration uncertainty . in each case , the rm values were output to the rm map only in pixels in with the rm uncertainty indicated by the fitting was less than 80 rad / m@xmath32 ; this value was chosen so that no spurious pixels were written to the output rm maps for any of the 200 realizations of the rm distribution . finally , an average rm map was derived by averaging together all 200 individual realizations of the rm distribution . this procedure was carried out for a number of model sources , all with a length of 1 mas and with transverse widths of 0.50 , 0.35 , 0.20 . 0.10 and 0.05 mas . a recent observation of 0716 + 714 with the _ _ r__adioastron space antenna and the european vlbi network has measured the size of a feature in the 6.2-cm core region to be 0.07 mas ( kardashev et al . 2013 ) , and our narrowest jet was designed to have a width somewhat smaller than this . we considered two types of monotonic transverse rm gradients : uni - directional along the entire source structure , and oriented in one direction in the `` core '' region and in the opposite direction in the `` jet '' region , i.e. , showing a reversal . these monte carlo simulations complement those carried out by hovatta et al . ( 2012 ) , in which simulated rm maps were made from model data that did not contain rm gradients , to determine the frequency of spurious transverse rm gradients appearing in the simulated rm maps . examples of the total intensity maps of the model sources used in the simulation are shown in fig . 7 in this appendix , and the results of the rm monte carlo simulations are shown in figs . 815 in this appendix ( we do not show the results for the jet width of 0.50 mas , since these are very similar for the 0.35-mas jet width ) . the panels in figs . 815 show ( i ) the rm map obtained by putting data without added thermal noise through the imaging procedure ( i.e. , the intrinsic rm distribution , but subject to errors due to the clean process and limited @xmath33 coverage ) ; ( ii ) two examples of the individual `` noisy '' rm maps obtained . note that the colour scales for the three maps in a corresponding set have been individually chosen to highlight the rm patterns present , and may differ somewhat in some cases . in all cases , the rm gradients that were introduced into the simulated data are visible in the `` noisy '' rm maps that were obtained , even when the intrinsic width of the jet is approximately 1/20 of the beam full - width at half - maximum ( fwhm ) . this may seem surprising , but it is clearly demonstrated by the simulated data . the magnitude of the rm gradient is reduced by the convolution more and more as the size of the beam relative to the intrinsic size of the jet width increases , but the rm gradients that were initially introduced into the simulated data remain visible . in the case of jet widths much less than the beam fwhm , the appearance of individual realizations can sometimes be fairly strongly distorted by noise ; however , in all cases , averaging together all the individual realizations confirms the presence of the rm gradients in the simulated images . these results essentially indicate that it may not be necessary to impose a restriction on the width spanned by an observed rm gradient , _ _ p__rovided that the difference between the rm values observed at opposite ends of the gradient is at least @xmath19 . this is consistent with the results of murphy & gabuzda ( 2012 ) , who investigated the effect of resolution on transverse rm profiles . it is also consistent with fig . 30 of hovatta et al . ( 2012 ) , which shows that the fraction of `` false positives '' , i.e. , spurious rm gradients , that were obtained in their monte carlo simulations did not exceed @xmath18 when a @xmath19 criterion was imposed for the rm gradient , even when the observed width of the rm gradient was less than 1.5 beamwidths . it becomes important to place some restriction on the width spanned by the gradient if the difference between the rm values being compared is less than @xmath19 , as was also shown clearly by the monte carlo simulations of hovatta et al . ( 2012 ) . ( top ) intrinsic total intensity image of the model core jet - like source with an intrinsic length of 1.0 mas ( 200 pixels ) and an intrinsic width of 0.20 mas ( 40 pixels ) , used for the monte carlo simulations . ( bottom ) one realization of a `` noisy '' intensity map produced during the simulations . the convolving beam is 1.28 [email protected] mas in pa = @xmath35 ( shown in the upp left - hand corner of the convolved image ) . the peak of the unconvolved image is @xmath36 jy , and the contours are 5 , 10 , 20 , 40 , and 80% of the peak . the peak of the convolved image is 1.11 jy / beam , and the contours are @xmath37 , 0.125 , 0.25 , 0.5 , 1 , 2 , 4 , 8 , 16 , 32 , and 64% of the peak . , width=328 ] ( top ) intrinsic total intensity image of the model core jet - like source with an intrinsic length of 1.0 mas ( 200 pixels ) and an intrinsic width of 0.20 mas ( 40 pixels ) , used for the monte carlo simulations . ( bottom ) one realization of a `` noisy '' intensity map produced during the simulations . the convolving beam is 1.28 [email protected] mas in pa = @xmath35 ( shown in the upp left - hand corner of the convolved image ) . the peak of the unconvolved image is @xmath36 jy , and the contours are 5 , 10 , 20 , 40 , and 80% of the peak . the peak of the convolved image is 1.11 jy / beam , and the contours are @xmath37 , 0.125 , 0.25 , 0.5 , 1 , 2 , 4 , 8 , 16 , 32 , and 64% of the peak . , width=328 ] results of monte carlo simulations using model core jet sources with uniformly directed transverse rm gradients . the intrinsic width of the jet ( rm gradient ) is 0.35 mas . the convolving beam ( 1.28 [email protected] mas in pa = @xmath35 ) is shown in the lower left - hand corner of each panel . the top panel shows the rm image obtained by processing the model data as usual , but without adding random noise or evpa calibration uncertainty ; pixels with rm uncertainties exceeding 10 rad / m@xmath32 were blanked . the remaining two panels show two examples of the 200 individual rm images obtained during the simulations ; pixels with rm uncertainties exceeding 80 rad / m@xmath32 were blanked . , width=347 ] results of monte carlo simulations using model core jet sources with uniformly directed transverse rm gradients . the intrinsic width of the jet ( rm gradient ) is 0.35 mas . the convolving beam ( 1.28 [email protected] mas in pa = @xmath35 ) is shown in the lower left - hand corner of each panel . the top panel shows the rm image obtained by processing the model data as usual , but without adding random noise or evpa calibration uncertainty ; pixels with rm uncertainties exceeding 10 rad / m@xmath32 were blanked . the remaining two panels show two examples of the 200 individual rm images obtained during the simulations ; pixels with rm uncertainties exceeding 80 rad / m@xmath32 were blanked . , width=347 ] results of monte carlo simulations using model core jet sources with uniformly directed transverse rm gradients . the intrinsic width of the jet ( rm gradient ) is 0.35 mas . the convolving beam ( 1.28 [email protected] mas in pa = @xmath35 ) is shown in the lower left - hand corner of each panel . the top panel shows the rm image obtained by processing the model data as usual , but without adding random noise or evpa calibration uncertainty ; pixels with rm uncertainties exceeding 10 rad / m@xmath32 were blanked . the remaining two panels show two examples of the 200 individual rm images obtained during the simulations ; pixels with rm uncertainties exceeding 80 rad / m@xmath32 were blanked . , width=347 ] results of monte carlo simulations using model core jet sources with oppositely directed transverse rm gradients in the core region and inner jet . the intrinsic width of the jet ( rm gradient ) is 0.35 mas . the convolving beam ( 1.28 [email protected] mas in pa = @xmath35 ) is shown in the lower left - hand corner of each panel . the top panel shows the rm image obtained by processing the model data as usual , but without adding random noise or evpa calibration uncertainty ; pixels with rm uncertainties exceeding 10 rad / m@xmath32 were blanked . the remaining two panels show two examples of the 200 individual rm images obtained during the simulations ; pixels with rm uncertainties exceeding 80 rad / m@xmath32 were blanked . , width=347 ] results of monte carlo simulations using model core jet sources with oppositely directed transverse rm gradients in the core region and inner jet . the intrinsic width of the jet ( rm gradient ) is 0.35 mas . the convolving beam ( 1.28 [email protected] mas in pa = @xmath35 ) is shown in the lower left - hand corner of each panel . the top panel shows the rm image obtained by processing the model data as usual , but without adding random noise or evpa calibration uncertainty ; pixels with rm uncertainties exceeding 10 rad / m@xmath32 were blanked . the remaining two panels show two examples of the 200 individual rm images obtained during the simulations ; pixels with rm uncertainties exceeding 80 rad / m@xmath32 were blanked . , width=347 ] results of monte carlo simulations using model core jet sources with oppositely directed transverse rm gradients in the core region and inner jet . the intrinsic width of the jet ( rm gradient ) is 0.35 mas . the convolving beam ( 1.28 [email protected] mas in pa = @xmath35 ) is shown in the lower left - hand corner of each panel . the top panel shows the rm image obtained by processing the model data as usual , but without adding random noise or evpa calibration uncertainty ; pixels with rm uncertainties exceeding 10 rad / m@xmath32 were blanked . the remaining two panels show two examples of the 200 individual rm images obtained during the simulations ; pixels with rm uncertainties exceeding 80 rad / m@xmath32 were blanked . , width=347 ] asada k. , inoue m. , uchida y. , kameno s. , fujisawa k. , iguchi s. & mutoh m. 2002 , pasj , 54 , l39 asada k. , inoue m. , kameno s. & nagai h. 2008a , apj , 675 , 79 asada k. , inoue m. , nakamura m. , kameno s. & nagai h. 2008b , apj , 682 , 798 asada k. , nakamura m. , inoue m. , kameno s. & nagai h. 2010 , apj , 720 , 41 attridge j. m. , roberts d. h. & wardle j. f. c. , 1999 , apj , 518 , l87 barlow r. j. 1989 , _ statistics : a guide to the use of statistical methods in the physical sciences _ , john wiley & sons . bisnovatyi - kogan g.s . 2007 , mnras , 376 , 1 , 457 blandford r. d. 1993 , _ astrophysical jets _ , ed . d. burgarella , m. livio , c. odea , cambridge university press , p. 15 blandford r. d. & knigl a 1979 , apj , 232 , 34 broderick , a. e. , & mckinney , j. c. 2010 , apj , 725 , 750 croke s. m. & gabuzda d. c. 2008 , mnras , 386 , 619 gabuzda d. c. & lisakov m. m. 2009 , astrep , 53 ( 1 ) , 51 gabuzda d. c. , murray . & cronin p. j. 2004 , mnras , 351 , l89 gabuzda d. c. , pushkarev . a. b , & cawthorne t. v. 2000 , mnras , 319 , 1109 gabuzda , d. c. , vitrishchak , v. m. , mahmud , m. & o sullivan , s. 2008 , mnras , 384 , 3 , 1003 hallahan d. r. and gabuzda d. c. , proceedings of science , + http://pos.sissa.it//archive/conferences/072/029 + /ix%20evn%20symposium_029.pdf kellermann , k. i. , lister , m. l. , homan , d. c. , et al . 2004 , apj , 609 , 539 hovatta t. , lister m. l. , aller m. f. , aller h. d. , homan d. c. , kovalev y. y. , pushkarev a. b. & savolainen t. 2012 , aj , 144 , 105 kardashev et al . 2013 , astron.rep . , in press . kovalev yu . , pushkarev a. , lobanov , a. , and sokolovsky k. 2008 , proceedings of science + http://pos.sissa.it//archive/conferences/072/007 + /ix%20evn%20symposium_007.pdf laing r. , 1996 , hardee p. e. , bridle a. h. , zensus j. a. , eds , asp conf . 100 , _ energy transport in radio galaxies and quasars_. astron . pacific , san francisco , 241 laing r. a. 1980 , mnras , 193 , 439 lister , m. l. & homan , d. c. 2005 , aj , 130 , 1389 lovelace r. v. e. , li h. , koldoba a. v. , ustyugova g. v. & romanova m. m. 2002 , apj , 572 , 445 lynden - bell , d. 1996 , mnras , 279 , 2 , 389 lyutikov , m. , pariev v. i. , gabuzda d. c. , 2005 , mnras , 360 , 869 nakamura , m. , uchida y. & hirose s. 2001 , new astronomy , 6 , 2 , 61 nakamura , m. , li , h. & li , s. 2006 , apj , 652 , 2 , 1059 mahmud m. , gabuzda d. c. & bezrukovs v. 2009 , mnras , 400 , 2 meier , d. l. , koide , s. , & uchida , y. 2001 , science , 291 , 84 murphy e. & gabuzda d. c. 2012 , journal of physics conference series , 355 , 012009 pacholczyk , a. g. , 1970 , radio astrophysics : _ nonthermal processes in galactic and extragalactic sources _ , ( san francisco : freeman ) pushkarev a. b. 2001 , astrep , 45(9 ) , 667 pushkarev a. b. , gabuzda d. c. , vetukhnovskaya yu . n. & yakimov v. e. 2005 , mnras , 356 , 3 , 859 reynolds , c. , cawthorne t.v . & gabuzda d.c . 2001 , mnras , 327 , 4 , 1071 roberts , d. h. , gabuzda , d. c. & wardle j. f. c. 1987 , apj , 323 , 536 taylor , g. b. , & zavala , r. 2010 , apj , 722 , l183 zavala r. t. & taylor g. b. 2005 , apj , 626 , l73	the idea that systematic faraday rotation gradients across the parsec - scale jets of active galactic nuclei ( agns ) can reveal the presence of helical magnetic ( * b * ) fields has been around since the early 1990s , although the first observation of this phenomenon was about ten years later . these gradients are taken to be due to the systematic variation of the line - of - sight * b * field across the jet . we present here the parsec - scale faraday rotation distributions for the bl lac objects 0716 + 714 and 1749 + 701 , based on polarization data obtained with the very long baseline array ( vlba ) at two wavelengths near each of the 2 cm , 4 cm and 6 cm bands ( 0716 + 714 ) and at four wavelengths in the range 1822 cm ( 1749 + 701 ) . the rotation measure ( rm ) maps for both these sources indicate systematic gradients across their jets , as expected if these jets have helical * b * fields . the significance of these transverse rm gradients is @xmath0 in all cases . we present the results of monte carlo simulations directly demonstrating the possibility of observing such transverse rm gradients even if the intrinsic jet structure is much narrower than the observing beam . we observe an intriguing new feature in these sources , a reversal in the direction of the gradient in the jet as compared to the gradient in the core region . this provides new evidence to support models in which field lines emerging from the central region of the accretion disk and closing in the outer region of the accretion disk are both `` wound up '' by the differential rotation of the disk . the net observed rm gradient will essentially be the sum effect of two regions of helical field , one nested inside the other . the direction of the net rm gradient will be determined by whether the inner or outer helix dominates the rm integrated through the jet , and rm gradient reversals will be observed if the inner and outer helical fields dominate in different regions of the jet . this potentially provides new insights about the overall configuration of the jet * b * fields . [ firstpage ]
the stable model semantics by gelfond and lifschitz @xcite is one of the two most widely studied semantics for normal logic programs , the other one being the well - founded semantics by van gelder , ross and schlipf @xcite . among 2-valued semantics , the stable model semantics is commonly regarded as the one providing the correct meaning to the negation operator in logic programming . it coincides with the least model semantics on the class of horn programs , and with the well - founded semantics and the perfect model semantics on the class of stratified programs @xcite . in addition , the stable model semantics is closely related to the notion of a default extension by reiter @xcite . logic programming with stable model semantics has applications in knowledge representation , planning and reasoning about action . it was also recently proposed as a computational paradigm well suited for solving combinatorial optimization and constraint satisfaction problems @xcite . before we proceed , we will recall the definition of a stable model of a logic program , and some related terminology and properties . the reader is referred to @xcite for a more detailed treatment of the subject . in the paper we deal only with the propositional case . a logic program _ rule _ is an expression @xmath10 of the form @xmath11 where @xmath12 , @xmath13s and @xmath14s are propositional atoms . the atom @xmath12 is called the _ head _ of @xmath10 and is denoted by @xmath15 . atoms @xmath13 and @xmath14 form the _ body _ of @xmath10 . the set @xmath16 is called the _ positive _ body of @xmath10 ( denoted by @xmath17 ) and the set @xmath18 is called the _ negative _ body of @xmath10 ( denoted by @xmath19 ) . a _ logic program _ is a collection of rules . for a logic program @xmath2 , by @xmath20 we denote the set of atoms occurring in its rules and by @xmath21 the set of atoms appearing as the heads of rules in @xmath2 . we will also denote the _ size _ of @xmath2 , that is , the total number of occurrences of atoms in @xmath2 , by @xmath22 . throughout the paper we use @xmath23 to denote the number of atoms in a logic program @xmath2 , and @xmath7 to denote the size of @xmath2 . a set of atoms @xmath24 _ satisfies _ a rule @xmath10 if @xmath25 , or if @xmath26 , or if @xmath27 . a set of atoms @xmath28 is a _ model _ of a program @xmath2 if @xmath29 satisfies all rules of @xmath2 . a logic program rule @xmath10 is called _ horn _ if @xmath30 . a _ horn program _ is a program whose every rule is a horn rule . the intersection of two models of a horn program @xmath2 is a model of @xmath2 . since the set of all atoms is a model of @xmath2 , it follows that every horn program @xmath2 has a unique least model . we will denote this model by @xmath31 . the least model of a horn program @xmath2 can be constructed by means of the van emden - kowalski operator @xmath32 @xcite . given a horn program @xmath2 and a set of atoms @xmath33 , we define @xmath34 we also define @xmath35 since the operator @xmath32 is monotone , the sequence @xmath36 is monotone and its union yields the least model of a horn program @xmath2 . that is , @xmath37 if @xmath2 is finite , the sequence stabilizes after finitely many steps . for a logic program rule @xmath10 , by @xmath38 we denote the rule obtained from @xmath10 by eliminating all negated atoms from the body of @xmath10 . if @xmath2 is a logic program , we define @xmath39 . let @xmath2 be a logic program ( possibly with rules containing negated atoms ) . for a set of atoms @xmath28 we define the _ reduct _ of @xmath2 with respect to @xmath29 to be the program obtained by eliminating from @xmath2 each rule @xmath10 such that @xmath40 ( we call such rules _ blocked _ by @xmath29 ) , and by removing negated atoms from all other rules in @xmath2 . the resulting program is a horn program . we will denote it by @xmath41 . as a horn program , @xmath41 has the least model @xmath42 . if @xmath43 , @xmath29 is a _ stable _ model of @xmath2 . clearly , if @xmath29 is a stable model of @xmath2 , @xmath44 . both the notion of the reduct and of a stable model are due to gelfond and lifschitz @xcite . in the paper we restrict our attention to programs whose rules do not contain multiple positive occurrences of the same atom nor multiple negative occurrences of the same atom in the body . it is clear that adopting this assumption does not limit the generality of our considerations . repetitive occurrences can be eliminated in linear time ( in the size of the program ) and doing so does not affect stable models of the program . if @xmath29 is a stable model of @xmath2 , each rule @xmath10 such that @xmath45 and @xmath46 ( that is , such that @xmath29 satisfies its body ) , is called a _ generating _ rule for @xmath29 . clearly , if @xmath29 is a stable model of @xmath2 , it is also a stable model of the program consisting of all rules in @xmath2 that are generating for @xmath29 . there are several ways to look at the search space of possible stable models of a program @xmath2 . the most direct way is to look for stable models by considering all candidate subsets of @xmath21 . for each candidate subset @xmath44 , one can compute the corresponding reduct @xmath41 , its least model @xmath42 , and check the equality @xmath43 to decide whether @xmath29 is stable . an alternative way is to observe that stable models are determined by subsets of the set of atoms appearing negated in @xmath2 . indeed , let us denote this set by @xmath47 and let us consider sets @xmath28 and @xmath48 . let @xmath49 . then , @xmath29 is a stable model of @xmath2 if and only if @xmath50 , @xmath51 and @xmath52 . thus , the existence of stable models can be decided by considering subsets of @xmath47 . finally , one can consider the search space of all subsets of @xmath2 itself , and regard each such subset as a candidate for the set of generating rules of a stable model . indeed , if @xmath28 and @xmath53 , then @xmath29 is a stable model of @xmath2 if and only if @xmath54 , @xmath55 is the set of all generating rules for @xmath29 in @xmath2 and @xmath56 . the problem with the stable model semantics is that , even in the propositional case , reasoning with logic programs under the stable model semantics is computationally hard . it is well - known that deciding whether a finite propositional logic program has a stable model is np - complete @xcite . consequently , it is not at all clear that logic programming with the stable model semantics can serve as a practical computational tool . this issue can be resolved by implementing systems computing stable models and by experimentally studying the performance of these systems . several such projects are now under way . niemel and simons @xcite developed a system , _ smodels _ , for computing stable models of finite function symbol - free logic programs and reported very promising performance results . for some classes of programs , _ smodels _ decides the existence of a stable model in a matter of seconds even if an input program consists of tens of thousands of clauses . encouraging results on using _ smodels _ to solve planning problems are reported in @xcite . another well - advanced system is deres @xcite , designed to compute extensions of arbitrary propositional default theories but being especially effective for default theories encoding propositional logic programs . finally , systems capable of reasoning with disjunctive logic programs were described in @xcite and @xcite . however , faster implementations will ultimately depend on better understanding of the algorithmic aspects of reasoning with logic programs under the stable model semantics . in this paper , we investigate the complexity of deciding whether a finite propositional logic program has stable models of some restricted sizes . specifically , we study the following two problems ( @xmath57 stands for the number of rules in a logic program @xmath2 ) : @xmath1 : : ( large stable models ) given a finite propositional logic program @xmath2 and an integer @xmath0 , decide whether there is a stable model of @xmath2 of size at least @xmath3 . @xmath4 : : ( small stable models ) given a finite propositional logic program @xmath2 and an integer @xmath0 , decide whether there is a stable model of @xmath2 of size no more than @xmath0 . inputs to the problems @xmath1 and @xmath4 are pairs @xmath58 , where @xmath2 is a finite propositional logic program and @xmath0 is a non - negative integer . problems of this type are referred to as _ parametrized _ decision problems . by fixing a parameter , a parameterized decision problem gives rise to its _ fixed - parameter _ version . in the case of problems @xmath1 and @xmath4 , by fixing @xmath0 we obtain the following two fixed - parameter problems ( @xmath0 is now no longer a part of input ) : @xmath59 : : given a finite propositional logic program @xmath2 , decide whether @xmath2 has a stable model of size at least @xmath60 . @xmath61 : : given a finite propositional logic program @xmath2 , decide whether @xmath2 has a stable model of size at most @xmath0 . the problems @xmath1 and @xmath4 are np - complete . it follows directly from the np - completeness of the problem of existence of stable models @xcite . but fixing @xmath0 makes a difference ! clearly , the fixed - parameter problems @xmath61 and @xmath59 can be solved in polynomial time ( unlike the problems @xmath4 and @xmath1 which , most likely , can not ) . indeed , consider a finite propositional logic program @xmath2 . then , there are @xmath62 subsets of @xmath20 ( in fact , as pointed out earlier , it is enough to consider subsets of @xmath21 or @xmath47 ) of cardinality at most @xmath0 ( we recall that in the paper @xmath23 stands for the number of atoms in @xmath2 ) . for each such subset @xmath29 , it can be checked in time linear in @xmath7 the size of @xmath2 whether @xmath29 is a stable model of @xmath2 . thus , one can decide whether @xmath2 has a stable model of size at most @xmath0 in time @xmath63 . similarly , there are only @xmath64 subsets of @xmath2 of size at least @xmath3 . each such subset is a candidate for the set of generating rules of a stable model of size at least @xmath3 ( and smaller subsets , clearly , are not ) . given such a subset @xmath65 , one can check in time @xmath66 whether @xmath65 generates a stable model for @xmath2 . thus , it follows that there is an algorithm that decides in time @xmath67 whether a logic program @xmath2 has a stable model of size at least @xmath3 . while both algorithms are polynomial in the size of the program , their asymptotic complexity is expressed by the product of the size of a program and a polynomial of order @xmath0 in the number of atoms of the program or in the number of rules of the program . even for small values of @xmath0 , say for @xmath68 , the functions @xmath69 and @xmath70 grow very fast with @xmath71 , @xmath72 and @xmath57 , and render the corresponding algorithms infeasible . an important question is whether algorithms for problems @xmath61 and @xmath59 exist whose order is significantly lower than @xmath0 , preferably , a constant independent of @xmath0 . the study of this question is the main goal of our paper . a general framework for such investigations was proposed by downey and fellows @xcite . they introduced the concepts of _ fixed - parameter tractability _ and _ fixed - parameter intractability _ that are defined in terms of a certain hierarchy of complexity classes known as the @xmath73 _ hierarchy_. in the paper , we show that the problem @xmath1 is fixed - parameter tractable and demonstrate an algorithm that for every fixed @xmath0 decides the problem @xmath59 in linear time a significant improvement over the straightforward algorithm presented earlier . on the other hand , we demonstrate that the problem @xmath4 is much harder . we present an algorithm to decide the problems @xmath61 , for @xmath74 , that is asymptotically faster than the simple algorithm described above but the improvement is rather insignificant . our algorithm runs in time @xmath75 , an improvement only by the factor of @xmath23 . the difficulty in finding a substantially better algorithm is not coincidental . we provide evidence that the problem @xmath4 is _ fixed - parameter intractable_. this result implies it is unlikely that there is an algorithm to decide the problems @xmath61 whose running time would be given by a polynomial of order independent of @xmath0 . the study of fixed - parameter tractability of problems occurring in the area of nonmonotonic reasoning is a relatively new research topic . another paper that pursues this direction is @xcite . the authors focus there on parameters describing structural properties of programs and show that in some cases , fixing these parameters leads to polynomial algorithms . our paper is organized as follows . in section [ ifp ] , we recall basic concepts of the theory of fixed - parameter intractability by downey and fellows @xcite . the following two sections present the algorithms to decide the problems @xmath1 and @xmath4 , respectively . the next section focuses on the issue of fixed - parameter intractability of the problem @xmath4 and contains the two main results of the paper . the last section contains conclusions and open problems . this section recalls basic ideas of the work of downey and fellows on fixed - parameter intractability . the reader is referred to @xcite for a detailed treatment of this subject . informally , a _ parametrized _ decision problem is a decision problem whose inputs are pairs of items , one of which is referred to as a _ parameter_. the graph colorability problem is an example of a parametrized problem . the inputs are pairs @xmath76 , where @xmath77 is an undirected graph and @xmath0 is a non - negative integer . the problem is to decide whether @xmath77 can be colored with at most @xmath0 colors . another example is the vertex cover problem in a graph . again , the inputs are graph - integer pairs @xmath76 and the question is whether @xmath77 has a vertex cover of cardinality @xmath0 or less . the problems @xmath4 and @xmath1 are also examples of parametrized decision problems . formally , a _ parametrized _ decision problem is a set @xmath78 , where @xmath79 is a fixed alphabet . by selecting a concrete value @xmath80 of the parameter , a parametrized decision problem @xmath81 gives rise to an associated _ fixed - parameter _ problem @xmath82 . for instance , by fixing the value of @xmath0 to 3 , we get a fixed - parameter version of the colorability problem , known as 3-colorability . inputs to the 3-colorability problem are graphs and the question is to decide whether an input graph can be colored with 3 colors . clearly , the problems @xmath61 ( @xmath59 , respectively ) are fixed - parameter versions of the problem @xmath4 ( @xmath1 , respectively ) . the interest in the fixed - parameter problems stems from the fact that they are often computationally easier than the corresponding parametrized problems . for instance , the problems @xmath4 and @xmath1 are np - complete yet , as we saw earlier , their parametrized versions @xmath61 and @xmath59 can be solved in polynomial time . similarly , the vertex cover problem is np - complete but its fixed - parameter versions are in the class p. to see this , observe that to decide whether a graph has a vertex cover of size at most @xmath0 , where @xmath0 is a fixed value and not a part of an input , it is enough to generate all subsets with at most @xmath0 elements of the vertex set of a graph , and then check if any of them is a vertex cover . a word of caution is in order here . it is not always the case that fixed - parameter problems are easier . for instance , the 3-colorability problem is still np - complete . as we already pointed out , the fact that a problem admits a polynomial - time solution does not necessarily mean that practical algorithms to solve it exist . an algorithm that runs in time @xmath83 , where @xmath84 is the size of the input , is hardly more practical than an algorithm with an exponential running time ( and may even be a worse choice in practice ) . the algorithms we presented so far to argue that the problems @xmath61 , @xmath59 and the fixed - parameter versions of the vertex cover problem are in p rely on searching through the space of @xmath85 possible solutions ( where @xmath84 is the number of atoms of a program , the number of rules of a program , or the number of vertices in a graph , respectively ) . thus , these algorithms are not practical , except for the very smallest values of @xmath0 . the key question is how fast those polynomial - time solvable fixed - parameter problems can really be solved . or , in other words , can one significantly improve over the brute - force approach ? a technique to deal with such questions is provided by the fixed - parameter intractability theory of downey and fellows @xcite . a parametrized problem @xmath78 is _ fixed - parameter tractable _ if there exist a constant @xmath86 , an integer function @xmath87 and an algorithm @xmath88 such that @xmath88 determines whether @xmath89 in time @xmath90 ( @xmath91 stands for the length of a string @xmath92 ) . the class of fixed - parameter tractable problems will be denoted by fpt . clearly , if a parametrized problem @xmath81 is in fpt , each of the associated fixed - parameter problems @xmath93 is solvable in polynomial time by an algorithm whose exponent does not depend on the value of the parameter @xmath94 . it is known ( see @xcite ) that the vertex cover problem is in fpt . there is substantial evidence to support a conjecture that some parametrized problems whose fixed - parameter versions are in p are not fixed - parameter tractable . to study and compare complexity of parametrized problems downey and fellows proposed the following notion of reducibility . a parametrized problem @xmath81 can be _ reduced _ to a parametrized problem @xmath95 if there exist a constant @xmath86 , an integer function @xmath96 and an algorithm @xmath88 that to each instance @xmath97 of @xmath81 assigns an instance @xmath98 of @xmath95 such that 1 . @xmath99 depends upon @xmath100 and @xmath94 and @xmath101 depends upon @xmath94 only , 2 . @xmath88 runs in time @xmath102 , 3 . @xmath89 if and only if @xmath103 . downey and fellows also defined a hierarchy of complexity classes called the _ w hierarchy _ : @xmath104 } \subseteq { \rm w[2 ] } \subseteq { \rm w[3 ] } \ldots\ ] ] the classes w[t ] can be described in terms of problems that are complete for them ( a problem @xmath105 is _ complete _ for a complexity class @xmath106 if @xmath107 and every problem in this class can be reduced to @xmath105 ) . let us call a boolean formula _ @xmath108-normalized _ if it is of the form of product - of - sums - of - products ... of literals , with @xmath108 being the number of products - of , sums - of expressions in this definition . for example , 2-normalized formulas are products of sums of literals . thus , the class of 2-normalized formulas is precisely the class of cnf formulas . we define the _ weighted @xmath108-normalized satisfiability problem _ as : @xmath109 : : given a @xmath108-normalized formula @xmath110 , decide whether there is a model of @xmath110 with exactly @xmath0 atoms ( or , alternatively , decide whether there is a satisfying valuation for @xmath110 which assigns the logical value * true * to exactly @xmath0 atoms ) downey and fellows show that for @xmath111 , the problems @xmath109 are complete for the class w[t ] . they also show that a restricted version of the problem @xmath112 : @xmath113 : : given a 3cnf formula @xmath110 and an integer @xmath0 ( parameter ) , decide whether there is a model of @xmath110 with exactly @xmath0 atoms is complete for the class @xmath114 $ ] . downey and fellows conjecture that all the implications in ( [ eq1 ] ) are proper . in particular , they conjecture that problems in the classes w[t ] , with @xmath115 , are not fixed - parameter tractable . in the paper , we relate the problem @xmath4 to the problems @xmath112 and @xmath116 to place the problem @xmath4 in the w hierarchy , to obtain estimates of its complexity and to argue for its fixed - parameter intractability . in this section we will show an algorithm for the parametrized problem @xmath1 that runs in time @xmath117 , where @xmath58 is an input instance and , as in all other places in the paper , @xmath71 . this result implies that the problem @xmath1 is fixed - parameter tractable and that there is an algorithm that for every fixed @xmath0 solves the problem @xmath59 in linear - time . given a logic program @xmath2 , denote by @xmath118 the logic program obtained from @xmath2 by eliminating from the bodies of the rules in @xmath2 all literals @xmath119 , where @xmath12 is not the head of any rule from @xmath2 . the following well - known result states the key property of the program @xmath118 . [ th-10 ] a set of atoms @xmath29 is a stable model of a logic program @xmath2 if and only if @xmath29 is a stable model of @xmath118 . lemma [ th-10 ] implies that the problem @xmath1 has a positive answer for @xmath58 if and only if it has a positive answer for @xmath120 . moreover , it is easy to see that @xmath118 can be constructed from @xmath2 in time linear in the size of @xmath2 . thus , when looking for algorithms to decide the problem @xmath1 we may restrict our attention to programs @xmath2 in which every atom appearing negated in the body of a rule appears also as the head of a rule ( that is , to such programs @xmath2 for which we have @xmath121 ) . by @xmath122 let us denote the program consisting of those rules @xmath10 in @xmath2 for which @xmath123 . we have the following lemma . [ lem-42 ] let @xmath2 be a logic program such that @xmath121 . let @xmath28 be a set of atoms such that @xmath124 . then : 1 . @xmath29 is a stable model of @xmath2 if and only if @xmath29 is a stable model of @xmath122 2 . if @xmath29 is a stable model of @xmath122 , then @xmath122 has no more than @xmath125 different negated literals appearing in the bodies of its rules . proof : ( 1 ) consider a rule @xmath126 . then @xmath127 and , consequently , @xmath40 . indeed , if @xmath128 , then @xmath129 . since @xmath121 , @xmath130 . in addition , ( both if we assume that @xmath29 is a stable model of @xmath2 and if we assume that @xmath29 is a stable model of @xmath122 ) , we have @xmath44 . thus , @xmath131 . now observe that @xmath132 . thus , @xmath133 , a contradiction . since for every rule @xmath126 we have @xmath27 , it follows that @xmath134 . hence , @xmath135 if and only if @xmath136 consequently , @xmath29 is a stable model of @xmath2 if and only if @xmath29 is a stable model of @xmath122 . \(2 ) let @xmath55 be the set of rules from @xmath122 such that @xmath137 if and only if @xmath46 ( the rules in @xmath55 contribute to the reduct @xmath138 ) and let @xmath139 be the set of the remaining rules in @xmath122 ( these are the rules that are eliminated when the reduct @xmath138 is computed ) . since @xmath121 , for every rule @xmath140 , @xmath130 . thus , @xmath141 . since @xmath44 ( as @xmath29 is a stable model of @xmath122 ) and @xmath132 , we have @xmath142 . further , since @xmath143 , it follows that @xmath144 . consequently , @xmath145 . hence , the second part of the assertion follows . @xmath146 let us now consider the following algorithm for the problem @xmath59 ( the input to this algorithm is a logic program @xmath2 ) . 1 . eliminate from the input logic program @xmath2 all literals @xmath119 , where @xmath12 is not the head of any rule from @xmath2 . denote the resulting program by @xmath147 . 2 . compute the set of rules @xmath148 consisting of those rules @xmath10 in @xmath147 for which @xmath123 . 3 . decide whether @xmath148 has a stable model @xmath29 such that @xmath149 . this algorithm reports yes if and only if the program @xmath148 has a stable model @xmath29 such that @xmath149 . by lemma [ lem-42 ] , that happens precisely if and only if @xmath147 has a stable model @xmath29 such that @xmath149 . this last statement , by lemma [ th-10 ] , is equivalent to the statement that @xmath2 has a stable model @xmath29 such that latexmath:[$\|m\|\geq @xmath59 . let us notice that steps 1 and 2 can be implemented in time @xmath66 , where the constant hidden by the `` big o '' notation does not depend on @xmath0 . to implement step 3 , let us recall that every stable model of a logic program is determined by some subset of the set of atoms that appear negated in the program ( each such subset uniquely determines the reduct , as we stated in the introduction ; see also @xcite ) . by lemma [ lem-42 ] , the set of such atoms in the program @xmath148 has cardinality at most @xmath151 . checking for each subset of this set whether it determines a stable model of @xmath148 can be implemented in time @xmath152 . consequently , our algorithm runs in time @xmath117 ( with the constant hidden by the `` big o '' notation independent of @xmath0 ) . the problem @xmath1 is fixed - parameter tractable . moreover , for each fixed @xmath0 there is a linear - time algorithm to decide whether a logic program @xmath2 has a stable model of size at least @xmath3 . in the introduction we pointed out that there is a straightforward algorithm to decide the problem @xmath61 that runs in time @xmath153 , where @xmath71 and @xmath72 . for @xmath74 ( the assumption we adopt in this section ) , this algorithm can be slightly improved . namely , we will now describe an algorithm for the problem @xmath61 that runs in time @xmath154 , where @xmath155 is some integer function . thus , if @xmath0 is fixed and not a part of the input , this improved algorithm runs in time @xmath75 . we present our algorithm under the assumption that input logic programs are _ proper_. we say that a logic program rule @xmath10 is _ proper _ if : ( p1 ) : : @xmath156 , and ( p2 ) : : @xmath157 we say that a logic program @xmath2 is _ proper _ if all its rules are proper . rules that violate at least one of the conditions ( p1 ) and ( p2 ) ( that is , rules that are not proper ) have no influence on the collection of stable models of a program as we have the following well - known result ( see , for instance , @xcite ) . [ l-11 ] a set of atoms @xmath29 is a stable model of a logic program @xmath2 if and only if @xmath29 is a stable model of the subprogram of @xmath2 consisting of all proper rules in @xmath2 . it is easy to see that rules that violate ( p1 ) or ( p2 ) can be eliminated from a logic program @xmath2 in time @xmath66 . thus , the restriction to proper programs does not affect the generality of our discussion . for a proper logic program @xmath2 and for a set @xmath158 of atoms , we define @xmath159 to be the program consisting of all those rules @xmath10 of @xmath2 that are not blocked by @xmath88 ( in other words , those that satisfy @xmath160 ) and whose positive body is contained in @xmath88 ( in other words , such that @xmath161 ) . let @xmath2 be a logic program and let @xmath158 be a set of atoms . a stable model @xmath29 of @xmath2 is called _ @xmath88-based _ if 1 . @xmath29 is of the form @xmath162 , where @xmath163 , and 2 . @xmath164 ( in other words , when computing @xmath42 , the derivation of @xmath88 does not require that @xmath12 be derived first ) . we have the following simple lemma . [ aa ] let @xmath0 be an integer such that @xmath74 . a proper logic program @xmath2 has a stable model of cardinality @xmath0 if and only if for some @xmath165 , with @xmath166 , @xmath2 has an @xmath88-based stable model . it follows from lemma [ aa ] that when deciding the existence of @xmath0-element stable models , @xmath74 , it is enough to focus on the existence of @xmath88-based stable models . this is the approach we take here . in most general terms , our algorithm for the problem @xmath61 consists of generating all subsets @xmath167 , with @xmath168 , and for each such subset @xmath88 , of checking whether @xmath2 has an @xmath88-based stable model . this latter task is the key . we will now describe an algorithm that , given a logic program @xmath2 and a set @xmath158 , decides whether @xmath2 has an @xmath88-based stable model . to this end , we define @xmath169 to be the program consisting of all those rules @xmath10 of @xmath2 such that : 1 . @xmath160 ( @xmath10 is not blocked by @xmath88 ) 2 . @xmath170 3 . @xmath171 consists of exactly one element ; _ we will denote it by @xmath172_. our algorithm is based on the following result allowing us to restrict attention to the program @xmath159 ( the statement of the lemma and its proof rely on the terminology introduced above ) . [ l-12 ] let @xmath88 be a set of atoms . a proper logic program @xmath2 has an @xmath88-based stable model if and only if @xmath159 has an @xmath88-based stable model @xmath173 , such that @xmath174 . proof : ( @xmath175 ) let @xmath29 be an @xmath88-based stable model of @xmath2 . assume that @xmath173 , for some @xmath176 . since @xmath177 , @xmath178 . since @xmath29 is @xmath88-based , we have that @xmath164 . it follows that @xmath29 is an @xmath88-based stable model of @xmath159 . let us assume that there is a rule @xmath179 such that @xmath180 . the rule @xmath181 is not blocked by @xmath88 . since @xmath182 , we have that @xmath183 ( we recall that all rules in @xmath2 are proper ) . hence , @xmath181 is not blocked by @xmath184 either . consequently , @xmath185 . since @xmath179 , the body of @xmath186 ( that is , @xmath187 ) is contained in @xmath29 . the set @xmath29 is a least model of @xmath41 . in particular , @xmath29 satisfies @xmath186 . thus , it follows that @xmath188 . in the same time , @xmath189 ( as @xmath181 is proper ) . thus , @xmath190 , a contradiction ( we recall that @xmath179 ) . it follows that @xmath174 . ( @xmath191 ) we will now assume that @xmath173 is an @xmath88-based stable model of @xmath159 such that @xmath174 . similarly as before , we have @xmath192 . let us assume that @xmath193 . then there is a rule @xmath108 in @xmath41 such that the body of @xmath108 is contained in @xmath29 and @xmath194 . let @xmath181 be a rule in @xmath2 that gives rise to @xmath108 when constructing the reduct . assume first that the body of @xmath108 ( that is , @xmath187 ) is contained in @xmath88 . then @xmath195 , @xmath196 and , consequently , @xmath197 , a contradiction . thus , the body of @xmath108 is not contained in @xmath88 . since the body of @xmath108 is contained in @xmath29 , it consists of @xmath12 and , possibly , some other elements , all of which are in @xmath88 . it follows that @xmath179 . consequently , @xmath180 and @xmath198 , a contradiction . thus , @xmath199 , that is , @xmath29 is a stable model of @xmath2 . since @xmath200 , it follows that @xmath29 is an @xmath88-based model of @xmath2 . @xmath146 let @xmath88 be a set of atoms . a logic program with negation , @xmath2 , is an _ @xmath88-program _ if @xmath201 , that is if for every rule @xmath140 we have @xmath202 and @xmath203 . clearly , the program @xmath159 , described above , is an @xmath88-program . we will now focus on @xmath88-programs and their @xmath88-based stable models . let @xmath88 be a set of atoms . we denote by @xmath204 the set of all proper horn rules over the set of atoms @xmath88 . clearly , the cardinality of @xmath204 depends on the cardinality of @xmath88 only . further , we define @xmath205 to be the set of all horn programs @xmath206 satisfying the condition @xmath207 . as in the case of @xmath204 , the cardinality of @xmath205 also depends on the size of @xmath88 only . we will now describe conditions that determine whether an @xmath88-program @xmath2 has an @xmath88-based stable model . to this end , with every atom @xmath208 , we associate the following values : * @xmath209 if there is a rule @xmath181 in @xmath2 with @xmath210 and @xmath183 ; @xmath211 , otherwise * @xmath212 the number of rules @xmath181 in @xmath2 with @xmath213 and @xmath214 . further , with every proper horn rule @xmath215 and every atom @xmath208 , we associate the quantity : * @xmath216 if there is a rule @xmath181 in @xmath2 with @xmath217 and @xmath183 ; @xmath218 , otherwise . the following lemma characterizes @xmath88-based stable models of an @xmath88-program . both the statement of the lemma and its proof rely on the terminology introduced above . [ l - main ] let @xmath88 be a set of atoms , let @xmath2 be an @xmath88-program and let @xmath12 be an atom such that @xmath163 . then @xmath162 is an @xmath88-based stable model of @xmath2 if and only if @xmath219 , @xmath220 , and for some program @xmath221 and for every rule @xmath222 , @xmath223 . proof : @xmath224 we denote @xmath225 and assume that @xmath29 is an @xmath88-based stable model for @xmath2 . it follows that @xmath135 . let @xmath226 be the subprogram of @xmath2 consisting of those rules of @xmath2 whose head belongs to @xmath88 . since @xmath29 is an @xmath88-based stable model of @xmath2 , we have @xmath227 . let @xmath147 be the program obtained from @xmath228 by removing multiple occurrences of rules . clearly , @xmath221 . it follows directly from the definition of the reduct that for every rule @xmath222 , @xmath229 . next , we observe that @xmath230 . thus , @xmath220 . let us assume that @xmath231 . let @xmath10 be a rule in @xmath2 such that @xmath232 and @xmath233 . since @xmath2 is an @xmath88-program , @xmath234 . thus , it follows that @xmath235 . we also have that @xmath236 . since @xmath29 is a model of @xmath41 , @xmath25 . however , in the same time we have that @xmath237 , a contradiction . it follows that @xmath238 . we now assume that for some @xmath163 , @xmath219 , @xmath220 and for some program @xmath221 and for every rule @xmath222 , @xmath229 . as before , we set @xmath173 . we will show that @xmath135 . first , since @xmath2 is an @xmath88-program and @xmath240 for every rule @xmath241 , it follows that @xmath242 . thus , @xmath243 . second , we have that @xmath244 . thus , there is a rule @xmath140 such that @xmath245 and @xmath233 . it follows that @xmath246 and @xmath235 . since @xmath242 , @xmath247 and @xmath161 , we obtain that @xmath248 . thus , @xmath164 . finally , since @xmath238 , we have that for every rule @xmath249 such that @xmath183 , @xmath188 . thus , @xmath42 does not contain any atom not in @xmath29 . consequently , @xmath250 and @xmath29 is a stable model of @xmath2 . since @xmath164 , @xmath29 is an @xmath88-based stable model of @xmath2 . @xmath146 we will discuss now effective ways to compute values @xmath251 , @xmath252 and @xmath253 . clearly , computing the values @xmath252 can be accomplished in time linear in the size of the program , that is , in time @xmath66 . indeed , we start by initializing all values @xmath252 to 0 . then , for each rule @xmath249 , we set @xmath254 if @xmath255 , and leave @xmath256 unchanged , otherwise . to decide which is the case requires that we scan all negated lierals in the body of @xmath181 . that takes time @xmath257 . thus , the overall time is @xmath66 . computing values @xmath251 and @xmath253 is more complicated . first , we prove the following lemma . [ vn ] let @xmath2 be an @xmath88-program , let @xmath163 and let @xmath258 . then 1 . @xmath259 if and only if @xmath260 . 2 . @xmath240 if and only if @xmath261 . proof : ( 1 ) let us assume first that @xmath259 . then there is a rule @xmath262 such that @xmath263 and @xmath183 . thus , @xmath264 . consequently , the identity @xmath265 follows . all the implications in this argument can be reversed . hence , we obtain the assertion ( 1 ) . \(2 ) let us assume that @xmath240 . then , there is a rule @xmath249 such that @xmath217 and @xmath183 . consequently , @xmath266 . as in ( 1 ) , all the implications are in fact equivalences and the assertion ( 2 ) follows . @xmath146 lemma [ vn ] shows that to compute all the values @xmath251 one has to compute the set @xmath267 to this end , for each atom @xmath12 we will compute the number of sets in @xmath268 that @xmath12 is a member of . we will denote this number by @xmath269 . we first initialize all values @xmath269 to 0 . then , we consider all sets in @xmath270 in turn . for each such set and for each atom @xmath12 in this set we set @xmath271 . the set @xmath272 is given by all those atoms @xmath12 for which @xmath269 is equal to the number of sets in @xmath273 . it is clear that the time needed for this computation is linear in the size of the program ( assuming appropriate linked - list representation of rules ) . thus , all the values @xmath251 can be computed in time linear in the size of the program , that is , in @xmath66 steps . to compute values @xmath253 we proceed similarly . first , we compute all the sets @xmath274 , where @xmath215 . to this end , we scan all rules in @xmath2 in order and for each of them we find the rule @xmath258 such that @xmath217 . then we include @xmath181 in the set @xmath275 . given @xmath181 , it takes @xmath276 steps to identify rule @xmath10 ( where @xmath277 is some function ) . indeed , the size of @xmath187 is bound by @xmath278 as @xmath2 is an @xmath88-program . moreover , the number of rules in @xmath204 depends on @xmath278 only . thus , the task of computing all sets @xmath275 , for @xmath215 , can be accomplished in @xmath279 steps . next , for each these sets of rules , we proceed as in the case of values @xmath251 , to compute their intersections . each such computation takes time @xmath66 , where @xmath71 ) . thus , computing all the values @xmath253 can be accomplished in time @xmath280 , for some function @xmath87 . we can now put all the pieces together . as a result of our considerations , we obtain the following algorithm for deciding the problem @xmath61 . = = = = = = = = = = + * algorithm to decide the problem @xmath61 , @xmath74 * + * input : * a logic program @xmath2 ( @xmath0 is _ not _ a part of input ) + + ( 0)if @xmath281 is a stable model of @xmath2 * then * return yes and exit ; + ( 1)@xmath282 the set of proper rules in @xmath2 ; + ( 2 ) every @xmath158 with @xmath168 * do * + ( 3)compute the set of rules @xmath204 and the set of programs @xmath205 ; + ( 4)compute the program @xmath159 ; + ( 5)compute the program @xmath169 and the set @xmath283 ; + ( 6)given @xmath159 and @xmath204 , compute tables @xmath155 , @xmath77 and @xmath284 ( as described above ) ; + ( 7 ) every @xmath285 * do * + ( 8) + ( 9)@xmath219 , @xmath220 * and * + ( 10)there is a program @xmath221 s. t. for every rule @xmath222 , @xmath223 + ( 11 ) report yes and exit ; + ( 12)report no and exit . the correctness of this algorithm follows from lemmas [ aa ] - [ l - main ] . we will now analyze the running time of this algorithm . clearly , line ( 0 ) can be executed in @xmath66 steps . as we already observed , rules that are not proper can be eliminated from @xmath2 in time @xmath66 . next , there are @xmath286 iterations of loop ( 2 ) . in each of them , line ( 3 ) takes time @xmath287 , for some function @xmath288 ( let us recall that @xmath289 and @xmath290 depend on @xmath278 only ) . further , lines ( 4 ) and ( 5 ) can be executed in time @xmath66 . line ( 6 ) , as we discussed earlier , can be implemented so that to run in @xmath291 steps . loop ( 7 ) is executed @xmath292 times and each iteration takes @xmath293 steps , for some function @xmath294 ( let us again recall that @xmath290 depends on @xmath0 only ) . thus , the running time of the whole algorithm is @xmath295 , for some integer function @xmath155 . consequently , we get the following result . there is an integer function @xmath155 and an algorithm @xmath296 such that @xmath296 decides the problem @xmath61 and runs in time @xmath295 ( the constant hidden in the `` big oh '' notation does not depend on @xmath0 ) . the algorithm outlined in the previous section is not quite satisfactory . its running time is still high . a natural question to ask is : are there significantly better algorithms for the problems @xmath61 ? in this section we address this question by studying the complexity of the problem @xmath4 . our goal is to show that the problem is difficult in the sense of the w hierarchy . we will show that the problem @xmath4 is @xmath5$]-hard and that it is in the class w[3 ] . to this end , we define the _ @xmath297-weighted @xmath108-normalized satisfiability problem _ as : @xmath298 : : given a @xmath108-normalized formula @xmath110 , decide whether there is a model of @xmath110 with at most @xmath0 atoms ( @xmath0 is a parameter ) . the problem @xmath298 is a slight variation of the problem @xmath109 . it is known to be complete for the class w[t ] , for @xmath111 ( see @xcite , page 468 ) . to show w[2]-hardness of @xmath4 , we will reduce the problem @xmath299 to the problem @xmath4 . given the overwhelming evidence of fixed - parameter intractability of problems that are @xmath5$]-hard @xcite , it is unlikely that algorithms for problems @xmath61 exist whose asymptotic behavior would be given by a polynomial of order independent of @xmath0 . to better delineate the location of the problem @xmath4 in the w hierarchy we also provide an upper bound on its hardness by showing that it can be reduced to the problem @xmath300 , thus proving that the problem @xmath4 belongs to the class @xmath9 $ ] . we will start by showing that the problem @xmath61 is reducible ( in the sense of the definition from section [ ifp ] ) to the problem @xmath300 . to this end , we describe an encoding of a logic program @xmath2 by means of a collection of clauses @xmath301 so that @xmath2 has a stable model of size at most @xmath0 if and only if @xmath301 has a model with no more than @xmath302 atoms . in the general setting of the class np , an explicit encoding of the problem of existence of stable models in terms of propositional satisfiability was described in @xcite . our encoding , while different in key details , uses some ideas from that paper . let us consider an integer @xmath0 and a logic program @xmath2 . for each atom @xmath96 in @xmath2 let us introduce new atoms @xmath303 , @xmath304 , @xmath305 , and @xmath306 , @xmath307 . intuitively , atom @xmath303 represents the fact that in the process of computing the least model of the reduct of @xmath2 with respect to some set of atoms , atom @xmath96 is computed no later than during the iteration @xmath308 of the van emden - kowalski operator . similarly , atom @xmath304 represents the fact that in the same process atom @xmath96 is computed exactly in the iteration @xmath309 of the van emden - kowalski operator . finally , atom @xmath306 , expresses the fact that @xmath96 is computed _ before _ the iteration @xmath309 of the van emden - kowalski operator . the formulas @xmath310 , @xmath307 , and @xmath311 describe some basic relationships between atoms @xmath303 , @xmath304 and @xmath306 that we will require to hold : @xmath312 @xmath313 let @xmath10 be a rule in @xmath2 with @xmath314 , say @xmath315 we define a formula @xmath316 , @xmath307 , by @xmath317 we define @xmath318 ( * false * is a distinguished contradictory formula in our propositional language ) if @xmath319 . otherwise , we define @xmath320 speaking informally , formula @xmath316 asserts that @xmath96 is computed by means of rule @xmath10 in the iteration @xmath309 of the least model computation process and that it has not been computed earlier . let @xmath321 be all rules in @xmath2 with atom @xmath96 in the head . we define a formula @xmath322 , @xmath323 , by @xmath324 intuitively , the formula @xmath322 asserts that when computing the least model of the reduct of @xmath2 , atom @xmath96 is first computed in the iteration @xmath309 . we now define the theory @xmath325 that encodes the problem of existence of small stable models : @xmath326 next , we establish some useful properties of the theory @xmath325 . first , we consider a set @xmath327 of atoms that is a model of @xmath325 and define @xmath328 [ lem-11 ] let @xmath327 be a model of @xmath325 and let @xmath329 . then there is a unique integer @xmath309 , @xmath323 , such that @xmath330 . proof : since @xmath327 is a model of a formula @xmath311 , there is an integer @xmath309 , @xmath323 , such that @xmath330 . to prove uniqueness of such @xmath309 , assume that there are two integers @xmath331 and @xmath332 , @xmath333 , such that @xmath334 and @xmath335 . since @xmath336 , it follows that there is a rule @xmath140 with @xmath314 and such that @xmath337 . in particular , @xmath338 . in the same time , since @xmath334 and @xmath339 , we have @xmath340 , a contradiction . @xmath146 for every atom @xmath329 define @xmath341 to be the integer whose existence and uniqueness is guaranteed by lemma [ lem-11 ] . define @xmath342 . next , for each @xmath309 , @xmath343 , define @xmath344_i=\{q\in m(u)\colon i_q = i\}.\ ] ] [ lem-12 ] let @xmath327 be a model of @xmath325 . under the terminology introduced above , for every @xmath309 , @xmath345 , @xmath346_i\not=\emptyset$ ] . proof : we will proceed by downward induction . by the definition of @xmath347 , @xmath346_{i_u}\not=\emptyset$ ] . consider @xmath309 , @xmath348 , and assume that @xmath346_i\not = \emptyset$ ] . we will show that @xmath346_{i-1}\not = \emptyset$ ] . let @xmath349_i$ ] . clearly , @xmath330 and , since @xmath350 , there is a rule @xmath351 such that @xmath352 . consequently , for every @xmath353 , @xmath354 , @xmath355 . assume that for every @xmath353 , @xmath354 , @xmath356 . since @xmath357 and since @xmath358 , it follows that @xmath359 . consequently , @xmath327 satisfies the formula @xmath360 and , so , @xmath361 . it follows that @xmath362 , a contradiction ( we recall that @xmath363 ) . hence , there is @xmath353 , @xmath354 , such that @xmath364 . it follows that @xmath365_{i-1}$ ] and @xmath346_{i-1}\not=\emptyset$ ] . @xmath146 [ lem-13 ] let @xmath327 be a model of @xmath325 and let @xmath366 . then 1 . @xmath367 , and 2 . @xmath368 is a stable model of @xmath2 . proof : ( 1 ) the assertion follows directly from the fact that @xmath369 and from lemma [ lem-12 ] . + ( 2 ) we need to show that @xmath370 . we will first show that @xmath371 . since @xmath372_i$ ] , we will show that for every @xmath309 , @xmath345 , @xmath346_i\subseteq lm(p^{m(u)})$ ] . we will proceed by induction . let @xmath349_1 $ ] . it follows that there is a rule @xmath10 such that @xmath373 . consequently , @xmath10 is of the form @xmath374 and @xmath375 . hence , for every @xmath353 , @xmath376 , @xmath377 . consequently , the rule @xmath378 is in @xmath379 and , so , @xmath380 . the inductive step is based on a similar argument . it relies on the inequality @xmath367 we proved in ( 1 ) . we leave the details of the inductive step to the reader . we will next show that @xmath381 . we will use the characterization of @xmath382 as the limit of the sequence of iterations of the van emden - kowalski operator @xmath383 : @xmath384 we will first show that for every integer @xmath309 , @xmath385 , we have : @xmath386 and for every @xmath387 , @xmath388 . clearly , @xmath389 . hence , the basis for the induction is established . assume that for some @xmath309 , @xmath390 , @xmath391 and that for every @xmath387 , @xmath388 . consider @xmath392 . if @xmath393 , then @xmath394 for some @xmath395 , @xmath396 . since @xmath397 , @xmath398 and @xmath399 . by lemma [ lem-11 ] , it follows that @xmath400 . hence , @xmath401 . thus , assume that @xmath402 . since @xmath403 , there is a rule @xmath404 in @xmath2 such that @xmath405 , for every @xmath353 , @xmath376 , and @xmath406 , @xmath407 . by the induction hypothesis , for every @xmath353 , @xmath354 , we have @xmath408 and @xmath409 . it follows that @xmath410 and , consequently , that @xmath411 . since @xmath412 , @xmath398 and @xmath399 . it also follows ( lemma [ lem-11 ] ) that @xmath413 . thus , we proved that @xmath414 . since @xmath415 , there is @xmath353 , @xmath416 such that @xmath417 . it follows that for every @xmath418 , @xmath419 , @xmath420 . consequently , @xmath421 for every non - negative integer @xmath309 . @xmath146 consider now a stable model @xmath29 of the program @xmath2 and assume that @xmath422 . clearly , @xmath423 . for each atom @xmath424 define @xmath425 to be the least integer @xmath181 such that @xmath426 . clearly , @xmath427 . moreover , since @xmath422 , it follows that for each @xmath424 , @xmath428 . now , define @xmath429 [ lem-14 ] let @xmath29 be a stable model of a logic program @xmath2 such that @xmath422 . under the terminology introduced above , the set of atoms @xmath430 is a model of @xmath325 . proof : clearly , @xmath431 for @xmath432 and @xmath307 , and @xmath433 for @xmath432 . we will now show that @xmath434 , for @xmath432 and @xmath435 . first , we will consider the case @xmath424 . there are three subcases here depending on the value of @xmath309 . we start with @xmath309 such that @xmath436 . then @xmath437 . it follows that @xmath438 for every rule @xmath140 such that @xmath314 . since @xmath439 , @xmath434 next , we assume that @xmath440 . then , there is a rule @xmath441 @xmath442 in @xmath2 such that @xmath443 , for every @xmath353 , @xmath376 , and @xmath444 , @xmath354 . clearly , @xmath445 . since @xmath446 , it follows that @xmath434 , for @xmath440 . finally , let us consider the case @xmath447 . assume that there is rule @xmath140 such that @xmath314 and @xmath445 . let us assume that @xmath441 @xmath442 . it follows that for every @xmath353 , @xmath376 , @xmath448 . consequently , for every @xmath353 , @xmath376 , @xmath443 and the rule @xmath449 belongs to the reduct @xmath41 . in addition , for every @xmath353 , @xmath354 , @xmath450 . thus , @xmath451 and @xmath452 . this latter property is equivalent to @xmath453 . thus , it follows that @xmath454 and @xmath455 a contradiction with the assumption that @xmath456 . hence , for every rule @xmath10 with the head @xmath96 , @xmath438 . since for @xmath456 , @xmath457 , @xmath434 . to complete the proof , we still need to consider the case @xmath458 . clearly , for every @xmath309 , @xmath459 , @xmath460 . assume that there is @xmath309 , @xmath459 , and a rule @xmath10 such that @xmath314 and @xmath445 . let us assume that @xmath10 is of the form @xmath461 . it follows that @xmath450 and , consequently , @xmath451 for every @xmath353 , @xmath354 . in addition , it follows that for every @xmath353 , @xmath376 , @xmath448 and , consequently , @xmath443 . thus , @xmath462 belongs to the reduct @xmath41 and , since @xmath29 is a model of the reduct , @xmath424 , a contradiction . it follows that for every @xmath309 , @xmath323 , @xmath434 . @xmath146 for each atom @xmath432 , let us introduce @xmath463 new atoms @xmath464 , @xmath465 , and define @xmath466 lemmas [ lem-11 ] - [ lem-14 ] add up to a proof of the following result . [ th-21 ] let @xmath0 be a non - negative integer and let @xmath2 be a logic program . the program @xmath2 has a stable model of size at most @xmath0 if and only if the theory @xmath301 has a model of size at most @xmath302 . proof : @xmath224 let @xmath29 be a stable model of @xmath2 such that @xmath422 . by lemma [ lem-14 ] , the set @xmath430 is a model of @xmath325 consequently , the set @xmath467 is a model of @xmath301 . moreover , it is easy to see that @xmath468 . hence , @xmath469 . conversely , let us assume that some set @xmath470 , consisting of atoms appearing in @xmath301 and such that @xmath471 , is a model of @xmath301 . let us define @xmath327 to consist of all atoms of the form @xmath303 , @xmath304 and @xmath306 that appear in @xmath470 . clearly , @xmath327 is a model of @xmath325 . let us assume that @xmath472 ( we recall that the notation @xmath368 was introduced just before lemma [ lem-11 ] was stated ) . then , there are at least @xmath302 atoms of type @xmath464 in @xmath470 . consequently , @xmath473 as it contains also at least @xmath308 atoms @xmath303 , where @xmath329 this is a contradiction . thus , it follows that @xmath366 . moreover , by lemma [ lem-13 ] , @xmath368 is a stable model of @xmath2 . @xmath146 let us now define the following sets of formulas . first , for each atom @xmath432 we define @xmath474 next , we define @xmath475 @xmath476 and @xmath477 where @xmath478 is the set of all rules in @xmath2 with @xmath96 in the head . clearly , the theory @xmath479 is equivalent to the theory @xmath301 . moreover , it is a collection of sums of products of literals . therefore , it is a 3-normalized formula . by theorem [ th-21 ] , it follows that the problem @xmath4 can be reduced to the problem @xmath300 . thus , we get the following result . the problem @xmath480 $ ] . next , we will show that the problem @xmath299 can be reduced to the problem @xmath4 . let @xmath481 be a collection of clauses . let @xmath482 be the set of atoms appearing in clauses in @xmath483 . for each atom @xmath484 , introduce @xmath0 new atoms @xmath485 , @xmath486 . by @xmath487 , @xmath486 , we denote the logic program consisting of the following @xmath23 clauses : @xmath488 + @xmath489 + @xmath490 define @xmath491 . clearly , each stable model of @xmath492 is of the form @xmath493 , where @xmath494 for @xmath495 . sets of this form can be viewed as representations of nonempty subsets of the set @xmath88 that have no more than @xmath0 elements . this representation is not one - to - one , that is , some subsets have multiple representations . next , define @xmath496 to be the program consisting of the clauses @xmath497 stable models of the program @xmath498 are of the form @xmath499 , where @xmath29 is a nonempty subset of @xmath88 such that @xmath422 and @xmath500 enumerate ( possibly with repetitions ) all elements of @xmath29 . finally , for each clause @xmath501 from @xmath483 define a logic program clause @xmath502 : @xmath503 where @xmath87 is yet another new atom . define @xmath504 and @xmath505 . a set of clauses @xmath483 has a nonempty model with no more than @xmath0 elements if and only if the program @xmath506 has a stable model with no more than @xmath507 elements . proof : let @xmath29 be a nonempty model of @xmath483 such that @xmath422 . let @xmath500 be an enumeration of all elements of @xmath29 ( possibly with repetitions ) . then the set @xmath508 is a stable model of the program @xmath498 . since @xmath29 is a model of @xmath483 , it follows that @xmath509 , where @xmath155 consists of the clauses of the form @xmath510 such that @xmath115 and for some @xmath353 , @xmath376 , @xmath511 . since @xmath512 , it follows that @xmath513 thus , @xmath514 is a stable model of @xmath506 . since @xmath515 , the `` only if '' part of the assertion follows . conversely , assume that @xmath514 is a stable model of @xmath506 . clearly , @xmath516 . consequently , @xmath517 that is , @xmath514 is a stable model of @xmath498 . as mentioned earlier , it follows that @xmath518 , where @xmath29 is a nonempty subset of @xmath20 such that @xmath422 and @xmath500 is an enumeration of all elements of @xmath29 . consider a clause @xmath519 from @xmath483 . since @xmath514 is a stable model of @xmath506 , it is a model of @xmath506 . in particular , @xmath514 is a model of @xmath502 . since @xmath520 , it follows that @xmath521 and , consequently , @xmath522 . hence , @xmath29 is a model of @xmath483 . @xmath146 now the reducibility of the problem @xmath299 to the problem @xmath4 is evident . given a collection of clauses @xmath483 , to check whether it has a model of size at most @xmath0 , we first check whether the empty set of atoms is a model of @xmath483 . if so , we return the answer yes and terminate the algorithm . otherwise , we construct the program @xmath506 and check whether it has a stable model of size at most @xmath507 . consequently , we obtain the following result . the problem @xmath4 is w[2]-hard . the paper established several results pertaining to the problem of computing small and large stable models . it also brings up interesting research questions . first , we proved that the problem @xmath1 is in the class fpt . for problems that are fixed - parameter tractable , it is often possible to design an algorithm running in time @xmath523 , where @xmath84 is the size of the problem , @xmath0 is a parameter , @xmath86 is a polynomial and @xmath87 is a function @xcite . such algorithms are often practical for quite large ranges of @xmath84 and @xmath0 . the algorithm for the @xmath1 problem presented in this paper runs in time @xmath524 . it seems plausible it can be improved to run in time @xmath525 , for some function @xmath87 . such an algorithm would most certainly be practical for wide range of values of @xmath7 and @xmath0 . we propose as an open problem the challenge of designing an algorithm for computing large stable models with this time complexity . there is a natural variation on the problem of computing large stable models : given a logic program @xmath2 and an integer @xmath0 ( parameter ) , decide whether @xmath2 has a stable model of size at least @xmath526 . this version of the problem @xmath1 was recently proved by zbigniew lonc and the author to be w[3]-hard ( and , hence , fixed - parameter intractable ) @xcite . the upper bound for the complexity of this problem remains unknown . in the paper , we described an algorithm that for every fixed @xmath0 , decides the existence of stable models of size at most @xmath0 in time @xmath527 , where @xmath23 is the number of atoms in the program and @xmath7 is its size . this algorithm offers only a slight improvement over the straightforward `` guess - and - check '' algorithm . an interesting and , it seems , difficult problem is to significantly improve on this algorithm by lowering the exponent in the complexity estimate to @xmath528 , for some constant @xmath529 . we also studied the complexity of the problem @xmath4 and showed that it is fixed - parameter intractable . our results show that @xmath4 is @xmath5$]-hard . this result implies that the problem @xmath4 is at least as hard as the problem to determine whether a cnf theory has a model of cardinality at most @xmath0 , and strongly suggests that algorithms do not exist that would decide problems @xmath61 and run in time @xmath530 , where @xmath8 is a constant independent on @xmath0 . for the upper bound , we proved in this paper that the problem @xmath4 belongs to class @xmath9 $ ] . recently , zbigniew lonc and the author @xcite showed that the problem @xmath4 is , in fact , in the class @xmath5 $ ] . the author thanks victor marek and jennifer seitzer for useful discussions and comments . the author is grateful to anonymous referees for very careful reading of the manuscript . their comments helped eliminate some inaccuracies and improve the presentation of the results . this research was supported by the nsf grants cda-9502645 , iri-9619233 and eps-9874764 . apt , k. , blair , h.a . , & walker , a. ( 1988 ) . towards a theory of declarative knowledge . minker , j. ( ed ) , _ foundations of deductive databases and logic programming . papers from the workshop held in washington , d.c . , august 1822 , 1986_. palo alto , ca : morgan kaufmann . bondarenko , a. , toni , f. , & kowalski , r.a . an assumption - based framework for non - monotonic reasoning . nerode , a. , & pereira , l. ( eds ) , _ logic programming and non - monotonic reasoning ( lisbon , 1993)_. cambridge , ma : mit press . gelfond , m. , & lifschitz , v. ( 1988 ) . the stable semantics for logic programs . kowalski , r. , & bowen , k. ( eds ) , _ proceedings of the 5th international symposium on logic programming_. cambridge , ma : mit press . gottlob , g. , scarcello , f. , & sideri , m. ( 1999 ) . fixed parameter complexity in ai and nonmonotonic reasoning . gelfond , m. , leone , n. , & pfeifer , g. ( eds ) , _ logic programming and nonmonotonic reasoning , proceedings of the 5th international conference , lpnmr99_. lecture notes in computer science , vol springer - verlag . marek , v.w . , & truszczyski , m. ( 1999 ) . stable models and an alternative logic programming paradigm . apt , k.r . , marek , w. , truszczyski , m. , & warren , d.s . ( eds ) , _ the logic programming paradigm : a 25-year perspective_. springer verlag . marek , w. , & truszczyski , m. ( 1989 ) . stable semantics for logic programs and default theories . e.lusk , & overbeek , r. ( eds ) , _ proceedings of the north american conference on logic programming_. mit press .	in this paper , we focus on the problem of existence and computing of _ small _ and _ large _ stable models . we show that for every fixed integer @xmath0 , there is a linear - time algorithm to decide the problem @xmath1 ( large stable models problem ) : does a logic program @xmath2 have a stable model of size at least @xmath3 ? in contrast , we show that the problem @xmath4 ( small stable models problem ) to decide whether a logic program @xmath2 has a stable model of size at most @xmath0 is much harder . we present two algorithms for this problem but their running time is given by polynomials of order depending on @xmath0 . we show that the problem @xmath4 is _ fixed - parameter intractable _ by demonstrating that it is @xmath5$]-hard . this result implies that it is unlikely an algorithm exists to compute stable models of size at most @xmath0 that would run in time @xmath6 , where @xmath7 is the size of the program and @xmath8 is a constant independent of @xmath0 . we also provide an upper bound on the fixed - parameter complexity of the problem @xmath4 by showing that it belongs to the class @xmath9 $ ] . [ firstpage ]
in many glass - forming system , like amorphous polymers or supercooled liquids near the glass transition temperature , the relaxation spectrum exhibits strongly non - exponential behavior . dielectric spectroscopy shows evidence of two relaxation processes : the so - called @xmath14 and @xmath15-relaxation . the asymmetric @xmath16-relaxation peak flattens into an @xmath15-relaxation at high frequency domain @xcite . in general , the @xmath16-relaxation corresponds to the atomic motion of the clusters of atoms themselves or atomic transport between clusters , while the @xmath15-relaxation corresponds to the motions within the clusters . in a phenomenological approach these two types of relaxation are usually described by the sum of empirical , non - debye laws , namely the sum of havriliak - negami ( hn ) @xcite relaxation functions @xcite or their sum with the stretched exponentials @xcite . the stretched exponentials ( named also the kohlrausch - williams - watts ( kww ) functions ) are extensively studied in many theoretical papers , see e.g. @xcite , and experimental papers , see e.g.@xcite . this behavior of the dielectric function can be also described by the excess wing @xcite . this paper is devoted to the hn function which was introduced in @xcite : @xmath17^{\beta } } , \quad 0 < \alpha < 1 , \quad \beta > 0.\ ] ] here , the parameters @xmath16 and @xmath15 do not denote the @xmath14 and @xmath15-relaxation . they are called the width and symmetry parameters , respectively . the symbol @xmath18 denotes the frequency and @xmath1 is an effective time constant . the symbols @xmath19 and @xmath20 denote the relative permittivity and the dielectric constant respectively . note that eq . generalizes the cole - cole ( cc ) @xcite relaxation ( eq . for @xmath9 ) and the cole - davidson ( cd ) @xcite relaxation ( eq . for @xmath21 ) . among experimental papers devoted to the hn functions we mention ref . @xcite where the hn function has been measured during the monitoring of contamination in sandstone . the hn relaxation is also observed in a complex system representing plant tissues of fresh fruits and vegetables in the frequency range @xmath22 hz , see @xcite . it can be also applied to model a thermal flux in the field of machining by turning , in the time domain @xcite several authors have investigated the hn relaxation from different points of view . for example , the relation between cc relaxation and the stretched exponential is established in @xcite . a transparent subordination approach to anomalous diffusion processes underlying the hn relaxation has been proposed in @xcite . the analytic expressions in the time domain for the hn relaxation in terms of the fox @xmath23 functions are presented in @xcite for the real values of parameters @xmath16 and @xmath15 . ( remark that the fox @xmath23 functions are defined via the mellin transform @xcite . ) the non - rational values of @xmath16 and @xmath15 are obtained by making the numerical fitting of the hn function with the experimentally obtained data . for this reason the solutions expressed via the fox @xmath23 functions are correct but impractical because of the fox @xmath23 functions are not yet implemented in the computer algebra systems . therefore the expression involving the fox @xmath23 functions for general parameter values are purely symbolic and , for the moment , can not be used in actual calculations . the purpose of this paper is to express the relevant functions , like the response functions , the probability densities and the relaxation functions , related to the hn relaxation in terms of the special functions implemented in the computer algebra systems . that can significantly simplify the numerical calculations . to realize this purpose we approach the non - rational value of parameter @xmath16 by the rationals such that @xmath8 . hence , everywhere in the paper we take @xmath24 with @xmath6 and @xmath7 relative prime integers . another aim of the paper is to propose the probability density related to the response function . that will hopefully allow one to find localization of the relaxed center in the given sample . the present paper is organized as follows . ii contains the basic facts about the relaxation theory , namely the relation between the response functions , the probability densities and the relaxation functions . in sec . iii , the response functions @xmath2 , @xmath8 and @xmath25 , are represented via the prabhakar function which contains the ( three - parameter ) generalized mittag - leffler functions . for @xmath4 , where @xmath6 and @xmath7 are integers , we represent the ( three - parameter ) generalized mittag - leffler functions as the finite sum of the generalized hypergeometric functions . next , we find the asymptotic properties of the response functions and the ( three - parameter ) generalized mittag - leffler functions . by expressing the response function in terms of the meijer g function in sec . iv we calculate the probability density @xmath26 connected with the hn relaxation . we show that for @xmath5 , @xmath26 is positively defined and normalized . thereafter , we find all the moments and asymptotics of @xmath3 . in sec . v the relaxation functions appropriate to the hn relaxation denoted as @xmath27_{\alpha , \beta}$ ] are derived . there the integral form of evolution equation is also found . the new relation between @xmath2 and the one - sided lvy stable densities is found in sec . the paper is concluded in sec . in the theory of relaxation the complex frequency - dependent absolute permittivity ( dielectric constant ) of the material is given by @xcite @xmath28.\ ] ] the symbol @xmath29 = \int_{0}^{\infty } e^{-p t } f(t/\tau_0 ) dt$ ] , @xmath30 , defines the laplace transform of @xmath31 @xcite . the response function @xmath32 ( the number of polarized center per unit time ) is related to the ratio of polarization at time @xmath33 to all possible polarization via @xmath34 . inserting this relation into the laplace transform in eq . , comparing with eq . , and using eq . ( 3 - 4 - 1 ) of @xcite we have @xmath35^{-\beta } = 1 -i\omega\ , \mathcal{l}\left[\frac{n(t)}{n_{0 } } ; i\omega\right].\ ] ] from eq . , hilfer calculates all possible relaxations ( called also the relaxation functions ) @xmath36 in @xcite using the fox @xmath23 function and the series . his relaxation function expressed via the fox @xmath23 function can be obtained by calculating the laplace transform in eq . and , thereafter , inverting it . on the other hand @xmath36 can be considered as the relaxation of the sample which contains @xmath37 centers of oriented polarization . each of these centers relaxes with a different relaxation time @xmath38 . the relaxation of all sample should be the weighted sum of the debye s relaxations . that is @xmath39 the probability distribution @xmath40 is a positive function which satisfies @xmath41 . taking the infinitesimally small @xmath42 and going with @xmath37 to infinity we get @xmath43 where @xmath44 . taking the derivative over time of the first equality in eq . the response function reads @xmath45 we point out that the equality of @xmath36 derived by hilfer ( see eq . where the information about the debye s relaxation is absent ) to @xmath36 given by eq . would mean that the hn relaxation is somehow built from the debye s relaxation . let us denote the response function in time domain related to the hn relaxation as @xmath46 . the subscripts @xmath16 and @xmath15 are the width and symmetry parameters in eq . . we calculate the function @xmath2 by comparing eqs . with , and , next , we invert the laplace transform . thus , by using eq . ( 2.5 ) of @xcite for @xmath47 , @xmath48 and @xmath49 the response function can be expressed via the so - called prabakhar function @xcite : @xmath50 ( the subscript @xmath51 , as the various subscripts below , is only added to emphasize the reference from which the formula was taken . ) the ( three - parameter ) generalized mittag - leffler function @xmath52 is commonly given through the series @xcite @xmath53 where @xmath54 denotes the pochhammer symbol . remark that for @xmath55 eq . gives @xmath56 which is the ( two - parameter ) mittag - leffler function @xcite . for @xmath57 eq . is equal to the ( classical ) mittag - leffler function @xmath58 @xcite , and for @xmath55 and @xmath59 we have @xmath60 . the extensive numerical calculation of @xmath61 is considered in @xcite , where the authors test the stability and the validity range for the parameters @xmath16 and @xmath62 . their algorithm is based on integral representations and exponential asymptotics . using this algorithm they simulate the behavior of @xmath61 also at zero and at infinity , and estimate the error of their method . we attempt to express the generalized mittag - leffler function of eq . through a finite sum of the generalized hypergeometric function @xmath63 @xcite . for this reason , we take the rational @xmath16 , @xmath64 , and change the summation index as follows : @xmath65 , where @xmath66 is the index which appears in eq . . here , we introduce the index @xmath67 which indicates how many generalized hypergeometric functions should be in the sum . thus , we get @xmath68 with @xmath69 . the first list of `` upper '' parameters is equal to @xmath70 followed by @xmath71 , whereas the second list of `` lower '' parameters is the union of @xmath72 and @xmath73 . let us now check if eq . reconstructs the exponential behavior of @xmath61 used in @xcite . starting from eq . and the last unnumbered formula on p. 155 of @xcite we can estimate @xmath74 for large values of @xmath75 . according to it @xmath76 function is equal to @xmath77 , where @xmath78 as @xmath79 , where we employed the gauss - legendre multiplication formula for gamma functions . substituting it to eq . we have @xmath80 which for @xmath24 and @xmath55 reproduces the dominant term of ( * ? ? ? * eq . ( 2.4 ) ) which presents the exponential behavior of @xmath81 . moreover , @xmath52 for @xmath82 and @xmath83 can be alternatively represented via the one - sided lvy stable distribution @xmath84 @xcite : @xmath85 eq . can be proved by writing the exponential function in the series form and using the explicit values of moments of @xmath86 , see @xcite . for closely related algebraic and transformation properties of lvy stable functions consult @xcite . note that eq . generalizes the known relation between the classical mittag - leffler function and the one - sided lvy stable distribution , see , e.g. , ( * ? ? ? * eq . ( 7 ) ) or ( * ? ? ? * eq . ( 11 ) ) . inserting the asymptotics of @xmath87 for large @xmath88 given by ( * ? ? ? * eq . ( 5 ) ) into eq . we get @xmath89 . after substituting this result into eq . we reconstruct the asymptotics at infinity given in the last unnumbered formula on p. 76 of @xcite , namely @xmath90 . the behavior of @xmath2 for @xmath91 looks like the asymptotics of @xmath86 for @xmath92 . that suggests the link between the prabhakar function and the one - sided lvy stable distribution . we investigate this link in some detail in sec . vi . from eqs . and with @xmath82 and @xmath93 we write @xmath94 as the finite sum of @xmath76 s functions . moreover , @xmath94 can be expressed through the standard special functions for @xmath95 only : @xmath96 where @xmath97 is the parabolic cylinder function @xcite . in the derivation of eq . we employ eqs . ( 7.11.1.9 ) and ( 7.11.1.10 ) on p. 579 of @xcite . to simplify the calculation in sec . iv , we insert eq . with @xmath84 given by ( * ? ? ? * eq . ( 2 ) ) into eq . and employ formula ( 2.24.3.1 ) on p. 295 of @xcite . thus , the prabakhar function can be expressed in terms of the meijer g function @xmath98 @xcite : @xmath99 the upper and lower lists of parameters are equal to the union of @xmath100 and @xmath101 , respectively . from eq . and eq . ( 2.24.2.1 ) on p. 293 of @xcite we calculate its @xmath102-th stieltjes moment @xmath103 which has the form @xmath104 where @xmath105 $ ] denotes the stieltjes moment of the the one - sided lvy stable distribution @xcite . the stieltjes moment @xmath106 is finite for @xmath107 and is infinite otherwise . note that @xmath108 is finite in the larger range of @xmath102 , namely for @xmath109 . @xmath108 does not exist for @xmath110 . ( color online ) plot of @xmath111 given by eq . for @xmath112 , @xmath113 and @xmath114 ( i ; red ) , @xmath115 ( ii ; blue ) , and @xmath116 ( iii ; green ) . ] eq . can be readily used to study the prabakhar functions graphically . the prabakhar functions @xmath117 given by eq . for @xmath118 are plotted in fig . [ fig0 ] . in the limit of @xmath119 they go to infinity for @xmath120 , @xmath121 for @xmath115 , and @xmath122 vanishes for @xmath123 . that exemplifies the asymptotic behavior of @xmath124 for @xmath119 @xcite which goes to @xmath70 for @xmath125 . it also shows that @xmath117 vanishes when @xmath33 tends to infinity . this confirms the asymptotics found below eq . . the inverse laplace transform allows one to pass from the time domain @xmath126 to the space domain @xmath127 and to address the question of emergence of probability distribution functions ( pdf ) in @xmath88 . the function @xmath3 is denoted here by @xmath128 . for @xmath129 , where @xmath7 and @xmath6 are integers such that @xmath130 , @xmath26 can be presented as the inverse laplace transform of eq . : @xmath131.\ ] ] substituting the prabhakar function given by eq . into eq . and employing eq . ( 3.38.1.2 ) on p. 393 of @xcite we represent @xmath26 in the form of the meijer g function @xcite : @xmath132 the symbol @xmath133 is defined after eq . . the numerical tests show that @xmath26 is a positive function for @xmath134 , whereas it has a negative part for @xmath135 , see fig . [ fig0a ] . ( the analytical confirmation of that fact is below eq . . ) ( color online ) plot of @xmath26 given by eq . for @xmath113 and @xmath114 ( i ; red ) , @xmath115 ( ii ; blue ) , and @xmath116 ( iii ; green ) . observe that line iii ( green ) is negative for @xmath136 . ] employing eq . ( 2.24.2.1 ) on p. 293 of @xcite we show that all their fractional moments @xmath137 are equal to @xmath138 for real @xmath139 , @xmath140 and @xmath141 , they are finite , and they are infinite otherwise . it means that @xmath26 are normalized and their higher moments like the mean value or variance do not exist . similar but not identical behavior is observed for the one - sided lvy stable distributions @xmath86 for which only fractional moments exist and all the higher moments are infinite @xcite . the existence of normalization and positivity of @xmath26 for @xmath134 permits to conclude that @xmath26 for @xmath5 , @xmath6 and @xmath7 integers , are the pdf . furthermore , it turns out that using eq . ( 8.2.2.4 ) on p. 617 of @xcite and the gauss - legendre multiplication formula for gamma functions in eq . , we can rewrite @xmath26 as a finite sum of @xmath6 generalized hypergeometric functions : @xmath142 \\ & \times { _ { 1+k}f_{k}}\left({1 , \delta(k , \beta+j ) \atop \delta(k , 1+j ) } ; \frac{(-1)^{l - k}}{u^{l}}\right ) . \end{split}\end{aligned}\ ] ] eq . gives a closed form of @xmath143 . for example , for the cc relaxation ( @xmath114 ) it can be observed that the appropriate cancelation in @xmath144 s gives @xmath145 which , after applying eq . ( 7.3.1.1 ) on p. 453 of @xcite , yields @xmath146 $ ] . employing eq . ( 1.353.1 ) on p. 38 of @xcite to the remaining sum over @xmath147 we get @xmath148 where @xmath149 . we point out that eq . was obtained in @xcite by using a different method , ( 3.24 ) on p. 245 there . all the distributions @xmath150 share the following features : ( i ) @xmath150 for @xmath8 and @xmath151 is positive ; ( ii ) @xmath150 goes to infinity at @xmath152 , and it vanishes for @xmath153 ; ( iii ) @xmath150 is an decreasing function for @xmath154 , it contains the flat sector near @xmath155 , and it has two extrema for @xmath156 . for @xmath155 the derivative @xmath157 is zero at the point @xmath158 . the numerical calculation shows that for the cc relaxation we have @xmath159 and @xmath160 ; ( iv ) @xmath150 has a maximum at @xmath161 and a minimum at @xmath162 for @xmath156 , where @xmath163/(1+\alpha)$ ] . the upper sign is for maximum value of @xmath88 and the lower sign is for minimum of @xmath88 offers an unlimited number of solutions for @xmath26 . for @xmath164 only it can be written down in terms of standard special functions . for example , for @xmath165 it is equal to @xmath166 which is plotted for @xmath167 in fig . [ fig1 ] , see line i in red , whereas @xmath26 for @xmath168 is equal to @xmath169 in the derivation of eq . we have used eqs . ( 7.3.3.1 ) and ( 7.3.3.2 ) on p. 486 of @xcite . . has been obtained by employing eqs . ( 7.4.1.30 ) - ( 7.4.1.32 ) on p. 499 of @xcite . from eq . , after analyzing the behavior of sine , we conclude that @xmath170 is positive for @xmath171 , whereas it is negative for @xmath172 and @xmath173^{2}$ ] . in fig . [ fig1 ] we exhibit the probability densities of @xmath26 given by eq . for @xmath174 and @xmath175 . ( color online ) plot of @xmath3 given by eq . for @xmath174 and @xmath176 ( i ; red ) , @xmath113 ( ii ; blue ) , and @xmath177 ( iii ; green ) . ] [ fig1 ] shows that for fixed value of @xmath15 there exists @xmath178 for which @xmath179 . for example , for @xmath174 it is @xmath180 and @xmath181 . for @xmath174 and @xmath182 the maximum is higher and it moves in the direction of larger values of @xmath88 . we now find the series representation of @xmath3 for rational @xmath24 . to realize that objective we use the series representation of the generalized hypergeometric function , see eq . ( 7.2.3.1 ) on p. 437 of @xcite . in this way we obtain the double sum : one over @xmath183 , @xmath184 , which is the index appearing in the series representation of @xmath63 , and another one over @xmath147 , @xmath67 , is from eq . . next , we change the summation index as follows @xmath185 . thus , @xmath3 has the form @xmath186}{u^{1+\alpha(\beta+r)}}.\ ] ] it turns out that writing sine as the imaginary part of @xmath187 , taking the integral representation of the gamma function ( * ? ? ? * eq . ( 8.312.2 ) ) , and applying eq . , we get the integral form of eq . : @xmath188 the auxiliary function @xmath189 written in terms of the ( three - parameter ) generalized mittag - leffler function is equal to @xmath190 using eq . in eq . it is easy to demonstrate the following features of @xmath189 . it can be shown that @xmath189 vanishes at @xmath191 . @xmath189 goes to @xmath70 in the limit of @xmath192 . the asymptotics at @xmath193 is presented in unnumbered formula for @xmath194 and @xmath195 on p. 76 of @xcite . for the cc relaxation ( @xmath9 ) , @xmath189 can be expressed via the classical mittag - leffler function , namely @xmath196 which is quoted in ( * ? ? ? * eq . ( 9 ) on p. 185 ) . for the cd relaxation ( @xmath197 ) we have @xmath198 @xcite , where @xmath199 is the incomplete gamma function . moreover , @xmath200 which results from eq . ( 1.9.6 ) on p. 46 of @xcite . by applying ( * ? ? ? ( 2.5 ) ) to eq . with eq . and by employing de moivre s formula to calculate the imaginary part of so obtained equation , we derive the modification of @xmath3 which is given by eq . ( 8) for @xmath201 , @xmath202 , and @xmath195 of @xcite . namely , if @xmath8 and @xmath25 we get @xmath203}{[u^{2\alpha } + 2u^{\alpha}\cos(\pi\alpha ) + 1]^{\beta/2}}\ ] ] with @xmath204 ( defining the range of @xmath88 , see below ) , and @xmath205.\ ] ] for @xmath206 eq . works for @xmath207 and @xmath208 . for @xmath209 eq . with @xmath207 defines @xmath3 for @xmath210^{1/\alpha}$ ] . eq . with @xmath211 gives @xmath3 for @xmath212^{1/\alpha}$ ] . in this way we can define @xmath3 for all @xmath127 . note that the presence of @xmath213 in eq . allows us to build the positively defined and normalized pdf @xmath3 for arbitrary real @xmath16 , @xmath8 and @xmath214 . this property is absent in the solution of ( * ? ? ? * eq . ( 8) ) for @xmath201 , @xmath215 , and @xmath216 , which is defined for @xmath8 and @xmath211 . for example , @xmath217 , where @xmath218 is given by ( * ? ? ? * eq . ( 8) ) , is negative for @xmath219 . introducing the term @xmath213 we improve ( * ? ? ? we remark also that eqs . , , and gives the same results . using the various but equivalent forms of @xmath3 make it easier to show many of its properties . for example , from eqs . and it can be shown that @xmath3 is positive for @xmath220 , whereas it contains the negative parts for @xmath221^{1/\alpha}$ ] for @xmath222 . from eq . , in a simple way , we get that @xmath3 goes to infinity in the limit of @xmath223 and @xmath3 vanishes for @xmath224 . the benefit of use eqs . or rather than eq . is that it is the only formula which defines @xmath3 for @xmath209 and @xmath208 . we define now the relaxation function @xmath225_{\alpha , \beta}$ ] whose explicit and exact form is obtained by calculating the laplace transform with @xmath3 given in eq . . here , we use the formulas from @xcite , i.e. eq . ( 2.24.3.1 ) on p. 350 , eq . ( 8.2.2.14 ) on p. 619 , and eq . ( 8.2.2.3 ) on p. 618 , and , then , we compare the results with eqs . and . thus , we get ( * ? ? ? ( 30 ) ) : @xmath226_{\alpha , \beta } = 1 - g_{\alpha , \beta}(\ulamek{t}{\tau_{0}}),\end{aligned}\ ] ] where @xmath227 is given in eq . with eq . . ( color online ) plot of @xmath27_{\alpha , \beta}(x)$ ] given in eqs . and for @xmath167 and @xmath176 ( i ; red ) , @xmath113 ( ii ; blue ) , and @xmath177 ( iii ; green ) . ] remark that eq . for @xmath228 is equal to one and it vanishes for @xmath33 going to infinity , see fig . eq . resembles the series representation of the relaxation function obtained in @xcite . it suggests that the hn relaxation could be explained by using the debye s relaxation . for instance , for @xmath9 the cc relaxation can be build from the debye s relaxations @xcite . in @xcite . is also represented in terms of the fox @xmath23 functions which however are not accessible in the computer algebra systems . the advantage of our solution , eq . with eqs . and , is clearly seen in practice . since in recent versions of the computer algebra systems the generalized hypergeometric functions @xmath63 as well as the meijer @xmath229 functions are fully implemented , their use permits high - precision calculations . the explicit form of relaxation function @xmath27_{\alpha , \beta}$ ] given by eq . and allow us to derive the self - similar properties of @xmath3 . rewriting the first equality in eq . , where instead of @xmath230 we take @xmath231 , @xmath232 , and taking into account eq . , we get @xmath233 where @xmath234 . an one - variable function @xmath3 is uniquely extended to a two - variable @xmath235 one : @xmath236 we stress that eq . is the self - similar property which is also obeyed by the ( classical ) mittag - leffler function @xcite and the lvy stable distribution @xcite . from the second equality in eq . with @xmath237 , @xmath238 , and for @xmath239 we get the laplace - like convolution properties : @xmath240 the similar property is fulfilled also by the one - sided lvy stable distribution , see ( * ? ? ? * eq . ( 12 ) ) and ( * ? ? ? * eq . ( 13 ) ) . . differs from the standard laplace convolution of _ one _ variable function . here , we have the integral form of evolution equation of _ two _ variables density distribution @xmath241 where the _ both _ variables are changed . . can be proved by employing eqs . for @xmath242 , @xmath238 , and . observe also that eq . is the evolution equation written in the integral form . following @xcite the relation between hn relaxation and the one - sided lvy density @xmath86 is usually understood in the form of ( * ? ? ? * eq . ( 7 ) ) or ( * ? ? ? * eq . ( 11 ) ) . here , we propose a new kind of link which will show the correlation of @xmath2 , for arbitrary @xmath8 and @xmath244 , with @xmath243 , @xmath8 . to achieve this task , we reparametrize in eq . @xmath10 , @xmath11 and @xmath245 throughout this section . the range of parameter @xmath13 , @xmath12 , assures that @xmath244 . the derivative of eq . over @xmath246 taken with the opposite sign is the probability density function @xmath247^{\frac{1}{1-q}}\ ] ] called the @xmath13-weibull distribution @xcite . we point out that eq . goes to the weibull distribution @xcite for @xmath248 and the weibull distribution is the derivative of the stretched exponential . thereafter , we take the prabhakar function given in eq . with the ( three - parameter ) generalized mittag - leffler function given via the finite sum of the generalized hypergeometric functions , see eq . . next , in fig . [ fig4 ] we compare @xmath249 for @xmath176 and @xmath250 with the @xmath251 . ( color online ) plot of @xmath252 given by eq . , where the prabhakar function eq . is defined for @xmath11 , @xmath253 , and @xmath254 ( i ; red ) , @xmath255 ( ii ; blue ) , and @xmath256 ( iii ; green ) . the black , dashed line shows the one - sided lvy stable distribution for @xmath176 , i.e. @xmath257/(2\sqrt{\pi } t^{3/2})$ ] . ] in fig . [ fig4 ] we observe that @xmath252 goes over to the one - sided lvy stable distribution @xmath251 when @xmath13 is approaching one . the behavior of @xmath258 in the limit of @xmath248 can be also shown by using in eq . and the series representation of the ( three - parameter ) generalized mittag - leffler function given in eq . . after applying the gauss - legendre multiplication formula to gamma functions in the denominator of eq . we get @xmath259^{\alpha(n + \frac{2-q}{q-1 } ) - 1 } \\ & \times \gamma[1-\alpha(n+\ulamek{2-q}{q-1})]\ , \sin[\alpha(n+\ulamek{2-q}{q-1 } ) ] . \end{split}\end{aligned}\ ] ] writing the sine function as the imaginary part of @xmath260 $ ] and employing eq . ( 8.312.2 ) on p. 892 of @xcite , the integral representation of @xmath258 is found in the form @xmath261^{-\frac{2-q}{q-1 } } \frac{dy}{\pi } \right\}.\ ] ] in the limit of @xmath248 eq . goes to the one - sided lvy stable distribution @xmath243 given in the first unnumbered equation in the last page of @xcite . the stieltjes moments of @xmath2 given in eq . go to the stieltjes moments of @xmath243 for @xmath248 . to show that we express @xmath1 and @xmath15 through @xmath13 , where @xmath12 , namely by taking @xmath11 and @xmath10 . thereafter , we apply the stirling formula for @xmath262 and @xmath263 where @xmath15 is appropriately modified . starting form the basic facts from relaxation theory we have introduced the response functions , the probability densities and the relaxation functions of the hn relaxation . the response functions are given via the prabhakar functions related to the ( three - parameter ) generalized mittag - leffler functions whose representation in terms of the finite sum of the generalized hypergeometric functions for rational @xmath16 is found . we also express the ( three - parameter ) generalized mittag - leffler functions via the one - sided lvy stable distributions . that relation for @xmath114 ( the cc relaxation ) goes over to the known relation between the ( classical ) mittag - leffler function and the one - sided lvy stable distribution . in this way it generalizes the latter mentioned relation . we have also identified the values of parameters @xmath15 , i.e. @xmath220 , for which the normalized function @xmath3 connected with the response function can be called the probability density . the moments of @xmath3 were also calculated . for @xmath220 the mean values , the variance and the higher moments of @xmath3 do not exist . similar properties characterize the one - sided lvy stable distribution . we also derived the laplace - like convolution properties which are the integral form of evolution equations . moreover , we present the evidence that @xmath2 and the appropriate @xmath243 present the identical asymptotic behavior at infinity . this suggests the existence of the link between @xmath2 and @xmath243 . this link has been also considered the main benefit of the paper is that , in addition to finding the explicit and exact forms of the relaxation functions , we represent them as the generalized hypergeometric functions which are implemented in the computer algebra systems . the use of the generalized hypergeometric functions have significantly simplified the calculation and in the fast way allowed us to obtain many results . another advantage is that we found restrictions on @xmath15 for which @xmath3 is the probability density . for this value of @xmath15 , @xmath3 gives localization of relaxing centers in the sample . that could give the intuition in which way we should prepare the experimentally measured sample . we have also elucidated the way in which the various versions of @xmath2 , @xmath3 , and @xmath27_{\alpha , \beta}$ ] appearing in the literature can be connected with each other . we have also shown the existence of new relations between the response function related to the hn relaxation , and the one - sided lvy stable distribution , which can allow us to find the differential equation related to the integral laplace - like convolution . k. g. , a. h. and k. a. p. were supported by the pan - cnrs program for french - polish collaboration . moreover , k. g. thanks for support from mnisw ( warsaw , poland ) , `` iuventus plus 2015 - 2016 '' , program no ip2014 013073 .	we study the functions related to the havriliak - negami frequency relaxation @xmath0^{-\beta}$ ] with @xmath1 characteristic time , measured in many experiments . we furnish exact and explicit expression for the response function @xmath2 in time domain and a probability density @xmath3 in space domain for @xmath4 and @xmath5 , with @xmath6 and @xmath7 positive integers . for @xmath8 and @xmath9 we reproduce the functions related to the cole - cole relaxation . we use the method of integral transforms . we show that @xmath2 with @xmath10 and @xmath11 , @xmath12 , goes over to the one - sided lvy stable distribution when @xmath13 tends to one . moreover , applying the self - similar property of @xmath3 we introduce two - variable density which satisfies the integral form of evolution equation .
discriminants are central mathematical objects that have applications in many fields . let @xmath0 be a field and suppose given integers @xmath1 and @xmath2 . let @xmath3 be the set of all @xmath4-uples of homogeneous polynomials @xmath5 in the polynomial ring @xmath6 $ ] of degree @xmath7 respectively . consider the subset @xmath8 of @xmath3 corresponding to those @xmath4-uples of homogeneous polynomials that define an algebraic subvariety in @xmath9 which is not smooth and of codimension @xmath4 . it is well - known that @xmath8 is an irreducible hypersurface provided @xmath10 for some @xmath11 , or provided @xmath12 ( in which case @xmath8 is nothing but the resultant variety ) @xcite . the discriminant polynomial is then usually defined as an equation of @xmath8 . it is a homogeneous polynomial in the coefficients of each polynomial @xmath13 whose vanishing provides a smoothness criterion @xcite . this geometric approach to discriminants yields a beautiful theory with many remarkable results ( e.g. @xcite ) . however , whereas there are strong interests in computing with discriminants ( e.g. @xcite ) , including in the field of number theory , this approach is not tailored to develop the required formalism . for instance , having the discriminant defined up to a nonzero multiplicative constant is an important drawback , especially when computing over fields of positive characteristic . another point is about the computation of discriminants : it is usually done by means of the famous cayley trick that requires to introduce new variables , which has a bad effect on the computational cost . in some cases , there exist an alternative to the above geometric definition of discriminants . in the case @xmath12 , which corresponds to resultants , there is a huge literature where the computational aspects are treated extensively . in particular , a vast formalism is available and many formulas allow to compute resultants , as for instance the well - known macaulay formula ( e.g. @xcite ) . when @xmath14 the theory becomes much more delicate . nevertheless , for both cases @xmath15 ( hypersurfaces ) and @xmath16 ( finitely many points ) discriminants can be defined rigorously and their formalism has been developed . the case @xmath15 goes back to demazure @xcite and the case @xmath16 has been initiated by krull @xcite . in both cases , the discriminant is defined by means of resultants , via a universal formula . this allows to develop the formalism , to obtain useful computational rules and also to compute it efficiently by taking advantage of the macaulay formula for resultants ; see @xcite for more details . the goal of this paper is to provide a similar treatment in the case @xmath17 . our approach relies on the characterization of this discriminant by means of a universal formula where resultants and discriminants of finitely many points appear . as far as we know , this formula is new and provide the first ( efficient ) method to compute the discriminant of a complete intersection curve over any ring . in particular , we provide a closed formula that allows to compute it as a ratio of determinants . we emphasize that the computations are done in dimension at most 3 , that is to say that there is no need to introduce new variables as with the cayley trick . we mention that the problem of studying and computing discriminants goes back to the remarkable paper @xcite of sylvester in 1864 . the case @xmath17 was the last remaining case to complete the picture in @xmath18 . before going into further details , we provide an example to illustrate the contribution of this paper . the clebsch cubic projective surface is defined by the homogeneous polynomial @xmath19.\ ] ] by ( * ? ? ? * definiton 4.6 ) and the macaulay formula , we get @xmath20 thus , @xmath21 and we recover that the clebsch surface is smooth except in characteristic 5 . now , consider the family of quadratic forms @xmath22[x_1,x_2,x_3,x_4].\ ] ] the formula we will prove in this paper allows to compute the discriminant of the intersection curve between the clebsch surface and these quadratic forms ; we get @xmath23 in characteristic 5 , the clebsch surface @xmath24 is singular at the point @xmath25 . so , if the surface defined by the equation @xmath26 goes through @xmath27 then their intersection curve will be singular at @xmath27 . in general , this is not the case . indeed , we have that @xmath28 now , if @xmath29 is specialized to @xmath30 then we force the surfaces defined by @xmath26 to go through @xmath27 . applying this specialization the above formula , we obtain @xmath31 so that this discriminant now vanishes modulo 5 as expected . the paper is organized as follows . in section [ sec : def ] we prove a new formula , based on resultants , that is used to provide a new definition of the discriminant of a complete intersection space curve . then , in section [ sec : formulas ] we give some properties and computational rules of this discriminant by relying on the existing formalism of resultants . finally , in section [ sec : geom ] we show that our definition is correct in the sense that it satisfies to the expected geometric property , in particular it yields a universal and effective smoothness criterion which is valid in arbitrary characteristic . in the sequel , we will rely heavily on the theory of resultants and its formalism , including the macaulay formula . we refer the reader to @xcite and ( * ? ? ? * chapter 3 ) . we will also assume some familiarity with the definition of discriminants in the case @xmath16 for which we refer the reader to @xcite . resultants and discriminants will be denoted by @xmath32 and @xmath33 respectively . suppose given two positive integers @xmath34 and consider the generic homogeneous polynomials in the four variables @xmath35 @xmath36 we denote by @xmath37 $ ] the universal ring of coefficients and we define the polynomial ring @xmath38 $ ] . the partial derivative of the polynomial @xmath13 with respect to the variable @xmath39 will be denoted by @xmath40 . moreover , given four homogeneous polynomials @xmath41 @xmath42 @xmath43 @xmath44 in the variables @xmath45 , the determinant of their jacobian matrix will be denoted by @xmath46 [ thm : gendef ] using the above notation , assume that @xmath47 . let @xmath48 be three linear forms @xmath49 and denote by @xmath50 the polynomial ring extension of @xmath51 with the coefficients @xmath52 s , @xmath53 s and @xmath54 s of the linear forms @xmath48 . then , there exists a unique polynomial in @xmath51 , denoted by @xmath55 and called the _ universal discriminant _ of @xmath56 and @xmath57 , which is independent of the coefficients of @xmath48 and that satisfies to the following equality in @xmath50 : @xmath58 by convention , if @xmath59 we set @xmath60 where @xmath61 . given a commutative ring @xmath62 and two homogeneous polynomials @xmath63 in @xmath64 $ ] of degree @xmath65 , @xmath66 respectively , the map of rings @xmath67 from @xmath68 $ ] to @xmath64 $ ] which sends @xmath69 to @xmath70 and leave each variable @xmath71 invariant , is called the _ specialization map _ of the universal polynomials @xmath72 to the polynomials @xmath73 , as @xmath74 . [ def : disc ] suppose given a commutative ring @xmath62 , two positive integers @xmath34 such that @xmath47 and two homogeneous polynomials @xmath73 in @xmath64 $ ] of degree @xmath34 respectively . denoting by @xmath67 the specialization map as above , we define the discriminant of the polynomials @xmath73 as @xmath75 to prove the claimed formula , one can assume that @xmath50 is the universal ring of the coefficients of the polynomials @xmath76 over the integers . our _ first step _ is to show that @xmath77 divides @xmath78 for that purpose , denote by @xmath79 the ideal of @xmath68 $ ] generated by @xmath80 and all the 3-minors of the jacobian matrix of the polynomials @xmath81 . we also define the ideal @xmath82 and we recall from ( * ? ? ? * theorem 3.23 ) that @xmath77 is a generator of the ideal of inertia forms of @xmath79 , i.e. the ideal @xmath83 now , from the similar characterization of the resultant by means of inertia forms ( * ? ? ? * proposition 2.3 ) , we deduce that there exists an integer @xmath84 such that @xmath85.\end{gathered}\ ] ] but @xmath86 and @xmath87 belong to @xmath79 , so we deduce that @xmath88 it follows that @xmath89 is an inertia form of @xmath79 and it is hence divisible by @xmath77 . our _ second step _ is to prove that the resultant @xmath90 divides @xmath89 . for all @xmath91 , we obviously have that @xmath92 by developing each of these determinants with respect to their first column , we get the linear system @xmath93 the matrix of this linear system is nothing but the transpose of the jacobian matrix of the polynomials @xmath48 . denote by @xmath94 any of its 3-minor . then , cramer s rules show that both polynomials @xmath95 and @xmath96 belong to the ideal generated by the polynomials @xmath97 and @xmath98 . therefore , the divisibility property of resultants @xcite implies that @xmath99 divides @xmath100 where @xmath101 ; observe that @xmath94 is independent of @xmath45 . as it is well - known , @xmath94 is an irreducible polynomial , being the determinant of a matrix of indeterminates . therefore , to conclude this second step we have to show that @xmath94 does not divide @xmath99 . for that purpose , we consider the specialization @xmath102 of the coefficients of @xmath56 and @xmath57 so that @xmath103 where the @xmath104 s and @xmath105 s are generic linear forms ; we add their coefficients as new variables to @xmath50 . using the multiplicativity property of resultants , a straightforward computation yields the following irreducible factorization formula @xmath106 where the last product runs over the integers @xmath107 , with @xmath108 and @xmath109 with @xmath110 . since @xmath111 and @xmath94 is not a factor in the above formula , we deduce that @xmath94 does not divide @xmath99 . the _ third step _ in this proof is to show that the discriminant @xmath112 and the resultant @xmath99 are coprime polynomials in @xmath50 . since @xmath113 is irreducible ( * theorem 3.23 ) ) , we have to show that it does not divide @xmath99 . consider again the specialization @xmath102 given by and assume that @xmath113 is a factor in @xmath99 . then , since @xmath113 is independent on the coefficients of the linear forms @xmath114 and @xmath115 , @xmath116 must contain some factors that depend on the coefficient of @xmath117 but not on @xmath114 and @xmath115 . however , the decomposition formula shows that @xmath116 contains only irreducible factors that do depend on three linear forms @xmath48 , or on none of them . therefore , we deduce that @xmath113 does not divide @xmath99 . to conclude this proof , we observe that the previous results show that @xmath118 divides @xmath89 . moreover , straightforward computations shows that @xmath118 and @xmath89 are both homogeneous polynomials with respect to the coefficients of @xmath117 of the same degree , and the same happens to be true with respect to the coefficients of @xmath114 and @xmath115 . to compute the discriminant it is much more efficient to specialize the formula in theorem [ thm : gendef ] by giving to the linear forms @xmath48 some specific values , for instance a single variable . consider the jacobian matrix associated to the polynomials @xmath72 @xmath119 and its minors that we will denote by @xmath120 in the sequel , given a ( homogeneous ) polynomial @xmath121 , for all @xmath122 we will denote by @xmath123 the polynomial @xmath124 in which the variable @xmath39 is set to zero . [ cor : def ] suppose given a commutative ring @xmath62 , two positive integers @xmath34 such that @xmath47 and two homogeneous polynomials @xmath73 in @xmath64 $ ] of degree @xmath34 respectively . then , @xmath125 straightforward by applying the formula in theorem [ thm : gendef ] with @xmath126 , @xmath127 , and @xmath128 . we notice that @xmath129 by property of the discriminant of three homogeneous polynomials in four variables ( * ? ? ? * proposition 3.13 ) . from a computational point of view the above formula allows to compute the discriminant of any couple of homogeneous polynomials @xmath130 $ ] as a ratio of determinants since all the other terms in can be expressed as ratio of determinants by means of the macaulay formula . there is no need to introduce new variables as in the cayley trick and the formula is universal in the coefficients of the polynomials over the integers . in this section , we provide some properties and computational rules of the discriminant @xmath55 as defined in the previous section . in particular , we give precise formulas regarding the covariance and invariance properties . we also provide a detailed computation of a particular class of complete intersection curves in order to illustrate how our formalism allows to handle the discriminant and simplify its computation and evaluation over any ring of coefficients . in what follows , @xmath62 denotes a commutative ring . from theorem [ thm : gendef ] , it is clear that the discriminant @xmath55 is homogeneous with respect to the coefficients of @xmath56 , respectively @xmath57 and that these degrees can easily be computed . as expected , we recover the degrees of the usual geometric definition of discriminant ( see @xcite ) . [ prop : homdeg ] the universal discriminant is homogeneous of degree @xmath131 with respect to the coefficient of @xmath13 where , setting @xmath132 and @xmath133 , @xmath134 this is a straightforward computation from the defining equality ( see theorem [ thm : gendef ] ) , since the degrees of resultants and discriminants of finitely many points are known ( see ( * ? ? ? * proposition 2.3 ) and ( * ? ? ? * proposition 3.9 ) ) . let @xmath135 $ ] be two homogeneous polynomials of degree @xmath65 and @xmath66 respectively , then @xmath136 this is a straightforward consequence of the similar property for resultants @xcite and discriminants of finitely many points ( * ? ? ? * proposition 3.12 i ) ) . [ prop : elemtransform ] let @xmath137 be four homogeneous polynomials in @xmath64 $ ] of degree @xmath138 respectively . then , @xmath139 this is a straightforward consequence of the invariance of resultants under elementary transformations @xcite and the invariance of discriminants of finitely many points under elementary transformations ( * ? ? ? * proposition 3.12 ) . in this section , we give precise statements about two important properties of the discriminant : its geometric covariance and its geometric invariance under linear change of variables . [ prop : covariance ] suppose given two homogeneous polynomials @xmath73 in @xmath64 $ ] of the same degree @xmath140 and a square matrix @xmath141 with coefficients in @xmath62 , then @xmath142 by definition , it is sufficient to prove this formula in the universal setting . for simplicity , we use the formula . setting @xmath143 and @xmath144 , we observe that @xmath145 so that @xmath146 in addition , by the covariance of resultants @xcite , @xmath147 so that we deduce that @xmath148 the covariance of resultants also shows that @xmath149 and the covariance property of discriminants of finitely many points ( * ? ? ? * proposition 3.18 ) yields @xmath150 from all these equalities and , we deduce the claimed formula . [ prop : changecoord ] let @xmath73 be two homogeneous polynomials in @xmath64 $ ] of degree @xmath151 and let @xmath152 be a square matrix with entries in @xmath62 . for all homogeneous polynomial @xmath153 $ ] we set @xmath154 then , we have that @xmath155 where @xmath156 , @xmath157 , @xmath158 . as always , to prove this formula we may assume that we are in the universal setting , @xmath56 and @xmath57 being the universal homogeneous polynomials of degree @xmath65 and @xmath66 respectively . we will also denote by @xmath48 three generic linear form and by @xmath159 the generic square matrix of size @xmath160 . applying theorem [ thm : gendef ] , we get the equality @xmath161 ( observe that @xmath162 are all linear forms in @xmath45 ) . now , by ( * ? ? ? * proposition 3.27 ) , we know that @xmath163 also , by the chain rule formula for the derivative of the composition of functions , we have the formulas @xmath164\cdot \det(\varphi ) \\ j(f_1\circ \varphi , f_2\circ \varphi , l\circ \varphi , n\circ \varphi))&= j(f_1,f_2,l , n ) \circ [ \varphi]\cdot \det(\varphi ) \\ j(f_i\circ \varphi , l\circ \varphi , m\circ \varphi , n\circ \varphi))&= j(f_i , l , m , n ) \circ [ \varphi]\cdot \det(\varphi ) \end{aligned}\ ] ] from we deduce , using the invariance of resultants @xcite and their homogeneity , that @xmath165 and @xmath166 from here , the claimed formula follows from the substitution of the above equalities in and the comparison with the formula given in theorem [ thm : gendef ] . the discriminant is invariant under permutation of the variables @xmath45 . it follows from proposition [ prop : changecoord ] since @xmath8 is even . given a plane curve , we prove that its discriminant as defined in section [ sec : def ] , is compatible with its discriminant as a plane hypersurface @xcite . [ lem : redvar ] let @xmath167 be a homogeneous polynomial in @xmath64 $ ] of degree @xmath140 . then , for all @xmath91 we have that @xmath168 by definition , it is sufficient to prove this equality in the case where @xmath167 is replaced by the generic homogeneous polynomial @xmath169 of degree @xmath170 . we apply theorem [ thm : gendef ] with @xmath171 , @xmath172 , @xmath173 that are chosen so that @xmath174 as sets . we obtain the equality @xmath175 since the degree of @xmath169 and one of its partial derivative are consecutive integers , their product is always an even integer . it follows by standard properties of resultants that @xmath89 does not depend on the sign of its entry polynomials , nor on their order , nor on the reduction of the variables , so that we have @xmath176 now , by property of discriminants , in particular and its invariance under permutation of variables ( * ? ? ? * proposition 3.12 ) , we have @xmath177 finally , ( * ? ? ? * proposition 4.7 ) shows that @xmath178 and the claimed equality is proved . [ prop : planecurve ] let @xmath179 $ ] be a homogeneous polynomial of degree @xmath180 and @xmath181 be a linear form in @xmath64 $ ] . then , for all @xmath91 we have that @xmath182 where @xmath183 stands for the kronecker symbol . we assume that we are in the generic setting , which is sufficient to prove this corollary . consider the linear change of coordinates given by the matrix @xmath184 defined as follows : its @xmath185 row is the vector @xmath186 and its other rows are filled with zeros except on the diagonal where we put @xmath187 . then , it is not hard to check that @xmath188 therefore , by proposition [ prop : homdeg ] we obtain @xmath189 on the other hand , since @xmath190 , proposition [ prop : changecoord ] yields @xmath191 ( notice that @xmath192 is even and @xmath193 ) . then , using lemma [ lem : redvar ] we deduce that @xmath194 compared with , this latter equality shows that @xmath195 since @xmath52 is not a zero divisor in the universal ring of coefficients . finally , to conclude we observe that @xmath196 where the last equality follows from the homogeneity of the discriminant of a single polynomial ( * ? ? ? * proposition 4.7 ) . in order to illustrate the gain we obtain with the new formalism we are developing , we give an explicit decomposition of the discriminant of a particular family of complete intersection space curves that are drawn on a generalized cylinder whose base is an arbitrary algebraic plane curve . [ prop : samplecomp ] suppose given an element @xmath197 and two homogeneous polynomials @xmath198 $ ] of degree @xmath65 and @xmath66 respectively . if @xmath47 then @xmath199}\\ \cdot \disc(f , g)^{d_1 - 1}\disc(g)^{d_1}.\end{gathered}\ ] ] because of the space limitation , we will only give the main lines to prove this formula . first , we notice that it is sufficient to assume that we are in the universal setting , that is to say to assume that the coefficients of @xmath200 and @xmath201 are indeterminates over the integers . set @xmath202 and @xmath203 . by corollary [ cor : def ] , we have that @xmath204 applying laplace s formula @xcite , we get @xmath205 and substituting this equality in , we deduce that @xmath206 now , applying again laplace s formula we get that @xmath207 in order to compute @xmath208 , we first observe that @xmath209 by multiplicativity of resultants . from the definition of the jacobian determinants we have @xmath210 and we deduce that @xmath211 but from the rule of permutation of polynomials for resultants @xcite and the definition of discriminants of finitely many points ( * definition 3.5 ) , we have @xmath212 similarly , from the rule of permutations of polynomials and the definition of discriminants of hypersurfaces ( * ? ? ? * definition 4.6 ) , we have @xmath213 and hence @xmath214 now , it remains to compute @xmath215 on the one hand we have @xmath216 on the other hand , @xmath217 and since @xmath218 by the euler formula , we deduce that @xmath219 finally , since @xmath220 the comparison of , and shows that @xmath221 now , coming back to the factor @xmath222 , we deduce from that @xmath223 and hence from that @xmath224 finally , we deduce from ( 3 ) that @xmath225 and the claimed formula follows . the aim of this section is to show that the discriminant @xmath55 defined in definition [ def : disc ] satisfies to the expected geometric property , namely that its vanishing corresponds to the existence of a singular point on the curve intersection of the two surfaces of equations @xmath24 and @xmath26 in @xmath18 . we start by recalling the precise meaning of this geometric property as we will work over coefficient rings which are not necessarily fields . let @xmath226 be a commutative ring . we consider the universal setting over @xmath226 , i.e. we suppose given two positive integers @xmath34 and we consider the ( generic ) homogeneous polynomials in the four variables @xmath35 @xmath227 that are polynomials in @xmath228 $ ] , where @xmath229 $ ] is the universal ring of coefficients over the base ring @xmath226 . if there is no possible confusion , we will omit the subscript @xmath226 in the above notation . we define the ideal @xmath230 generated by the variables @xmath45 , the ideal @xmath231 generated by all the 2-minors of the jacobian matrix of @xmath56 and @xmath57 , and the ideal @xmath232 . thus , using notation , we have that @xmath233 the quotient ring @xmath234 is a graded ring with respect to the variables @xmath45 . as such , it gives rise to the projective scheme @xmath235 that corresponds to the points @xmath236 such that the corresponding polynomials @xmath72 and all the 2-minors of their jacobian matrix vanish simultaneously at @xmath27 . the canonical projection of @xmath237 onto @xmath238 is a closed subscheme @xmath94 of @xmath238 whose support is precisely what is commonly called the _ discriminant locus_. by definition , the defining ideal of @xmath94 is the ideal @xmath239 where @xmath240 is the so - called ideal of _ inertia forms _ the notation @xmath241 denotes the localization of @xmath242 with respect to the variable @xmath71 and @xmath243 is the product of the canonical quotient maps . in what follows , we will show that @xmath244 , as defined by definition [ def : disc ] , is a generator of @xmath245 if @xmath226 is a ufd , so that it satisfies to the expected geometric property . before going into the details , we recall the following important and well - known result ( see e.g. @xcite ) : if @xmath226 is a field , then the reduced scheme of @xmath246 is an irreducible hypersurface , i.e. the radical of @xmath245 is a principal and prime ideal , so that it is generated by an irreducible polynomial @xmath247 . this polynomial is not unique ; it is unique up to multiplication by a nonzero element in @xmath226 . in addition , @xmath248 is homogeneous of degree @xmath131 ( see proposition [ prop : homdeg ] for the definition of @xmath131 ) with respect to the coefficients of @xmath13 . we begin with some preliminary results on the jacobian minors and the ideal @xmath249 they generate . [ lem : jdet ] for any @xmath250 we have that @xmath251 using the euler formula , we have that @xmath252 and the claim follows . [ lem : enrel ] for any integer @xmath253 and any triple of distinct integers @xmath254 in @xmath255 we have that @xmath256 develop the determinant of the 3-minor corresponding to the columns @xmath254 in the jacobian matrix of @xmath72 and @xmath257 . [ lem : primenessj ] if @xmath226 is a domain , then for all @xmath91 the ideal @xmath258 is a prime ideal . for simplicity , we will assume that @xmath259 , the other cases being similar . in order to emphasize some particular coefficients of @xmath56 and @xmath57 we rewrite them as follows : @xmath260 we consider the @xmath51-algebra morphism @xmath261 \rightarrow c[x_4^{-1 } ] : u_{i , j } \mapsto -\partial_jf_i / x_4^{d_i-1}\ ] ] which leaves invariant all the variables @xmath45 and all the coefficients of @xmath72 , except the @xmath262 s . as @xmath263 , @xmath102 is surjective . moreover , setting @xmath264 and denoting by @xmath265 the 2-minor of @xmath266 corresponding to the column number @xmath267 , we have that @xmath268 @xmath269 considering the map @xmath270 induced by @xmath102 , @xmath271 \rightarrow c[x_4^{-1}]/\left(\uu_{1,2},\uu_{2,3},\uu_{1,3}\right),\ ] ] we deduce that @xmath272\subset \ker(\overline{\eta}).$ ] actually , this inclusion is an equality . indeed , if @xmath273 then @xmath274 but since @xmath275 , applying again @xmath102 to we deduce that @xmath276.\ ] ] it follows that @xmath270 induces a graded isomorphism @xmath277/\left(\uu_{1,2},\uu_{2,3},\uu_{1,3}\right).\ ] ] from here , if @xmath226 is a domain then the ideal generated by the 2-minors of @xmath266 is a prime ideal ( see ( * ? ? ? * theorem 2.10 ) ) and hence @xmath278 is a domain . the above lemma is the key result to deduce the following properties of the ideal of inertia forms @xmath279 . [ prop : bxi - domain ] if @xmath226 is a domain then @xmath241 is a domain for all @xmath91 . we prove the claim for @xmath259 , the other cases being similar . let @xmath280 be two polynomials in @xmath281 so that @xmath282 in @xmath283 , i.e. @xmath284 belongs to the ideal @xmath79 up to multiplication by a power of @xmath285 . using this fact and lemma [ lem : jdet ] , we deduce that there exists an integer @xmath286 such that @xmath287 in order to emphasize the leading coefficients of @xmath56 and @xmath57 with respect to the variable @xmath285 , we rewrite them as @xmath288 denote by @xmath289 the polynomial ring @xmath281 in which the variables ( coefficients ) @xmath290 are removed and consider the surjective graded morphism @xmath291 \rightarrow \overline{c}[x_4^{-1 } ] : u_{i,0 } \mapsto -q_i / x_4^{d_i}\ ] ] which leaves invariant all the variables @xmath45 and all the coefficients of @xmath72 , except @xmath290 . it induces an isomorphism @xmath292.\ ] ] now , by we deduce that @xmath293 belongs to the ideal @xmath294.$ ] therefore , using lemma [ lem : primenessj ] we deduce that either @xmath295 or @xmath296 belongs to this ideal , say @xmath295 . this implies that there exists an integer @xmath297 such that @xmath298 in turns , this implies precisely that @xmath299 in @xmath283 , which concludes the proof . [ cor : tf ] for all @xmath91 we have that @xmath300 thus , both @xmath279 and @xmath301 are prime ideals if @xmath226 is a domain . using proposition [ prop : bxi - domain ] , the proof of ( * ? ? ? * corollary 3.21 ) applies verbatim to show that @xmath71 is not a zero divisor in @xmath302 for all @xmath303 . from here , we deduce that the canonical maps @xmath304 , @xmath303 , are all injectives maps and hence the claimed equalities follow . we are now ready to prove the main result of this section . if @xmath226 is a ufd then @xmath305 is a generator of @xmath245 . it is hence an irreducible polynomial in @xmath306 . first , let @xmath0 be a field . from the geometric property we recalled previously , we know that the radical of @xmath307 is generated by an irreducible polynomial @xmath308 . using corollary [ cor : tf ] , we deduce that @xmath308 is actually a generator of @xmath307 . now , assume that @xmath226 is a domain and take again the notation of theorem [ thm : gendef ] . the resultant @xmath309 is an inertia form of its four input polynomials and hence , by developing the jacobian determinants , we see that it belongs to @xmath310 . therefore , theorem [ thm : gendef ] shows that @xmath311 we claim that @xmath77 does not belong to @xmath312 . indeed , assume the contrary . by extension to the fraction field @xmath0 of @xmath226 , we would have that the square - free part of @xmath313 belongs to the prime ideal @xmath307 . but @xmath307 is generated by @xmath308 so we get a contradiction since @xmath313 is homogeneous of degree @xmath314 with respect to the coefficients of @xmath56 , and similarly with respect to the coefficients of @xmath57 ( * ? ? ? * proposition 3.9 ) . with a similar argument , we also get that the resultant @xmath315 does not belong to @xmath245 since it is homogeneous of degree @xmath316 with respect to the coefficients of @xmath56 , and similarly with respect to the coefficients of @xmath57 . finally , as @xmath245 is a prime ideal we deduce from that @xmath305 belongs to @xmath245 ( recall that @xmath226 is here assumed to be a domain ) . as a first consequence , since @xmath244 and @xmath248 are homogeneous polynomials of the same degree with respect to the coefficients of each @xmath13 , we conclude that this theorem is proved if @xmath226 is a assumed to be a field . let @xmath317 be an inertia form and set @xmath318 . our next aim is to show that @xmath319 divides @xmath320 . for that purpose , using the definition of inertia forms and lemma [ lem : jdet ] , we deduce that there exists an integer @xmath286 such that @xmath321 then , lemma [ lem : enrel ] shows that @xmath322 so that we get that @xmath323 now , by the divisibility property of resultants @xcite we obtain that @xmath324 divides @xmath325 applying computational rules of resultants and choosing @xmath326 we get that @xmath327 where , in addition , @xmath328 by the definition of discriminants of finitely many points ( * ? ? ? * definition 3.5 ) . combining the above equalities and using , we deduce that @xmath55 divides the product @xmath329 with a similar degree inspection as above and after extension to @xmath330 , we deduce that @xmath331 , which is an irreducible polynomial , can not divide the discriminant and the two resultants in . then , we claim that the discriminant and the two resultants in are primitive polynomials . this is a known property for the first two ones . for the third one , namely @xmath332 , we argue by specialization : for instance , @xmath333 is a primitive polynomial since both discriminants on the right side are known to be primitive polynomials . therefore , we conclude that @xmath334 divides @xmath320 . finally , from what we proved we deduce that @xmath245 and the ideal generated by @xmath244 have the same radicals . since we assume that @xmath226 is a ufd , @xmath245 is prime and we deduce that there exist an irreducible polynomial @xmath335 , an invertible element @xmath336 and a positive integer @xmath337 such that @xmath338 . by extension to @xmath0 we deduce immediately that @xmath339 , which concludes the proof . the above theorem shows that @xmath334 is a primitive and irreducible polynomial in @xmath340 . it also shows that the discriminant formula we gave provides an _ effective _ smoothness criterion ( as the criterion in @xcite that applies verbatim ) . part of this work was done while the second author was visiting imsp at benin , supported by the daad . both authors warmly thank the ictp for its hospitality and are very grateful to alicia dickenstein , marie - franoise roy and fernando rodrigues villegas for their continued support .	in this paper , we develop a new approach to the discriminant of a complete intersection curve in the 3-dimensional projective space . by relying on the resultant theory , we first prove a new formula that allows us to define this discriminant without ambiguity and over any commutative ring , in particular in any characteristic . this formula also provides a new method for evaluating and computing this discriminant efficiently , without the need to introduce new variables as with the well - known cayley trick . then , we obtain new properties and computational rules such as the covariance and the invariance formulas . finally , we show that our definition of the discriminant satisfies to the expected geometric property and hence yields an effective smoothness criterion for complete intersection space curves . actually , we show that in the generic setting , it is the defining equation of the discriminant scheme if the ground ring is assumed to be a unique factorization domain .
wannier functions ( wfs ) can be obtained by a unitary transformation of the extended wavefunctions of a periodic system . @xcite so far , the effective diffusion / application of wfs in electronic structure calculations was hindered by their intrinsic non - uniqueness . @xcite in 1997 marzari and vanderbilt proposed a useful approach @xcite to overcome this drawback : the proposed methodology allows one to extract , from a selected manifold of bands , the set of wfs with the maximum spatial localization , _ i.e. _ the _ maximally localized wannier functions _ ( mlwfs ) . on one hand , mlwfs are attractive because they constitute a complete and orthonormal basis set in the real space . on the other hand , with respect to other numerical real - space basis sets , they carry also the physical information of the starting bloch functions . indeed , mlwfs may yield the chemical view of molecular bond orbitals , and they can be exploited for the computation of the spontaneous polarization in periodic systems,@xcite becoming very popular to tackle these issues in advanced materials . @xcite in addition , mlwfs have most recently been proposed to calculate the transport properties of nano - size conductors connected to external electrodes.@xcite the calculation of mlwfs was originally implemented @xcite for a projection from bloch orbitals expanded on a plane - wave basis set in density functional theory ( dft ) calculations , within the norm - conserving pseudopotential ( ncpp ) framework . norm - conserving pseudopotentials , @xcite that allow to neglect core electrons in the evaluation of physical observables , are usually characterized by a high transferability of an element to a variety of chemical environments . however , the norm conservation in the core region is a strong constraint , that affects the computational effort of the dft calculations . as a consequence , only a part of the periodic table elements results numerically accessible : some chemical species , such as first - row elements ( e.g. c , n , o , f ) and especially transition metals ( e.g. mn , fe , co , ni , cu ) and rare earths ( e.g. la , gd , yb ) , would require an extremely high number of basis functions ( _ e.g. _ plane waves ) , in order to be described with a satisfactory accuracy . unfortunally , the typical systems of interest in nanoscience and specifically in molecular electronics contain atoms of those critical species . this problem was brilliantly solved within a frozen - core approach by introducing the ultrasoft pseudopotentials ( uspps),@xcite that relax the norm conservation constraint , by compensating with a pseudized charge . since the computational requirement in the calculation of the dft bloch functions affects also the evaluation of the mlwfs , the generalization of the original marzari - vanderbilt s procedure to the case of uspps becomes necessary , in order to make the mlwfs a powerful tool to tackle realistic systems of high technological and fundamental relevance . this is the focal aim of the paper . in pursuing this extension , we also noted that recent studies mapped @xcite the uspp procedure into the paw theory . the latter @xcite approach was developed by blchl to combine an all - electron description of the system with the simplicity of the frozen - core pseudopotential methods . the reader is referred to the original articles for a description of the paw scheme and for the matching between paw and uspp . the uspp can indeed be recast in the paw formalism as an approximation , as we will discuss in the following . given this equivalence , we developed a theoretical framework within the paw theory to compute wfs from uspp bloch orbitals , as we show by expressing the necessary matrix elements . the paper is organized as follows : in section [ sec : formalism ] we first write down the salient quantities of the uspp and paw methods that enter the computation of the wfs , and then obtain and discuss the wf computation ; in section [ sec : applications ] we report the results of the application of our method to several test cases , that explore both the chemical bonding and the electrical polarization in the pertinent cases ; finally we draw our conclusions in section [ sec : conclusions ] . in this section we give a brief introduction to the theory of maximally localized wannier functions . a more detailed description can be found in the original papers . @xcite in the case of an isolated band , wannier functions can be defined @xcite as a combination of the bloch orbitals @xmath0 corresponding to different @xmath1points as follows : @xmath2 where @xmath3 is a @xmath4-dependent phase factor . this definition has been generalized @xcite to a group of bands leading to the expression : @xmath5 here the extra degrees of freedom related to the phases of the bloch eigenstates are collected in the unitary matrix @xmath6 . in the one - band case it has been demonstrated that a suitable choice of the phases @xmath3 leads to wfs which are real and exponentially decaying in a real space representation . @xcite in the many - band case @xcite this theorem does not hold anymore but the arbitrariness in the unitary ( gauge ) transformation @xmath6 can be exploited . following marzari and vanderbilt @xcite we define a _ spread _ functional @xmath7 $ ] , which gives a measure of the degree of localization of the wf set . it reads : @xmath8= \sum_n \ , \left [ \ , \langle \hat{r}^2 \rangle_n - \langle \hat{\mathbf{r } } \rangle_n^2 \ , \right ] \ , , \ ] ] where @xmath9 is the expectation value of a given operator on the @xmath10-th wf calculated using the @xmath6 gauge transformation . it is therefore possible to define the _ maximally localized _ wannier functions ( mlwfs ) as the wfs resulting from eq . ( [ eq : wannier_definition ] ) by means of the unitary transformation @xmath6 that minimizes the spread functional . according to the formal analysis of the @xmath11 and @xmath12 terms given by blount , @xcite it is possible to demonstrate that the dependence of @xmath13 on the gauge transformation is determined only by the so - called overlap integrals @xmath14 : @xmath15 @xmath16 being the periodic part of the bloch states @xmath17 . the detailed form of the position expectation values and of the spread functional ( including its gradient wrt to @xmath6 ) in terms of the overlap integrals is reported in appendix [ app : wannier ] . since the representation of bloch eigenstates enters only the calculation of the @xmath14 integrals , these quantities are the main objects to deal with when using a uspp or paw formalism . the detailed treatment is reported in section [ sec : wannier_uspp ] . the paw formalism has been introduced by blchl @xcite and it has also been demonstrated @xcite that the vanderbilt ultrasoft pseudopotential ( uspp ) theory @xcite can be obtained within the paw approach . blchl s starting point is to partition the volume of the system by setting spherical regions ( atomic spheres ) around each atom . in each sphere two complete sets of wavelets @xcite ( @xmath18 where @xmath19 the index @xmath20 runs over different atoms ( _ e.g. _ different atomic spheres ) , while @xmath21 are the projectors @xcite related to the pseudo wavelets @xmath22 . the @xmath23 operator acts on the pseudized functions to reconstruct @xcite the ae ones . matrix elements and expectation values of a generic operator @xmath24 on the physical ae states can be written as : @xmath25 one of the main results of ref . [ ] is the explicit expression for @xmath26 , which we report here for the case of local and semilocal operators : @xmath27 \ , \| \beta_{ii}^ { } \rangle \langle \ , \beta_{ij}^{}\| \ , .\end{aligned}\ ] ] the second term ( @xmath28 ) in the rhs of eq . ( [ eq : expectation_value ] ) takes into account the corrections due to the use of the pseudo functions instead of the ae ones , and from here on it will be called augmentation term . at this point it is useful to define the quantities @xmath29 and @xmath30 ( augmentation densities and charges respectively ) as : @xmath31 within these definitions , eq . ( [ eq : expectation_value ] ) for local operators @xmath32 can be recast in a more convenient form : @xmath33 \ , \| \beta_{ii}^ { } \rangle \langle \ , \beta_{ij}^{}\| \ , .\ ] ] setting @xmath24 to the identity in eqs . ( [ eq : expectation_value],[eq : expectation_value_local ] ) , scalar products are given by @xmath34 ( that characterizes also the uspp formalism @xcite ) is given by : @xmath35 in the same way , by setting @xmath36 we obtain an expression for the density : @xmath37 where @xmath38 is the density contribution of the pseudo wavefunctions . since we are able to express all the quantities of interest in terms of the soft pseudo - states , the quantum problem can be solved directly in this representation . in order to do this , it is necessary to _ augment _ the hamiltonian operator : the procedure leads to additional terms which have exactly the same role as that of the pseudopotentials in standard pw calculations . moving to the uspp framework , the generalization introduced by vanderbilt @xcite is twofold . ( i ) more than one projector per angular momentum channel can be taken into account : the inclusion of multiple projectors per channel enlarges the energy range @xcite over which logarithmic derivatives are comparable with the full potential case , thus increasing the overall portability of the pseudopotential . ( ii ) by relaxing the norm - conservation constraint of the pseudo reference - states , the pseudo wavefunctions are smoothened : thus , the required cutoff energy for pw representation can be drastically lowered . the fact that properties ( i ) and ( ii ) are verified for the paw wavelets @xmath39 establishes the connection between the paw and uspp methods . in fact , ( i ) is naturally valid for the wavelets @xmath40 , because these form a basis set , and therefore have in principle an infinite number of states for each angular momentum channel . ( ii ) is valid as well , and the possibility of non norm - conservation in passing from @xmath41 to @xmath42 is accounted by non null @xmath43 terms in eqs . ( [ eq : augmentation_densities],[eq : augmentation_charges ] ) . consequently , the paw theory for wavefunction reconstruction can be basically adopted also in the case of uspp . @xcite while the paw method is in principle an exact ae ( frozen - core ) approach , @xcite the uspp method adopts a further approximation @xcite represented by the requirement of pseudizing the augmentation densities @xmath44 [ eq . ( [ eq : augmentation_densities ] ) ] . since these terms contain the ae reference states , they are not simply writable on a pw basis : the uspp pseudization is done to make them suitable for a pw representation . this also means that the total density from eq . ( [ eq : density ] ) would be pw representable within such an approach . even though the augmentation densities are pseudized , they must capture some features of the physical ae density . consequently , their pw cutoff energy may be larger than the one associated to the pseudo wavefunction density [ @xmath38 in eq . ( [ eq : density ] ) ] . as mentioned in sec . [ sec : formalism_wf ] , bloch wavefunctions enter the calculation of mlwf s only through the overlap matrix elements @xmath14 , defined in eq . ( [ eq : overlap ] ) . therefore , the reconstruction of these integrals from the knowledge of pseudo ( ultrasoft ) functions completely solves the problem of computing mlwf s within a paw / uspp formalism . once overlap matrices have been calculated , we do not longer distinguish whether the parent bloch wavefunctions were pseudized or not . being the overlaps @xmath45 [ eq . ( [ eq : overlap ] ) ] the matrix elements of the local operator @xmath46 , we built up the corresponding augmentation using eq . ( [ eq : expectation_value_local ] ) . overlaps can be written as : @xmath47 where we have defined @xmath48 and @xmath49 are @xmath4-point symmetrized projectors ( bloch sums ) . summation over ions @xmath20 is done in a single unit cell . details about the calculation of these quantities are reported in appendix [ app : radial_ft ] . we stress that the scalar product of @xmath50 functions can not be simply augmented by the @xmath51 [ eq . ( [ eq : number_operator ] ) ] operator as those involving @xmath52 s . in order to work with the periodic part @xmath53 of the bloch functions , _ first _ we have to reconstruct the ae bloch states by means of @xmath23 and _ then _ we can obtain the required @xmath54 states by applying the local operator @xmath55 . since this last operator does not commute with @xmath23 , we are not allowed to directly work on @xmath53 with the reconstruction operator . in the first scalar product of eq . ( [ eq : overlap_augmentation ] ) the number operator @xmath56 has been therefore substituted by the augmented operator @xmath57 . we thus need to introduce the fourier transform of the augmentation densities @xmath58 instead of the augmentation charges @xmath30 . we also note that in the thermodynamic limit the @xmath4-point grid becomes a continuum , so that @xmath59 . this limit leads to identical @xmath51 and @xmath60 operators . therefore , within discrete @xmath4-meshes the use of @xmath51 instead of @xmath60 is an approximation expected to give the best performance in the limit of a large number of @xmath4-points . we will refer to this approximation as the thermodynamyc limit approximation ( tla ) . we will comment more on its numerical aspects in sec . [ sec : applications ] . the problem of calculating wannier functions within uspp has also been faced elsewhere in the literature . @xcite in a first attempt , vanderbilt and king - smith @xcite extended the calculation of the spontaneous polarization through the berry phase @xcite to the uspp procedure . since the berry phase @xcite is directly related to overlap integrals , an expression for its calculation is reported in eq . ( 23 ) of ref . [ ] . this expression adopts the @xmath51 number operator instead of @xmath60 . therefore , the result does not completely agree with the one presented here [ eq . ( [ eq : overlap_augmentation ] ) ] , but it can be read as an approximating formula having the right thermodynamic limit in view of the above discussion . bernasconi and madden @xcite derived instead the formalism for mlwfs with uspp in a simplified approach @xcite valid only in the case of @xmath61-sampled supercells . although the basic ingredients ( _ e.g. _ overlaps ) are the same , our treatment is valid for generic periodic systems , and recovers the @xmath61-only calculation as a special case . furthermore , the proof given in ref . [ ] is not based on a paw reconstruction as we did . the authors generalized the augmentation operator for the density [ eq . ( [ eq : density ] ) ] also to the case of the density matrix and then derived the augmentation for the @xmath62 operator . here instead the @xmath63 are the generators of the reciprocal lattice . we note that the result by bernasconi and madden is coherent with our treatment , while the first one by vanderbilt and king - smith @xcite is not . following bernasconi and madden , thygesen and coworkers @xcite recently derived an expression for overlap matrices within uspp for periodic systems , considering a @xmath61-only supercell containing the whole crystal . closing this section we wish to underline that matrix elements of the form @xmath64 enter also other physical problems . a particularly appealing case is the calculation of one - particle green function in the gw approximation . @xcite current implementations @xcite of the method experience dft wavefunctions mainly through the above defined matrix elements ( evaluation of the polarizability ) , while no direct access to the density is required . the extension of gw calculations to the case of paw @xcite or uspp is therefore feasible along the same lines we presented here for wannier functions . we note , however , that the numerical cost is extremely higher due to the larger number of overlaps to be computed . in this section we would like to describe some issues related to the numerical performance of the method . we implemented this formalism in the freely - available want code , @xcite for the calculation of electronic and transport properties with wfs . we also took advantage of the complete integration of the want code with the pwscf package , @xcite which explicitly treats the dft problem using uspps . from here on we focus on the uspp formalism , even though a large part of the discussion is still valid also in the paw case . the most important advantage of using the uspp construction for wannier functions is the scaling of the original dft calculations , which has been described elsewhere @xcite and we do not repeat now . however , this scaling has also an effect on the actual computation of wfs and we analyze this aspect in detail . as we described above , almost the whole amount of changes induced by the uspp description ( relative to ncpp ) in the the calculation of mlwfs is related to the implementation of eq . ( [ eq : overlap_augmentation ] ) . details on how to compute @xmath65 and @xmath66 are reported in appendix [ app : radial_ft ] . since no reference to the charge is made , only the very smooth wavefunction grid is used throughout the calculation . while a linear scaling with the number of plane waves is exploited in the first term ( pseudo overlaps ) of eq . ( [ eq : overlap_augmentation ] ) , scalar products between projectors and pseudo states in the second term ( augmentation overlaps ) are the price to pay for introducing uspps . when we consider that the number of @xmath67-projectors @xmath68 is of the same order of the number of bands @xmath69 ( but usually larger by a factor between one and two ) we see that both pseudo and augmentation overlap terms have the same scaling , namely @xmath70 . however , the pseudo overlaps turn out to have a larger prefactor @xcite and represent the leading term . usually , uspps allow for a reduction of the pw cutoff by a factor of 2 to 3 for first - row elements up to 5 or even more for atoms with @xmath71- or @xmath72-states . this leads to a reduction of the pw number @xmath73 by a factor of around 3 to 10 or more . even if the scaling wrt @xmath73 is linear , it more than compensates the effort for augmenting overlaps and makes the introduction of uspps numerically advantageous . our experience shows that uspps avoid the creation of bottlenecks in the computation of overlaps and make the wf localization the leading part of the calculation . finally , we note that in order to give a guess for the iterative minimizations involved in the mlwf method , it is sometimes required to compute the projections of bloch states onto some starting localized functions . @xcite the augmentation of scalar products is performed as usual accounting for the @xmath51 number operator [ eq . [ eq : number_operator ] ] . projections on the @xmath67-functions are required as well , but they have already been computed and it turned out that scaling is linear wrt the pw number as before . therefore , no implications on the above discussion arise . in this section we apply the above described formalism to some test cases , ranging from periodic crystals to isolated molecules . precisely , we study _ fcc _ copper bulk , wurtzite aln , and watson - crick dna base pairs . we address a number of physical properties connected to wannier functions : interpolation of the electronic structure , calculation of dipole moments and spontaneous polarization , analysis of the chemical bonding . all the calculations are performed with both norm - conserving and ultra - soft pseudopotentials . the numerical implications in using the uspp - tla approach for the augmentation of overlaps are also discussed . we compute mlwfs for _ fcc_-copper , which has already been used as a test case in the literature @xcite of wfs . we adopt a @xmath74 mesh of @xmath4-points to sample the brillouin zone and compute six wfs corresponding to the lowest @xmath75 manyfold . in the disentangling procedure @xcite ( used to get the optimal subspace for wf localization ) we _ freeze _ the bloch eigentates below the fermi energy : this means that the subspace is constructed by these selected states plus a mixture of the states above the fermi level . we adopt a kinetic energy cutoff for wavefuntions of 120 ry ( 25 ry ) when using ncpp ( uspp ) and of 480 ry ( 200 ry ) for the density . in fig . [ fig : copper - bands ] we superimpose the band structure directly computed from a uspp - dft calculation and that obtained from wannier function interpolation @xcite on the adopted 8@xmath768@xmath768 uniform @xmath4-point grid . these two sets of bands are almost superimposed below the fermi energy and some slight differences arise only at higher energies . this is expected due to the choice of the energy window for the frozen states : while the eigenvalues for the @xmath4-points in the adopted regular mesh are the same as those from the dft calculation by construction , it is not trivial that the band structure along a generic brillouin zone line is well reproduced . this is indeed the case here , being a signature of proper localization of the computed wfs . the band structure interpolation obtained from ncpp is essentialy the same as the one in fig . [ fig : copper - bands ] and it is not reported . some small descrepancies between ncpp and uspp interpolated bands are also present in the starting dft calculations and are not of our interest in this context . in tab . [ tab : copper - data ] we report a more detailed description of quantities related to wfs ( spreads , real - space decay of the hamiltonian matrix elements ) in order to compare the ncpp and uspp approaches . a measure of the hamiltonian decay is defined as : @xmath77 where we defined @xmath78 . p1 mm l p0 mm r p0 mm r p0 mm r & & & * ncpp * & & * uspp * & & * uspp - tla * + + & @xmath13 & & 22.434 & & 23.010 & & 18.886 + & @xmath79 & & 14.767 & & 15.629 & & 11.303 + & @xmath80 & & 7.667 & & 7.381 & & 7.583 + + & @xmath81 & & 0.4876 & & 0.4795 & & 0.4779 + & @xmath82 & & 0.0493 & & 0.0473 & & 0.0476 + & @xmath83 & & 0.0203 & & 0.0207 & & 0.0205 + uspp results appear to be in very good agreement with those related to ncpp , most of the differences being reasonably due to the pseudopotential generation and not to the wf computation . the average number of iterations to converge the disentanglement and the localization procedures are almost the same , as well as the singular spread values for each of the wfs . figure [ fig : copper - plot ] reports the spatial distribution of wfs from uspp calculations . as in previous works , @xcite we find one interstitial @xmath84-like wf with the largest spread [ fig . [ fig : copper - plot](a ) ] , and five more localized @xmath71-like wfs [ fig . [ fig : copper - plot](b d ) ] , correctly reproducing the physical @xmath75 picture of copper . fcc_-copper _ : charge distribution for wfs computed using uspp . ( a ) interstitial wf , @xmath85 bohr@xmath86 ; ( b d ) @xmath71-character wfs , @xmath87 bohr@xmath86 , @xmath88 bohr@xmath86 , @xmath89 bohr@xmath86 . the two @xmath71-like missing wfs are strictly similar to ( b ) and ( c ) and are not reported . , scaledwidth=38.0% ] as a last remark for the case of copper , we analyze the effect of neglecting the @xmath90 term in eq . ( [ eq : overlap_augmentation ] ) , _ i.e. _ the uspp thermodynamic limit approximation ( uspp - tla ) . the theoretical background has already been discussed in sec . [ sec : wannier_uspp ] , here we focus on the numerical aspects . in tab . [ tab : copper - data ] the third column reports the resukts of the calculation performed within this approximation : it is evident that the numerical values of the uspp - tla spreads deviate from the ncpp ones much more than the uspp do . on the contrary , the interpolated band structure and the real - space decay hamiltonian matrix elements are definitely well - suited and comparable with those obtained in the full uspp treatment . since the tla is known to become exact in the thermodynamic limit , we expect it to work better when incresing the dimension of the @xmath4-point mesh . we checked the behavior of the approximation with respect to different meshes but no convergence could be reached for grids ranging from @xmath91 to @xmath92 @xmath4-points . wannier functions have been widely used to characterize the electrostatic properties of several molecular systems ranging from _ e.g. _ water @xcite and small molecules @xcite to large biomolecules , such as proteins,@xcite nucleic acids , @xcite enzymes @xcite and ionic channels . @xcite in fact , the wannier transformation allows one to partition the charge density into localized distributions of charges sitting on the so - called wannier centers @xmath93.@xcite in the case of isolated molecular systems , the dipole moment is a well defined quantity given by @xmath94 with @xmath95 where @xmath96 and @xmath97 are the ionic and electronic component respectively ; @xmath98 is the electron charge ; the @xmath20 summation is over the ionic sites @xmath99 and @xmath100 is the valence charge of the @xmath101 atom , as defined by the corresponding psudopotential ; the @xmath10 summation is over the doubly occupied valence states . p1 mm l p0 mm c p0 mm c p0 mm c p0 mm c & & & * uspp * & & * ncpp * & & * hf/6 - 31g@xmath102 * & & * exp . * + + & g & & 7.1 & & 7.2 & & 7.1 & & 7.1@xmath103 + & c & & 6.7 & & 6.9 & & 7.1 & & 7.0@xmath104 + & a & & 2.3 & & 2.3 & & 2.5 & & 2.5@xmath103 + & t & & 4.2 & & 4.4 & & 4.6 & & 4.1@xmath105 + & g - c & & 4.9 & & 4.9 & & 6.5 & & + & a - t & & 1.7 & & 1.8 & & 2.0 & & + @xmath106devoe and tinoco ( ref . ) + @xmath107weber and craven ( ref . ) + @xmath108kulakowski _ et al . _ ( ref . ) as a key test , we calculated the dipole moments of the four isolated dna bases : guanine ( g ) , cytosine ( c ) , adenine ( a ) and thymine ( t ) , and of the two watson - crick base pairs g - c and a - t , whose structures are reported in fig . [ fig : dnawf](a ) . we simulated each system in a large ( @xmath109 ) @xmath110 supercell , which allows us to avoid spurious interactions among neighbor replicas . for isolated systems , the uniform k - point grid reduces to the case of @xmath61-point only , and the connecting vectors @xmath111 [ eq . ( [ eq : overlap ] ) ] correspond to the generators of the reciprocal lattice vectors . we expanded the electronic wavefuncions with the kinetic energy cutoff of 25 and 80 ry , using uspp and ncpp respectively . we first optimized the atomic structure of the two base pairs g - c and a - t until forces on all atoms were lower than 0.03 ev / , using ultra - soft pseudopotentials . then , keeping atoms fixed in the relaxed geometry , we calculated the electronic structure and the corresponding mlwfs for both the single bases and the base pairs . we maintained the same geometries also for the corresponding ncpp calculations . in this case we have checked that forces on single atoms never exceeded the value of 0.05ev / . our results for both sets of calculations are reported in table [ tab : dna - dipole ] . we note a very good internal agreement between the uspp and ncpp cases , as well as in comparison with previous quantum chemistry hf/6 - 31g@xmath102 calculations @xcite and experimental results . the total spread @xmath13 and its components @xmath79 , @xmath112 ( not shown ) are also very similar in the two set of calculations , while the diagonal term @xmath113 is , by definition , @xcite identically zero for isolated systems . finally , we can take advantage from the calculated wannier functions to further investigate the electronic distribution and the bonding pattern in the molecules . @xcite for example , as shown in fig . [ fig : dnawf](b ) for the case of the g - c base pair , we are able to characterize different kinds of bonds . the sb wannier function represents a single @xmath114 bond : it is centered in the middle point of the c - c bond . the two db wfs describe instead a double n = c bond : this bond is partially polarized with a slight charge accumulation near the nitrogen atom . finally , the lp wfs represents two electron lone pairs localized around the oxygen atom of the guanine molecule . note also that the distribution of single / double bonds correctly reflects the theoretical one reported in fig . [ fig : dnawf](a ) . let us remark here that our benchmark to assess the success of the newly developed uspp - wf methodology is its relative performance with respect to the ncpp - wf ( already established ) framework , namely the comparison between columns 2 and 3 in table ii . here we move to the calculation of polarization in periodic systems . we focus on the case of aluminum nitride and compute the spontaneous polarization @xmath115 of the wurtzite phase . this is a particularly appealing test - case since a large debate exists in the literature and many results are present . @xcite moreover , nitrogen may be more easily described ( in terms of pw kinetic - energy cutoff ) with uspps than with ncpps , allowing for an advantageous application of the current formalism . following refs . [ ] we evaluate the polarization ( electronic and ionic contributions ) for the wurtzite ( wz ) phase taking the zinc - blend ( zb ) structure as a reference . our calculations adopt the gga - pbe parametrization for the exchange - correlation functional , and uses cutoff energies of 60 ry ( 25 ry ) for ncpp ( uspp ) for wavefunctions and 240 ry ( 200 ry ) for the density . we relax both the cell dimensions and the atomic positions of the wz phase . the zb reference is assumed to have ideal atomic positions and the cell is taken equal to the one computed for the wz polytype ( six bilayers are included in the cell ) . these structural calculations have been performed using uspp and the obtained ( lattice and ionic ) parameters have been used also in the ncpp simulations . using a @xmath116 monkhorst - pack @xcite mesh of @xmath4-points we obtained the @xmath117 and @xmath118 parameters of the exagonal lattice as @xmath117=3.1144 and @xmath118=1.6109 ( for the standard wz cell including two bilayers , @xmath118=4.8326 for the actual cell we adopted ) . in tab . [ tab : aln - polarization ] we report the comparison of uspp and ncpp results in our calculations as well as other results from the literature . p1 mm l p0 mm c p0 mm c p0 mm c & p@xmath119 & & * ncpp * & & * uspp * & & * uspp - tla * + + & this work & & -0.095 & & -0.094 & & -0.094 + & ref . [ ] & & & & & & -0.090 ( -0.099 ) + & ref . [ ] & & & & -0.120 & & + + & @xmath120 & & 80.895 & & 80.756 & & 80.271 + & @xmath121 & & 67.683 & & 67.857 & & 67.282 + the values of spontaneous polarization computed using ncpps are almost identical to those obtained with uspps ( @xmath122 - 0.094 c / m@xmath86 ) . the uspp - tla behaves very accurately in this case and no difference can be found with respect to the full uspp calculation . our results are also in very nice agreement with the uspp - tla calculation by bernardini _ et al . _ , @xcite who used a berry - phase formalism @xcite and found a value of @xmath122 - 0.090 c / m@xmath86 . we thus conclude that the uspp - tla approximation performs well for the computation of the spontaneous polarization in nitrides , relative to a pure uspp treatment without the thermodynamic limit approximation . these results for the p@xmath119 in aln wurtzite are also in agreement with indirect experimental evidencies as reported in refs . [ ] . while some earlier experimental fits @xcite claim for much lower values of p@xmath119 ( ranging from -0.040 to -0.060 c / m@xmath123 ) , later @xcite works explain this discrepancy as due the neglecting of bowing effects ( non - linearity ) of the p@xmath119 with respect to the composition of the alloy al@xmath124ga@xmath125n employed in the measurements . in this paper we presented an approach to calculate _ maximally localized wannier functions _ in the _ ab initio _ plane - wave ultrasoft pseudopotential scheme . our methodology is formulated in the general framework of the paw theory and recovers the uspp framework as a special case . the main advantage using uspp is the well known reduction of the computational effort in the evaluation of the electronic wavefunctions at the dft level . this leads to a consequent reduction of computational load also in the calculation of mlwfs . we demonstrated that the extension to the uspp case does not introduce further approximations in the computation of the mlwfs with respect to the ncpp case . furthermore , the reformulation within the paw scheme allows us to interface the computation of mlwfs to other popular approaches for the electronic structure calculation . finally , our method is formulated in the case of a uniform @xmath4-point mesh for the sampling of the brillouin zone , generalizing previous attempts based on @xmath61-only calculations . this allows us to treat periodic solid - state systems ( such as crystals , surfaces and interfaces ) , which require a full description of the bz , as well as molecular , finite or amorphous systems which are well described with the @xmath61-only representation . as a first illustration of the capability of this methodology , we presented the calculation of the mlwfs for a few selected test cases , easily referable to well established theoretical and experimental results . for each selected system we also compared the results for both uspp and ncpp calculations , underlying a very good agreement between the two cases . the reduction of the computational cost resulting from uspp calculations opens the way to the exploitation of the mlwfs as a powerful tool to analyze the electronic structure of larger and more realistic nanoscale systems , in particular for transport in nano - junctions . we acknowledge discussions with giovanni bussi , marco buongiorno nardelli and elisa molinari for the treatment of uspp . funding was provided by the ec through tmr network `` exciting '' , by infm through `` commissione calcolo parallelo '' , by the italian miur through prin 2004 , and by the regional laboratory of em `` nanofaber '' . part of the figures has been realized using the xcrysden package . for sake of completeness we report here the main relations @xcite entering the expression ( [ eq : spread_functional ] ) for the spread functional @xmath126 $ ] and its first derivative wrt the unitary transformation @xmath127 [ eq . ( [ eq : wannier_definition ] ) ] . these are all the quantities involved in the minimization procedure for the calculation of maximally localized wannier functions . the expectation values of the position operator are : @xmath128 + \left [ \text{im } \ , \text{ln } m^{\mathbf{k},\mathbf{b}}_{nn } \right]^2 \},\end{aligned}\ ] ] where @xmath111 vectors connect nearest - neighbor @xmath4-points and @xmath129 are their weights according to appendix b of ref . the spread functional can be divided into three terms , the invariant ( i ) , the diagonal ( d ) and the off - diagonal ( od ) components : @xcite @xmath130 = \omega_i + \omega_d [ u ] + \omega_{od } [ u],\ ] ] their definitions are , respectively : @xmath131 we report the explicit expression for the calculation of the reciprocal space representation of the paw / uspp projectors and the fourier transform of the augmentation densities : @xmath133 these tasks are also required in the evaluation _ e.g. _ of the density in reciprocal space and are therefore performed in standard plane - waves dft codes . the index @xmath134 in @xmath135 and @xmath136 stand for radial and angular numbers , _ e.g. _ @xmath137 . projectors and augmentation densities are explicitly written as a product of a radial part times ( real ) spherical harmonics : @xmath138 first we focus on the expression for @xmath67 projectors . functions of the form @xmath139 have a known semi - analytical fourier transform which is given by : @xmath140 where @xmath141 is the spherical bessel function of order @xmath142 . the problem for @xmath67 projectors is therefore directly solved once we add the structure factors accounting for the acutal positions of the ion : @xmath143 moving to the case of the augmentation densities , we note that the product of two spherical harmonics can be expressed as a linear combination of single spherical harmonics using clebsch - gordan coefficients : @xmath144 this allows to follow the same strategy as before also for eq . ( [ eq : augmentation_densities_expanded ] ) . putting eqs . ( [ eq : clebsch - gordan]-[eq : radial_ft ] ) together , the final expression for @xmath145 reads : @xmath146 where @xmath147 indexes run as in eq . ( [ eq : clebsch - gordan ] ) . this fact comes first due to the presence of the sum over @xmath111-vectors ( giving a prefactor from 3 to 6 ) and secondly because the coupling of different @xmath4-points for pseudo overlaps makes wavefunction managing more cumbersome . o. ambacher , j. smart , j. r. shealy , n. g. weimann , k. chu , m. murphy , w. j. schaff , l. f. eastman , r. dimitrov , l. wittmer , m. stutzmann , w. rieger , and j. hilsenbeck , j. appl . phys . * 85 * , 3222 ( 1999 ) .	we report a theoretical scheme that enables the calculation of maximally localized wannier functions in the formalism of projector - augmented - waves ( paw ) which also includes the ultrasoft - pseudopotential ( uspp ) approach . we give a description of the basic underlying formalism and explicitly write all the required matrix elements from the common ingredients of the paw / uspp theory . we report an implementation of the method in a form suitable to accept the input electronic structure from uspp plane - wave dft simulations . we apply the method to the calculation of wannier functions , dipole moments and spontaneous polarizations in a range of test cases . comparison with norm - conserving pseudopotentials is reported as a benchmark .
heterogeneous wireless networks , comprising of small cell base stations ( sbss ) underlying the macrocellular network , is a promising solution to improve capacity , coverage and users quality - of - experience ( qoe ) @xcite . at the same time , the unplanned deployment of dense small cells further renders these problems very challenging @xcite @xcite . recently , content caching at the network edge ( i.e. , sbss , user devices etc . ) has been proposed as a cost - efficient solution to offload cellular / backhaul traffic , while satisfying users qoe and alleviating network congestion @xcite @xcite . although content caching and spectrum sharing received significant attention in recent years , most of the existing works consider spectrum allocation and caching separately , and little work has been done in exploring their joint benefits . in this respect , @xcite proposes a collaborative caching mechanism for wireless multimedia streaming exploiting storage and bandwidth auctions . however , the distribution of caches is considered fixed . meanwhile , @xcite studies the gains of caching in a stochastically distributed small cell scenario , where the effects of storage size and file popularity are investigated . furthermore , the work in @xcite characterizes the outage probability in small cell networks for uniform content distribution . however , the impact of spectrum has not been addressed in @xcite @xcite . + unlike @xcite , this letter characterizes the outage probability in serving user requests over a coverage area by jointly exploiting spectrum and caching . by considering the distribution of macro cellular base stations ( mbss ) ( with large cache ) and sbss ( with limited cache ) as poisson point processes ( ppps ) , we derive a closed form expression of the outage probability for a given content distribution . the outage probability is defined as the probability of not satisfying users requests over a given coverage area , as a function of signal - to - interference ratio ( sir ) , cache size and sbss density . we emphasize that the proposed model is pessimistic in the sense that it provides a lower bound on the overall network performance and focuses on the regime with large sbs density . for spectrum allocation , we define the spectrum access factor as a measure of spectrum accessed by the sbss . we also show the interplay of caching and spectrum sharing in such heterogeneous networks , drawing insights on the tradeoff between cache size and spectrum access factor , for different content popularity models . consider the downlink transmission of a two - tier cellular network comprising of sbss , underlying mbss in a two - dimensional euclidean plane @xmath0 as shown in fig . [ fig : net_diagram5 ] . hereafter , we will use the subscripts `` @xmath1 '' and `` @xmath2 '' to refer to the system variables associated to the small- and macro - base - stations , respectively . we assume that mbss are deployed according to a homogeneous ppp , @xmath3 with intensity @xmath4 ( mbss per square meter ) , where @xmath5 represents the spatial location of the @xmath6-th mbs ( in relation to the origin ) . similarly , sbss distribution is modeled as a homogeneous ppp , @xmath7 , with intensity @xmath8 ( sbss per square meter ) , such that @xmath9 and @xmath10 denotes the spatial location of the @xmath11-th sbs . we consider a spectrum configuration where the bandwidth @xmath12 is divided into @xmath13 equal sub - channels and each sbs randomly ( and independently ) accesses @xmath14 sub - channels , such that @xmath15 $ ] . hereafter , @xmath16 is referred to as spectrum access factor . the transmit power of mbs and sbs is represented by @xmath17 and @xmath18 , respectively where @xmath19 and @xmath20 is the maximum transmit power of mbs and sbs , respectively . + let each mbs be equipped with a large storage to cache contents from a given library @xmath21 , with size @xmath22 . all library contents are of equal size ( in bits ) . each sbs , in turn , is equipped with a cache storage of size @xmath23 and stores a subset of library contents . for simplicity , we express the cache size in terms of normalized cache size where @xmath24 . let @xmath25 represent the specific content(s ) cached at the @xmath11-th sbs . we assume that sbss cache contents following two different policies : a uniform caching policy ( ucp ) , where the sbss randomly caches contents regardless of their popularity , or popularity - based caching policy ( pcp ) , where the sbss hold the @xmath23 most popular contents . + we assume that a reference user a mobile user is located at the origin @xmath26 and analyze the system performance for the reference link assuming it requests a given content @xmath27 independently at each time slot . we analyze two different distribution of users requests : ( i ) uniform , where the contents are equally popular , and ( ii ) zipf where the contents popularity follows a power - law distribution ( * ? ? ? + [ fig : net_diagram5 ] we define a threshold distance @xmath28 and @xmath29 that specifies the maximum distance over which the mbs and sbs serves the user s content request respectively . the reference user is associated to the nearest sbs with the cached content within @xmath29 . if no sbs has cached the requested content , mbss may fulfill the user request . in this scenario , the user attempts to associate to the nearest mbs within @xmath28 such that @xmath30 . at each transmission , we assume a standard power loss propagation model with path loss exponent @xmath31 . the multi - path effects are modeled as rayleigh fading whose gains @xmath32 are exponentially distributed with mean @xmath33 . furthermore , we assume an interference - limited regime and therefore , neglect the effects of additive white gaussian noise ( awgn ) . then , the sir is computed as : @xmath34 where @xmath35 is the transmit power of @xmath36 , @xmath37 is the ( random ) distance of the reference link and @xmath38 is the interference experienced by the reference user . the interference @xmath38 at the reference user is the interference from all the mbss and sbss transmitting on the same frequency . as the interference from sbss is a thinned ppp , @xmath39 with density @xmath40 , then the interference experienced by the reference user served by an sbs and mbs respectively is given by : [ eq : interference ] & i_o , = _ x_i _ p _ h_o , x_i^- + _ y_j _ \{o } p _ h_o , y_j^- , & + & i_o , = _ x_i _ h_o , x_i^- + _ y_j _ p _ h_o , y_j^- . in this section , we derive a closed - form expression of the outage probability for serving user requests . the analysis considers the content distribution in a static network where a single network realization considers @xmath41 , @xmath42 and @xmath43 independently . [ ssec : cache_hit_probability ] in order to define the cache hit probability , we first introduce a content replication @xmath44 defined as the probability that content @xmath27 is cached ( replicated ) at the sbs , expressed as : @xmath45 where @xmath27 is ordered by the content popularity ( i.e. @xmath46 is the most popular content , @xmath47 the second most popular and so on ) . as mentioned earlier , sbss employ two replication strategies , namely ucp and pcp . for ucp , @xmath44 is the fraction of library contents stored by sbs ( i.e. , @xmath48 ) . for pcp , the sbss cache @xmath23 most popular contents . in this case , @xmath44 is given by : @xmath49 user requests , represented by @xmath50 , are modeled as uniform or zipf distribution . while in the former user randomly requests library contents , the latter considers different file popularity . for the zipf distribution , content requests are ranked from the most popular to the least such that the request for a content @xmath51 is : @xmath52 note that @xmath53 models the skewness of the popularity profile . for @xmath54 the popularity profile is uniform over files , and becomes more skewed as @xmath53 grows larger . is 0.8 and 1.2 for user - generated contents and video - on - demand contents respectively @xcite . ] + based on content replication , the cache hit probability is defined as the probability of existence of the content within @xmath29 or @xmath28 . therefore , for a given reference user , the existence of the closest sbs within @xmath29 is given by : @xmath55 for the mbss , we proceed in a similar way , but considering that they have all contents in their library . then , _ & = 1 - ( r _ ) + & = 1 - e^- _ ( r_)^2 . by considering the fact that the reference user always associates to the closest sbs having the requested content , the probability density function @xmath56 of the distance @xmath57 , given that the content @xmath27 exists within @xmath29 , is @xcite : @xmath58{0,0,1}{\beta p_c > 0}. \vspace{-0.1cm}\ ] ] for mbs , the probability density function @xmath59 is : @xmath60 let @xmath61 be the event that the requested content @xmath27 has been transmitted successfully given that @xmath27 has been cached by a sbs within @xmath29 . then , the outage probability of transmitting content @xmath27 over @xmath29 , denoted by @xmath62 is : @xmath63,\ ] ] where , for a given sir threshold @xmath64 , @xmath65 = \mathbb{p}(\mathrm{sir}_{o,\mathrm{sbs } } ( c)>\gamma)$ ] . note that , due to the association procedure , the sir implicitly depends on the content requested . + for the mbs , all the contents are cached and therefore the outage probability is the same for all @xmath51 , given that there exists at least one mbs within @xmath28 . in this case , @xmath66 = \mathbb{p}(\mathrm{sir}_{o,\mathrm{mbs}}>\gamma ) . \vspace{-0.1cm}\ ] ] the probability that transmission is successful , conditioned on the random variable @xmath67 , representing the distance between the reference user and the base - station associated with it , is : ( _ o > \| r ) & = = & + = & _ i_o & + = & _ i_o \| r = _ i_o \| r the laplace transform of the interference experienced by the reference user served by an sbs or mbs over the random distance @xmath68 and @xmath69 respectively , defined in , is given by : [ eq : laplace_transform ] _ i_o , ( ) = ( - r_1 ^ 2 ( _ k_1 + _ k_2 ) ) , & + _ i_o , ( ) = ( - r_2 ^ 2 ( _ k_3 + _ k_4 ) ) . & where @xmath70 and @xmath71 are given by : @xmath72 considering the conditional laplace transform ; [ eq - int-1 ] _ i_o , ( t ) & = _ i_o , ( e^-t i_o , ) , & + & _ _ , h_o , & + & _ _ , h_o , , & where @xmath73 and @xmath74 is the definition of laplace transform . by using the definition of probability generating functional of ppp , i.i.d of @xmath42 and @xmath41 and exponential interference distribution , the simplification yields @xcite . the laplace tranform of interference experienced by a reference user served by an mbs follows the same procedure as above . [ prop : poutsbs ] given the sir threshold @xmath64 , the outage probability of content @xmath27 requested by reference user over @xmath29 is : @xmath75 } { \big(k_1 \lambda_{\mathrm{mbs } } + ( k_2 + p_c b ) \beta \lambda_{\mathrm{sbs } } \big ) ( 1 - e^{-\beta b \lambda_{\mathrm{sbs } } p_c \pi r_{\mathrm{sbs}}^2})},\ ] ] the outage probability in serving the user request for @xmath27 over the distance @xmath67 , whose pdf is given in , is : _ , ( c ) & = _ r[(_o , ( c ) < ) \| r = r ] & + & = _ 0^r _ f_c,(r ) r. & using to solve the above integral , we find . the outage probability of content @xmath27 requested by the reference user over a threshold distance @xmath28 , is : @xmath76 } { \left ( \lambda_{\mathrm{mbs } } ( k_3 + 1 ) + \beta \lambda_{\mathrm{sbs } } k_4 \right ) \big(1 - e^{-\lambda_{\mathrm{mbs } } \pi r_{\mathrm{mbs}}^2 } \big)},\ ] ] this proposition s proof follows the same steps as before . [ coro_1 ] for a given @xmath44 , @xmath29 and @xmath28 , the outage probability in serving a user requesting content @xmath27 is given by : _ ( c ) = & _ ( c ) _ , ( c ) + ( 1 - _ ( c ) ) . theorem [ coro_1 ] imply that if the content has been cached by a sbs within @xmath29 , the outage probability of serving the user request is given by . otherwise , the mbs within @xmath28 serves the reference user with the desired content , given by . therefore , the mean outage probability is the linear combination of and . meanwhile , the average outage probability is given by : @xmath77 we present here a numerical analysis of the outage probability , validating our analytical results with monte - carlo simulations for a network grid of size @xmath78 . we generate @xmath79 requests from a library of size @xmath80 according to the request distribution described in sec . [ ssec : cache_hit_probability ] . for each request , the outage probability is evaluated over @xmath81 network realizations . in addition , we assume @xmath82 while the maximum transmit powers of mbs and sbs are set to 43dbm and 23dbm , respectively . [ fig : sbs_tradeoff ] shows the outage probability for various caching strategies as a function of sbs density by considering uniform and zipf content request distributions . it can be clearly seen that the analytical results match very well the simulation results , thus validating the analytical expression . without caching , increasing @xmath8 only increases interference , resulting in an increased outage probability . when sbss cache contents , the outage probability varies with increasing @xmath8 depending on the content distribution . for each of the content distribution , the outage probability first increases before levelling off . under zipf distribution , the outage probability maximizes for smaller @xmath8 as sbss only cache the most popular contents . under uniform distribution , the outage probability is maximized for a relatively larger @xmath8 , thus requiring more sbss to fulfill user requests . in addition , the cache hit probability given by ( 7 ) is an increasing function of @xmath8 . + [ fig : sbs_tradeoff ] fig . [ fig : storage_bwtradeoff ] shows the tradeoff between cache size and bandwidth for different content distributions . in case of ucp , the maximum outage probability is 0.2 for a large cache size . for a decreasing cache size , the outage probability increases under uniform distribution . however , for pcp , the outage probability increases much slower as sbss cache the most popular contents . meanwhile , for pcp , the maximum outage probability is 0.46 albeit utilizing the whole spectrum ( @xmath83 ) while uniform caching attains the maximum outage probability of 0.5 . therefore , caching smartly requires less amount of spectrum to achieve the same cache hit probability . hence , pcp makes use of the network resources more efficiently for a constrained cache size . furthermore , pcp is preferred for a fixed bandwidth as it achieves the minimum outage probability . finally , fig . [ fig : threshold_tradeoff ] shows the impact of sir threshold on outage probability , showing that pcp relaxes the sir threshold requirement for a given outage probability . furthermore , spectrum utilization without caching induces much higher outage probability than the joint cache - spectrum allocation . [ fig : threshold_tradeoff ] in this letter , we examined a cache - enabled small cell network underlying the macro cellular network , in which sbss strategically store contents . by considering the distribution of sbss to be a ppp , we analytically derived the outage probability of serving the requested content by jointly considering spectrum allocation and storage constraints . in addition , we characterized the number of sbss required to satisfy a given cache hit probability for different content distributions . this letter highlights the key tradeoff between cache size , node density and spectrum , underscoring the fact that larger cache size can be leveraged in conjunction with smaller spectrum access factor .	caching contents at the network edge is an efficient mean for offloading traffic , reducing latency and improving users quality - of - experience . in this letter , we focus on aspects of storage - bandwidth tradeoffs in which small cell base stations are distributed according to a homogeneous poisson point process and cache contents according to a given content popularity distribution , subject to storage constraints . we provide a closed - form expression of the cache - miss probability , defined as the probability of not satisfying users requests over a given coverage area , as a function of signal - to - interference ratio , cache size , base stations density and content popularity . in particular , it is shown that for a given minimum cache size , the popularity based caching strategy achieves lower outage probability for a given base station density compared to uniform caching . furthermore , we show that popularity based caching attains better performance in terms of cache - miss probability for the same amount of spectrum . caching , heterogeneous networks , latency
the rapidly oscillating ap ( roap ) stars are magnetic main sequence stars that pulsate in high radial overtone p modes with periods in the range of @xmath2min . they show broad - band photometric amplitudes less than 0.01mag , whereas rapid radial velocity variations in rare earth element lines can reach several kms@xmath1 ( e.g. , @xcite ; @xcite ) . the roap stars are important targets for the study of the interactions among chemical anomalies , magnetic field and pulsations . these stars show abnormal atmospheric structure with chemical stratification ( @xcite ; @xcite ) . the pulsations of these stars also make them promising objects for interior model testing using asteroseismology ( @xcite ) . the roap stars were discovered by @xcite ; at present more than 40 such stars are known . interesting and surprising discoveries have been made in recent years , which give new insight in the study of pulsating ap stars . new ground is being broken with @xmath3mag precision photometric data from the _ kepler _ mission . @xcite discovered a roap star that pulsates in both high overtone pmodes and a low- frequency gmode , opening up the possibility of better modelling of the interiors of these most peculiar stars . in 2007 we started a high resolution survey of cool chemically peculiar stars based mostly on the photometric catalogue of @xcite . one of the goals of the survey is to select stars with high peculiarity and strong magnetic field , and to determine their fundamental parameters to select the most promising candidates to be roap stars . nearly 400 stars have been observed and many of them are good roap candidates for further high time resolution spectroscopic and photometric observations . for several of these stars we have obtained such observations and here we present the discovery of pulsations for two objects , hd69013 and hd96237 . both stars were in the list of stars with magnetically split spectral lines found by @xcite . hd69013 is a typical cool ap star ; hd96237 is more impressive with significant spectral variability . the physical parameters @xmath4 and @xmath5 indicate that hd69013 and hd96237 are both main sequence stars , situated in the hr diagram where the instability strip crosses the main sequence . for many ap stars in this region of main sequence rapid oscillations have been detected , therefore both stars were promising objects for pulsation testing . high time resolution spectroscopic observations were carried out at the european southern observatory ( eso ) using the ultraviolet and visual echelle spectrograph ( uves ) installed at unit telescope 2 ( ut2 ) of the very large telescope ( vlt ) . for hd69013 , data were obtained during two high time resolution observing runs on 2008 january 17 and february 6 . for each run 34 spectra were obtained with exposure times of 80s and readout and overhead times of @xmath6s , corresponding to a time resolution of @xmath7101s . for hd96237 we obtained 34 spectra on 2008 march 15 with the same exposure and readout times . the wavelength region observed is @xmath8 , with a small gap in the region around 6000 caused by the space between the two ccds . the average spectral resolution is about @xmath9 . the ccd frames were processed using the uves pipeline to extract and merge the echelle orders to 1d spectra that were normalised to the continuum . we also obtained photometric observations of hd69013 and hd96237 in january , february and may 2010 . these observations were obtained at the south african astronomical observatory ( saao ) in johnson @xmath10 filter with the 1-m telescope and saao ccd and ste4 detector and with the modular photometer at the 0.5-m telescope . the reduction of photometric observations was done with eso - midas software and with software developed at saao . the lists of the observations are shown in tables1 and 2 . .a journal of observations of hd69013 . the columns give the julian date ( jd ) of the start of exposure , the observation time and the number of spectra or photometric measurements . [ cols="<,^,^ , < " , ] for roap stars lines of rare earth elements show higher pulsation amplitudes , while the lines of other chemical species , including light elements and iron peak elements , show much smaller pulsation amplitude , or show none at all ( see , e.g. , @xcite ; @xcite ) . this strange behaviour is explained by stratification where rare earth elements concentrate in the upper layers of the stellar atmosphere where oscillation amplitudes reach a maximum , while most other chemical elements tend to concentrate in deeper layers where the pulsation amplitude is lower . lines of iron peak elements in roap stars show very low pulsation amplitude or none at all ( @xcite , @xcite ) . to search for rapid radial velocity variability we performed cross - correlation of sections of the spectrum using eso - midas software . we also measured the central positions for profiles of individual spectral lines by the centre of gravity method . frequency analyses of radial velocity and photometric time series were performed using eso - midas s time series analysis and a discrete fourier transform programme by @xcite . cross correlation for the spectral band @xmath11 with an average spectrum taken as a template shows obvious rapid oscillations for hd69013 , as can be seen in fig.[69013:freq1 ] . for the two independent observing data sets that we obtained with the eso vlt telescope we find in the amplitude spectra highest peaks at @xmath12mhz and @xmath13mhz , correspondingly , with a full - width - half - maximum uncertainty of 0.07mhz . hence these two independent peaks are at the same frequency within the frequency error . the spectral lines of the rare earth elements also show pulsation with different amplitudes . the highest amplitude we detected was obtained for lines of priii , shown in fig.[69013:priii ] , while lines of ndiii reveal a smaller amplitude as shown in fig.[69013:ndiii ] . the other lines which belong to euii , ceii , laii also show pulsations with significant peaks in the amplitude spectra . the pulsation amplitude is smaller for the second observing run , which suggests either rotational modulation or multiperiodicity . while many ap stars show rotational light variations caused by abundance spots usually associated with their magnetic poles , there is no such evidence yet found for hd69013 @xcite . this constraint is weak and does not rule out rotational modulation pulsation amplitudes . further observations are needed to study this question . the photometric observations of this star obtained at saao also show rapid oscillations , as can be seen in fig.[69013:saao ] . previous photometric observations by @xcite and @xcite did not detect variations in this star . the amplitude spectra of our saao photometric observations are shown in fig.4 . we obtained data on two nights . the first data sets shows a signal at the same frequency as for the radial velocity data , but there is no significant peak in the second data set , as can be seen in fig.4 . the case for pulsation in hd96237 is not as strong as for hd69013 , but all of the evidence together gives confidence that pulsation has been detected and this star is a roap star . the five panels of fig.[96237:ampspec ] show amplitude spectra for spectroscopic analysis of hd96237 . clear pulsation peaks were obtained from cross correlation for the spectral region @xmath14 using an average spectrum as a reference template , and for the core of the h@xmath0 line . as mentioned above , the spectrum of hd96237 is very rich in rare earth element lines . we tried to measure pulsations for individual spectral lines which were identified by comparison with synthetic spectra calculated with the synth code of @xcite , but found that the pulsation amplitude is too low for clear detection in most individual lines , given the noise level in our spectra and relatively short observing run . in a few spectral lines of rare earth elements a peak corresponding the pulsation frequency detected by cross correlation is visible in the amplitude spectrum , as can be seen in the third panel of fig.[96237:ampspec ] for a single ceii line . the combination of several good spectral lines reduces the noise level and produces a more reliable picture . the two lower panels in fig.[96237:ampspec ] show amplitude spectra for combinations of five rare earth elements lines giving a clear peak . for twelve lines of fei and feii no pulsation is detectable , as is typical of the roap stars . as a further test we examined a combination of several telluric lines and found that there is no signal above a noise level of 12ms@xmath1 . we conclude that the case that hd96237 is a roap star is good . further studies particularly at the rotation phase where the pulsation amplitude is highest will give more confidence . photometric observations were obtained at saao for additional testing of pulsations in hd96237 . the frequency analysis of the photometric data listed in table 1 is presented in fig.[96237:phot ] . the upper panel of this figure supports the pulsation period found by spectroscopy , but other photometric observations shown in the bottom four panels and by @xcite do not detect pulsation in this star . this can be understood in terms of rotational modulation in an oblique pulsator , and/or beating of multiple frequencies . the pulsation amplitudes in many roap stars are modulated with the rotation period of the star . thus some of the photometric observations may not have been in best aspect . to judge this the rotational period and ephemeris of hd96237 needs to be determined . from the all sky automated survey ( asas ) and hipparcos photometry , @xcite found photometric variations with a period of 20.91d , which may be the rotational period . using two seasons of data from the wasp ( wide angle search for planets ) project ( @xcite ) covering the intervals 2007 january 4 to 2007 june 3 and 2008 january 5 to 2008 may 28 we obtained a similar period . combining the wasp ( passband from 400 to 700 nm ) and asas ( v - band ) photometry we find the following rotational ephemeris for the photometric maximum : @xmath15 the rotational phases calculated from this ephemeris are also shown in fig.[96237:phot ] . the bottom panel is for data observed in the same rotation period as the upper panel , but does not display any pulsations . the spectral observations presented at fig.[96237:ampspec ] were also obtained at a similar rotation phase using the above ephemeris . assuming our case for pulsation in this star to be good , we suggest two possible solutions : 1 ) the rotation period may be not 20.91d , but double that . this ambiguity happens for some peculiar stars ( see for example @xcite ) . longitudinal magnetic field measurements over the rotational period can resolve this ; 2 ) the star may be multiperiodic . there is some hint in fig.[96237:phot ] of a peak at 0.79mhz . additional observations are required to resolve these questions . the chemically peculiar magnetic star hd96237 is an interesting object which has a very peculiar spectrum with significant spectral variability . the star demonstrates large overabundances of rare earth elements . a high resolution spectrum obtained with the eso 2.2-m telescope and feros spectrograph resembles the spectrum of another highly peculiar star , hd101065 . abundances of ndii and ndiii determined from this spectrum are even higher than in hd101065 . other rare earth elements also show large overabundances similar to those found in hd101065 . another spectrum of the star obtained with vlt uves was significantly different from the feros spectrum with much less intense spectral lines of the rare earth elements ( @xcite ) . the similarity of the spectra and chemical abundances of hd96237 to those of hd101065 , which was the first detected rapidly oscillating ap star , increases the interest of this star . it is not known what determines the pulsation amplitude of the roap stars . while the principal periods of hd96237 and hd101065 are similar ( 13.6min and 12.1min , respectively ) , their photometric and spectroscopic radial velocity amplitudes are very different . our results for hd69013 demonstrate clearly that this is a new roap star with a low pulsation amplitude . this star shows possible rotation modulation and may be a useful target to study pulsation behaviour over rotation period . dwk and vge acknowledge support for this work from the science and technology facilities council ( stfc ) of the uk . this research has made use of nasa s astrophysics data system and simbad database , operated at cds , strasbourg , france . this paper uses observations made at the south african astronomical observatory ( saao ) .	we report the detection of short period variations in the stars hd69013 and hd96237 . these stars possess large overabundances of rare earth elements and global magnetic fields , thus belong to the class of chemically peculiar ap stars of the main sequence . pulsations were found from analysis of high time resolution spectra obtained with the eso very large telescope using a cross correlation method for wide spectral bands , from lines belonging to rare earth elements and from the h@xmath0 core . pulsation amplitudes reach more than 200ms@xmath1 for some lines in hd69013 with a period of 11.4min and about 100ms@xmath1 in hd96237 with periods near 13.6min . the pulsations have also been detected in photometric observations obtained at the south african astronomical observatory . stars : chemically peculiar stars : oscillations stars : magnetic .
over the past decade there has been enormous interest in reaction - diffusion systems ( see [ 112 ] and references therein ) , with particular emphasis on the effects of fluctuations in low spatial dimensions . most attention has been paid to reactions of the form @xmath0 and @xmath2 with a variety of different initial / boundary conditions . at or below an upper critical dimension @xmath5 , these systems exhibit fluctuation induced anomalous kinetics , and the straightforward application of traditional approaches , such as mean field rate equations , breaks down . attempts to understand the role played by fluctuations for @xmath6 have involved several techniques , including smoluchowski type approximations @xcite and field theoretic methods @xcite . in this paper we set out to study these fluctuation effects in a system with three competing reactions : @xmath7 the reactions are irreversible , and we choose homogeneous , though not necessarily equal , initial densities for the two species at @xmath8 . our goal is to calculate density decay exponents and amplitudes , taking into account fluctuation effects . in pursuit of this aim , we analyse the system using both the smoluchowski approximation and the field theory approach , and we show that the two methods are closely related . however , whereas it is unclear how the smoluchowski approach may be improved , the field theory provides a systematic way to obtain successively more accurate values for the asymptotic density decay exponents and amplitudes . we shall concentrate on situations where one of the two species is greatly in the majority ( as is almost always the case asymptotically ) - so , for example , if the a species is predominant , then we can safely neglect the reaction @xmath1 . this kind of assumption will lead to a considerable simplification in our analysis . previous work on this problem includes use of the smoluchowski approximation @xcite , as well as exact @xmath9 results obtained by derrida _ @xcite for the special case of _ immobile _ minority particles . derrida _ were , in fact , studying a different problem , namely the probability that a given spin has never flipped in the zero temperature glauber dynamics of the q - state potts model in one dimension . by solving that model exactly @xcite they showed that this probability decreased as a power law : @xmath10 for the ising ( @xmath11 ) case . however , in one dimension , the ising spin flip problem and the decay rate for the immobile impurity in our reaction - diffusion system are exactly equivalent problems , and hence this exact decay rate also holds in our case . we also mention one other previous result for the immobile impurity problem , due to cardy @xcite . using renormalisation group methods similar to those employed in this paper , it was shown that the density of the minority species decays away as a universal power law : @xmath12 for @xmath4 , where @xmath13 and @xmath14 . the case where the _ majority _ species is immobile has also been solved ( see @xcite ) . in this case the decay rate for the minority species is dominated by minority impurity particles existing in regions where there happen to be very few of the majority particles . since these majority particles are strictly stationary , this situation is not describable using a rate equation approach , and it turns out that the minority species decays away as @xmath15 , a result which is not accessible by perturbative methods . in this paper , using a field theory formalism and techniques from the renormalisation group , we will obtain decay rates and amplitudes for the general case of arbitrary diffusivities - a regime previously only accessible using the smoluchowski approximation . our basic plan is to map the microscopic dynamics , as described by a master equation , onto a quantum field theory . this theory is then renormalised ( for @xmath3 ) , and the couplings ( reaction rates ) are shown to have @xmath16 fixed points , whose values depend only on the ratio of the species diffusion constants . note that this system ( with irreversible reactions ) is particularly simple in that only the couplings ( and not the diffusivities ) are renormalised . the next step is to group together feynman diagrams which are of the same order in the renormalised couplings - i.e. diagrams with the same number of loops . these diagrams are then evaluated and a callan - symanzik equation used to obtain improved asymptotic @xmath17 expansions for the densities . in this fashion , quantities of interest may be systematically calculated by successively including higher order sets of diagrams ( with more loops ) in the perturbative sum . one consequence of the theory is that the asymptotic decay rates and amplitudes for @xmath18 will be independent of the reaction rates - a result which is in accordance with the smoluchowski approach . in fact , all physical quantities below the upper critical dimension asymptotically depend only on the diffusivities and the initial densities , and in this sense they display universality . we now present a summary of our results for the density decay rates . in what follows we define @xmath19 , @xmath20 to be the initial density of a , b particles , and @xmath21 to be the ratio of the diffusion constants . for @xmath22 , @xmath23 and @xmath24 ( where @xmath25 is a crossover time derived in section 4 ) , we have ( as in @xcite ) : @xmath26 for the minority species , we find , from the rg improved tree level approximation in the field theory : @xmath27 where @xmath28 these decay exponents are identical with the smoluchowski results . performing a strict @xmath17 expansion on this rg improved tree level result gives an exponent @xmath29 for the immobile impurity case ( @xmath30 ) . this is in agreement with previous rg calculations by cardy @xcite . if we now go beyond the tree level calculation by including one loop diagrams , then we obtain an improved value for the exponent @xmath31 using an @xmath17 expansion : @xmath32 \right.\right.\ ] ] @xmath33\right)\right]\right ) + o(\epsilon^2),\ ] ] where @xmath34 is the dilogarithmic function @xcite . this exponent is found to be in good agreement with simulations @xcite and exact results @xcite in @xmath35 . however , for @xmath36 , the system crosses over to a second regime where @xmath37 . this situation is similar to the case where we begin with @xmath38 . in that regime , at times @xmath39 , and for @xmath38 , @xmath40 and @xmath4 , we have : @xmath41 for the majority species . using the rg improved tree level result for the minority species , we obtain : @xmath42 with @xmath43 the exponent is again in agreement with the smoluchowski result . if we attempt to improve this calculation to one loop accuracy , then we obtain : @xmath44 \right.\right.\ ] ] @xmath45\right)\right]\right ) + o(\epsilon^2).\ ] ] this exponent is only valid for @xmath46 quite close to unity , and even in this region it may be less accurate than the ( non @xmath17-expanded ) rg improved tree level result given above . this point will be discussed further in section 4.2 . we next give results valid for @xmath47 , where we find extra logarithmic factors multiplying the power law decay rates . treating first the case @xmath48 and @xmath49 , we have , from the rg improved tree level , an initial regime with : @xmath50 @xmath51 however , for @xmath36 , the system again crosses over to a second regime where @xmath52 . in this second regime the density decay exponents ( though not the amplitudes ) are the same as for the case where we begin with @xmath38 . in that case we have , for @xmath40 : @xmath53 @xmath54 crossover times for these cases are given in section 4.3 . we now give a brief description of the layout of this paper . in the next section we analyse the system using the mean field / smoluchowski approach . we then set up the necessary formalism for our field theory in section 3 , and use it to perturbatively calculate values for the density exponents and amplitudes in section 4 . finally , we give some conclusions and prospects for future work in section 5 . the simplest description of a reaction - diffusion process is provided by the mean field rate equations . for the system we are considering with densities @xmath55 and @xmath56 , they take the form : @xmath57 where @xmath58 , @xmath59 , and @xmath60 are the reaction rates , and where we impose initial conditions of the form @xmath61 and @xmath62 . in this approach we have completely neglected the effects of fluctuations - in other words we have made assumptions of the form @xmath63 etc . , where the angular brackets denote averages over the noise . below the critical dimension , where fluctuations become relevant , this sort of approximation will break down . nevertheless , even at the mean field level , the complete solution set for these rate equations is quite complicated . in what follows we shall restrict our analysis to the case where @xmath64 . the solution for this particular parameter set will be required for our later field theoretic analysis . following @xcite , it is easy to show ( by forming a rate equation for the concentration ratio ) that @xmath65 as @xmath66 . thus if we begin with initial conditions where @xmath23 , we can identify two distinct regimes - an early time regime where @xmath67 and , after a crossover , a late time ( true asymptotic ) regime where @xmath68 . treating the early time regime first , we find ( after some algebra ) : @xmath69 note that the a particles are decaying away more quickly than the b s , so eventually we crossover to a second regime : @xmath70 alternatively , if we begin with @xmath38 , then we have a single asymptotic regime : @xmath71 however , if we now wish to extend our results at or below the upper critical dimension , we must attempt to include some of the fluctuation effects . the simplest way in which this can be done is to employ the smoluchowski approximation @xcite . the essential idea of this approach is to relate the effective reaction rates @xmath72 to the diffusion constants @xmath73 . suppose we want to calculate the reaction rate @xmath74 . we begin by choosing a ( fixed ) a species target `` trap '' , which is surrounded by b particles . when a b particle approaches within a distance @xmath75 of the target , a reaction is deemed to have occurred . consequently , the reaction rate may be obtained by solving a diffusion equation with boundary conditions of fixed density as @xmath76 , and absorption at @xmath77 . the flux of b particles across the @xmath78 dimensional sphere of radius @xmath75 is then proportional to an effective microscopic reaction rate . if we now generalise to the case where both the a and b species are mobile , then we find ( in dimension @xmath4 and in the large time limit ) : @xmath79 for @xmath47 we obtain logarithmic corrections : @xmath80 the smoluchowski reaction rates for @xmath81 and @xmath82 are obtained in a similar fashion . note that above @xmath47 the reaction rate approaches a limiting ( constant ) value , and we see that the smoluchowski approach predicts a critical dimension of @xmath83 for this system . this is simply related to the reentrancy property of random walks in @xmath3 . it is the inclusion of this effect which accounts for the improvement introduced by the smoluchowski approach . if we now substitute these modified reaction rates into the rate equations , we can obtain the smoluchowski improved density exponents . for the case where @xmath84 , we find an initial regime with @xmath85 once again , since the a particles are decaying away faster than the b s , we cross over to a second regime , where ( for @xmath86 ) @xmath87 this second set of exponents is the same as for the case where we begin with @xmath38 and @xmath40 . in this situation no crossover occurs and the exponents are valid for all asymptotic times . these exponents can be compared favourably with both simulations @xcite , and exact results @xcite . for example , the decay rate for an immobile minority impurity is given by smoluchowski to be @xmath88 . this compares well with the exact decay rate of @xmath89 . turning to the case @xmath90 and @xmath23 , we obtain , for the initial regime : @xmath91 we again eventually crossover to a second regime , where ( for @xmath86 ) : @xmath92 this second set of exponents is again valid ( for all asymptotic times ) in the case where we begin with @xmath38 and @xmath40 . note that the smoluchowski approach can also be employed for @xmath93 , where again we will find ( time independent ) reaction rates which depend on the diffusion constants . however , in the general case , our later field theoretic analysis shows that there is no real justification for this procedure . one exception to this occurs in the case where we have heterogeneous single species annihilation , as considered in @xcite . in this situation we have only one fundamental reaction process , but different reaction rates may still arise , for example , by having two or more different particle masses ( and hence two or more different diffusion constants ) . in this case it is physically reasonable to suppose that the exponents ( which are ratios of reaction rates ) may depend only on the diffusivity ratios , with any other parameters canceling out . however , in the general case , where the reaction processes are genuinely distinct this will not be the case . overall , we have seen that the smoluchowski approach is a simple way to incorporate some fluctuation effects into the rate equation approach . unfortunately , it is not at all clear how these methods may be systematically improved . it is for this reason that we turn to the main purpose of this paper - the development of an alternative field theoretic framework . fluctuation effects in reaction - diffusion systems have previously been successfully tackled using techniques borrowed from quantum field theory and also from the renormalisation group . examples include studies of the diffusion limited reactions @xmath0 @xcite and @xmath2 @xcite . the first step in this analysis is to write down a master equation , which exactly describes the microscopic time evolution of the system . using methods developed by doi @xcite and peliti @xcite , this can be mapped onto a schroedinger - like equation , with the introduction of a second quantised hamiltonian , and then onto a field theory , with an action @xmath94 . these steps have been described in detail elsewhere @xcite , and consequently we shall simply give the resulting action appropriate for our theory : @xmath95 -\bar an_a -\bar bn_b\right ) . \nonumber\end{aligned}\ ] ] here we have defined @xmath21 and also introduced the response fields @xmath96 and @xmath97 . in addition time @xmath98 , together with the reaction rates @xmath99 have been rescaled to absorb the diffusion constant @xmath100 . averaged quantities are then calculated according to @xmath101 where @xmath102 notice that in the path integral @xmath103 integration over the fields @xmath96 , @xmath55 and @xmath97 , @xmath56 whilst neglecting the quartic terms , leads to a recovery of the mean field rate equations . performing power counting on the action @xmath94 , we can now give the natural canonical dimensions for the various parameters appearing in the action : @xmath104\sim k^{-2}\qquad [ a],[b],[n_a],[n_b]\sim k^d \qquad [ \bar a],[\bar b]\sim k^0 \qquad [ \lambda_{\{ij\}}]\sim k^{2-d}.\ ] ] notice that the reaction rates become dimensionless in @xmath47 , which we therefore postulate as the upper critical dimension for the system , in agreement with the smoluchowski prediction . from the action @xmath94 , we can see that the propagators for the theory are given by @xmath105 diagrammatically , we represent @xmath106 by a thin solid line and @xmath107 by a thin dotted line . the vertices for the theory are given in figure 1 . one of the most important features of this theory , as mentioned in the introduction , is the relative simplicity of its renormalisation . examination of the vertices given in figure 1 reveals that it is not possible to draw diagrams which dress the propagators . hence the bare propagators are the full propagators for the theory . consequently , the only renormalisation needed involves the reaction rates @xmath99 , and in particular the diffusion constants ( or @xmath46 ) are _ not _ renormalised . the temporally extended vertex functions for the reaction rates are given by the diagrammatic sums given in figure 2 . as is the case in similar theories @xcite , these sums may be evaluated exactly , using laplace transforms : @xmath108 where @xmath109 and @xmath110 is the laplace transformed time variable . we can now use these vertex functions to define the three dimensionless renormalised and bare couplings , with @xmath111 , @xmath112 as the normalisation point : @xmath113 consequently , we can define three @xmath31 functions : @xmath114 and three fixed points @xmath115 : @xmath116 finally , we see from ( [ tevf1 ] ) , ( [ tevf2 ] ) , and ( [ tevf3 ] ) that the expansion of @xmath117 in powers of @xmath118 is given by : @xmath119 we now exploit the fact that physical quantities calculated using the field theory must be independent of the choice of normalisation point . this leads us to a callan - symanzik equation : @xmath120\langle a\rangle_r=0.\ ] ] however dimensional analysis implies @xmath121\langle a\rangle_r(t , n_a , n_b , g_{r_{\{ij\}}},\delta,\kappa)=0.\ ] ] exactly similar equations hold for @xmath122 . eliminating the terms involving @xmath123 and solving by the method of characteristics , we find : @xmath124 with the characteristic equations : @xmath125 and initial conditions : @xmath126 @xmath127 these equations have the exact solutions : @xmath128 and @xmath129 in the large @xmath98 limit @xmath130 , a relationship which will allow us to relate an expansion in powers of the renormalised couplings @xmath118 to an @xmath17 expansion using ( [ css ] ) . in our later density calculations we will assume that this asymptotic regime has been reached . in order to perform systematic @xmath17 expansion calculations we now need to identify the leading and subleading terms in an expansion in powers of @xmath117 . in calculating @xmath131 and @xmath132 , contributions from tree diagrams are of order @xmath133 , for integer @xmath134 , and densities @xmath135 . however , diagrams with @xmath136 loops will be of order @xmath137 . the addition of loops makes the power @xmath117 higher relative to the power of the densities - so we conclude that the number of loops gives the order of the diagram . the lowest order diagrams contributing to @xmath138 and @xmath132 are the tree diagrams shown in figure 3 . we represent the classical ( tree level ) density @xmath139 by a wavy solid line , and @xmath140 by a wavy dotted line . these sets of diagrams are equivalent to the mean field rate equations , as may be seen by acting on each by their respective inverse green functions . the second tree level quantities appearing in the theory are the response functions : @xmath141 which we represent diagrammatically by the thick lines shown in figure 4 . these functions can be evaluated analytically , but only in the limit @xmath48 , or @xmath142 . the details of this calculation are presented in appendix a , where the following results are derived ( for @xmath48 ) : @xmath143 an extra check on validity of these response functions is provided by the relations : @xmath144 which follow from the definition of the response functions , and from the initial condition terms in the action @xmath94 . it is easy to check that the above response functions do indeed satisfy these relations . for the opposite situation where @xmath38 ( and hence @xmath145 ) , we could use a formalism similar to the above for the density calculations . however , it is much simpler to map this case onto the @xmath146 regime by swapping the labels on the a and b particles , and then relabeling : @xmath147 we can then obtain the exponents and amplitudes for this second regime with no extra work . this concludes our discussion of the field theory formalism . the framework we have built up allows ( in principle ) the systematic calculation of fluctuation effects in all circumstances . however , it is only in the case where one of the species is greatly in the majority where the equations ( for the tree level densities and response functions ) are sufficiently simple for analytic progress to be made . we now turn to use of the field theory in calculating the fluctuation modified densities . the first step in using our field theory to include fluctuation effects is to insert the mean field ( tree level ) solution into the callan - symanzik solution ( [ css ] ) , using the results for the running densities / couplings ( [ rd ] ) , ( [ rc ] ) . since the fixed points for the couplings obey @xmath148 ( when @xmath36 ) it is appropriate to use the mean field solutions derived in section 2 . for the case where @xmath23 , this gives : @xmath149 and @xmath150 with @xmath151 valid for @xmath24 , where @xmath152 these modified crossover times are obtained by using the expressions for the running couplings / densities in the mean field crossovers . notice that the density decay exponents derived here are the same as those obtained from the smoluchowski approach . however , as we are performing an @xmath17 expansion , we are only strictly justified in retaining leading order @xmath17 terms . consequently we find , for the minority species density decay exponent and amplitude : @xmath153 eventually , however , as the a particles are decaying away more quickly than the b particles ( due to their greater diffusivity when @xmath36 ) , we crossover to a second regime where @xmath52 . for @xmath86 , we have : @xmath154 @xmath42 with @xmath155 where @xmath156n_a\over [ \delta^{d/2}-((1+\delta)/2)^{d/2}]n_b}\right)^{-1-\left({1 + \delta^{-1}\over 2}\right)^{d/2}\left({((1+\delta)/2)^{d/2}- \delta^{d/2}\over ( ( 1+\delta)/2)^{d/2}-1}\right)}.\ ] ] this result is valid for @xmath157 , where @xmath158 note that for @xmath159 the first crossover time @xmath160 - in this case the two species decay away at the same rate , and so no further crossover occurs . alternatively if @xmath30 , then the first regime is left , but the second crossover time @xmath161 . in that case the minority species finally decays away in the exponential fashion predicted in @xcite . for the intermediate case where @xmath46 is small , but nonzero , the decay exponent for the minority species becomes large in the final regime . the explanation for this result lies in the relatively large diffusivity of the minority a species ( if @xmath100 is large ) and/or the increased density amplitude for the majority b particles ( if @xmath162 is small ) . both these effects will lead to an increased rate of decay for the a species . finally , if the initial conditions are changed such that now @xmath163 , with @xmath40 , then we obtain the same results as for the second of the above regimes for @xmath164 , with @xmath165 . we now describe the one loop improvements to the tree level result . in the regime @xmath166 , the dominant diagrams will be those where the minimum possible number of @xmath140 insertions are made . for the majority a species the appropriate diagram is shown in figure 5 , where there are no @xmath140 insertions . this is identical to the one loop diagram for @xmath0 evaluated in @xcite , which gives , in conjunction with the subleading terms from the tree level : @xmath167 in addition , for that subset of diagrams with no @xmath168 insertions , the decay exponent is exact . more details of this calculation , including a demonstration of the cancellation of divergences , can be found in @xcite . turning now to the one loop calculation for the minority species , the appropriate diagrams are the three shown in figure 6 , each of which contains just one @xmath140 insertion : @xmath169 } \label{la1}\\ ( ii ) & & \qquad { -2\lambda_{ab}^2\lambda_{aa}n_a^2n_b\over ( 2\lambda_{aa}n_at)^{\lambda_{ab}/2\lambda_{aa}}}\int{d^dk\over ( 2\pi)^d } \int_0^t dt_2\int_0^{t_2 } dt_1\int_{t_1}^{t_2 } dt'{(1 + 2\lambda_{aa}n_at_1)^2\over ( 1 + 2\lambda_{aa}n_at_2)^2 } \nonumber \\ & & \qquad\qquad\times{1\over ( 1 + 2\lambda_{aa}n_at')^2 } \exp{[-k^2(t_2(1+\delta)-2t_1+(1-\delta)t ' ) ] } \label{la2 } \\ ( iii ) & & \qquad { \lambda_{ab}^2n_an_b\over ( 2\lambda_{aa}n_at)^{\lambda_{ab}/2\lambda_{aa}}}\int{d^dk\over ( 2\pi)^d}\int_0^t dt_2\int_0^{t_2 } dt_1 { ( 1 + 2\lambda_{aa}n_at_1)\over ( 1 + 2\lambda_{aa}n_at_2)^2 } \nonumber \\ & & \qquad\qquad\qquad\qquad \qquad\qquad\qquad \times\exp{[-k^2(1+\delta)(t_2-t_1)]}. \label{la3}\end{aligned}\ ] ] the detail of the evaluation of these diagrams is rather subtle . essentially we are interested in extracting the most divergent parts of these integrals , which will turn out to be pieces of @xmath170 and @xmath171 . however , we must be careful not to confuse genuine bare divergences ( of @xmath170 which must be removed by the renormalisation of the theory ) , with logarithmic pieces , which we must retain . the divergences arise in diagrams ( i ) and ( iii ) as the difference in time @xmath172 between the beginning and end of the loops tends to zero ( in @xmath47 ) . after the process of renormalisation we find corrections of the form : @xmath173 if this series is identified as the expansion of an exponential , then we find that our one loop diagrams ( together with subleading components from the tree level ) have provided @xmath16 corrections to the exponents . diagrams ( i ) and ( iii ) are relatively straightforward to evaluate . the @xmath174 and @xmath175 integrals are elementary , and the final @xmath176 integrals can be done by parts to extract the necessary most divergent pieces ( up to @xmath171 ) . the second diagram of figure 6 is more complicated , and we perform its evaluation in appendix b - although we are only able to extract the logarithmic piece of @xmath177 . there will be corrections to this of @xmath178 ( contributing to a modified amplitude ) which we have been unable to calculate . we find asymptotically : @xmath179 @xmath180 @xmath181 @xmath182+o(\epsilon)\right ) t^{\epsilon/2}\ln(2\lambda_{aa}n_at)\ ] ] @xmath183 @xmath184 to one loop accuracy we can make the replacement : @xmath185 . these results must now be combined with the subleading terms from the tree level . using ( [ gexp ] ) , we find @xmath186 if we now insert explicit @xmath17 expanded values for the fixed points @xmath187 , then we discover that the bare divergences cancel between ( [ q1 ] ) , ( [ q2 ] ) , and ( [ q3 ] ) . with insertion into the callan - symanzik solution ( [ css ] ) , we also find that the pieces we have left as integrals in ( i ) and ( iii ) ( which are @xmath188 ) also mutually cancel . eventually we find : @xmath189 @xmath190\ln(({\rm const.})t^{d/2})+o(\epsilon^2)\right ) , \label{bfirst}\ ] ] where we have neglected @xmath16 pieces which , aside from the prefactor , are time _ independent_. these terms contribute only to the density amplitude . we now evaluate the integral in ( [ bfirst ] ) , using @xmath191 @xmath192 where @xmath193 is the dilogarithm function @xcite . the other parts of the integral are elementary . the next step is to @xmath17 expand the rg improved tree level result : @xmath194 then , exponentiating the @xmath17 expansion in ( [ bfirst ] ) , we find @xmath195 , where @xmath196 \right.\right . \nonumber \\ & & \left.\left.\qquad\qquad -{1\over 4}(\delta^2 - 1)\left(1+(1+\delta)\left[f\left\{{2\over 1+\delta}\right\}-{\pi^2\over 6}\right]\right)\right]\right ) + o(\epsilon^2 ) . \label{bsecond}\end{aligned}\ ] ] @xmath31 is plotted as a function of @xmath46 for @xmath197 ( @xmath35 ) in figure 9 . for the case where @xmath159 , we recover the decay rate @xmath198 . this is to be expected , as when @xmath159 we are effectively again dealing with a single species reaction - diffusion system ( at least for @xmath4 ) . in that case the density decay exponent is known to all orders in perturbation theory @xcite , and is in agreement with our result . for the case where @xmath30 and @xmath35 , the decay exponent is also known exactly to be @xmath199 @xcite . this can be compared with our result , where we find @xmath200 consequently , this answer is a modest improvement over the smoluchowski result derived in section 2 , and also in @xcite . for the case @xmath38 ( and hence @xmath201 ) , we could follow the same route as described above , by evaluating the one loop diagrams shown in figures 7 and 8 . however , as we mentioned in the last section we can much more easily obtain these corrections by swapping the labels on the a and b particles , and then relabeling : @xmath202 following this procedure , the majority species amplitude / exponent can be found by taking @xmath203 in equation ( [ maxexp ] ) : @xmath204 we can obtain the one loop minority species exponent by substituting @xmath205 in equation ( [ bsecond ] ) : @xmath206 where @xmath44 \right.\right.\ ] ] @xmath45\right)\right]\right ) + o(\epsilon^2 ) . \label{olalph}\ ] ] notice , however , that in forming the one loop corrections for the minority species exponent , we have had to expand the rg improved tree level result : @xmath207 the error arising from this expansion will become large as @xmath46 becomes small . eventually this inaccuracy will cause the exponent to reach a maximum and then _ decrease _ as @xmath46 is further reduced - behaviour which is clearly unphysical . in order to reduce the error , and to ensure that the expansion in equation ( [ treexp1 ] ) is qualitatively correct , we need to retain the @xmath208 terms . hence the one loop exponent in equation ( [ olalph ] ) should be treated with some caution - terms of order @xmath208 will probably be required for precise results . consequently the ( non @xmath17 expanded ) rg improved tree level result given in the last section may be more accurate in this regime . in figure 10 we have plotted the one loop exponent @xmath209 as a function of @xmath46 , for @xmath35 ( @xmath197 ) , in the region @xmath210 , where the exponent is still _ increasing _ for decreasing @xmath46 . in principle , calculations can also be made for the case with @xmath23 , but where we have crossed over to the regime @xmath52 ( for @xmath86 and times @xmath157 ) . however , a rigorous evaluation of the one loop diagrams is now much more difficult , as the functional forms for the densities and response functions will change over time . nevertheless , since the above corrections to the exponents come from asymptotic logarithmic terms , it is plausible to suppose that the new exponent corrections will be dominated by contributions from the final asymptotic regime . if this is indeed the case , then the one loop exponents ( though _ not _ the amplitudes ) will be unchanged from the previous results ( equations ( [ ola ] ) to ( [ olalph ] ) ) . this calculation will , however , suffer from the same problem as described above . for the case @xmath90 we expect logarithmic corrections to the decay exponents , as the reaction rates @xmath99 are marginal parameters at the critical dimension . we can find the running couplings from the characteristic equation ( [ gchar ] ) by taking the limit @xmath212 in equations ( [ b1 ] ) , ( [ b2 ] ) , and ( [ b3 ] ) : @xmath213 @xmath214 @xmath215 where we have taken the asymptotic limits . corrections to the asymptotic running couplings will be an order @xmath216 smaller , and consequently these asymptotic expressions will only be correct at very large times . hence our expressions for the densities will only be valid when both this condition , and the crossover time constraints given below , are satisfied . in what follows we shall assume the validity of the first of these two conditions . notice that the asymptotic running couplings are still ordered @xmath217 for @xmath36 , so we can use the mean field solutions derived in section 2 as the basis for the rg improved tree level exponents and amplitudes . making use of the callan - symanzik solution ( [ css ] ) and the above running couplings , we find for @xmath218 : @xmath219 where @xmath220 is a non - universal amplitude correction . note that the next order terms for the minority species are suppressed by a factor of only @xmath221 . using our expressions for the running couplings / densities in the mean field crossovers , we find that these expressions are valid for times @xmath222 , where @xmath223 for the case @xmath36 the system will eventually enter a second regime , where now the b species will be in the majority . we have ( for @xmath40 ) : @xmath224 with @xmath225 this is valid for times when @xmath226 , where @xmath227 alternatively , if we begin with @xmath38 , then for @xmath40 and @xmath228 , we have the same results as for the second of the above cases , with @xmath229 . interestingly , the logarithmic corrections we have derived in this section using the rg approach differ slightly from the smoluchowski results given in section 2 . in this paper we have made a comparison of two methods for treating fluctuation effects in a reaction - diffusion system . we have found that the smoluchowski and field theory approaches are rather similar - the smoluchowski approximation , for @xmath4 , giving the same exponents as the renormalisation group improved tree level in the field theory . in addition , we have gone on to calculate the field theoretic one loop corrections , which have yielded improved values for the exponents . the advantage of the field theory is that it provides a systematic way to calculate these corrections - a procedure which is lacking in the smoluchowski approach . furthermore the use of renormalisation group techniques has demonstrated universality in the asymptotic amplitudes and exponents , in that , for @xmath4 , they only depend on the diffusivities and the initial densities , and not on the reaction rates . the theory we have developed in this paper can easily be extended to slightly different situations . consider first an annihilation / coagulation reaction - diffusion system , where the following reactions occur : @xmath230 the smoluchowski approach differs from before only in the absence of factors of @xmath231 in the rate equation terms describing the same species reactions . consequently , if we begin with @xmath23 then the minority species will decay as @xmath232 on the other hand , the field theory description lacks only the factors of @xmath231 in the action ( [ action ] ) . if this difference is followed through then the decay exponent in the rg improved tree level is seen to be the same as in the smoluchowski approach . however , this difference of a factor of @xmath231 has a major effect on the response functions ( where this factor appears as a power ) , and as a result the new one loop corrections will be different from those calculated in section 4.2 . these results should be compared with the exact solution @xcite for the minority species decay rate @xmath233 , where : @xmath234 note that in this case , although the smoluchowski answer is qualitatively correct , it deviates considerably from the exact answer . hence we can see that application of the smoluchowski approach does not always lead to accurate exponents . another possible extension is to consider reaction - diffusion systems with more than two species of particle . for example , examining a three species system , we could have the reactions : @xmath235 @xmath236 analysis of this situation is very similar to before , and we merely remark that in the appropriate asymptotic regimes the smoluchowski and rg improved tree level exponents ( consisting of ratios of diffusion constants ) are once again identical . hence the convergence between the smoluchowski exponents and those obtained from the rg improved tree level is fairly robust , and is not simply confined to the two species systems we have previously been considering . a further possibility is to analyse the case where we have a continuous distribution of diffusivities , but with only a _ single _ reaction channel . this has been studied from the smoluchowski point of view by krapivsky _ @xcite , and it would be interesting to extend our rg methods to include this situation . our theory could also be employed to consider clustered immobile reactants - a generalisation of the @xmath30 case included in our calculations . this situation has been analysed by ben - naim @xcite , using the smoluchowski approach , where the dimension of the cluster @xmath237 was found to substantially affect the kinetics . specifically , for codimensionality @xmath238 ( in a space of dimension @xmath78 ) a finite fraction of the impurities was found to survive , whereas for @xmath239 the clusters decayed away indefinitely . the formalism we have presented in this paper could be adapted to study this clustered impurity problem , where calculations could be made without reliance on the smoluchowski approach . * acknowledgments . * the author thanks john cardy for suggesting this problem and for many useful discussions . financial support from the epsrc is also acknowledged . obtaining an exact analytic expression for the response functions is , in general , very hard . suppose we define the `` trunk '' to be the line of propagators onto which the density lines are attached , as shown at the bottom of figure 4 . difficulties arise from diagrams where the `` trunk '' changes from one propagator into the other , and then back again , as shown in last of the diagrams for the @xmath240 response function in figure 4 . if diagrams of this type are initially excluded then progress can be made . consider first the two subseries shown in figure 11 , for the functions @xmath241 and @xmath242 , where diagrams of the above kind have been excluded . these series can be summed exactly ( using the same technique as described in @xcite ) , giving : @xmath243 @xmath244 the full response functions are now given by the diagrammatic equations shown in figure 12 , where all possible diagrams are included . written out explicitly these give : @xmath245 in general this set of coupled integral equations is intractable - however we can make progress in the limit where @xmath246 , or @xmath247 . considering the case where @xmath246 , the dominant contributions to the response functions come from diagrams with the minimum possible number of @xmath140 density line insertions . accordingly , we can now truncate the full diagrammatic equations , as shown in figure 13 . notice that to this order @xmath240 , @xmath248 , and @xmath249 contain no @xmath140 density insertions , whereas @xmath250 must contain one such insertion . in this approximation we can now perform the integrals inside the @xmath251 and @xmath252 functions , using the appropriate mean field density : @xmath253 @xmath254 and therefore @xmath255 @xmath256 using these expressions , it is now straightforward to derive the response functions given in equations ( [ r1l ] ) , ( [ r1n ] ) , ( [ r1p ] ) , and ( [ r1 m ] ) . for the case where @xmath166 the hardest of the three diagrams of figure 6 to evaluate is ( ii ) - see equation ( [ la2 ] ) . we shall evaluate it first in @xmath47 , and then deduce its form in @xmath257 . notice that the extra integration resulting from the @xmath140 insertion in the loop ensures that this diagram is not divergent . taking the asymptotic part of the @xmath175 and @xmath258 pieces , we find : @xmath259 @xmath260 the @xmath174 and @xmath258 integrals are elementary , giving @xmath261 @xmath262+{1-\delta\over ( t_2(1+\delta)-2t_1)^2}\ln\left({2t_1\over ( 1+\delta)t_2}\right)\right).\ ] ] although the first part of the @xmath175 integral is straightforward , the second piece involving the logarithm is more difficult if we make the transformation @xmath263 we find : @xmath264 where all time dependency has been removed from the integral limits . the final @xmath176 integral is then easy to perform , and we end up with : @xmath265 @xmath266\right ) \ln(2\lambda_{aa}n_at).\ ] ] however , we now need to extend this analysis to determine the behaviour of the integral in @xmath257 . if we take the asymptotic part of all the pieces inside the integral , and perform power counting , we find that it should scale as @xmath267 . however , this procedure is not strictly valid , as in moving to the asymptotic version a false @xmath268 divergence is created . nevertheless , the integral is dominated by contributions from late times where arguments based on power counting should be valid . hence in @xmath257 we find : @xmath265 @xmath269+o(\epsilon)\right)t^{\epsilon/2}\ln(2\lambda_{aa}n_at).\ ] ] further subleading corrections ( in time ) , which we have not calculated , will lack the logarithm factor , and so will contribute to the _ amplitude _ for the minority species density . 99 toussaint d and wilczek f 1983 _ j. chem . phys . _ * 78 * 2642 kang k and redner s 1985 _ phys . rev . a _ * 32 * 435 kuzovkov v and kotomin e 1988 _ rep . phys . _ * 51 * 1479 glfi l and rcz z 1988 _ phys . rev . a _ * 38 * 3151 ovchinnikov a , timashev s and belyy a 1990 _ kinetics of diffusion controlled chemical processes _ ( new york : nova science publishers ) ben - naim e and redner s 1992 _ j. phys . a : math . _ * 25 * l575 cornell s and droz m 1993 _ phys . rev . lett . _ * 70 * 3824 lee b 1994 _ j. phys . a : math . * 27 * 2633 krapivsky p , ben - naim e and redner s 1994 _ phys . e _ * 50 * 2474 lee b and cardy j 1995 _ j. stat . phys . _ * 80 * 971 howard m and cardy j 1995 _ j. phys . a : math . _ * 28 * 3599 ben - naim e 1994 _ preprint cond - mat 9412048 _ smoluchowski m 1917 _ z. phys . * 92 * 215 chandrasekhar s 1943 _ rev . phys . _ * 15 * 1 derrida b , bray a and godrche c 1994 _ j. phys . a : math . * 27 * l357 derrida b 1995 _ j. phys . a : math . _ * 28 * 1481 derrida b , hakim v and pasquier v 1995 _ phys . lett . _ * 75 * 751 cardy j 1995 _ j. phys . a : math . _ * 28 * l19 grassberger p and procaccia i 1982 _ j. chem . phys . _ * 77 * 6281 abramowitz m and stegun i 1965 _ handbook of mathematical functions _ ( new york : dover ) doi m 1976 _ j. phys . a : math . gen . _ * 9 * 1465 , 1479 peliti l 1985 _ j. physique _ * 46 * 1469 fisher m e 1984 _ j. stat . _ * 34 * 667 fisher m e and gelfand m p 1988 _ j. stat . phys . _ * 53 * 175 ben - avraham d 1988 _ j. chem . phys . _ * 88 * 941	we study fluctuation effects in a two species reaction - diffusion system , with three competing reactions @xmath0 , @xmath1 , and @xmath2 . asymptotic density decay rates are calculated for @xmath3 using two separate methods - the smoluchowski approximation , and also field theoretic / renormalisation group ( rg ) techniques . both approaches predict power law decays , with exponents which asymptotically depend only on the ratio of diffusion constants , and not on the reaction rates . furthermore , we find that , for @xmath4 , the smoluchowski approximation and the rg improved tree level give identical exponents . however , whereas the smoluchowski approach can not easily be improved , we show that the rg provides a systematic method for incorporating additional fluctuation effects . we demonstrate this advantage by evaluating one loop corrections for the exponents in @xmath4 , and find good agreement with simulations and exact results . pacs numbers : 02.50 . -r , 05.40 . + j , 82.20 . -w . = 148 mm = 231 mm
it is well - known that wang - landau sampling ( wls ) @xcite faces difficulties for continuous systems such as atomic clusters @xcite , polymers and proteins @xcite , liquid crystals @xcite , and spin models @xcite . in continuous systems , the volume of phase space near the ordered ( low entropic ) states is infinitesimally small compared to that of the disordered ( high entropic ) regions . nevertheless , the traditional wls uses the same random trial moves for the whole range of energies , even though the phase space volume between the ordered and disordered states can differ by many orders of magnitude in different energy domains . this makes it very hard for the random walker of wls to perform statistically significant visits to the low entropic states . an energy - independent random trial move naturally favors diffusion into the voluminous and disordered regions of phase space , whereas visits to the ordered regions are forced " upon the random walker solely by the acceptance - rejection criterion . as a result , one needs to perform long simulations to properly sample the rare ordered states . such difficulties are indeed well - documented in the literature . on the theoretical side , the classic paper by zhou and bhatt@xcite showed that the statistical error of wls progresses as @xmath0 , where @xmath1 is the modification factor used in wls , and is @xmath2 is a constant . this constant was later shown by morozov and lin @xcite in a careful analysis of discrete systems to be proportional to the rate of change of entropy with energy @xmath3 . if we apply their result to continuous systems where the entropy gradient at the ground state diverges , it means that the statistical error of wls diverges . in numerical simulations , such problems have been reported in many complex and challenging continuous systems such as protein molecules @xcite and liquid crystals @xcite . perhaps the most telling example is that even for a simple and well - understood system such as the ferromagnetic @xmath4 model , traditional wls faces difficulties sampling the ordered states @xcite . there have been previous studies addressing the sampling of low entropic states in wls . xu and ma @xcite studied the two dimensional @xmath4 model where the density of states ( dos ) is known to change very steeply near the ground state energy . they first analytically derived the low temperature approximation of the partition function and then made a laplace transform to obtain the approximate dos near the ground state energy . using this as the initial approximation , they performed wls in a narrow region of low energy space to refine their dos . however , their approach can not be applied to more general systems such as spin glasses where the ground state is not known _ _ a priori__@xcite . furthermore , restricting the random walker to only a limited energy range makes it non - ergodic in frustrated systems . al . proposed updating and smoothing the dos with a continuous kernel @xcite . although the effects of smoothing does indeed help in the sampling of the dos at low entropic regions , this method is heuristic , and the width of the kernel might affect the outcome . actually , the difficulty of sampling the low entropic regions of phase space is not restricted just to wls , and has indeed been studied previously within the general context of monte carlo simulations by bouzida , kumar , and swendsen @xcite . the main idea is to strike a balance between choosing a good step size for the trial move and rapid exploration of the entire phase space . using smaller step sizes for the trial move can improve the sampling of ordered states this is because small moves allow the system to make minor adjustments to fine - tune itself into a highly specific ordered configuration . however , the problem with making small steps is that it leads to slow exploration of phase space . the acceptance ratio method of bouzida , kumar , and swendsen is a systematic way of achieving high computational efficiency by balancing a good step - size with fast exploration of phase space . in this method , one updates the step size @xmath5 as @xmath6 where @xmath7 and @xmath8 are the current and optimum ( i.e. desired ) acceptance rate , @xmath9 and @xmath10 are the current and new ( i.e. improved ) step sizes , and @xmath11 are constants to protect against singularities when @xmath12 or 1 . given @xmath7 , eq . ( [ eq : bks intro ] ) iteratively adjusts the step size to achieve @xmath8 . @xcite a systematic study by the original authors has found the best @xmath8 for systems in various dimensions @xcite . in this paper , we propose two ideas to circumvent the difficulties faced by the wls in sampling the low entropic regions . the first is to generalize the acceptance ratio method by bouzida et . such that the step size @xmath5 and the acceptance rate @xmath7 in eq . ( [ eq : bks intro ] ) become energy - dependent . more precisely , we would like @xmath5 to be small in the ordered regions of phase space , but large in the disordered regions . this will enable the random walker to make small moves at the low entropic regions to sample rare states , but also make larges moves to quickly diffuse through the easily sampled disordered ones . by making the acceptance rate @xmath7 energy - dependent as well , we can use eq . ( [ eq : bks intro ] ) to adjust @xmath5 at a particular energy based on the acceptance rate of that energy . our second contribution is to generalize the updating the dos . in the original wls , the dos is updated with the same modification factor @xmath1 for the entire energy range . we propose multiplying @xmath1 by an energy - dependent factor . as discussed above , generalizing the acceptance ratio method will provide us with an optimized trial move step size that reflects the entropic structure of phase space at that energy . a large step size means that at that energy , the phase space is large , whereas a small step size will imply that the phase space at that energy is small . hence , we propose multiplying the modification factor by the optimized trial move step size . our physical motivation is that the modification factor should be large at high entropic states to quickly accumulate the estimated dos , whereas for small entropic states , the accumulation should be more gradual to avoid sudden increments that usually leads to overestimation of visits to these small regions of phase space . ideally , we want more frequent visits to the low entropic region but a slower and careful accumulation of dos through the use of smaller modification factors . we shall refer to our proposed method as the adaptive wang - landau sampling ( adawl ) . actually , our proposed strategy constitutes a significant departure from the original wls . it might be questioned if biasing the wls in an energy - dependent fashion might lead to an erroneous dos . we shall show numerically by comparing with benchmark calculations that our generalization of wls does lead to the correct dos , and indeed , it improves dramatically upon the original wls . the rest of the paper is organized as follows . in section [ sec : adawl ] , we describe our algorithm in detail . section [ sec : model ] introduces our test model , the two - dimensional square lattice @xmath4 model , as a testbed for our method . section [ sec : numerical ] presents results of numerical simulations . in particular , we look at three different measures to assess the performance of adawl compared to wls : the specific heat , the first visit time , and the saturation error of the dos . details about these measures will be described in the respective subsections . we discuss and conclude in section [ sec : discussion ] . wang - landau sampling performs a random walk in energy space and seeks to provide an accurate estimate of the microcanonical density of states . in the traditional wls , a trial move with a fixed step size is used to sample a new configuration @xmath13 from the current configuration @xmath14 , i.e. @xmath15=q_{0}(x ) , \label{eq : ver025:original trial move}\ ] ] where @xmath16 $ ] is the probability of making the trial move from @xmath14 to @xmath13 , the random variable @xmath17 gives the change from @xmath14 to @xmath13 , and @xmath18 is a probability distribution for generating @xmath17 using a constant step size which remains fixed during simulation . for instance , @xmath18 can be a gaussian distribution with the standard deviation being the step size . then @xmath19 can be how much to move the position of a particle , where @xmath20 and @xmath21 are the positions of the particle before and after the trial move . note that apart from having a fixed step size , @xmath18 is also independent of the configuration of the system . in other words , @xmath22 is the same for every point in the entire phase space . trial moves can in general depend on the system configuration , an example being the swendsen - wang @xcite and other cluster algorithms @xcite where the flipping of a cluster of spins depends on the current existing spin clusters . the traditional wls , however , usually employs configuration - independent trial moves . using a trial move like eq . ( [ eq : ver025:original trial move ] ) , wls accepts the new state @xmath13 with probability @xmath23 where @xmath24 and @xmath25 are respectively the energies of the current and proposed configurations , and @xmath26 is the estimated dos at energy @xmath24 . note that as the trial move does not depend on system configuration , @xmath18 does not appear in eq . ( [ eq : wl ] ) . after each move , wls modifies the dos as @xmath27 by means of a modification factor @xmath28 . the subscript @xmath29 indicates the @xmath29th @xmath1 stage of the wang - landau algorithm . in their original formulation , wang and landau proposed reducing this factor as @xmath30 based on the flatness of the accumulated histogram . however , detailed investigations by various authors have found that histogram flatness is not a satisfactory criterion @xcite . here we shall adopt a different criterion based on the saturation of the dos error , which will be described in section [ sec : ver024 : saturation ] . for continuous system , the energies are discretized , and the estimated @xmath26 is a piecewise constant function , i.e. @xmath31 within each energy bin @xmath32 . in adawl , to generate the proposed new configuration @xmath33 , our trial moves will be more general and depend on the current configuration @xmath14 . let us first define the adjustable probability distribution @xmath34 whose width can be tuned using @xmath35 . the actual form of @xmath34 will depend on the system and the kinds of moves one wishes to make . we can choose to make the distribution narrow or wide using @xmath35 . in practice , @xmath35 will be substituted by the step size of the trial move . in this paper , we use an energy - dependent step size @xmath36 and set @xmath37 . if we consider just single - site update so that @xmath14 and @xmath33 differ by one site , our trial move is given by @xmath38 = \frac{1}{n } q(x ; \delta(e ) ) , \label{eq : trial}\ ] ] where @xmath39 $ ] is the probability of making the trial move from @xmath14 to @xmath13 with step size @xmath36 . the step size @xmath36 is the size of the move at the energy @xmath24 . note that since the energy in @xmath36 is a function of the configuration @xmath14 , the trial move eq . ( [ eq : trial ] ) is now dependent on system configuration , unlike eq . ( [ eq : ver025:original trial move ] ) which is not . the factor @xmath40 is to account for the probability of selecting one site out of @xmath41 ( e.g. the total number of spins ) . in numerical calculation , @xmath36 is represented as a piecewise constant function of energy , i.e. @xmath42 for @xmath32 . the challenge now is to optimize the step sizes @xmath43 for the most efficient simulation . we extend the acceptance ratio method of bouzida et . al . @xcite and update @xmath43 at the energy bin @xmath44 according to @xmath45 where @xmath46 is the optimal acceptance rate , and @xmath2 , @xmath47 are constants to protect against singularities . the choice of @xmath46 depends on the dimension of the system , and we shall use the value recommended by bouzida et . al .. the parameters @xmath46 , @xmath2 , and @xmath47 we used in this paper for our simulations are given in the caption of table [ table : simulation_parameters ] . during simulations , we first initialize @xmath43 to a constant value for all energy bins . in addition to the usual histogram , we also accumulate the counts of accepted and rejected moves at bin @xmath44 , @xmath48 and @xmath49 . after a certain number of monte carlo moves , we compute the acceptance rate at @xmath44 as @xmath50 and use eq . ( [ eq : arm ] ) to update the step sizes . we now describe the transition probability from the old configuration @xmath14 to the new one @xmath13 . unlike eq . ( [ eq : wl ] ) for the wls , our trial moves depend on the system configuration . hence , the transition probability has to be modified to obtain an unbiased sampling : @xmath51}{t[\sigma^{\prime}\|\sigma;\delta(e)]\ , } \cdot \frac{\tilde{g}(e)}{\tilde{g}(e^{\prime } ) } \right ) , \label{eq : db}\ ] ] where @xmath39 $ ] is the probability of making the forward move , @xmath52 $ ] that of making the backward one , and both are given by eq . ( [ eq : trial ] ) . @xmath53 is a linearly - interpolated estimate of the dos . the ratio @xmath54 is used to account for the energy - dependent accumulation of the dos which we will now describe . as mentioned in the introduction , adawl adopts an energy - dependent modification factor , @xmath55 where @xmath28 is as defined in wls , and @xmath56 is our new modification factor . to accommodate the possibility of using non - uniform intervals between energy levels , the updating of @xmath57 and histogram @xmath58 at each step has to take into account the actual size of the bins , @xmath59 where @xmath60 is the size of the bin width at @xmath44.@xcite this completes the description of adawl . a summary of the algorithm is given in appendix [ sec : summary adawl ] . to test our new algorithm , we consider the two - dimensional @xmath61 square lattice @xmath4 model , @xmath62 where @xmath14 is now a vector of @xmath41 spins @xmath63 , @xmath64 , @xmath65 denotes summation over nearest - neighbor pairs , and periodic boundary condition is used for both lattice dimensions . @xmath66 is the total number of spins . the @xmath4 model , although simple , has been shown to contain the essential difficulties encountered in many continuous systems , and hence is a good testbed for our method @xcite . we first specify the adjustable distribution @xmath34 , @xmath67 @xmath68 is symmetric and piecewise linear in @xmath17 . @xmath69 is the adjustable width . from the normalization condition , we get the height of the distribution @xmath70 . fig . [ fig : q ] shows plots of @xmath34 for some values of @xmath35 . for the trial move , first pick at random a lattice site @xmath71 , then draw a random variable @xmath17 from the distribution @xmath34 , and then update the spin as @xmath72 the width of the distribution @xmath35 is specified by the step size @xmath36 . when step size is small , i.e. @xmath73 , @xmath74 is a delta function sharply centered at @xmath75 , and the new configuration @xmath76 is close to the current one @xmath77 . when the step size is large , i.e. @xmath78 , @xmath74 approximates the uniform distribution , and the new configuration is uncorrelated with the current one . @xmath34 satisfies our requirements for an adjustable distribution and is simple enough to allow us to sample @xmath17 efficiently @xcite . we also bin the energy levels non - uniformly . the top panel of fig . [ fig : logg ] shows the dos of the @xmath4 model for @xmath79 . the dos is symmetric , is relatively flat around @xmath80 , and drops abruptly near the minimum and maximum energies @xmath81 and @xmath82 . hence , in both our wls and adawl simulations , we assign smaller energy bins near @xmath81 and @xmath82 in order to represent the dos near the edges more accurately . this is accomplished by using the following formula to assign the negative energies , @xmath83 the initial energy level is given by @xmath84 . the gaussian - like exponential term in eq . ( [ eq : bin_width ] ) is to make neighboring energies close near @xmath81 where the dos drops abruptly , but far apart near @xmath80 . the constants @xmath85 , @xmath86 , and width of the initial bin @xmath87 are set manually . @xmath85 is a positive even integer that controls the rate of increase of the exponential term . @xmath86 is the width at @xmath80 . @xmath88 is determined once @xmath85 , @xmath86 , and @xmath89 have been specified . the binning parameters we used are listed in table [ table : simulation_parameters ] . the negative energies are reflected about @xmath80 to obtain the positive energies . the bottom graph in fig . [ fig : binwidth ] shows the bin widths we used for @xmath90 in the simulations of this paper . the procedure for our numerical simulation of wls and adawl is as follows . the binning of energies are set using eq . ( [ eq : bin_width ] ) . at the start of the simulation , @xmath91 . the modification factor is reduced in stages as @xmath92 , where in the final stage @xmath93 we have @xmath94 . for each stage , we perform simulation until the error of the dos saturates for that stage before reducing the modification factor . the error of the dos will be discussed in detail in section [ sec : ver024 : saturation ] . for each system size @xmath95 , we computed a total of @xmath96 independent trajectories where each trajectory is started using a different random seed . the details of the simulation parameters are summarized in table [ table : simulation_parameters ] . for wls , our trial moves are also given by eq . ( [ eq : q ] ) with @xmath35 being a constant @xmath97 . we have experimented with several constant step sizes and found @xmath98 to perform the best . the numerical results supporting this claim are presented in subsections [ sec : ver024 : ergordicity ] and [ sec : ver024 : saturation ] and the insets of figs . [ fig : fvt ] and [ fig : dh ] . in the rest of the paper , unless otherwise stated , we shall be comparing adawl with wls of step size 0.05 . for adawl , the step sizes are @xmath99 for all @xmath44 at the start of the simulation . during simulation , we also accumulate @xmath100 and @xmath101 . once every @xmath102 single site updates per spin , we use eqs . ( [ eq : arm ] ) and ( [ eq : ver023:a_r_p ] ) to update the step sizes . @xmath100 and @xmath101 are then reset to zero , and their accumulation restarted for the next iteration of step size update . during simulations , the curves for @xmath103 and @xmath36 converged very quickly ( i.e. after a few iterations of eqs . ( [ eq : arm ] ) and ( [ eq : ver023:a_r_p ] ) ) . fig . [ fig : logg ] shows the dos , @xmath103 , and @xmath36 of our adawl simulation for @xmath79 . as can be seen , eq . ( [ eq : arm ] ) adjusts the step sizes such that the acceptance rate is @xmath104 . in the high dos energy range between -50 and 50 , the acceptance rate did not reach @xmath104 because the step size has already saturated to the maximum of 1 and the acceptance rate can not be further optimized . the dos is updated quickly with the maximal modification factor in this energy range . near the edges of the dos , the step size and modification factors are both small , and the dos is updated gradually . in the following , we compare the performance between wls and adawl using three different measures : ( 1 ) the specific heat capacity , ( 2 ) the first visit time , and ( 3 ) the saturation of dos error . we first demonstrate that adawl computes the correct dos , and that it is more accurate than wls . to do that , we compute the specific heat capacity . we first divide @xmath96 into four equal portions . for each portion , we compute the mean of the dos ( i.e. we average over the final dos s of the @xmath105 trajectories ) . this average dos is used to compute the specific heat capacity per spin @xmath106 at temperature @xmath107 using @xmath108 where the thermal average of @xmath109 is given by @xmath110 the @xmath106 is then further averaged over the four portions . we denote this specific heat averaged over the four portions as @xmath111 . [ fig : cvall ] shows the results of @xmath111 for @xmath90 and @xmath112 . the left ( right ) panels are for adawl ( wls ) . @xmath111 is given by the solid curve . the standard error at some temperatures is also indicated using error bars . the size of the error bars show that the precision of the specific heat calculated by adawl is better than that of wls . for @xmath113 , it is apparent that wls produces a grossly incorrect curve for @xmath111 . to check the accuracy of the adawl results , we performed metropolis simulations to generate accurate specific heat capacity values at selected temperatures , and these are also plotted for comparison in fig . [ fig : cvall ] using solid circles . the @xmath111 curves from adawl agree very well with the results of metropolis calculations . the actual @xmath111 values from all three methods are also listed in tables [ table : cv_l16 ] ( @xmath90 ) and [ table : cv_l32 ] ( @xmath113 ) . we see that adawl is consistently closer to the benchmarked metropolis numbers compared to wls . the simulation parameters of our metropolis calculations are listed in table [ table : simulation_parameters ] . one way to measure the performance of a random sampler is its ergodicity . the more ergodic the sampler , the more efficient it is in exploring representative parts of phase space . for the wang - landau algorithm , some authors have used the so - called tunneling time as a measure of ergodicity @xcite . this is the time it takes for the random walker to go from one energy minimum configuration to another . the shorter the tunneling time , the more ergordic is the random walker . here , we adopt a related measure of ergodicity which is much easier to compute the first visit time . at the start of each @xmath1 stage of the wls or adawl simulation when the modification factor has just been decreased , the histogram is zero for all energy bins . the first visit time is defined as the time it takes for all the bins of the histogram to be visited at least once by the random walker . for each trajectory , we compute one first visit time for each @xmath1 stage of the simulation . we then average over @xmath96 trajectories . [ fig : fvt ] shows the results . the average first visit times is plotted against @xmath114 for adawl and wls for various system size . generally , adawl ( filled symbols ) visits all energy levels much faster than wls ( empty symbols ) at all stages and for all system sizes , implying better ergodicity . the insert is a similar plot comparing the results for wls with different constant step sizes ; a constant step size of @xmath115 performs the best for wls . the first visit time at small @xmath1 for @xmath90 is similar for adawl and wls . we attribute this to binning effects which will be discussed in section [ sec : ver026:discuss nonuniform binning ] . in their study of the @xmath4 model , sinha and roy@xcite reported that the random walker of wls frequently does not visit energy bins near @xmath81 and @xmath82 . here , we mention that our bins near the edges are much smaller and also nearer to @xmath81 and @xmath82 compared to what sinha and roy had used . that adawl has no difficulty sampling all energy bins is indicative that it performs better than wls . we now consider the error in the dos . in wang and landau s original formulation , the ` flatness of histogram ' criterion was used as a measure of convergence of the wls . each stage of the sampling was performed until the accumulated histogram becomes sufficiently flat before the modification factor is reduced . however , it is now known that this is not a good measure of convergence because the height of the histogram increases linearly with time and will ultimately reach flatness regardless of whether the simulation for that stage has converged or not . detailed studies by various authors on the dos error of wls have revealed that the error is related to the modification factor instead of histogram flatness @xcite . also , the use of arbitrary histogram flatness as a criterion has been shown to lead to non - convergence of wls by @xcite . the correct convergence of wls has also been proposed by morozov and lin @xcite . in a separate investigation , lee et . @xcite formulated a more precise measure of the convergence of wls which is shown to agree with the @xmath116 analysis by zhou and bhatt . details will be presented in appendix [ sec : ver024:dh ] . here we shall present the main idea . denote the histogram for the @xmath29th stage of the simulation as @xmath117 . we define a new histogram @xmath118 obtained by subtracting away the minimum value of @xmath117 , i.e. @xmath119 hence , @xmath118 is not plagued by the problem of linear growth . the area under @xmath118 @xmath120 is conjectured by lee , okabe , and landau to be a measure of the error in the dos @xcite . ( the @xmath121 in eq . ( [ eq : ver024:dhk ] ) is to account for the non - uniform energy bin widths . ) during each stage of the simulation , @xmath122 first increases and then saturates to around some mean value . this means that further sampling will not help to reduce the error in the dos , and therefore the modification factor should be reduced . note that an increasing @xmath122 does not mean increasing error in the dos , because the actual error has to take into account the smallness of the modification factor ( c.f . ( [ eq : error01 ] ) ) . the key observation is the saturation of @xmath122 during each stage of the simulation . @xcite applied @xmath122 to study the dos error of wls in the two - dimension ising model where the exact numerical solution for the dos is available , and found that it is a good measure of the dos convergence . it has also been applied by sinha and roy to study wls of the @xmath4 model @xcite . [ fig : dh_compare ] shows an example of the saturation of @xmath122 for the @xmath4 model . it compares the saturation curves for adawl and wls at the modification factor @xmath123 for @xmath79 . each curve @xmath124 is obtained by averaging over @xmath96 trajectories . it can be seen that both wls and adawl curves saturate to some constant value after a certain number of monte carlo steps . however , the saturation value of adawl is much smaller than wls , implying a smaller error for adawl . for the simulations in this paper , we ran the simulation at each stage long enough to obtain accurate saturation values of @xmath124 . although in practice @xmath28 should be decreased as soon as saturation is reached , as our purpose here is to compare the performance of adawl and wls , we ran each stage much longer than is necessary to obtain reliable measurements of @xmath125 . we now describe how we compare the dos saturation error of adawl and wls . the @xmath96 trajectories are first divided into four equal portions . for each portion , at each stage @xmath29 , we compute @xmath124 curves similar to that of fig . [ fig : dh_compare ] by averaging @xmath126 over @xmath105 trajectories . using this averaged curve @xmath124 , we estimate its saturation value by averaging over the time steps in the flat part ( say last 10 percent ) of the curve . this gives us the saturation value of @xmath124 of that stage for that one portion . we then average the saturation value over all four portions . the results are shown in fig . [ fig : dh ] . the average saturation value of @xmath127 is plotted against @xmath28 for adawl and wls for various system size . adawl ( filled symbols ) has significantly smaller saturation values than wls ( empty symbols ) , implying a smaller error in the dos . the insert is a similar plot comparing the results for wls with different constant step sizes ; a constant step size of 0.05 gives the smallest saturation value for wls . lastly , we briefly comment on the use of non - uniform energy bin widths . when using non - uniform bin widths in wang - landau simulations , there is the freedom to choose large energy spacings at certain energies . however , to compute thermodynamic quantities such as the specific heat capacity accurately , the spacings between energy levels has to be small enough to enable a good representation of the distribution @xmath128 at the temperatures of interests . hence , it is recommended that one first check by making a rough plot of @xmath128 to ensure that it is represented with a sufficient number of energy levels at the temperatures concerned . this is especially important for large system size because the appearances of singularities or cusps usually require finer energy spacings to resolve . of course , the spacings also can not be too small otherwise each bin will not accumulate enough visits by the random walker . in this paper , we have used eq . ( [ eq : bin_width ] ) to set our energy levels . it might be tempting to choose @xmath85 and @xmath86 to be quite large , thereby greatly reducing the number of energy levels used , especially near @xmath80 . however , we found that this will lead to an insufficient number of energy levels representing @xmath128 at lower temperatures . our choice of binning parameters in table [ table : simulation_parameters ] ensures a good representation of @xmath128 . we have also studied the effects of different bin widths on adawl and wls , and found that there might be rare instances where wls appears to give similar performances as adawl . but these rare cases are usually due to effects of bin widths . if one uses coarse bins , wls can reach all bins easily , whereas if a finer set of bin widths near the ground state is used , wls will have difficulty visiting those small bins . adawl , however , will not show such dependence because its step size is designed to be adaptively adjusted according to the energies . in fig . [ fig : fvt ] , wls shows signs of smaller first visit time than adawl towards the smaller @xmath1 for @xmath90 . we have found that using even finer bin widths will increase the first visit time for wls , but not for adawl . however , since we have already obtained a more accurate specific heat capacity for adawl at that bin width , we did not pursue to further accentuate the performance between the two methods . as another example , fig . [ fig : binwidthl16 ] compares the saturated dos error of adawl and wls for the coarse and fine bin widths shown in fig . [ fig : binwidth ] . adawl gives the same results for both sets of bin widths , whereas the error for wls increases for the fine bin widths . to summarize , we proposed an adaptive variant of the wang - landau sampling , which is effective for sampling dos that ranges many orders of magnitude . the main contributing factors to this increase in efficiency are adaptive step sizes and adaptive modification factors . adaptive step sizes sample the configuration space well , while adaptive modification factors accumulate the dos effectively and accurately . we have tested the effectiveness of adawl for system sizes up to @xmath95=32 . for larger sizes , we may break into several energy regions @xcite , where the method to avoid boundary effect " should be taken into account @xcite . in such a case , the present adaptive method is still effective for treating dos that has many orders of magnitude . for future work , adawl should be tested on different continuous systems , especially frustrated ones . in fig . [ fig : logg ] , we see that adawl is not yet fully optimized because the acceptance rate in the middle energy range has not been adjusted to 0.5 due to the saturation of @xmath129 to the maximum value of 1 . at larger lattice sizes , where the energy range is larger , one might consider going beyond single site updates ( e.g. global moves ) to enable even larger step sizes to be used . this might make the sampling of adawl even more efficient . recently , there have been many works on improving wls both for discrete @xcite and continuous @xcite systems . to obtain better convergence , the @xmath130 algorithm @xcite was proposed . moreover , _ tomographic _ entropic sampling scheme @xcite was proposed as an algorithm to calculate dos . the convergence of wls was discussed with paying attention to the difference of density of states by komura and okabe @xcite . it will be interesting to combine the present work with the recent progress . finally , we make a note that our idea of using an adaptive modification factor could potentially be used for simulating discrete systems as well as continuous systems . this will also be part of our future work . this work was supported ( in part ) by the biomedical research council of astar ( agency for science , technology and research ) , singapore . our adawl algorithm is as follows . 1 . initialize the bin sizes @xmath131 according to eq . ( [ eq : bin_width ] ) . initialize the system configuration @xmath14 , the dos @xmath132 , the histogram @xmath133 , modification factor @xmath134 , and step sizes @xmath43=constant . [ it : init ] 2 . sample a new configuration @xmath13 from @xmath135 and accept the move as given by eq . ( [ eq : db ] ) . [ it : accept ] 3 . update the dos and histogram according to eq . ( [ eq : update ] ) . update the acceptance and rejection counts @xmath48 and @xmath49 . [ it : update ] 4 . repeat steps [ it : accept ] and [ it : update ] for some predefined number of monte carlo steps and update the step size according to eq . ( [ eq : arm ] ) . set @xmath136 . [ it : arm ] 5 . reduce @xmath28 ( e.g. , @xmath137 , after the dos error saturates ) and set @xmath138 ; else , repeat steps [ it : accept ] to [ it : arm ] . [ it : flat ] 6 . repeat steps [ it : accept ] to [ it : flat ] until the modification factor @xmath28 is smaller than some tolerance threshold . the contents of this appendix was first given in lee et . al . @xcite . the reader is referred there for a more complete presentation . here , for completeness , we outline the main idea presented there , and also update the analysis to take into account the use of non - uniform energy bin widths . the dos @xmath139 accumulated after the @xmath140th stage can be written as @xmath141 where @xmath117 is the accumulated histogram and @xmath28 is the modification factor for the @xmath29th stage of simulation . ( [ eq : logg_n ] ) holds for both wls and adawl . calculation of thermodynamics quantities are not affected if we subtract a constant from @xmath117 , hence we subtract the minimum of @xmath117 , @xmath142 and define a new but equally valid density of states , @xmath143 to introduce our histogram measure , we observe that it is reasonable to estimate the error between the computed density of states @xmath144 and the true one @xmath145 as @xmath146 = \sum_{e } \sum_{k = n+1}^{\infty } w(e ) \tilde{h}_k(e ) \ln(f_k ) \label{eq : error01}\ ] ] an intuitive view of eq . ( [ eq : error01 ] ) is that if an infinite number of stages were performed ( i.e. @xmath147 ) , then the exact dos will be obtained . this statement was made formal by the conjecture of lee , okabe and landau @xcite . if just @xmath140 stages were done instead , the error of @xmath144 will be the sum of all the rest of the stages that were not carried out explicitly . we denote the fluctuation of @xmath118 as @xmath148 note that the summation over @xmath24 in eq . ( [ eq : error02 ] ) includes the binwidth @xmath121 . this is a slight modification from the original formulation . swapping the order of summation , the rhs of eq . ( [ eq : error01 ] ) becomes @xmath149 hence , the error depends only on @xmath122 and the sequence of modification factors @xmath28 . if @xmath28 are predetermined , then @xmath122 becomes the only determining factor of the error . hence , when we see that it saturates ( for a certain @xmath29 ) , it is an indication that enough statistics has been accumulated for this @xmath28 value and simulation for the next value @xmath150 should begin . finally , it is important to note that smaller @xmath122 values indicates that the accumulated histogram is flatter . swetnam and m.p . allen , j. comput . chem . 32 * , 816 ( 2010 ) ; d.t . seaton , t. wst , and d.p . landau , phys . e * 81 * , 011802 ( 2010 ) ; s. . jnsson , s. mohanty , and a. irbck , j. chem . phys . * 135 * , 125102 ( 2011 ) . we choose eq . ( [ eq : q ] ) for the trial move transition probability because its cumulative and inverse cumulative distributions can easily be derived analytically , facilitating the numerical calculation of the random variable @xmath17 . .summary of parameters used in adawl , wls , and metropolis simulations . @xmath97 : constant step size used for wls ( @xmath35 in eq . ( [ eq : q ] ) ) . @xmath94 : smallest ( i.e. final ) modification factor used in simulation . @xmath157 : no . of single site updates per spin used for @xmath94 ( the final stage ) . @xmath158 : no . of single site updates per spin used for metropolis simulation . parameter values for eq . ( [ eq : arm ] ) : @xmath159 , @xmath160 , and @xmath161 @xcite . [ cols="^,^,^,^,^,^,^,^,^,^,^",options="header " , ] & + @xmath107 & @xmath162 & @xmath163 & @xmath162 adawl & @xmath162 wls + 0.1 & 0.5112 & 8 & 0.3 & 5 + 0.2 & 0.5266 & 8 & 0.3 & 4 + 0.3 & 0.5446 & 6 & -1 & 0.2 + 0.4 & 0.5664 & 5 & 0.8 & 5 + 0.5 & 0.5948 & 7 & 0.03 & -5 + 0.75 & 0.7358 & 7 & -1 & -0.7 + 1.0 & 1.2200 & 20 & 0.05 & 3 + 1.075 & 1.4467 & 20 & 2 & -8 + 1.1 & 1.4796 & 30 & 1 & -6 + 1.13 & 1.4690 & 10 & 3 & -11 + 1.75 & 0.4483 & 4 & 0.5 & -8 + & + @xmath107 & @xmath162 & @xmath163 & @xmath162 adawl & @xmath162 wls + 0.1 & 0.5132 & 9 & 10 & -20 + 0.2 & 0.5283 & 6 & 10 & -30 + 0.3 & 0.5459 & 5 & -4 & -100 + 0.4 & 0.5683 & 7 & 5 & 10 + 0.5 & 0.5966 & 7 & -8 & 20 + 0.8 & 0.7966 & 5 & -10 & 20 + 1.0 & 1.336 & 30 & -2 & 20 + 1.025 & 1.448 & 30 & -7 & -10 + 1.05 & 1.519 & 30 & -8 & -90 + 1.075 & 1.521 & 20 & -1 & -200 + 1.1 & 1.465 & 30 & 4 & -100 + 1.15 & 1.314 & 10 & 6 & -70 + 1.2 & 1.182 & 20 & 3 & 30 + 1.8 & 0.4174 & 2 & 2 & -30 + model ( @xmath95=8 ) obtained using adawl . step sizes are adjusted to keep an optimum acceptance ratio of @xmath104 . between energies @xmath167 to @xmath168 , step sizes saturate to a maximum value of @xmath166 . some representative error bars are shown for the acceptance rate . , width=453 ] model for @xmath90 . the widths are larger near @xmath80 and smaller near @xmath169 . the lower graph ( fine bins ) is the binning scheme given in table [ table : simulation_parameters ] and used throughout this paper ( for @xmath90 ) . the upper graph ( coarse bins ) is discussed in the text . ] for adawl and wls ( constant step size 0.05 ) . errorbar when not shown is smaller than the size of the symbol . insert : first visit times of wls for different step sizes . the most efficient step size for wls is 0.05 , with the smallest first visit time . symbols for insert are as follows . for @xmath95=4 : @xmath170 for step size=0.01 , @xmath171 for 0.05 , @xmath172 for 0.1 , and @xmath173 for 0.5 . for @xmath95=8 : @xmath174 for step size=0.01 , @xmath175 for 0.05 , @xmath176 for 0.1 , and @xmath177 for 0.5 . , width=453 ] versus @xmath28 for adawl and wls ( constant step size=0.05 ) . insert : for wls with different step sizes . symbols have the same meaning as fig . [ fig : fvt ] . the most efficient step size for wls is 0.05 , which has the lowest saturation values . , width=453 ]	the density of states of continuous models is known to span many orders of magnitudes at different energies due to the small volume of phase space near the ground state . consequently , the traditional wang - landau sampling which uses the same trial move for all energies faces difficulties sampling the low entropic states . we developed an adaptive variant of the wang - landau algorithm that very effectively samples the density of states of continuous models across the entire energy range . by extending the acceptance ratio method of bouzida , kumar , and swendsen such that the step size of the trial move and acceptance rate are adapted in an energy - dependent fashion , the random walker efficiently adapts its sampling according to the local phase space structure . the wang - landau modification factor is also made energy - dependent in accordance with the step size , enhancing the accumulation of the density of states . numerical simulations show that our proposed method performs much better than the traditional wang - landau sampling .
traditionally , the production of the electroweak gauge vector bosons is considered as a benchmark for understanding the dynamics of the strong and the electroweak interactions in the standard model . it is also an important test to assess the validity of collider data . many collaborations have reported numerous sets of measurements , probing different events in variant dynamical regions , in direct or indirect relation with such processes , to count a few see the references @xcite . among the most recent of these reports are the measurements of the production of @xmath2 bosons at the @xmath0 and @xmath1 collaborations , for proton - proton collisions at the @xmath16 for @xmath17 , with different kinematical regions @xcite . the @xmath0 data are in the forward pseudorapidity region ( @xmath18 ) while the @xmath1 measurements are in th central domain ( @xmath19 ) . in our previous work @xcite , we have successfully utilized the transverse momentum dependent ( @xmath20 ) unintegrated parton distribution functions ( @xmath5 ) of the @xmath12-factorization ( the references @xcite ) , namely the @xmath6-@xmath7-@xmath8 ( @xmath9 ) and @xmath7-@xmath8-@xmath10 ( @xmath11 ) formalisms in the leading order ( @xmath21 ) and the next - to - leading order ( @xmath4 ) to calculate the inclusive production of the @xmath22 and the @xmath2 gauge vector bosons , in the proton - proton and the proton - antiproton inelastic collisions @xmath23 in order to increase the precision of the calculations , we have used a complete set of @xmath24 @xmath4 partonic sub - processes , i.e. @xmath25 where @xmath26 represents the produced gauge vector boson . @xmath27 and @xmath28 , @xmath29 are the 4-momenta of the incoming and the out - going partons . the results underwent comprehensive and rather lengthy comparisons and it was concluded that the calculations in the @xmath9 formalism are more successful in describing the existing experimental data ( with the center - of - mass energies of @xmath30 and @xmath31 tev ) from the @xmath13 , @xmath14 , @xmath15 and @xmath1 collaborations @xcite . the success of the @xmath9 scheme ( despite being of the @xmath21 and suffering from some misalignment with its theory of origin , i.e. the @xmath32-@xmath33-@xmath34-@xmath35-@xmath36 ( @xmath37 ) evolution equations , @xcite ) can be traced back to the particular physical constraints that rule its kinematics . to find extensive discussions regarding the structure and the applications of the @xmath5 of @xmath12-factorization , the reader may refer to the references @xcite . meanwhile , arriving the new data from the @xmath0 and @xmath1 collaborations , the references @xcite , gives rise to the necessity of repeating our calculations at the @xmath38 . this is in part due to the very interesting rapidity domain of the @xmath0 measurements , since in the forward rapidity sector ( @xmath39 ) , one can effectively probe very small values of the bjorken variable @xmath40 ( @xmath40 being the fraction of the longitudinal momentum of the parent hadron , carried by the parton at the top of the partonic evolution ladder ) , where the gluonic distributions dominate and hence the transverse momentum dependency of the particles involving in the partonic sub - processes becomes important . in the present work , we intend to calculate the transverse momentum and the rapidity distributions of the cross - section of production of the @xmath2 boson using our @xmath4 level diagrams ( from the reference @xcite ) and the @xmath5 of the @xmath9 formalism . the @xmath5 will be prepared using the @xmath41 of @xmath42 , @xcite . in the following section , the reader will be presented with a brief introduction to the @xmath43 framework ( i.e. @xmath4 @xmath44 matrix elements and @xmath21 @xmath5 ) that is utilized to perform these computations . the section [ sec : ii ] also includes the main description of the @xmath9 formalism in the @xmath12-factorization procedure . finally , the section [ sec : iii ] is devoted to results , discussions and a thoroughgoing conclusion . generally speaking , the total cross - section for an inelastic collision between two hadrons ( @xmath45 ) can be expressed as a sum over all possible partonic cross - sections in every possible momentum configuration : @xmath46 in the equation ( [ eq3 ] ) , @xmath47 and @xmath48 respectfully represent the longitudinal fraction and the transverse momentum of the parton @xmath49 , while @xmath50 are the density functions of the @xmath51 parton . the second scale , @xmath52 , are the ultra - violet cutoffs related to the virtuality of the exchanged particle ( or particles ) during the inelastic scattering . @xmath53 are the partonic cross - sections of the given particles . for the production of the @xmath2 boson , the equation ( [ eq3 ] ) comes down to ( for a detailed description see the reference @xcite ) @xmath54 @xmath55 @xmath56 @xmath57 are the rapidities of the produced particles ( since @xmath58 in the infinite momentum frame , i.e. @xmath59 ) . @xmath60 are the azimuthal angles of the incoming and the out - going partons at the partonic cross - sections . @xmath61 represent the matrix elements of the partonic sub - processes in the given configurations . the reader can find a number of comprehensive discussions over the means and the methods of deriving analytical prescriptions of these quantities in the references @xcite . @xmath62 is the center of mass energy squared . additionally , in the proton - proton center of mass frame , one can utilize the following definitions for the kinematic variables : @xmath63 defining the transverse mass of the produced particles , @xmath64 , we can write , @xmath65 furthermore , the density functions of the incoming partons , @xmath66 ( which represent the probability of finding a parton at the semi - hard process of the partonic scattering , with the longitudinal fraction @xmath40 of the parent hadron , the transverse momentum @xmath12 and the hard - scale @xmath67 ) can be defined in the framework of @xmath12-factorization , through the @xmath9 formalism : @xmath68 , \label{eq7}\ ] ] the @xmath69 form factor , @xmath70 , factors over the virtual contributions from the @xmath21 @xmath37 equations , by defining a virtual ( loop ) contributions as : @xmath71 with @xmath72 . @xmath73 is the @xmath21 @xmath44 running coupling constant , @xmath74 are the so - called splitting functions in the @xmath21 , parameterizing the probability of finding a parton with the longitudinal momentum fraction @xmath40 to be emitted form a parent parton with the fraction @xmath75 , while @xmath76 , see the references @xcite . the infrared cutoff parameter , @xmath77 , is a visualization of the angular ordering constrtaint ( @xmath78 ) , as a consequanse of the color coherence effect of successive gluonic emittions @xcite , defined as @xmath79 . limiting the upper boundary on @xmath80 integration by @xmath77 , excludes @xmath81 form the integral equation and automatically prevents facing the soft gluon singularities , @xcite . additionally , the @xmath82 are the single - scaled parton distribution functions ( @xmath41 ) , i.e. the solutions of the @xmath21 @xmath37 evolution equation . the required @xmath41 for solving the equation ( [ eq7 ] ) are provided in the form of phenomenological libraries , e.g. the @xmath83 libraries , the reference @xcite , where the calculation of the single - scaled functions have been carried out using the deep inelastic scattering data on the @xmath84 structure function of the proton . now , one can carry out the numerical calculation of the equation ( [ eq4 ] ) using the @xmath85 algorithm in the monte - carlo integration , @xcite . to do this , we have chosen the hard - scale of the @xmath5 as : @xmath86 and set the upper bound on the transverse momentum integrations of the equation ( [ eq4 ] ) to be @xmath87 , with @xmath88 one can easily confirm that since the @xmath5 of @xmath9 quickly vanish in the @xmath89 domain , further domain have no contribution into our results . also we limit the rapidity integrations to @xmath90 $ ] , since @xmath91 and according to the equation ( [ eq6 ] ) , further domain has no contribution into our results . the choice of above hard scale is reasonable for the production of the z bosons , as has been discussed in the reference @xcite . finally , we choose @xmath92 to define the density of the incoming partons in the non - perturbative region , i.e. @xmath93 with @xmath94 . this appears to be a natural choice , since ( see the references @xcite ) @xmath95 using the theory and the notions of the previous sections , one can calculate the production rate of the @xmath2 gauge vector boson for the center - of - mass energy of @xmath96 @xmath97 . the @xmath41 of martin et al @xcite , @xmath42 , are used as the input functions to feed the equations ( [ eq7 ] ) . the results are the double - scale @xmath5 of the @xmath9 schemes . these @xmath5 are in turn substituted into the equation ( [ eq4 ] ) to construct the @xmath98 cross - sections in the framework of @xmath12-factorization . one must note that the experimental data of the @xmath0 collaboration , @xcite , and the preliminary data of the @xmath1 collaboration , @xcite , are produced in different dynamical setups ; the @xmath0 data are in the forward rapidity region , @xmath99 , while @xmath1 data are in a central rapidity sector , i.e. @xmath100 . we have imposed the same restrictions in our calculations . thus , in the figure [ fig1 ] we present the reader with a comparison between the different contributions into the differential cross - sections of the production of @xmath2 , ( @xmath101 ) , as a function of the transverse momentum ( @xmath102 ) of the produced particles , in the @xmath9 scheme . one readily notices that the contributions from the @xmath103 ( the so - called gluon - gluon fusion process ) dominate the the production . the share of other production vertices is small ( but not entirely negligible ) compared to these main contributions . this is to extent different from our observations in the smaller center - of - mass energies ( see the section v of the reference @xcite ) . also , differential cross - sections are considerably larger at the central rapidity region compared to the results in the forward sector . the total differential cross - section of the production of @xmath2 vector boson is calculated within the figure [ fig2 ] , as the sum of the constituting partonic sub - processes ( see the relation ( [ eq2 ] ) ) . the calculations are carried out for the center - of - mass energy @xmath38 and plotted as a function of the transverse momentum of the produced particle . in the panels ( a ) and ( c ) , the contributions from the individual sub - processes have been compared to each other . the results in these panels respectfully correspond to the forward rapidity region , @xmath99 ( with the addition of @xmath104 and @xmath105 constraints , corresponding for the experimental measurements of the @xmath0 collaboration , the reference @xcite ) and to the central rapidity region , @xmath100 ( with the addition of @xmath106 and @xmath105 constraints , corresponding for the perlimanary measurements of the @xmath1 collaboration , the reference @xcite ) . the calculations have been performed , using the @xmath9 @xmath5 and the @xmath41 of @xmath83 . the panels ( b ) and ( d ) illustrate our results in their corresponding uncertainty bounds , compared to the data of the @xmath0 and the @xmath1 collaborations . the uncertainty bounds have been calculated , by means of manipulating the hard - scale , @xmath67 , of the @xmath5 by a factor of 2 , since this is the only free parameter in our framework . also , as expected for the both regions , the contributions from the @xmath107 sub - process dominate , @xmath108 the figure [ fig3 ] presents the differential cross - section of the production of @xmath2 vector boson , @xmath109 , as a function of the rapidity of the produced boson ( @xmath110 ) at the center - of - mass energy of @xmath38 in the @xmath9 formalism . the notion of the figure is similar to that of the figure [ fig2 ] : the panels ( a ) and ( c ) illustrate the contributions of each of the sub - processes into the total production rate , while the total results have been subjected to comparison with the experimental data of the @xmath0 and the @xmath1 collaborations ( the references @xcite ) , within their corresponding uncertainty bounds , in the panels ( b ) and ( d ) . one finds that our calculations are in general agreement with the experimental measurements . overall , it appears that our @xmath43 framework is generally successful in describing the corresponding experimental measurements in the explored energy range . this success if by part owed to the @xmath5 of @xmath9 , which as an effective model , has been very successful in producing a realistic theory in order to describe the experiment , see the references @xcite . one however should note that having a semi - successful prediction from the framework of @xmath12-factorization by itself is a success , since our calculations utilizing these @xmath5 have inherently a considerably larger error compared to those from the @xmath111 @xmath44 or even the @xmath4 @xmath44 , presented here by the relatively large uncertainty region . this is because we are incorporating the single - scaled @xmath41 ( with their already included uncertainties ) to form double - scaled @xmath5 with additional approximations and further uncertainties . being able to provide predictions with a desirable accuracy would require a thorough universal fit for these frameworks , see the reference @xcite . nevertheless , the @xmath12-factorization framework , despite its simplicity and its computational advantages , see the reference @xcite , can provide us with a valuable insight regarding the transverse momentum dependency of various high - energy @xmath44 events . in summary , throughout the present work , we have calculated the production rate of the @xmath2 gauge vector boson in the framework of @xmath12-factorization , using a @xmath43 framework and the @xmath5 of the @xmath9 formalism . the calculations have been compared with the experimental data of the @xmath0 and the @xmath1 collaborations . our calculation , within its uncertainty bounds , are in good agreement with the experimental measurements . we also reconfirm that the @xmath9 prescription , despite its theoretical disadvantages and its simplistic computational approach , has a remarkable behavior toward describing the experiment . @xmath112 would like to acknowledge the research council of university of tehran and the institute for research and planning in higher education for the grants provided for him . @xmath113 sincerely thanks n. darvishi for valuable discussions and comments . lhcb collaboration , r. aaij et al . , jhep 06 ( 2012 ) 058 . lhcb collaboration , r. aaij et al . , jhep 02 ( 2013 ) 106 . lhcb collaboration , r. aaij et al . , jhep 08 ( 2015 ) 039 . lhcb collaboration , jhep 05 ( 2015 ) 109 , arxiv:1503.00963 . lhcb collaboration , r. aaij et al . , jhep 01 ( 2015 ) 155 . atlas collaboration , phys.rev.lett . 109 ( 2012 ) 012001 . atlas collaboration , phys.rev.d 91 ( 2015 ) 052005 . atlas collaboration , georges aad et al . phys . j. c 76(5 ) ( 2016 ) 1 - 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neutrino oscillation measurements @xcite have established evidence for tiny neutrino masses , which are supposed to be zero in the standard model ( sm ) of particle physics . it seems mysterious that the scale of neutrino masses is much smaller than that of the other fermion masses . the simplest way to obtain tiny neutrino masses is the seesaw mechanism @xcite where right - handed neutrinos are introduced . due to suppression with huge majorana masses of the right - handed neutrinos as compared to the electroweak scale , neutrino masses can be very small even though dirac yukawa coupling constants for neutrinos are of @xmath7 . however testability of the mechanism seems to be a problem because key particles ( right - handed neutrinos ) with such huge masses would not be accessible in future experiments . a possible solution for the problem is radiative generation of dirac yukawa couplings for neutrinos . in order to explain tiny neutrino masses , the right - handed neutrinos with masses of @xmath8 are acceptable naturally without assuming excessive fine tuning among coupling constants by virtue of loop - suppressed dirac yukawa couplings . radiative generation of dirac yukawa couplings has been discussed in various frameworks such as the left - right symmetry @xcite , supersymmetry ( susy ) @xcite , extended models within the sm gauge group @xcite , and an extra @xmath2 gauge symmetry @xcite ( see also ref . . on the other hand , existence of dark matter has been indicated by zwicky @xcite , and its thermal relic abundance has been quantitatively determined by the wmap experiment @xcite . if the essence of the dark matter is an elementary particle , the weakly interacting massive particle ( wimp ) would be a promising candidate . a naive power counting shows that the wimp dark matter mass should be at the electroweak scale . this would suggest a strong connection between the wimp dark matter and the higgs sector . it is desired to have viable candidate for dark matter in models beyond the standard model . usually stability of the dark matter candidate is ensured by imposing a @xmath1 parity where the dark matter is the lightest @xmath1-odd particle . it is well known that such @xmath1 odd particles are compatible with radiative neutrino mass models @xcite . usually in such models , however , the origin of the @xmath1 parity has not been clearly discussed . it seems better if a global symmetry to stabilize the dark matter is not just imposed additionally but obtained as a remnant of some broken symmetry which is used also for other purposes @xcite . in this paper we propose a new model in which tiny neutrino masses and the origin of dark matter are naturally explained at the tev - scale . we introduce the @xmath0 gauge symmetry to the sm gauge group which is spontaneously broken at multi - tev scale @xcite . its collider phenomenology has been studied @xcite . in our model , dirac yukawa couplings for neutrinos are forbidden at the tree level by the @xmath0 . they are generated at the one - loop level after the @xmath0 breaking . simultaneously , right - handed neutrino masses are generated at the tree level by spontaneous breaking of the @xmath0 . as a result , light neutrino masses are obtained effectively at the two - loop level without requiring too small coupling constants ( @xmath9 ) from the tev - scale physics . furthermore it turns out that the dark matter in our model is stabilized by an unbroken global @xmath2 symmetry which appears automatically in the lagrangian with appropriate assignments of the @xmath0-charges for new particles . the mass of a dark matter candidate @xmath3 , which is a dirac fermion , is also generated by the @xmath0 breaking ( see also ref . we show that the model is viable under the current experimental constraints . prospects in collider experiments are also discussed . @xmath10 & @xmath11 & @xmath12 & @xmath13 & @xmath14 & @xmath15 + su(3)@xmath16 & * * & * * & * * & * * & * * & * * + su(2)@xmath17 & * * & * * & * * & * * & * * & * * + u(1)@xmath18 & 0 & @xmath19 & 0 & 0 & 0 & 0 + u(1)@xmath20 & @xmath19 & @xmath19 & @xmath21 & @xmath22 & @xmath23 & @xmath24 + in our model , the @xmath0 gauge symmetry is added to the sm gauge group . new particles and their properties under gauge symmetries of the model are shown in table [ table : particle ] . fields @xmath10 , @xmath11 , and @xmath15 are complex scalars while @xmath25 , @xmath26 , and @xmath27 are weyl fermions . all of them except @xmath11 [ @xmath28 are singlet fields under the sm gauge group . the sm higgs doublet field @xmath29 [ @xmath30 and @xmath11 have different @xmath0-charges although their representations for the sm gauge group are the same . notice that mass terms of @xmath31 , @xmath32 , and @xmath33 are forbidden by the @xmath0 symmetry . yukawa interactions are given by @xmath34 where @xmath35 stands for the yukawa interactions in the sm and @xmath36 are the lepton doublet fields of flavor @xmath37 . superscript @xmath38 means the charge conjugation and @xmath39 ( @xmath40 ) are the pauli matrices . we take a basis where yukawa matrices @xmath41 and @xmath42 are diagonalized such that their real positive eigenvalues satisfy @xmath43 and @xmath44 . scalar potential in this model is expressed as @xmath45 where @xmath46 , @xmath47 , @xmath48 , and @xmath49 are positive values . the coupling constant @xmath50 of the trilinear term can be taken as a real positive value by redefinition of phase of @xmath10 . the @xmath50 is the breaking parameter for a global @xmath51 which conserves difference between the @xmath11 number and the sm lepton number . some coupling constants in the potential are constrained by the tree - level unitarity @xcite . notice that this model has a global @xmath2 symmetry ( we refer to it as @xmath52 ) of which @xmath10 , @xmath11 , @xmath31 , and @xmath32 have the same charge and others have no charge . because of the global @xmath52 symmetry , the lagrangian is not changed by an overall shift of @xmath0-charges with an integer for the @xmath52-charged particles . the @xmath0 is broken spontaneously by the vacuum expectation value ( vev ) of @xmath15 , @xmath53 [ @xmath54 while the @xmath55 is broken to @xmath56 by the vev of @xmath57 , @xmath58 [ @xmath59 . by imposing the stationary condition , @xmath53 and @xmath58 are determined as @xmath60 the gauge boson @xmath6 of @xmath0 acquires its mass as @xmath61 , where @xmath62 denotes gauge coupling constant of @xmath0 . a constraint @xmath63 is given by precision tests of the electroweak interaction @xcite . furthermore , right - handed neutrinos @xmath14 obtain majorana masses @xmath64 [ @xmath65 while @xmath25 and @xmath26 for each @xmath66 become a dirac fermion @xmath67 with its mass @xmath68 [ @xmath69 . since the global @xmath52 is not broken by @xmath53 , the lightest @xmath52-charged particle is stable . notice that there is no anomaly for the @xmath52 because @xmath25 and @xmath26 have the same @xmath52-charge . if the particle is electrically neutral ( @xmath3 or a mixture of @xmath10 and @xmath70 ) , it becomes a candidate for the dark matter . after symmetry breaking with @xmath53 and @xmath58 , mass eigenstates of two cp - even scalars and their mixing angle @xmath71 are given by @xmath72 where @xmath73 and @xmath74 . it is needless to say that @xmath75 and @xmath76 are nambu - goldstone bosons which are absorbed by @xmath77 and @xmath6 , respectively . masses of @xmath78 and @xmath79 are defined by @xmath80 on the other hand , since @xmath10 and @xmath70 are @xmath52-charged particles , they are not mixed with @xmath15 and @xmath57 . mass eigenstates of these @xmath52-charged scalars and their mixing angle @xmath81 are obtained as @xmath82 mass eigenvalues @xmath83 and @xmath84 of these neutral complex scalars are defined by @xmath85 where @xmath86 and @xmath87 . finally , the mass of charged scalars @xmath88 is @xmath89 + in this model , dirac mass terms for neutrinos are generated by a one - loop diagram in fig . [ fig:1-loop ] . this diagram is used also in a model in ref . @xcite in which lepton number is conserved . via the seesaw mechanism , tiny majorana masses of light neutrinos are induced at the two - loop level as shown in fig . [ fig : neutrinomass](a ) ( see also refs . in addition , there are one - particle - irreducible ( 1pi ) diagrams also at the two - loop level ( figs . [ fig : neutrinomass](b ) and [ fig : neutrinomass](c ) ) . the majorana mass matrix is calculated as @xmath90 \nonumber\\ & & \hspace{40 mm } { } + \sum_{i , j , a } f_{\ell i}\ , ( y_3^\dagger)_{ia}\ , ( m_r^{})_a\ , ( y_3^\ast)_{aj}\ , ( f^t)_{j{{\ell^\prime } } } \left\ { i_3 \right\}_{ija } \biggr\ } , \label{eq : neutrinomass}\end{aligned}\ ] ] where dimensionless functions @xmath91 , @xmath92 , and @xmath93 are defined by @xmath94 \nonumber\\ & & \hspace{40 mm } \times \left [ \int\frac{d^4k_2}{(2\pi)^4 } \frac{1 } { k_2 ^ 2 - m_{\psi_j}^2 } \left\ { \frac{1 } { k_2 ^ 2 - m_{s^0_1}^2 } - \frac{1 } { k_2 ^ 2 - m_{s^0_2}^2 } \right\ } \right ] , \\ ( i_2)_{ija } & = & ( 8\pi^2 \sin2\theta ) ^2 m_{\psi_i}^ { } m_{\psi_j}^ { } \nonumber\\ & & \times \int\!\!\!\int\frac{d^4k_1}{(2\pi)^4}\frac{d^4k_2}{(2\pi)^4 } \left\ { \frac{1 } { k_1 ^ 2 - m_{s^0_1}^2 } - \frac{1 } { k_1 ^ 2 - m_{s^0_2}^2 } \right\ } \frac{1}{k_1 ^ 2-m_{\psi_i}^2 } \nonumber\\ & & \times \frac{1}{(k_1-k_2)^2-(m_r^{})_a^2 } \left\ { \frac{1 } { k_2 ^ 2 - m_{s^0_1}^2 } - \frac{1 } { k_2 ^ 2 - m_{s^0_2}^2 } \right\ } \frac{1}{k_2 ^ 2 - m_{\psi_j}^2 } , \\ ( i_3)_{ija } & = & ( 8\pi^2 \sin2\theta ) ^2 \int\!\!\!\int\frac{d^4k_1}{(2\pi)^4}\frac{d^4k_2}{(2\pi)^4 } k_1\cdot k_2 \left\ { \frac{1 } { k_1 ^ 2 - m_{s^0_1}^2 } - \frac{1 } { k_1 ^ 2 - m_{s^0_2}^2 } \right\ } \frac{1}{k_1 ^ 2-m_{\psi_i}^2 } \nonumber\\ & & \times \frac{1}{(k_1-k_2)^2-(m_r^{})_a^2 } \left\ { \frac{1 } { k_2 ^ 2 - m_{s^0_1}^2 } - \frac{1 } { k_2 ^ 2 - m_{s^0_2}^2 } \right\ } \frac{1}{k_2 ^ 2 - m_{\psi_j}^2 } , \end{aligned}\ ] ] which correspond to the diagrams in figs . [ fig : neutrinomass](a ) , [ fig : neutrinomass](b ) , and [ fig : neutrinomass](c ) , respectively . if there is only one @xmath95 ( one @xmath32 and one @xmath31 ) , the mass matrix @xmath96 is proportional to @xmath97 . then two of three eigenvalues of @xmath96 are zero and the mass matrix conflicts with the oscillation data . therefore two @xmath67 are introduced in this model . we also introduce two @xmath14 in order for an easy search of parameter sets which satisfy experimental constraints , one @xmath33 is sufficient for that the rank of @xmath96 is two . ] . then the rank of @xmath96 is two , for which one neutrino becomes massless . hereafter degeneracy of right - handed neutrino masses , @xmath98 , is assumed for simplicity . the mass matrix @xmath96 is diagonalized by the maki - nakagawa - sakata ( mns ) matrix @xmath99 as @xmath100 . the standard parametrization of the mns matrix is @xmath101 where @xmath102 and @xmath103 stand for @xmath104 and @xmath105 , respectively . mixing angles @xmath106 and @xmath107 are constrained by neutrino oscillation measurements @xcite . in our analyses we use the following values as an example ; @xmath108 notice that there is no difficulty to use nonzero values of @xmath109 @xcite in our analyses . in table [ table : parameter ] , we show two examples ( set a and set b ) for the parameter set which reproduces the values given in eqs . and for @xmath110 . these sets satisfy also other experimental constraints as shown below . @xmath111 & ' '' '' @xmath112 & @xmath113 + @xmath114 & ' '' '' @xmath115 & @xmath116 + @xmath117 & ' '' '' @xmath118 & @xmath119 + @xmath98 & @xmath120 & @xmath121 + @xmath122 & @xmath123 & @xmath124 + @xmath125 & @xmath126 & @xmath127 + @xmath128 & @xmath129 & @xmath130 + @xmath131 & @xmath132 & @xmath133 + @xmath62 & @xmath134 & @xmath134 + @xmath135 & @xmath136 & @xmath136 + charged scalar bosons @xmath88 , which also have a @xmath52-charge , contribute to the @xmath137 process . the branching ratio of @xmath137 is calculated as @xmath138 where @xmath139 for set a and set b , we obtain @xmath140 and @xmath141 , respectively . they satisfy the current upper bound ; @xmath142 ( 90% cl ) @xcite . these values for set a and set b could be within the future experimental reach . the dark matter candidate is @xmath3 ( for set a ) or @xmath4 ( for set b ) . the relic abundance of dark matter is constrained stringently by the wmap experiment as @xmath143 @xcite . the dark matter candidate in this model pair - annihilates into a pair of sm fermions @xmath144 by @xmath145-channel processes mediated by @xmath78 and @xmath79 for both of set a and set b. the @xmath146-channel diagram is highly suppressed due to the small values of @xmath111 , which are required by the @xmath137 search results . we first consider the case where @xmath3 is the dark matter , i.e. set a , whose mass is given by @xmath147 . because @xmath63 , the magnitude of @xmath148 is @xmath149 for @xmath150 . the annihilation cross section is suppressed by the small @xmath148 because it is proportional to @xmath151 . in order to enhance the cross section for the appropriate relic abundance , a large mixing between @xmath152 ( which couples with @xmath3 ) and @xmath153 ( which couples with sm fermion @xmath144 ) is required @xcite . thus we take @xmath154 for set a. then the wmap result gives a constraint on the dark matter mass @xmath155 for fixed @xmath156 and @xmath157 ( @xmath158 and @xmath159 for set a , respectively ) . figure [ fig : wmap](a ) shows dependence of the relic abundance on @xmath155 where other parameters are the same as values of set a. it can be confirmed in the figure that our choice @xmath160 is consistent with the wmap result for set a. next , we consider the case where @xmath4 is the dark matter . a coupling constant @xmath161 for @xmath162 is constrained by the wmap result . figure [ fig : wmap](b ) shows the relic abundance of @xmath4 as functions of @xmath83 for several values of @xmath161 . in the figure , we used @xmath163 and @xmath164 which are the values for set b. contribution of @xmath79 to the annihilation cross section is negligible because @xmath157 is taken to be away from @xmath165 for simplicity . in order to satisfy the wmap constraint for set b , we find that @xmath166 this constraint can be satisfied easily because @xmath161 can be taken to be arbitrary depending on several parameters in the scalar potential . finally , we discuss the constraint from direct search experiments for the dark matter . if @xmath4 is mainly composed of @xmath70 , it can not be a viable candidate for the dark matter even if it is the lightest @xmath52-charged particle . this is because the scattering cross section with a nucleon @xmath167 ( @xmath168 ) becomes too large due to the weak interaction . thus @xmath4 should be dominantly made from singlet @xmath10 , and this is the reason why we take @xmath169 for set b. scattering cross sections of dark matter candidates ( @xmath3 and @xmath4 ) with a nucleon @xmath167 ( @xmath168 ) are given by @xmath170 where @xmath171 is the mass of the nucleon and we use @xmath172 , @xmath173 @xcite , and @xmath174 @xcite . our results for the @xmath6 mediation are consistent with those in ref . @xcite . difference between @xmath175 and @xmath176 is neglected . we have @xmath177}$ ] for @xmath160 for set a. the value is dominantly given by the @xmath6 mediation while scalar mediations give only @xmath178}$ ] . on the other hand , for @xmath179 for set b , we have @xmath180}$ ] by taking into account eq . as the wmap constraint . contributions from @xmath6 and @xmath78 are @xmath181}$ ] and @xmath182}$ ] , respectively . these values of cross sections for two sets are just below the constraint from the xenon100 experiment @xcite . notice that even if such values are excluded in near future this model is not ruled out because the @xmath6 contribution ( @xmath183 ) can be easily suppressed by a little bit larger @xmath53 . we have seen that set a and set b satisfy experimental constrains in the previous section . let us discuss the collider phenomenology by using these parameter sets . since @xmath0 is dealt with as the origin of neutrino masses etc . , @xmath6 is an important particle in this model . the @xmath6 can be the mother particle to produce the @xmath52-charged particles and @xmath33 in the model at collider experiments because they are all @xmath0-charged . for @xmath184 and @xmath185 , the production cross section of @xmath6 at the cern large hadron collider ( lhc ) with @xmath186 is @xmath187 @xcite . the number of @xmath6 produced with @xmath188 is @xmath189 . branching ratios of the @xmath6 decay are shown in table [ table : brzp ] for set a and set b. similar @xmath190 are predicted for the two sets because @xmath6 is sufficiently heavier than the others . the large @xmath191 ( cf . @xmath192 in the sm ) could be utilized for discovery of the @xmath6 at the lhc . the @xmath5 nature of the @xmath6 would be tested if @xmath193 would be confirmed . & @xmath194 & @xmath195 & @xmath196 & @xmath197 & @xmath198 & @xmath199 & @xmath200 & @xmath201 & @xmath202 + set a & @xmath203 & @xmath204 & @xmath205 & @xmath206 & @xmath207 & @xmath208 & @xmath209 & @xmath210 & @xmath210 + set b & @xmath211 & @xmath212 & @xmath213 & @xmath206 & @xmath214 & @xmath214 & @xmath215 & @xmath209 & @xmath209 + since each @xmath52-charged particle decays finally into a dark matter , @xmath216 of the @xmath6 gives a pair of the dark matter ( @xmath198 ) for set a while @xmath217 of the @xmath6 produces @xmath200 for set b. since @xmath111 are preferred to be small in order to satisfy the constraint on @xmath218 , @xmath219 for set a ( @xmath3 and @xmath219 for set b ) decays into @xmath220 with @xmath114 and @xmath117 . subsequently @xmath33 dominantly decays into @xmath221 or @xmath222 ( see table [ table : brnr ] ) , and thus it gives a visible signal . for set a , @xmath223 ( @xmath224 ) decays invisibly into @xmath225 with @xmath226 , and the @xmath4 ( @xmath227 ) decays also invisibly into @xmath225 with @xmath228 . for set b , @xmath223 decays into @xmath229 with @xmath50 . it is clear that @xmath88 provide visible signals with @xmath230 for set a and @xmath231 for set b. as a result , about @xmath232 ( @xmath233 ) of the @xmath6 for set a ( b ) is invisible and the constraint on @xmath234 becomes milder . if a light @xmath6 ( @xmath235 ) is discovered at the lhc by @xmath236 , this model could be tested further at future linear colliders by measuring the amount of invisible decay of the @xmath6 . & @xmath221 & @xmath237 & @xmath238 & @xmath239 + set a & @xmath240 & @xmath241 & @xmath206 & @xmath242 + set b & @xmath243 & @xmath241 & @xmath211 & @xmath244 + it is a good feature of this model that a light @xmath33 is acceptable naturally because of loop - suppressed dirac mass terms . for set a , we see that @xmath245 and @xmath246 followed by @xmath247 produce about 1200 pairs of @xmath33 from @xmath189 of @xmath6 . for set b , the number of @xmath33 pairs increases to about 1700 because of an additional contribution from the decay of @xmath3 . the @xmath33 decays into @xmath221 , @xmath237 , @xmath238 , and @xmath239 through mixing due to the 1-loop induced dirac yukawa coupling . for set a and set b , @xmath248 is the main decay mode as shown in table [ table : brnr ] . then , the majorana mass of @xmath33 can be reconstructed by observing the invariant mass of the @xmath249 @xcite . the @xmath78 produced from @xmath33 will be energetic due to the helicity structure . therefore it would be possible to test the existence of @xmath33 by search for energetic @xmath250 . if @xmath3 is the dark matter , the yukawa coupling constant @xmath251 [ @xmath252 for the decay @xmath253 should be small because of small @xmath155 . for set a , we have @xmath254 . then main decay mode of @xmath78 @xmath255 is @xmath256 similarly to the sm case . since @xmath257 is required to obtain the appropriate relic abundance of @xmath3 , the main decay mode of @xmath79 is also @xmath258 . their decay widths are about a half of the width of the sm higgs boson . the large mixing prefers that @xmath156 and @xmath157 are of the same order of magnitude . thus we would find two sm - like higgs bosons whose masses are @xmath8 , e.g. @xmath259 and @xmath260 for set a. on the other hand , if @xmath4 is the dark matter , the interaction of @xmath78 with dark matter @xmath4 should satisfy eq . . then the invisible decay @xmath261 dominates for @xmath262 . we have @xmath263 for set b where @xmath78 is @xmath264 sm - like . therefore , even if @xmath78 is not discovered at the lhc within a year , a light @xmath78 might exist because the constraint on @xmath156 is relaxed from the one for the sm higgs boson . if such a light @xmath78 is discovered late with smaller number of signals than the sm expectation , this model would be confirmed at the lhc @xcite and future linear colliders @xcite by `` observing '' the @xmath78 invisible decay , the higgs portal dark matter such as the @xmath4 in this scenario would be able to be tested at the lhc @xcite and future linear colliders @xcite . ] . in our paper , we have assumed the mass of the @xmath265 boson to be @xmath266 . it is expected that the lower bound on the mass will be more and more stringent due to the new results from the lhc and that the above value of the mass would be excluded in near future . in such a case , the mass should be taken to be higher values than @xmath266 . then , the branching ratio of @xmath267 becomes closer values to that of @xmath268 because the effect of their mass difference becomes less significant . heavy @xmath265 is achieved by assuming larger values of @xmath62 or assuming larger values of @xmath269 . for the former case , most of the phenomenological analyses are unchanged . for the latter case , our phenomenological analysis would be changed slightly . even in this case , the experimental constraints from neutrino experiments and the @xmath270 results can be satisfied by using smaller values of @xmath271 and @xmath272 which keep @xmath64 and @xmath68 unchanged from values in our sets a and b. in the scenario of set a , a smaller value of @xmath148 results in a larger value of the dm abundance which is proportional to @xmath273 . even if the red curve in fig . [ fig : wmap](a ) goes up by about a factor of 10 , the wmap result can still be explained by @xmath274 . therefore , we can accept three times larger @xmath53 ( namely , three times smaller @xmath148 ) than the value ( @xmath275 ) we used . a shortage is about the gauge anomaly for @xmath0 . it is well known that @xmath0 is free from anomaly if three singlet fermions of @xmath276 ( usual right - handed neutrinos ) are added to the sm . however , we have introduced not three but only two @xmath14 , and their @xmath5 is not @xmath277 but @xmath278 . there are also extra @xmath0-charged fermions ( @xmath26 and @xmath25 ) . therefore the @xmath0 gauge symmetry has anomalies for the triangle diagrams of @xmath279 ^ 3 $ ] and @xmath279\times [ \text{gravity}]^2 $ ] . they would be resolved by some heavy singlet fermions with appropriate @xmath5 ( see e.g. , ref . @xcite ) . another is the way to reproduce the baryon asymmetry of the universe . since we have used only particles below the tev - scale , leptogenesis @xcite does not work in a natural way . heavy fermions to eliminate the @xmath0 gauge anomaly might help . the electroweak baryogenesis @xcite would be accommodated by the introduction of an additional higgs doublet to the model so that new source of cp - violation appears in the higgs potential . we have presented the tev - scale seesaw model in which @xmath0 gauge symmetry can be the common origin of neutrino masses , the dark matter mass ( if @xmath3 is the one ) , and stability of the dark matter . light neutrino masses are generated by the two - loop diagrams which are also contributed by the dark matter , a dirac fermion @xmath3 or a complex scalar @xmath4 . the symmetry to stabilize the dark matter appears in the @xmath0-symmetric lagrangian without introducing additional global symmetry ( ex . a @xmath1 symmetry ) . it has been shown that this model can be compatible with constraints from the neutrino oscillation data , the search for @xmath137 , the relic abundance of the dark matter , and the direct search results for dark matter . it should be emphasized that these constraints are satisfied with sizable coupling constants ( @xmath280 ) and new particles ( including @xmath33 ) whose masses are at or below the tev - scale . we have discussed collider phenomenology in this model by using two sets of parameters which satisfy constraints from the current experimental data . the @xmath0 gauge boson @xmath6 can be discovered at the lhc by observing @xmath236 if it is not too heavy . in our model , since the dark matter has @xmath0 charge , @xmath6 partially decays into a pair of the dark matter . as a result , more than @xmath232 of produced @xmath6 is invisible for both the sets . then a lighter @xmath6 is allowed than the usual experimental bound . detailed studies for such a @xmath6 would be performed at future linear colliders . since masses of @xmath14 and @xmath67 are obtained by the @xmath0 breaking , it would be natural that @xmath3 is light and becomes the dark matter when we assume @xmath33 masses are of @xmath8 . in this case , a large mixing between @xmath78 and @xmath79 is required for the appropriate relic abundance because the yukawa coupling constant is small . decay branching ratios of @xmath78 and @xmath79 are almost the same as the one for the sm higgs boson . therefore two sm - like higgs bosons with similar masses ( ex . @xmath259 and @xmath260 for set a ) would be discovered at the lhc . on the other hand , if @xmath4 is the dark matter , the decay of the lighter higgs boson @xmath78 can be dominated by invisible @xmath261 . for set b , we obtain @xmath281 . then this model would be tested at future linear colliders by measuring the amount of the invisible decay . this work was supported in part by grant - in - aid for scientific research , japan society for promotion of science ( jsps ) , nos . 22244031 and 23104006 ( s.k . ) and no . 23740210 ( h.s . ) . the work of h.s . was supported by the sasakawa scientific research grant from the japan science society . b. t. cleveland _ et al . _ , astrophys . j. * 496 * , 505 ( 1998 ) ; 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di girolamo b and neukermans l , 2003 atlas note atl - phys-2003 - 006 .	we show a tev - scale seesaw model where majorana neutrino masses , the dark matter mass , and stability of the dark matter can be all originated from the @xmath0 gauge symmetry . dirac mass terms for neutrinos are forbidden at the tree level by @xmath0 , and they are induced at the one - loop level by spontaneous @xmath0 breaking . the right - handed neutrinos can be naturally at the tev - scale or below because of the induced dirac mass terms with loop suppression . such right - handed neutrinos would be discovered at the cern large hadron collider ( lhc ) . on the other hand , stability of the dark matter is guaranteed without introducing an additional @xmath1 symmetry by a remaining global @xmath2 symmetry after the @xmath0 breaking . a dirac fermion @xmath3 or a complex neutral scalar @xmath4 is the dark matter candidate in this model . since the dark matter ( @xmath3 or @xmath4 ) has its own @xmath5 charge , the invisible decay of the @xmath0 gauge boson @xmath6 is enhanced . experimental constraints on the model are considered , and the collider phenomenology at the lhc as well as future linear colliders is discussed briefly .
the campbell - magaard theorem@xcite@xcite states that any analytic solution of einstein s equations in @xmath8-dimensions can be locally embedded in a @xmath9-dimensional ricci - flat manifold . it was noted@xcite@xcite that this theorem provides a mathematical support to the @xmath10-dimensional space - time - matter ( stm ) theory@xcite@xcite in which the @xmath4 manifold is ricci - flat with @xmath11 while the @xmath1 matter is induced from the @xmath4 vacuum . many solutions of this type have been studied within the framework of the stm theory . here , in this paper , we are going to study a class of @xmath4 ricci - flat cosmological solutions which was firstly presented by liu and mashhoon@xcite and restudied latter by liu and wesson@xcite . this class of solutions is algebraically rich because it contains two arbitrary functions of the time @xmath5 . it was shown@xcite that two major properties characterize these @xmath4 models : firstly , the @xmath1 induced matter could be described by a perfect fluid plus a variable cosmological term , and by properly choosing the two arbitrary functions both the radiation and matter dominated standard frw models could be recovered . secondly , the big bang singularity of the @xmath1 standard cosmology could be replaced by a big bounce at which the size of the universe is finite and the universe contracts before the bounce and expands after the bounce . many further studies could be found in literature such as those about the embeddings of the @xmath4 solutions to brane modelsseahra@xcite , about the accelerating expansion and dark energy of the @xmath4 universe@xcite@xcite , and about the isometry between the @xmath4 solutions and @xmath4 black holes@xcite . we aware that it is the two arbitrary functions that makes the @xmath4 solutions mathematically rich in giving various cosmological models and various normal and non - normal induced matters . therefore , it is useful to look for physical constraints to fix these two arbitrary functions and to study physical properties of various different solutions . this paper is organized as follows : in section ii , we will use the @xmath0 split technique to derive the @xmath1 induced matter . in section iii , we will use four known energy conditions ( strong , weak , null , and dominant ) to study properties of the @xmath1 induced matter . section iv is a short conclusion . the @xmath4 ricci - flat cosmological solutions read @xmath12@xmath13@xmath14where @xmath15 , @xmath16 and @xmath17 are two arbitrary functions , @xmath18 is the 3d curvature index ( @xmath19 ) , @xmath20 is a constant . because the @xmath4 manifold ( [ line element])-([b ] ) is ricci - flat , we have @xmath21 , @xmath22 , and@xmath23so @xmath20 is related to the @xmath4 curvature . the general @xmath0 split in stm theory has been given in ref . , where it was shown that the @xmath24 @xmath4 equations @xmath11 decompose into @xmath25 @xmath1 equations : ten for the @xmath1 ricci tensor , four for the gauss - codazzi equations , and one for the scalar field . for our @xmath4 metric ( [ line element ] ) we see that the @xmath1 hypersurfaces @xmath26 are @xmath27 constant and the normal vectors to @xmath26 are@xmath28so the projection tensor is @xmath29the extrinsic curvature tensor @xmath30 , which describes the rate of changes of @xmath26 as it moves in the normal direction , is defined by @xmath31where @xmath32 is the @xmath4 covariant derivative operator . meanwhile , the lapse function @xmath33 and the shift vector @xmath34 . these greatly simplifies the calculations for @xmath30 . using ( [ n - a ] ) and ( h - ab ) we find@xmath35with @xmath36the @xmath1 ricci tensor @xmath37 on the @xmath1 hypersurface @xmath26 can be expressed in terms of @xmath38 by @xmath39where @xmath40 . substituting the exact @xmath4 solutions ( [ line element ] ) - ( [ b ] ) into ( [ kab3 ] ) to calculate @xmath41 and then using @xmath38 in ( [ 4dricci ] ) , we obtain the non - vanishing components of @xmath37 being @xmath42 . \label{r0123}\end{aligned}\]]so the @xmath1 einstein tensor @xmath43 @xmath44 become @xmath45according to einstein theory of general relativity , this @xmath46 implies an effective or induced @xmath1 energy - momentum tensor @xmath47 . let @xmath48 and suppose @xmath49 being described by a perfect fluid with density @xmath2 and pressure @xmath3 , @xmath50where @xmath51 is the @xmath52-velocity with @xmath53 and @xmath54 , then we find @xmath55thus we rederived the same results for the @xmath1 induced matter as those in ref . . from the @xmath4 solutions ( [ line element ] ) - ( [ b ] ) one can see that if the two arbitrary functions @xmath56 and @xmath57 are specified , the two scale factors @xmath58 and @xmath59 and then the evolution of the universe are fixed immediately . then the cosmic induced matter @xmath2 and @xmath3 can be calculated with use of ( [ rho , p ] ) . meanwhile , the @xmath1 bianchi identities @xmath60 lead to the @xmath1 conservation laws @xmath61 which give@xmath62as is in the standard @xmath1 cosmology . the energy conditions of einstein s general relativity are designed to extract as much information as possible from the field equations . they were used in deriving many theorems such as the singularity theoremshowking , the censorship theorem@xcite and so on . in this section , we wish to use the energy conditions to study the properties of the induced matter of the @xmath4 ricci - flat cosmological solutions ( [ line element ] ) - ( [ b ] ) . the standard classical energy conditions are the null energy condition ( nec ) , weak energy condition ( wec ) , strong energy condition ( sec ) , and dominant energy condition ( dec ) . basic definitions of these energy conditions can be found in ref . . for the case in cosmology they are@xcite@xcite @xmath63here , for simplicity , we are not going to discuss models which contain more than one components of matter . we just consider the case where the universe is dominated by just one fluid with @xmath2 and @xmath3 . without loss of generality , we write the equation of state as@xmath64because both @xmath2 and @xmath3 are functions of @xmath5 and @xmath65 in the @xmath4 model , so @xmath6 is , in general , a function of @xmath5 and @xmath65 as well , @xmath66 . substituting ( [ eos ] ) into the four energy conditions ( [ nec ] ) - ( dec ) , we get@xmath67 following previous usage ( see , for example , ref . - ) we call matter that satisfies all the four energy conditions `` normal '' and call matter that specifically violates the sec `` abnormal '' . and we call matter that violates any one of the four energy conditions `` non - normal '' . in the following we will discuss both the normal and non - normal matter separately . for normal matter all the four standard energy conditions should be satisfied . we can easily show that ( [ nec - w ] ) - ( [ dec - w ] ) are equivalent to@xmath68with use of these constraints in the @xmath4 induced matter ( [ rho , p ] ) we obtain@xmath69meanwhile , with use of the definition of the proper time , @xmath70 , and the relation @xmath71 , the hubble and deceleration parameters of the @xmath4 solutions are found to be@xmath72using ( [ h , q ] ) in ( [ mu-1 ] ) - ( [ mu-3 ] ) we get@xmath73we find that all the results in ( [ nm ] ) and ( [ hq-1 ] ) - ( [ hq-3 ] ) are completely the same as those in @xmath1 standard fwr cosmology for models dominated by normal matter . and we also arrive , from ( [ hq-2 ] ) , at a conclusion that _ the expansion of the universe is decelerating if the universe is dominated by normal matter , and so the gravitational force of normal matter is attractive_. this conclusion valid for both the @xmath1 standard frw cosmology and the @xmath4 ricci - flat cosmology . as we mentioned above that @xmath6 in ( [ eos ] ) is in general not a constant . now we assume @xmath6 being a constant for simplicity . ( [ rho , p ] ) lead to@xmath74integrating with respect to @xmath5 , we obtain@xmath75and the conservation law([conserv ] ) gives@xmath76let us return to the two arbitrary functions @xmath56 and @xmath57 of the @xmath4 solutions . from the @xmath4 solution ( [ a ] ) we can see that the second arbitrary function @xmath57 can always be solved out in terms of @xmath77 if we substituting ( [ mu2+k ] ) in ( [ a ] ) . this leaves us a freedom to consider the equation ( [ mu2+k ] ) alone . recall that the campbell - magaard theorem just says that the @xmath1 solutions of einstein equations can be embedded in a @xmath4 ricci - flat manifold _ _ locally ( _ _ not_globally)_. this reminds us to choose the time @xmath5 in such a way that the coordinate time @xmath5 approaches to the proper time asymptotically for a very large @xmath5 . consider the @xmath78 case for simplicity . then , to make @xmath79 , we must have , from ( [ mu2+k]),@xmath80then we get an approximate @xmath4 metric@xmath81this is a class of approximate solutions of the @xmath4 ricci - flat universe while the corresponding exact solutions are given in ( [ mu2+k ] ) and ( rho - a ) . now we calculate the deceleration parameter @xmath82 . we find that for both the exact ( with @xmath78 ) and approximate solutions , we get the same result,@xmath83as we mentioned above that the constant @xmath6 takes values from @xmath84 to @xmath85 for normal matter for which @xmath82 takes the values from @xmath86 to @xmath87 . so the expansion of the universe is decelerating for normal matter . we should also point out that this approximate solution is actually valid for any values of @xmath6 as we will discuss in the next subsection . thus we see from ( metricapp ) that the @xmath1 part of the @xmath4 universe asymptotically approaches to that of the standard frw cosmology in the case for a normal induced matter . let us consider the @xmath4 approximate solution ( [ mu - t ] ) - ( [ metricapp ] ) and their corresponding exact solutions in ( [ mu2+k ] ) and ( [ rho - a ] ) where @xmath6 takes values from @xmath84 to @xmath85 for normal matter . now we discuss the case for @xmath88 , for which at least one of the four energy conditions must be violated and so the induced matter is non - normal . from the four energy conditions ( [ nec - w ] ) - ( [ dec - w ] ) we can see that if @xmath7 , sec is violated while nec , wec , and dec are satisfied . this kind of matter is usually called abnormal matter . we can also see that if @xmath89 , all the four energy conditions , nec , wec , sec , and dec , are violated . in what follows we will discuss these two cases separately . for this case , eq . ( [ q - nonnor ] ) gives that @xmath90 . so the expansion of the universe is accelerating . thus we recover the @xmath4 cosmological scaling solution@xcite in which the @xmath1 induced matter is of a quintessence model of scalar field . in the quintessence dark energy model@xcite@xcite , @xmath6 is in the range [ @xmath91 , @xmath85 ] . so quintessence might be a normal matter if @xmath93 or abnormal matter if @xmath94 . this conclusion valid in both the @xmath1 standard and @xmath4 ricci - flat cosmological models . for this case we have @xmath95 . note that from ( [ mu - t ] ) we find that @xmath96 is negative and @xmath97 is decreasing , contrary to the assumption for an expanding universe . to resolve this problem , we find that , without violating their correctness as an approximate solution , the forms of ( [ mu - t ] ) and ( [ metricapp ] ) can be changed to@xmath98and@xmath99thus we recover the @xmath4 attractor solution@xcite in which the @xmath1 induced matter is described by a phantom model of the dark energy . as @xmath5 tends to @xmath100 , the scale factor @xmath77 tends to infinity implying that the universe will undergo a big rip and will expand to infinity within a finite time . here we see that all the four energy conditions are violated for phantom , the @xmath4 ricci - flat cosmological models are characterized by having a big bounce instead of a big bang . now we wish to know what kind of non - normal matter dominated the universe before and during the bounce . to make the discussion easier , we list an exact solution@xcite in the following . @xmath101 where @xmath102 is a constant . substituting this equation into ( [ a ] ) , ( b ) and ( [ rho , p ] ) , we obtain @xmath103 , \label{abounce}\]]@xmath104 ^{2}% \left [ 1+\left ( \frac{y+t_{c}}{t}\right ) ^{2}\right ] ^{-1 } , \label{bbounce}\]]and @xmath105 } , \label{rhobounce}\]]@xmath106 } -\frac{1}{% t^{2}\left [ 1+\left ( ( y+t_{c})/t\right ) ^{2}\right ] } . \label{pbounce}\ ] ] from eq.([abounce ] ) we can show that the scalar factor @xmath107 has a minimum point at @xmath108at which we have @xmath109here @xmath110 is the big bounce singularity . in this solution , the time @xmath5 is defined in the range @xmath111 , @xmath112 . generally , @xmath5 is not the proper time . however , when @xmath113 , we have @xmath114 and @xmath115 . so the coordinate time @xmath5 tends to the proper time for very large @xmath5 . the bounce is at @xmath116 . before the bounce means for @xmath117 . when @xmath118 , we have @xmath119 and @xmath120 . when changed to the proper time @xmath121 , we find @xmath122 . so @xmath118 corresponds to @xmath123 and @xmath119 . from this solution we can show that before the bounce ( for @xmath117 ) we have@xmath124we find that before the bounce all the four energy conditions , nec , wec , sec , and dec , are violated . meanwhile , because @xmath125 and @xmath126 , so @xmath127 . so , before the bounce , the non - normal induced matter has a large negative value and generates a repulsive force . then the bounce could be explained as due to this repulsive force and the induced matter could be explained as a phantom . we can also shown that for @xmath128 , we have @xmath125 , @xmath129 , @xmath130 , @xmath131 , and @xmath132 . so nec , wec and sec are satisfied while dec is violated . during this period , the induced cosmic matter is perhaps a super - luminal acoustic matter . in this paper we have used the @xmath0 split to derive the @xmath1 induced energy - momentum tensor of the @xmath4 ricci - flat cosmological solutions . then we have used the four energy conditions , nec , wec , sec , and dec , to discuss the physical properties of the induced matter and the various solutions of the @xmath4 models . firstly , we have shown that if the universe is dominated by normal matter , the expansion of the universe is decelerating and the @xmath1 part of the @xmath4 ricci - flat universe approaches asymptotically to the @xmath1 standard frw models in late times of the universe . if the universe is dominated by quintessence , the induced matter could be normal or abnormal , depending on the value of @xmath6 . if @xmath133 , all the four energy conditions are satisfied and the quintessence is a normal matter . if @xmath134 , only the sec is violated and so in this range the quintessence is abnormal and the expansion of the universe is accelerating . for phantom , all the four energy conditions are violated . we also find that before the bounce all the four energy conditions are violated , and so the induced matter could be explained as phantom and the bouncing could be explained as due to the repulsive force of the phantom . in the early time after the bounce , dec is violated while the other three are satisfied , so the induced cosmic matter in this period could be explained as a super - luminal acoustic matter . we would like to thank nef ( 10573003 ) and nbrp ( 2003cb716300 ) of p.r . china for financial support . sanjeev s. seahra , _ phys . d _ * 68 * , 104027 ( 2003 ) , hep - th/0309081 . j. ponce de leon , _ mod.phys.lett . a _ * 16 * , 2291 ( 2001 ) , gr - qc/0111011 . h.y . liu , _ phys . b _ * 560 * , 149 ( 2003 ) , hep - th/0206198 . l. x. xu , h. y. liu and b. l. wang , _ chin . lett . _ * 20 * , 995 ( 2003 ) , gr - qc/0304049 . b. l. wang , h. y. liu and l. x. xu , _ mod . phys . a _ * 19 * , 449 ( 2004 ) , gr - qc/0304093 . lixin xu and hongya liu , _ int . d _ * 14 * , 883 ( 2005 ) , astro - ph/0412241 .	we use @xmath0 split to derive the @xmath1 induced energy density @xmath2 and pressure @xmath3 of the @xmath4 ricci - flat cosmological solutions which are characterized by having a bounce instead of a bang . the solutions contain two arbitrary functions of time @xmath5 and , therefore , are mathematically rich in giving various cosmological models . by using four known energy conditions ( null , weak , strong , and dominant ) to pick out and study physically meaningful solutions , we find that the @xmath1 part of the @xmath4 solutions asymptotically approach to the standard @xmath1 frw models and the expansion of the universe is decelerating for normal induced matter for which all the four energy conditions are satisfied . we also find that quintessence might be normal or abnormal , depending on the parameter @xmath6 of the equation of state . if @xmath7 , the expansion of the universe is accelerating and the quintessence is abnormal because the strong energy condition is violated while other three are satisfied . for phantom , all the four energy conditions are violated . before the bounce all the four energy conditions are violated , implying that the cosmic matter before the bounce could be explained as a phantom which has a large negative pressure and makes the universe bouncing . in the early times after the bounce , the dominant energy condition is violated while the other three are satisfied , and so the cosmic matter could be explained as a super - luminal acoustic matter .
the technological aspects of high-@xmath3 superconductors in strong magnetic fields and especially of their ability to preserve superconductivity by flux pinning @xcite have led to intense theoretical studies of the properties of a flux line array in a type - ii superconductor with random point - like pinning centers @xcite . it has been conjectured @xcite that the flux lines in a superconductor with point disorder form a new thermodynamic phase , the vortex glass phase . it is supposed that in this phase the flux lines are collectively pinned by the point defects and energy barriers between different metastable states of the flux line array occur which diverge with increasing length scale @xmath4 leading to a glassy dynamics and zero linear resistivity @xcite . but there is still too little conclusive evidence to confirm this scenario by analytic means . due to the absence of topological defects @xcite in @xmath5 dimensions , the planar @xmath5-dimensional flux line array can be well treated analytically in an elastic approach . actually , the system of flux lines in a type - ii superconducting plane with parallel magnetic field and point disorder is the only system for which the existence of a vortex glass phase has been shown analytically @xcite by applying various different methods . on the other hand , the predictions for important physical features of the disordered @xmath5 dimensional flux line array obtained by the different analytical methods differ significantly . in particular , there are still many competing conjectures concerning the correlations in the vortex glass phase . moreover , the results of numerical simulations confirm neither of the analytical predictions @xcite . essentially three analytic approaches have been applied to the problem : ( i ) after using the replica trick and mapping onto a 2d xy - model with random anisotropy but without vortices @xcite , a renormalization group ( rg ) calculation @xcite has been carried out with the replicated hamiltonian not taking into account the possibility of replica symmetry breaking ( rsb ) . ( ii ) the replicated hamiltonian has also been studied by a variational treatment admitting of continuous rsb finding that a one - step breaking is realized @xcite . ( iii ) without making use of the replica trick , the corresponding kinetic equation has been treated by a dynamical rg analysis @xcite . in the present paper we want to study how the concept of rsb could enter into a rg analysis . for this purpose we map the disordered planar flux line array onto the 2d xy - model with random anisotropy and perform a rg calculation with the replicated hamiltonian where we generalize the set of coupling constants such that we can take into account a one - step rsb . our aim is to show that an instability with respect to one - step rsb can also be found in the rg treatment . this leads to a more unified view of the approaches ( i ) and ( ii ) , and results obtained by the variational approach can partly be reproduced in our calculation . on scales larger than the flux line distance @xmath6 the planar flux line array is described by an elastic model with the positions of the flux lines given by a scalar transverse displacement field @xmath7 . the field @xmath8 itself can be regarded as a phase field giving the phase shift of the superconducting phase caused by the flux line displacements , because an increase in the flux line displacement by @xmath6 induces a shift of @xmath9 in the field @xmath8 . on large length scales , the planar flux line array interacting with the random pinning centers can then be described by the hamiltonian @xcite @xmath10 = \int d^2{\bf r } \left\ { \frac{1}{2}k(\nabla\phi)^2 + v_{1}({\bf r})\sin(2\phi({\bf r}))+v_{2}({\bf r})\cos(2\phi({\bf r } ) ) \right\}.\ ] ] the second term containing the random potential @xmath11 with zero average and short range correlations ( @xmath12 ) @xmath13 models the interaction of the flux lines with the point disorder in the continuum description ( @xmath14 includes a factor @xmath15 ) . the function @xmath16 is a delta - like function of the small width @xmath17 given by the maximum of the flux line core radius and the impurity size . a crucial feature of this term is to respect the periodicity of the lattice , i.e. it is invariant under a uniform shift @xmath9 of the field @xmath8 . by rescaling of one coordinate the isotropic first term for the elastic energy is obtained with one elastic constant k ( including a factor @xmath18 ) . each of the three approaches sketched above @xcite as well as the numerical simulation @xcite , yield a phase transition at @xmath19 or @xmath20 with @xmath21 which serves as a small parameter measuring the distance from the transition and controlling expansions around the transition . for @xmath22 the system is in a high - temperature phase , disorder is not relevant and does not alter the correlations induced by thermal fluctuations on large length scales @xmath23 . however , the results concerning the correlations for @xmath24 in the glassy phase differ significantly . the rg analysis carried out on the replica hamiltonian in a replica symmetric way yields correlations @xmath25 at large length scales @xmath26 @xcite . the same result is obtained in the dynamical rg calculation @xcite . besides , correlations diverging like the square of the logarithm follow from a real - space rg procedure @xcite . on the other hand , in the variational approach with one - step rsb , correlations @xmath27 are found @xcite to diverge logarithmically but with a prefactor increasing with decreasing temperature . logarithmically diverging correlations have also been found in the numerical simulation @xcite but the prefactor of the logarithm does not accord with the analytical prediction in @xcite . our calculation including one - step rsb in the rg analysis of the replicated hamiltonian can reproduce both correlations diverging with a simple logarithm and correlations diverging with the square of the logarithm depending on the choice of the block size parameter @xmath0 in the one - step rsb scheme . introducing @xmath28 replicas and averaging over the disorder gives the effective replicated hamiltonian ( with replica indices @xmath29,@xmath30 running from 1 to n ) @xmath31 = \int d^2{\bf r } \left\ { \frac{1}{2 } \sum_{\alpha\beta } k_{\alpha\beta } \nabla\phi_{\alpha } \cdot \nabla\phi_{\beta } - \sum_{\alpha\beta } g_{\alpha\beta } \cos{(2(\phi_{\alpha}-\phi_{\beta } ) ) } \right\}\ ] ] with matrices @xmath32 and @xmath33 taking on their bare values @xmath34 this hamiltonian is equivalent to the replica hamiltonian of a 2d xy - model with random anisotropy but without vortices @xcite . the variational studies allowing for continuous rsb performed so far on this hamiltonian @xcite have shown that in the 2-dimensional model considered here a one - step rsb is realized . our calculation is restricted so far to a one - step rsb scheme but the results from the variational approach suggest that our results may stay valid even if it is possible to extend the calculation to higher steps of rsb or a continuous rsb scheme . introducing one - step rsb and following the resulting rg flow , it is necessary to admit matrices @xmath32 and @xmath33 of the form @xmath35 and @xmath36 ; the elements of the matrix @xmath37 are 1 if @xmath29 and @xmath30 belong to the same block of size @xmath0 and 0 otherwise . because of ( [ 4a ] ) , ( [ 4b ] ) we have initially @xmath38 , @xmath39 and @xmath40 . to derivate the full rg flow equations , we perform an analysis technically similar to that of cardy and ostlund @xcite but with significant extensions to take into account the one - step rsb . the calculation is based on the mapping onto a coupled coulomb gas . we want to consider only weak disorder , so initially the disorder strength @xmath14 will be small ; also throughout the rg procedure the matrix elements of @xmath33 stay sufficiently small to use standard methods @xcite to transform the cos - couplings in the partition sum and to integrate out the fields @xmath41 in favour of integer charges @xmath42 ( @xmath43 ) with fugacity @xmath33 . the replicated disorder averaged partition sum @xmath44 factors then into @xmath45 where the factor @xmath46 represents the purely elastic part of the partition sum ; this factor plays a role only in deriving the rg equation for the free energy density and will be considered later on in detail . in the coulomb gas factor @xmath47 of the partition sum , it has to be summed over all spatial configurations of interacting charges , which can take on any integer value , but because the charge fugacities @xmath33 are sufficiently small , only positive and negative unit charges have to be considered , which obey in addition a neutrality condition . switching from the continuum description with a short wavelength cutoff @xmath17 to a description on a square lattice with lattice constant @xmath17 and lattice vectors @xmath48 for easier notation , one obtains the following expression for the replicated disorder averaged partition sum : @xmath49 where @xmath50 and @xmath51 ( @xmath43 ) . the matrix @xmath52 has the same block form as @xmath32 with @xmath53 , @xmath54 in the limit @xmath55 . the block form of the matrix @xmath33 implies that two kinds of charges exist differing in their fugacities and , moreover , in their interactions with other charges due to the block form of the matrix @xmath56 . taking this into account , a rg calculation in the style of cardy and ostlund ( co ) @xcite can be performed , which yields the following rg recursions in the limit @xmath55 upon a change of scale by a factor @xmath57 ( henceforth we always include a factor @xmath58 in @xmath59,@xmath60 and @xmath61 and a factor @xmath62 in @xmath63 , @xmath64 . ) : @xmath65 the parameter @xmath0 is a free parameter in these equations with @xmath66 in the limit @xmath55 ; possible choices for @xmath0 will be discussed later on . ( [ flow1 ] ) , ( [ flow2 ] ) show that @xmath67 is not renormalized ; this result is exact to all orders in the @xmath68 ( @xmath69 ) due to a statistical invariance under tilt @xcite . in the special replica symmetric cases @xmath70 and @xmath71 , we get back the flow equations of co : in the rg equations for @xmath70 ( @xmath71 ) , @xmath64 ( @xmath63 ) plays the role of the single disorder strength parameter in @xcite , the diagonal matrix elements @xmath72 ( @xmath59 ) of @xmath73 are not renormalized , and also the off - diagonal matrix elements @xmath61 ( @xmath74 ) and the fugacity @xmath64 ( @xmath63 ) renormalize as in @xcite . the system exhibits the known co fixed points @xmath75 ( @xmath76 ) and @xmath77 ( @xmath78 ) . for @xmath70 the rg flow is sketched in fig . @xmath63 ( @xmath64 ) does not feed back into the rg flow of the other quantities and does therefore not enter physical results like correlation functions ( see below ) . for this reason the introduction of a small initial replica asymmetric perturbation @xmath79 has no effect on physical results if @xmath70 ( @xmath71 ) , although @xmath80 turns out to be a relevant perturbation under rg ( see below ) . starting as in ( [ 4b ] ) with replica symmetric initial conditions @xmath81 , replica symmetry is preserved throughout the rg procedure independently of @xmath0 , and @xmath59 , @xmath60 are not renormalized ; therefore the co scenario with the trivial fixed point @xmath82 and the non - trivial co fixed point @xmath83 is reproduced if replica symmetry holds initially . however , introducing a small initial replica asymmetry @xmath84 contrary to ( [ 4b ] ) , the rg flow develops for @xmath24 an instability with respect to rsb . the system flows for @xmath85 to a regime with @xmath86 and for @xmath87 to a regime @xmath88 ( entering on large length scales the unphysical regime of negative fugacities @xmath63 ) . in particular the replica symmetric co fixed point @xmath89 , @xmath90 is unstable against small replica asymmetric perturbations . a linear stability analysis of the co fixed point yields ( @xmath91 ) @xmath92 these equations describe an instability with eigenvalue @xmath93 of the co fixed point with respect to perturbations @xmath94 . to avoid entering the unphysical regime of negative fugacities , we consider only perturbations @xmath85 . as it is seen from ( [ ls1 ] ) , ( [ ls2 ] ) , such a perturbation causes the charge interaction strength parameter @xmath59 to decrease and the asymmetry @xmath94 to increase ; finally , @xmath59 renormalizes to 0 following ( [ flow1 ] ) . this flow towards the fixed point @xmath95 implies that one non - interacting type of unit charge with fugacity @xmath63 appear on large length scales . furthermore , we can find from ( [ flow4 ] ) , ( [ flow5 ] ) two additional non - trivial rsb fixed points ( [ 13a ] ) and ( [ 13b ] ) with @xmath96 , @xmath97 for each of them : @xmath98 at a certain @xmath99 the fixed points ( [ 13a ] ) and ( [ 13b ] ) fall exactly together . only for @xmath100 the fixed point ( [ 13a ] ) is in the physical regime @xmath101 of non - negative fugacities . moreover , the fixed point ( [ 13a ] ) is in this range of @xmath0 stable with respect to perturbations in @xmath63 and @xmath64 ( getting marginal with respect to perturbations in @xmath64 at @xmath102 where it coincides with ( [ 13b ] ) ) , whereas the fixed point ( [ 13b ] ) is unstable with respect to perturbations @xmath103 . therefore the fixed point ( [ 13a ] ) is attractive for all rg trajectories starting with @xmath104 ( as illustrated in fig . 1b ) while the fixed point ( [ 13b ] ) is attractive only for rg trajectories with @xmath105 . for @xmath106 ( [ 13b ] ) is the only rsb fixed point in the physical regime of non - negative fugacities @xmath107 . it is in this range of @xmath0 the attractive fixed point for all rg trajectories with @xmath108 ( see fig . 1c,1d ) ; furthermore , it is stable with respect to perturbations in @xmath63 and @xmath64 . as pointed out in ( [ 4a ] ) , ( [ 4b ] ) , the proper initial values @xmath109 and @xmath110 are replica symmetric with @xmath111 . it remains unclear in this approach how the initial asymmetry @xmath112 necessary for the development of an instability with respect to rsb can be obtained from physical reasons . one hint is given in the next section where it is shown by comparison with the replica symmetric co flow for @xmath111 that contributions to the free energy from large scale fluctuations can be optimized ( which means maximized in the limit @xmath55 ) if a small perturbation @xmath112 is introduced . in the high - temperature phase for @xmath22 the system flows to the stable trivial replica symmetric fixed point @xmath82 regardless of an initial asymmetry @xmath113 . in this phase the trivial replica fixed point is stable with respect to the rsb perturbation @xmath114 so that rsb can not occur in the high - temperature phase as it is expected . for @xmath20 the trivial fixed point stays marginally stable . we want to proceed with a discussion of energetic aspects of the instability in the rg flow upon introducing a replica asymmetric perturbation @xmath85 in the low - temperature phase . this enables us to fix the so far undetermined block size parameter @xmath0 if @xmath85 and to compare the free energy in the rsb case with @xmath85 with the free energy in the replica symmetric co case @xmath115 . the standard procedure to determine @xmath0 is to maximize the free energy density per replica in the limit @xmath116 with respect to the additional free parameter @xmath0 . as a consequence of the factor @xmath126 appearing in the integrand of ( [ 19 ] ) , the main contribution to @xmath127 comes from the short scales . starting at @xmath124 , @xmath125 and evaluating ( [ 19 ] ) straightforwardly to the leading order in @xmath128 , one obtains a maximum of @xmath127 at the replica symmetric @xmath71 . this is because the main contribution to @xmath129 comes on short scales from the term @xmath130/2\xi^2 $ ] . the @xmath0-dependent part of @xmath130/2\xi^2 $ ] can be approximated by means of a linear stability analysis of the flow equations ( [ flow1])([flow5 ] ) at the co fixed point enlarging on ( [ ls1 ] ) , ( [ ls2 ] ) as @xmath131/2\xi^2 $ ] with a maximum at @xmath71 . on the other hand , the maximization of the asymptotic large scale contributions leads to a quite different result . examining the large scale contributions , one has to investigate the asymptotics of the integrand in ( [ 19 ] ) and to maximize @xmath129 in the limit of @xmath132 . for this purpose it is necessary to derive the asymptotics of @xmath61 which is determined by the flow equation ( [ flow3 ] ) . from ( [ flow3 ] ) follows that for @xmath133 @xmath61 is asymptotically linear divergent with an asymptotics @xmath134 where @xmath135 is taken in the stable rsb fixed point ( [ 13a ] ) , which is @xmath136 to a good approximation . however , we find in the regime @xmath137 from ( [ flow3 ] ) a saturation of @xmath61 to a value @xmath138 because the stable rsb fixed point is in this regime given by ( [ 13b ] ) with @xmath75 . to obtain an estimate for @xmath138 one has to determine the characteristic scale @xmath139 on which @xmath64 renormalizes towards 0 ; a linear stability analysis of the flow equations ( [ flow1 ] ) , ( [ flow4 ] ) , ( [ flow5 ] ) at the co fixed point extending ( [ ls1 ] ) , ( [ ls2 ] ) reveals that @xmath139 can be approximated as @xmath140 . from ( [ flow3 ] ) it follows @xmath141 for the leading order contribution in @xmath142 . moreover , it is seen from the flow equation ( [ flow1 ] ) that @xmath143 on large scales . using these results for @xmath61 and @xmath59 , one can verify easily from ( [ 18 ] ) that the most divergent contributions to @xmath129 come from @xmath144/2\xi^2 $ ] for large @xmath6 . maximization of these terms yields @xmath145 because the maximization of the second term restricts @xmath0 to values @xmath137 to avoid the occurrence of the linear divergence in the regime @xmath146 and the maximization of the only logarithmically diverging first term singles out the greatest value @xmath102 of the interval @xmath137 . ( [ 20 ] ) is in fairly good agreement with @xcite . rsb is a large scale effect associated with the existence of diverging energy barriers generating metastable states . therefore it seems to be more reasonable to consider only the large scale contributions to the free energy in ( [ 19 ] ) although the expression ( [ 19 ] ) for @xmath127 is dominated by its short scale part . this is equivalent to considering the free energy of the renormalized but not rescaled hamiltonian on large scales but discarding a constant energy shift depending on @xmath0 which comes from short scales . this energy shift , which is essentially replica symmetric , may describe the free energy of single metastable states . in the presence of an initial asymmetric perturbation @xmath85 , maximization of the large scale contributions to the free energy yields then a maximum at @xmath102 as derivated above . comparison of these large scale contributions for the flow to the rsb fixed point when ( @xmath85 ) and for the replica symmetric flow to the co fixed point ( @xmath115 ) shows that this part of the free energy is greater in the rsb case . this is because the most divergent contribution to @xmath129 is in the replica symmetric case as in the rsb case given by @xmath130/2\xi^2 $ ] with @xmath147 but the replica symmetric co fixed point value @xmath148 for @xmath135 is always greater than or equal to ( @xmath70 ) the rsb fixed point values given by ( [ 13a ] ) , ( [ 13b ] ) . therefore it is energetically favorable on large scales to break the replica symmetry by introducing a perturbation @xmath85 . this energy gain can occur on scales larger than @xmath149 . in the high - temperature phase the trivial fixed point is stable with respect to the introduction of a perturbation @xmath113 . for this reason the large scale contributions to the free energy are the same as in the replica symmetric case . for the short scale contributions to the free energy the same argumentation applies as in the low - temperature phase leading to a maximum at the replica symmetric @xmath70 if @xmath113 . so rsb is energetically not favorable in the high - temperature phase as it is expected . the rg flow and fixed point structure changes significantly upon introducing an energetically favorable , replica asymmetric perturbation @xmath85 in the low - temperature phase as outlined above . as well the behaviour of the @xmath150-correlations changes drastically depending on the value of @xmath0 . the fourier transformed correlations between replicas on large scales can be calculated by using a gaussian approximation to the renormalized but not rescaled replica hamiltonian , which yields for small @xmath151 @xcite @xmath152 so that the large scale correlations depend essentially only on the asymptotic rg flow of the matrix elements @xmath59 , @xmath60 and @xmath61 . from expression ( [ 15 ] ) one can verify that the connected correlation function @xmath153 does not change its form at the transition , independently from the introduction of a nonzero @xmath114 due to the non - renormalization of @xmath154 , contrary to the @xmath150-correlations @xcite . for the @xmath150-correlations ( [ 15 ] ) yields @xmath155 in the high - temperature phase no rsb takes place , even if @xmath112 , and there is essentially no renormalization of @xmath59 , @xmath60 and @xmath61 ; the connected @xmath156-correlation function and the @xmath150-correlation function coincide then and @xmath157 . in the low - temperature phase the asymptotics of @xmath61 , which is determined by the flow equation ( [ flow3 ] ) , is of special interest because in the replica symmetric case , i.e. without a replica asymmetric perturbation ( @xmath158 ) , the asymptotics @xmath159 diverging linearly gives correlations @xmath160 diverging with @xmath2 @xcite . in the rsb case with an initial @xmath161 , @xmath61 has also a linear divergent asymptotics @xmath147 for @xmath146 ( see above ) , where @xmath136 is taken in the stable rsb fixed point ( [ 13a ] ) . this entails @xmath150-correlations diverging also with @xmath2 for this range of @xmath0 but with a prefactor reduced by a factor @xmath162 compared to the replica symmetric case ; in particular , we get back the replica symmetric co result choosing @xmath70 . the situation changes significantly in the regime @xmath137 because the stable rsb fixed point is in this regime given by ( [ 13b ] ) with @xmath75 . therefore @xmath61 saturates on large scales to a value @xmath163 for the leading order contribution in @xmath142 ( see above ) and we obtain from ( [ 16 ] ) only logarithmically divergent @xmath150-correlations with a prefactor @xmath164 which is greater than in the high - temperature phase . to the leading order in @xmath142 this yields correlations @xmath165 ; with our above choice ( [ 20 ] ) of @xmath166 we get @xmath167 . this implies that the prefactor of the logarithm increases with @xmath142 in the low - temperature phase . our results for the low - temperature phase show that within the one - step rsb rg approach with a small initial replica asymmetry @xmath161 , it is possible to obtain the known replica symmetric result for the @xmath150-correlations diverging like @xmath2 @xcite if @xmath70 as well as @xmath150-correlations diverging in the same way but with a smaller prefactor if @xmath168 and logarithmically divergent @xmath150-correlations with a prefactor increasing with decreasing temperature if @xmath169 . the latter possibilities are of interest with regard to the numerical simulations @xcite and the results of the variational approaches @xcite . to summarize we have shown an instability in the rg flow of the disordered planar flux line array , which is equivalent to the 2d xy - model with random anisotropy but without vortices , with respect to a one - step rsb . the flow approaches new rsb fixed points if an initial replica asymmetric perturbation is introduced . the system can optimize its free energy contributions from large length scale ( @xmath170 ) fluctuations by breaking the replica symmetry and approaching the rsb fixed point where the energetical optimal choice of @xmath0 is @xmath171 . introducing the initial perturbation , the @xmath150-correlations show a @xmath2-divergence on large length scales in the range @xmath168 returning to the replica symmetric result at @xmath70 ; for @xmath169 the correlations diverge only as @xmath1 , which is especially for @xmath102 the case . during completion of this work p. le doussal and t. giamarchi have submitted a letter @xcite in which they find independently from our results an instability with respect to rsb in the 2d xy model in a random field . the author thanks t. nattermann , s.e . korshunov and t. hwa for discussions and sfb 341 ( b8 ) for support . 99 g. blatter _ et al _ , rev . 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honeycomb arrays were introduced by golomb and taylor @xcite in 1984 , as a hexagonal analogue of costas arrays . examples of honeycomb arrays are given in figures [ honey137_figure ] to [ honey45_figure ] below . a honeycomb array is a collection of @xmath0 dots in a hexagonal array with two properties : * * ( the hexagonal permutation property ) * there are three natural directions in a hexagonal grid ( see figure [ fig : lee_example ] ) . considering ` rows ' in each of these three directions , the dots occupy @xmath0 consecutive rows , with exactly one dot in each row . * * ( the distinct differences property ) * the @xmath1 vector differences between pairs of distinct dots are all different . golomb and taylor found @xmath2 examples of honeycomb arrays ( up to symmetry ) , and conjectured that infinite families of honeycomb arrays exist . blackburn , etzion , martin and paterson @xcite recently disproved this conjecture : there are only a finite number of honeycomb arrays . unfortunately , the bound on the maximal size of a honeycomb array that blackburn et al . provide is far too large to enable an exhaustive computer search over all open cases . in this paper , we prove a theorem that significantly limits the possibilities for a honeycomb array with @xmath0 dots . ( in particular , we show that @xmath0 must be odd . ) we report on our computer searches for honeycomb arrays , and give two previously unknown examples with @xmath3 dots . we now introduce a little more notation , so that we can state the main result of our paper more precisely . we say that a collection of dots in the hexagonal grid is a _ hexagonal permutation _ if it satisfies the hexagonal permutation property . a collection of dots is a _ distinct difference configuration _ if it satisfies the distinct difference property . so a honeycomb array is a hexagonal permutation that is a distinct difference configuration . we say that hexagons are _ adjacent _ if they share an edge , and we say that two hexagons @xmath4 and @xmath5 are _ at distance @xmath6 _ if the shortest path from @xmath4 to @xmath5 ( travelling along adjacent hexagons ) has length @xmath6 . lee sphere of radius @xmath7 _ is a region of the hexagonal grid consisting of all hexagons at distance @xmath7 or less from a fixed hexagon ( the _ centre _ of the sphere ) . the region in figure [ fig : lee_example ] is a lee sphere of radius @xmath8 . note that a lee sphere of radius @xmath7 intersects exactly @xmath9 rows in each of the three natural directions in the grid . a _ honeycomb array of radius @xmath7 _ is a honeycomb array with @xmath9 dots contained in a lee sphere of radius @xmath7 . there are many other natural regions of the hexagonal grid that have the property that they intersect @xmath0 rows in each direction . one example , the tricentred lee sphere of radius @xmath7 , is shown in figure [ fig : tricentred_example ] : it is the union of three lee spheres of radius @xmath7 with pairwise adjacent centres , and intersects exactly @xmath10 rows in any direction . does there exist a honeycomb array with @xmath10 dots contained in a tricentred lee sphere of radius @xmath7 ? golomb and taylor did not find any such examples : they commented ( * ? ? ? * page 1156 ) that all known examples of honeycomb arrays with @xmath0 dots were in fact honeycomb arrays of radius @xmath7 , but stated `` we have not proved that this must always be the case '' . we prove the following : [ thm : main ] let @xmath0 be an integer , and suppose there exists a hexagonal permutation @xmath11 with @xmath0 dots . then @xmath0 is odd , and the dots of @xmath11 are contained in a lee sphere of radius @xmath12 . since any honeycomb array is a hexagonal permutation , the following result follows immediately from theorem [ thm : main ] : [ cor : honeycomb ] any honeycomb array is a honeycomb array of radius @xmath7 for some integer @xmath7 . in particular , a honeycomb array must consist of an odd number of dots . so if we are looking for honeycomb arrays , we may restrict ourselves to searching for honeycomb arrays of radius @xmath7 . the structure of the remainder of the paper is as follows . in section [ sec : anticodes ] , we state the results on the hexagonal grid that we need . in section [ sec : brooks ] , we remind the reader of the notion of a brook , or bee - rook , and state a theorem on the maximum number of non - attacking brooks on a triangular board . in section [ sec : honeycomb ] we prove theorem [ thm : main ] , we summarise our computer searches for honeycomb arrays , and we provide a list of all known arrays . this last section also contains a conjecture , and some suggestions for further work . because the hexagonal grid might be difficult to visualise , we use an equivalent representation in the square grid ( see figure [ fig : hex_to_square ] ) . in this representation , we define each square to be adjacent to the four squares it shares an edge with , and the squares sharing its ` north - east ' and ` south - west ' corner vertices . the map @xmath13 in figure [ fig : hex_to_square ] distorts the centres of the hexagons in the hexagonal grid to the centres of the squares in the square grid . the three types of rows in the hexagonal grid become the rows , the columns and the diagonals that run from north - east to south - west . for brevity , we define a _ standard diagonal _ to mean a diagonal that runs north - east to south - west . ( 150,55)(-10,-40 ) ( 10,5)(10,0)3 ( -10,0)(0,1)5.77350269 ( 0,0)(0,1)5.77350269 ( 0,5.77350269)(-1732,1000)5 ( -10,5.77350269)(1732,1000)5 ( 5,-3.66033872)(10,0)4 ( -10,0)(0,1)5.77350269 ( 0,0)(0,1)5.77350269 ( 0,5.77350269)(-1732,1000)5 ( -10,5.77350269)(1732,1000)5 ( 0,-12.3206774)(10,0)5 ( -10,0)(0,1)5.77350269 ( 0,0)(0,1)5.77350269 ( 0,5.77350269)(-1732,1000)5(-10,5.77350269)(1732,1000)5 ( 5 , -20.9810162)(10,0)4 ( -10,0)(0,1)5.77350269 ( 0,0)(0,1)5.77350269 ( 0,5.77350269)(-1732,1000)5 ( -10,5.77350269)(-1732,1000)5 ( -10,5.77350269)(1732,1000)5 ( 0,5.77350269)(1732,1000)5 ( 10 , -29.6413549)(10,0)3 ( -10,0)(0,1)5.77350269 ( 0,0)(0,1)5.77350269 ( 0,5.77350269)(-1732,1000)5 ( -10,5.77350269)(-1732,1000)5 ( -10,5.77350269)(1732,1000)5 ( 0,5.77350269)(1732,1000)5 ( 15 , -38.3016936)(10,0)2 ( 0,5.77350269)(-1732,1000)5 ( -10,5.77350269)(-1732,1000)5 ( -10,5.77350269)(1732,1000)5 ( 0,5.77350269)(1732,1000)5 ( 15,-9.43392605)(0,0)@xmath14 ( 5,-9.43392605)(0,0)@xmath15 ( 25,-9.43392605)(0,0)@xmath8 ( 10,-0.77358733)(0,0)@xmath16 ( 20,-0.77358733)(0,0)@xmath17 ( 10,-18.0942648)(0,0)@xmath18 ( 20,-18.0942648)(0,0)@xmath19 ( 60,-16.43392605)(0,0)@xmath20 ( 80,-38.3014395)(1,0)34.6410162 ( 80,-26.7544341)(1,0)46.1880215 ( 80,-15.2074287)(1,0)57.7350269 ( 80.5470054,-3.66042332)(1,0)57.7350269 ( 91.5470054,7.88658206)(1,0)46.1880215 ( 103.094011 , 19.4335874)(1,0)34.6410162 ( 80,-38.3014395)(0,1)34.6410162 ( 91.5470054,-38.3014395)(0,1)46.1880215 ( 103.094011,-38.3014395)(0,1)57.7350269 ( 114.641016,-38.3014395)(0,1)57.7350269 ( 126.188022,-26.7544341)(0,1)46.1880215 ( 137.735027,-15.2074287)(0,1)34.6410162 ( 108.867513,-9.43392605)(0,0)@xmath14 ( 97.3205081,-9.43392605)(0,0)@xmath15 ( 120.414519,-9.43392605)(0,0)@xmath8 ( 108.867513,2.11307933)(0,0)@xmath16 ( 97.3205081,-20.9809314)(0,0)@xmath18 ( 120.414519,2.11307933)(0,0)@xmath17 ( 108.867513,-20.9809314)(0,0)@xmath19 , width=264 ] for non - negative integers @xmath0 and @xmath21 such that @xmath22 , define the region @xmath23 of the square grid as in figure [ fig : anticode ] . note that @xmath24 and @xmath25 are essentially the same region : one is obtained from the other by a reflection in a standard diagonal . the regions @xmath26 are important in the hexagonal grid , as they are the maximal anticodes of diameter @xmath27 ; see blackburn et al . * theorem 5 ) . note that the region @xmath28 is a lee sphere of radius @xmath7 . regions of the form @xmath29 or @xmath30 are tricentred lee spheres of radius @xmath7 . also note that the regions @xmath23 as @xmath21 varies are precisely the possible intersections of an @xmath31 square region with @xmath0 adjacent standard diagonals , where each diagonal intersects the @xmath31 square non - trivially . in the lemma below , by a ` region of the form @xmath32 ' , we mean a region that is a translation of @xmath32 in the square grid . [ lem : anticode ] let @xmath11 be a hexagonal permutation with @xmath0 dots , and let @xmath33 be the image of @xmath11 in the square grid . then the dots in @xmath33 are all contained in a region of the form @xmath24 for some @xmath21 in the range @xmath22 . * proof : * let @xmath34 be the set of squares that share a row with a dot of @xmath33 . similarly , let @xmath35 and @xmath36 be the sets squares sharing respectively a column or a standard diagonal with a dot of @xmath33 . the dots in @xmath33 are contained in @xmath37 . since @xmath11 is a hexagonal permutation , @xmath34 consists of @xmath0 adjacent rows and @xmath35 consists of @xmath0 adjacent columns . hence @xmath38 is an @xmath31 square region . ( since there is exactly one dot in each row and column of the square @xmath38 , the dots in @xmath33 correspond to a permutation ; this justifies the terminology ` hexagonal permutation ' . ) now , @xmath36 consists of @xmath0 adjacent standard diagonals ; each diagonal contains a dot in @xmath33 , and so each diagonal intersects @xmath38 non - trivially . hence @xmath37 is a region of the form @xmath23 , as required.@xmath39 a _ brook _ is a chess piece in the square grid that moves like a rook plus half a bishop : it can move any distance along a row , a column or a standard ( north - east to south - west ) diagonal . brooks were first studied by bennett and potts @xcite , who pointed out connections to constant - sum arrays and hexagonal lattices . a set of brooks in a square grid is _ non - attacking _ if no two brooks lie in a row , a column or a standard diagonal . under the correspondence @xmath13 between the square and hexagonal grids mentioned in the previous section , brooks in the square grid correspond to _ bee - rooks _ in the hexagonal grid : pieces that can move any distance along any row , where a row can go in each of the three natural directions . a set of bee - rooks is therefore _ non - attacking _ if no two bee - rooks lie in the same row of the hexagonal grid . in particular , bee - rooks placed on the dots in a hexagonal permutation @xmath11 are non - attacking , and so the corresponding set @xmath33 of brooks in the square grid is non - attacking . a _ triangular board of width @xmath40 _ is the region @xmath41 in the square grid depicted in figure [ fig : triangular_board ] . let @xmath42 be the maximum number of non - attacking brooks that can be placed in the triangular board of width @xmath40 . the following theorem is proved by nivasch and lev @xcite and in vaderlind , guy and larson ( * ? ? ? * p252 and r252 ) : , width=188 ] [ thm : brooks ] for any positive integer @xmath40 , @xmath43 . three of the present authors have found an alternative proof for this theorem , using linear programming techniques @xcite . see bell and stevens @xcite for a survey of similar combinatorial problems . we begin this section with a proof of theorem [ thm : main ] . we then describe our searches for honeycomb arrays . we end the section by describing some avenues for further work . * proof of theorem [ thm : main ] : * let @xmath11 be a hexagonal permutation with @xmath0 dots . by lemma [ lem : anticode ] , the dots of @xmath33 are contained in a region of the form @xmath23 where @xmath22 . when @xmath44 ( so @xmath0 is odd and @xmath26 is a lee sphere ) the theorem follows . suppose , for a contradiction , that @xmath45 . by reflecting @xmath11 in a horizontal row in the hexagonal grid , we produce a hexagonal permutation @xmath46 such that @xmath47 is contained in a region of the form @xmath48 . by replacing @xmath11 by @xmath46 if necessary , we may assume that @xmath49 . consider the triangular board of width @xmath50 in figure [ fig : covering_triangle ] containing @xmath23 . since no two dots in @xmath33 lie in the same row , column or standard diagonal , the dots in @xmath33 correspond to @xmath0 non - attacking brooks in this triangular board . but this contradicts theorem [ thm : brooks ] , since @xmath51 this contradiction completes the proof of the theorem.@xmath39 theorem [ thm : main ] tells us that the only honeycomb arrays are those of radius @xmath7 for some non - negative integer @xmath7 . a result of blackburn et al ( * ? ? ? * corollary 12 ) shows that @xmath52 . we now report on our computer searches for examples of honeycomb arrays . the known honeycomb arrays are drawn in figures [ honey137_figure ] , [ honey9_figure ] , [ honey15_figure ] , [ honey2127_figure ] and [ honey45_figure ] . this list includes two new examples not known to golomb and taylor @xcite , namely the second and third examples of radius @xmath53 ; we found these examples as follows . a _ costas array _ is a set of @xmath0 dots in an @xmath31 region of the square grid , with the distict difference property and such that every row and column of the array contains exactly one dot . golomb and taylor observed that some costas arrays produce honeycomb arrays , by mapping the dots in the costas array into the hexagonal grid using the map @xmath54 given by figure [ fig : hex_to_square ] . indeed , it is not difficult to see that all honeycomb arrays must arise in this way . we searched for honeycomb arrays by taking each known costas array with @xmath55 or fewer dots , and checking whether the array gives rise to a honeycomb array . for our search , we made use of a database of all known costas arrays with 200 or fewer dots that has been made available by james k. beard @xcite . this list is known to be complete for costas arrays with @xmath56 or fewer dots ; see drakakis et al . @xcite for details . so our list of honeycomb arrays of radius @xmath57 or less is complete . it is a remarkable fact that all known honeycomb arrays possess a non - trivial symmetry ( a horizontal reflection as we have drawn them ) . indeed , apart from a single example of radius @xmath19 ( the first radius @xmath19 example in figure [ honey137_figure ] ) all known honeycomb arrays possess a symmetry group of order @xmath16 : the group generated by the reflections along the three lines through opposite ` corners ' of the hexagonal sphere . we implemented an exhaustive search for honeycomb arrays with @xmath58 having this @xmath16-fold symmetry : we found no new examples . we also checked all constructions of honeycomb arrays from costas arrays in golomb and taylor @xcite ( whether symmetrical or not ) for @xmath59 , and again we found no new examples . theorem [ thm : main ] shows that hexagonal permutations are always contained in some lee sphere . but such permutations have been prevously studied in several contexts : bennett and potts @xcite study them as non - attacking configurations of bee - rooks and as the number of zero - sum arrays ; kotzig and laufer @xcite study them as the number of @xmath61-permutations ; bebeacua , mansour , postnikov and severini @xcite study them as x - rays of permutations with maximum degeneracy . let @xmath62 be the number of hexagonal permutations with @xmath63 dots . the on - line encyclopedia of integer sequences ( * ? ? ? * sequence a002047 ) quotes a computation due to alex fink that computes the first few terms of the sequence @xmath62 : @xmath64 kotzig and laufer ask : how big can @xmath62 be ? it seems that the sequence grows faster than exponentially with @xmath0 . we ask a more precise question : is it true that @xmath65 tends to a constant as @xmath66 ? part of this work was completed under epsrc grant ep / d053285/1 . the authors would like to thank tuvi etzion for discussions , funded by a royal society international travel grant , which inspired this line of research . 99 james k. beard , http://jameskbeard.com/jameskbeard/ cecilia bebeacua , toufik mansour , alex postnikov and simone severini , ` on the x - rays of permutations ' , _ elec . notes disc . _ * 20 * ( 2005 ) 193203 . jordan bell and brett stevens , ` a survey of known results and research areas for @xmath0-queens ' , _ discrete math . _ * 309 * ( 2009 ) 131 . bennett and r.b . potts , ` arrays and brooks ' , _ j. australian math . * 7 * ( 1967 ) 2331 . simon r. blackburn , tuvi etzion , keith m.martin and maura b. paterson , ` two - dimensional patterns with distinct differences constructions , bounds and maximal anticodes ' , _ ieee trans . theory _ , to appear . see http://arxiv.org/abs/0811.3832 for the preprint version . simon r. blackburn , maura b. paterson and douglas r. stinson , ` putting dots in triangles ' , preprint , http://arxiv.org/abs/0910.4325 . konstantinos drakakis , scott rickard , james k beard , rodrigo caballero , francesco iorio , gareth obrien and john walsh , ` results of the enumeration of costas arrays of order 27 ' , _ ieee transactions on information theory _ , to appear . solomon w. golomb and herbert taylor , ` constructions and properties of costas arrays ' , _ proc . ieee _ * 72 * ( 1984 ) 11431163 . a. kotzig and p.j . laufer , ` when are permutations additive ? ' , _ the american math . monthly _ bf 85 ( 1978 ) , 364 - 365 . gabriel nivasch and eyal lev , ` nonattacking queens on a triangle ' , _ math . magazine _ * 78 * ( 2005 ) 399403 . sloane ( ed . ) , _ the on - line encyclopedia of integer sequences _ , http://www.research.att.com/~njas / sequences/. paul vaderlind , richard guy and loren larson , _ the inquisitive problem solver _ ( mathematical assoc . of america , washington , 2002 ) .	a honeycomb array is an analogue of a costas array in the hexagonal grid ; they were first studied by golomb and taylor in 1984 . a recent result of blackburn , etzion , martin and paterson has shown that ( in contrast to the situation for costas arrays ) there are only finitely many examples of honeycomb arrays , though their bound on the maximal size of a honeycomb array is too large to permit an exhaustive search over all possibilities . the present paper contains a theorem that significantly limits the number of possibilities for a honeycomb array ( in particular , the theorem implies that the number of dots in a honeycomb array must be odd ) . computer searches for honeycomb arrays are summarised , and two new examples of honeycomb arrays with 15 dots are given .
controversial experimental hints on the existence of three - neutron ( @xmath10 ) resonances [ 1 - 4 ] and more recently even on four - neutron resonances [ 5 ] occurred again and again . no definite experimental conclusion on @xmath10 resonances seem to exist . on the theoretical side we are aware of various attempts to investigate states of @xmath10 s but also here the situation is pretty unsettled . obviously the optimal theoretical present day insight would be if the most modern neutron - neutron ( @xmath11 ) forces from the set of high precision @xmath12 potentials together with @xmath10 forces of the @xmath13-exchange type ( tucson - melbourne [ 6 ] or urbana type [ 7 ] ) would be used and the energy eigenvalues of the @xmath10 schrdinger equation according to resonance boundary conditions would be determined in the unphysical energy sheet adjacent to the positive real energy axis . unfortunately we can not provide such an insight right now but think we can improve at least on the existing studies we are aware of . one of the possibly first studies on that list is a solution of the @xmath10 faddeev equation based on a yamaguchi rank 1 @xmath11 force in the state @xmath14 [ 8 ] . only the @xmath10 states of total angular momenta @xmath15 and @xmath16 ( degenerate in this case ) have been studied . the @xmath11 force was artificially enhanced such that two and three neutrons were bound . then the enhancement was reduced which has the consequence that both bound state energies move towards @xmath3 . for the forces used it happened that the @xmath10 binding energy moves faster than the @xmath17 binding energy and thus hits the dineutron - neutron threshold to the left of @xmath3 . the trajectory of the @xmath10 bound state energy continues then into the unphysical sheet below the dineutron - neutron cut . in this manner a @xmath10 resonance occurs , below the @xmath10 break - up threshold at @xmath3 . further decreasing the enhancement factor the @xmath10 resonance trajectory moves up again exactly towards @xmath3 where it meets with the @xmath17 bound state energy . both energies coincide exactly at @xmath3 corresponding still to an enhancement factor larger than 1 . in further decreasing the potential strength till one reaches the physical value 1 the vanishing @xmath17 bound state energy turns into the well established @xmath17 virtual state energy in the second sheet on the negative real energy axis and the @xmath10 resonance disappears in a unphysical sheet which is different from the sheet adjacent to the positive real axis . in other words the complex resonance energy of that @xmath10 resonance for @xmath15 and @xmath16 has no positive real part and therefore can not be felt for positive three - neutron energies . we have to emphasise that this refers to a @xmath11 force acting only in the state @xmath14 and it will change if additional @xmath11 force components will be added as will be shown in this paper . a very much related study has been performed in [ 9 ] with essentially the same result as in [ 8 ] . using just bound state techniques [ 10 ] the necessary enhancement factors on the @xmath11 forces in @xmath0- and @xmath18-waves have been determined to bind 3 neutrons near zero energy . based on the reid potential [ 11 ] enhancement factors of the order of 4 have been found which make low lying @xmath10 resonances quite unlikely . another theoretical investigation [ 12 ] we are aware of for the state @xmath15 has been performed on the basis of a hyperspherical harmonic expansion . local forces like the ones of mafliet tjon have been used and this only in the @xmath11 state @xmath14 . the expansion was truncated and the zeros of the jost function were determined . it resulted a resonance energy around @xmath19 mev in an unphysical sheet . the authors found a strong sensitivity to the choice of the potential . a further investigation [ 13 ] relied on a variational ansatz and used the complex scaling method which allows to treat a resonance problem like a bound state one . the minnesota effective @xmath11 force [ 14 ] together with tensor forces [ 15 ] have been applied . the authors find as the only candidate for a @xmath10 resonance the state @xmath20 and a prediction for its energy of @xmath21 mev . finally in a more recent work [ 16 ] the faddeev equation in configuration space has been solved for the states @xmath15 and @xmath22 using the more realistic forces by gogny _ [ 17 ] and the reid93 [ 18 ] potential in the @xmath17 states @xmath14 , @xmath23 and @xmath24 - @xmath25 . choosing proper boundary conditions according to @xmath10 resonances the complex energy eigenvalues are determined starting again from artificially enhanced forces and reducing their strengths gradually . unfortunately with increasing negative imaginary parts of the complex resonance energies numerical instabilities occurred and the trajectories could not be followed up until the physical values for the enhancement factors have been reached . now we would like to improve on all that by solving the faddeev equations with low rank @xmath11 forces in all relevant partial wave states exactly and determining the final resonance positions for the actual force strength as prescribed by @xmath11 phase shifts ( assumed to be identical to the strong @xmath26 phase shifts ) . in the light of our results we shall also comment on the previous findings . this paper is organised as follows . in section ii we briefly review the set of coupled faddeev eigenvalue equations for a @xmath10 system based on finite rank forces and mention the steps needed for an analytical continuation into the unphysical energy sheet adjacent to the positive real axis . the dynamical @xmath11 force input and the resulting @xmath10 resonance trajectories for the states @xmath27 and @xmath22 are presented in section iii . in section iv we show how the part of the @xmath10 resonance trajectory close to the threshold @xmath3 can be predicted with the help of pad approximant s from a set of @xmath10 bound state energies . this serves as a test for our results obtained through the analytical continuation of the faddeev equation . though the final @xmath10 resonance positions could not be reached with that method , we think this illustration should be of interest since it works beautifully for resonances close to the first threshold and requires only bound state techniques . we summarise in section v. the faddeev equation for three identical nucleons has the well known form [ 19 ] @xmath28 where @xmath29 is the free @xmath30 propagator , @xmath31 the @xmath12 @xmath31-matrix and @xmath18 the sum of a cyclical and anti - cyclical permutation of 3 objects . we regard that equation in a momentum space representation and partial wave decomposed . it results a set of coupled equations in two variables , p and q , which are the magnitudes of two relative momenta . in the present investigation we shall use @xmath31-matrices of finite rank . the set of two - dimensional equations changes then into a set of coupled one - dimensional ones . they have the well known form @xmath32 and @xmath33 is connected to @xmath34 via @xmath35 the underlying two - body @xmath31-matrices are @xmath36 the corresponding form factors @xmath37 depend on the obvious two - body quantum numbers @xmath38 . the two - body subsystem energy @xmath39 is given in terms of the total energy @xmath40 and the kinetic energy of relative motion of the third particle . further @xmath41 stands for the string of three - body quantum numbers @xmath42 for all those by now standard notations we refer to [ 19 ] . finally @xmath43 are geometrical coefficients arising from the partial wave representation of the permutation operator @xmath18 and @xmath44 and @xmath45 are shifted arguments related to @xmath18 and depend on @xmath46 , @xmath47 and @xmath48 , see [ 19 ] for details . we are interested in @xmath10 bound state and resonance energies , the latter ones emerging from bound state energies when the artificially increased interaction strength is reduced . since two neutrons are not bound in nature the only unphysical sheet of interest is the one which can be reached through the cut along @xmath49 . it will be denoted in the following simply as the unphysical sheet . while for @xmath50 there is no singular denominator , for @xmath51 the free propagator in eq . ( 2 ) is not defined . the well established manner to analytically continue the coupled set of equations in the energy into the unphysical sheet is to deform the path of integration into the lower half plane . this has been described for instance in [ 8 ] and [ 20 ] . the singularities which are avoided in this manner arise from the free propagator and the form factors . the @xmath52-function is not singular for energies @xmath53 in the unphysical sheet under investigation . a typical path in the complex @xmath46-plane ( for both @xmath46 and @xmath47 ) together with domains of singularity arising from the free propagator for the example @xmath54 mev is shown in fig . 1 . the strategy now as pointed out in the introduction is to first enhance the @xmath11 forces such that a @xmath10 bound state exists . at the same time also two - neutron bound states might exist . then by suitably chosen enhancement factors one can achieve that the two - body bound state energies coincide and move together towards @xmath3 , when the enhancements are reduced . in all cases we studied the @xmath10 bound state energies stayed always to the left of the two - body bound state energies until they reached @xmath3 . this was not the case in [ 8 ] , where for the degenerate states @xmath55 and @xmath16 only the @xmath14 force had been kept . here we include on top @xmath18- and @xmath56-wave forces and such a coincidence of @xmath10 and @xmath17 bound state energies does not occur . once the two - neutron bound state energies have disappeared we still have a @xmath10 bound state . then reducing the enhancement factors further the @xmath10 bound state energy will approach @xmath3 and then the energy eigenvalue @xmath40 will dive into the lower half plane of the unphysical sheet . there we shall follow its trajectory until the physical strengths of the @xmath11 forces are reached . this defines the final positions of the @xmath10 resonances . it is well known and easily seen that what we call @xmath10 resonance energies are poles of the @xmath10 @xmath0-matrix . we use @xmath11 forces of rank 2 . they are given in appendix a. they describe @xmath26 phase shifts ( without coulomb force ) reasonably well as documented in the appendix . we restrict our study to the states @xmath14 , @xmath23,@xmath24 and @xmath57 . the coupling between @xmath24 and @xmath25 is neglected . the @xmath11 force in the state @xmath58 is repulsive and therefore we did not include it in the main investigation , since it has to be expected , that it will move the @xmath10 resonance position further away from the positive real energy axis . at the end we shall consider its effect and show that its influence has indeed repulsive character but only a marginal one and could therefore indeed be neglected . we shall investigate now the 4 possible states @xmath59 and @xmath60 in turn . we use up to 10 three - neutron channels which are displayed in table 1 . to reach a @xmath10 bound state it is sufficient to enhance just the @xmath24 @xmath11 force and to keep all the other force components at their physical values . as an example for an enhancement factor we quote @xmath61 where the 3 neutrons in a 10 channel calculation are bound at @xmath62 mev . the threshold energy @xmath3 is reached for @xmath63 . for the sake of future comparisons we provide a few intermediate @xmath10 resonance positions in table 2 . the @xmath10 resonance trajectory is shown in fig . 2 for a 10 channel calculation . the trajectories for a smaller number of channels are very similar . the final resonance positions for an increasing number of channels in the order given in table 1 are displayed in table 3 . adding more @xmath11 force components will certainly not change the final position in a significant manner in the sense that the position would come somewhere close to the positive real energy axis . if one adds the @xmath58 @xmath11 force , which is of repulsive nature , we have 13 channels and the final resonance position shifts by @xmath64 mev , which is marginal . in ref.[16 ] which can be considered to be the most realistic approach towards three - neutron resonances carried out so far the rather realistic @xmath11 forces by gogny [ 17 ] and reid93 [ 18 ] have been used . in contrast to our simplified forces they also include the tensor force induced coupling @xmath65 . for the case of the gogny potential the enhancement factor for the @xmath65 @xmath11 force component was about 4.4 to have @xmath10 s bound at zero energy , while for our simplified force this was the smaller value of @xmath66 . in [ 16 ] reducing @xmath67 to @xmath68 yields a resonance energy of @xmath69 mev , while at this value our @xmath10 system is still bound . thus one might conjecture that in the more realistic case the @xmath10 resonance would be even further shifted away from the positive real energy axis . on the other hand the motion of our resonance energy for a change of @xmath70 from the threshold @xmath3 ( corresponding to an enhancement factor @xmath71 ) is @xmath72 mev , which is slightly larger . therefore a direct quantitative comparison is not possible , though qualitatively the results are similar . this is also true for the case of reid93 , also considered in [ 16 ] , where the enhancement factor @xmath73 yields a resonance position of @xmath74 mev . .the partial wave quantum number for the three neutron state @xmath75 . [ cols="^,^,^,^,^,^ " , ] we would like to thank y. koike for very helpful hints at the beginning of the study . a. hemmdan would like to thank the institut fr theoretische physik ii of the ruhr - universitt bochum for the very kind hospitality . this research was supported by the egyptian government under a fellowship in the channel system . the finite rank potential we use has the following form [ 25 ] the @xmath79-dependent parameters of the potential form factors are displayed in table 19 . the resulting phase shifts of the nn system for the partial waves @xmath14 , @xmath23 , @xmath58 , @xmath24 and @xmath57 are compared to @xmath26 phase shift values ( without coulomb ) given in said [ 21 ] in fig . 6 . in table 20 we show the scattering length @xmath80 and the effective range @xmath81 for the @xmath11 system in the state @xmath14 .	the pending question of the existence of three - neutron resonances near the physical energy region is reconsidered . finite rank neutron - neutron forces are used in faddeev equations , which are analytically continued into the unphysical energy sheet below the positive real energy axis . the trajectories of the three - neutron @xmath0-matrix poles in the states of total angular momenta and parity @xmath1 and @xmath2 are traced out as a function of artificial enhancement factors of the neutron - neutron forces . the final positions of the @xmath0-matrix poles removing the artificial factors are found in all cases to be far away from the positive real energy axis , which provides a strong indication for the nonexistence of nearby three - neutron resonances . the pole trajectories close to the threshold @xmath3 are also predicted out of auxiliary generated three - neutron bound state energies using the pad method and agree very well with the directly calculated ones . * indications for the nonexistence of three - neutron resonances near the physical region * a. hemmdan@xmath4 , w. glckle @xmath5 , and h.kamada@xmath6 @xmath7_institut fr theoretische physik ii , ruhr - universitt bochum , d-44870 bochum , germany _ + @xmath8_department of physics , faculty of science , south valley university , aswan , egypt _ + @xmath9_department of physics , faculty of engineering , kyushu institute of technology , + 1 - 1 sensuicho , tobata , kitakyushu 804 - 8550 , japan _ pacs numbers : 21.45.+v,24.30.gd,25.70.ef
the production of hadrons is due to the strong interactions of quarks and gluons . quantum chromodynamics ( qcd ) , the gauge theory of the strong interactions , provides a quantitative description of the transitions from quarks and gluons to jets of hadrons , which may be tested experimentally . when jets are produced at colliders , they can be initiated either by a quark or a gluon . the two types of jets are expected to exhibit different properties , above all because quarks and gluons carry different color charges and spin . in fact , a gluon jet is typically broader and contains a larger amount of hadrons . jets with different mother partons can also be studied by looking for the jet charge distribution as discussed in ref . @xcite , with important consequences for the physics at the cern large hadron collider ( lhc ) . to understand the interplay of quarks and gluons in a jet and to predict testable consequences thereof lies at the very core of qcd . the typical way to depict the production of a jet from a parton ( quark or gluon ) is the following . an initial parton starts radiating gluons , which in turn can radiate further gluons or split into secondary quark - antiquark pairs . this so - called parton showering process causes the virtualities of the parent partons to decrease . finally , when the virtuality falls below a certain cutoff , the cascade stops and the final - state partons hadronize into color - neutral hadrons , a process usually described by phenomenological models . this happens because the production of hadrons is a typical process where nonperturbative phenomena are involved . however , for particular observables , this problem can be avoided . in particular , the _ counting _ of hadrons in a jet that is initiated at a certain scale @xmath5 belongs to this class of observables . in this case , one can adopt with quite high accuracy the hypothesis of local parton - hadron duality ( lphd ) , which simply states that parton distributions are renormalized in the hadronization process without changing their shapes @xcite . hence , if the scale @xmath5 is large enough , this would in principle allow perturbative qcd to be predictive without the need to consider phenomenological models of hadronization . nevertheless , such processes are dominated by soft - gluon emissions , and it is a well - known fact that , in such kinematic regions of phase space , fixed - order perturbation theory fails , rendering the usage of resummation techniques indispensable . as we shall see , the computation of avarage jet multiplicities indeed requires small-@xmath0 resummation , as was already realized a long time ago @xcite . in ref . @xcite , it was shown that the singularities for @xmath6 , which are encoded in large logarithms of the kind @xmath7 , spoil perturbation theory , and also render integral observables in @xmath0 ill - defined , disappear after resummation . usually , resummation includes the singularities from all orders according to a certain logarithmic accuracy , for which it _ restores _ perturbation theory . small-@xmath0 resummation has recently been carried out for timelike splitting fuctions in the @xmath8 factorization scheme , which is generally preferable to other schemes , yielding fully analytic expressions . in a first step , the next - to - leading - logarithmic ( nll ) level of accuracy has been reached @xcite . in a second step , this has been pushed to the next - to - next - to - leading - logarithmic ( nnll ) , and partially even to the next - to - next - to - next - to - leading - logarithmic ( n@xmath9ll ) , level @xcite . thanks to these results , we are able to analytically compute the nnll contributions to the evolutions of the average gluon and quark jet multiplicities with normalization factors evaluated to next - to - leading ( nlo ) and approximately to next - to - next - to - next - to - order ( n@xmath9lo ) in the @xmath10 expansion . the previous literature contains a nll result on the small-@xmath0 resummation of timelike splitting fuctions obtained in a massive - gluon scheme . unfortunately , this is unsuitable for the combination with available fixed - order corrections , which are routinely evaluated in the @xmath8 scheme . a general discussion of the scheme choice and dependence in this context may be found in refs . @xcite . the average gluon and quark jet multiplicities , which we denote as @xmath11 and @xmath12 , respectively , represent the avarage numbers of hadrons in a jet initiated by a gluon or a quark at scale @xmath5 . in the past , analytic predictions were obtained by solving the equations for the generating functionals in the modified leading - logarithmic approximation ( mlla ) in ref . @xcite through n@xmath9lo in the expansion parameter @xmath10 , i.e. through @xmath13 . however , the theoretical prediction for the ratio @xmath14 given in ref . @xcite is about 10% higher than the experimental data at the scale of the @xmath15 boson , and the difference with the data becomes even larger at lower scales , although the perturbative series seems to converge very well . an alternative approach was proposed in ref . @xcite , where a differential equation for the average gluon - to - quark jet multiplicity ratio was obtained in the mlla within the framework of the colour - dipole model , and the constant of integration , which is supposed to encode nonperturbative contributions , was fitted to experimental data . a constant offset to the average gluon and quark jet multiplicities was also introduced in ref . @xcite . recently , we proposed a new formalism @xcite that solves the problem of the apparent good convergence of the perturbative series and does not require any ad - hoc offset , once the effects due to the mixing between quarks and gluons are fully included . our result is a generalization of the result obtained in ref . @xcite . in our new approach , the nonperturbative informations to the gluon - to - quark jet multiplicity ratio are encoded in the initial conditions of the evolution equations . motivated by the excellent agreement of our results with the experimental data found in ref . @xcite , we propose here to also use our approach to extract the strong - coupling constant @xmath16 at some reference scale @xmath17 and thus extend our analysis by adding an apropriate fit parameter . the paper is organized as follows . in section [ ffs ] , we introduce the equations governing the evolution of the average gluon and quark jet multiplicities with the scale @xmath5 at which the jet is initiated , develop a formalism to solve them , and improve our results by resummation . in section [ multiplicities ] , we explain how we can predict the average - jet - multiplicity evolutions in our framework adding as much as possible available information on small-@xmath0 timelike resummation . in section [ analysis ] , we fit our resummed formulae to the available experimental data exctracting the initial conditions for the evolutions , and discuss the uncertainties coming from both the statistical analysis of the data and the missing higher - order terms . in section [ coupling ] , we inject the strong - coupling constant into our analysis and extract it . finally , in section [ conclusions ] , we summarize our conclusions and present an outlook . when one considers average multiplicity observables , the basic equation is the one governing the evolution of the fragmentation functions @xmath18 for the gluon quark - singlet system @xmath19 . in mellin space , it reads : @xmath20 where @xmath21 , with @xmath22 , are the timelike splitting functions , @xmath23 , with @xmath24 being the standard mellin moments with respect to @xmath0 , and @xmath25 is the couplant . the standard definition of the hadron multiplicities in terms of the fragmentation functions is given by their integral over @xmath0 , which clearly corresponds to the first mellin moment , with @xmath26 ( see , e.g. , ref . @xcite ) : @xmath27_{\omega=0}=d_a(\omega=0,q^2 ) , \label{multdef2}\ ] ] where @xmath19 for a gluon and quark jet , respectively . the timelike splitting functions @xmath21 in eq . ( [ ap ] ) may be computed perturbatively in @xmath28 , @xmath29 the functions @xmath30 for @xmath31 in the @xmath8 scheme may be found in refs . @xcite through nnlo and in refs . @xcite with small-@xmath0 resummation through nnll accuracy . in the remainder of this section , we explain in detail our new approach to solve eq . ( [ ap ] ) in order to use its solution in eq . ( [ multdef2 ] ) to obtain the average gluon and quark jet multiplicities . to this end , we first discuss how eq . ( [ ap ] ) can be diagonalized and then how to implement resummation to improve it , so as to obtain well - defined quantities at @xmath26 . it is not in general possible to diagonalize eq . ( [ ap ] ) because the contributions to the timelike - splitting - function matrix do not commute at different orders . the usual approach is then to write a series expansion about the leading - order ( lo ) solution , which can in turn be diagonalized . one thus starts by choosing a basis in which the timelike - splitting - function matrix is diagonal at lo ( see , e.g. , ref . @xcite ) , @xmath32 with eigenvalues @xmath33 . in one important simplification of qcd , namely @xmath34 super yang - mills theory , this basis is actually more natural than the @xmath35 basis because the diagonal splitting functions @xmath36 may there be expressed in all orders of perturbation theory as one universal function with shifted arguments @xcite . it is convenient to represent the change of basis for the fragmentation functions order by order for @xmath37 as @xcite : @xmath38 this implies for the components of the timelike - splitting - function matrix that @xmath39 where @xmath40 our approach to solve eq . ( [ ap ] ) differs from the usual one in that we write the solution expanding about the diagonal part of the all - order timelike - splitting - function matrix in the plus - minus basis , instead of its lo contribution . for this purpose , we rewrite eq . ( [ pmbasis ] ) in the following way : @xmath41 in general , the solution to eq . ( [ ap ] ) in the plus - minus basis can be formally written as @xmath42 where @xmath43 denotes the path ordering with respect to @xmath44 and @xmath45 as anticipated , we make the following ansatz to expand about the diagonal part of the timelike - splitting - function matrix in the plus - minus basis : @xmath46z(\mu_0 ^ 2 ) , \label{ansatz}\ ] ] where @xmath47 is the diagonal part of eq . ( [ sfdec ] ) and @xmath48 is a matrix in the plus - minus basis which has a perturbative expansion of the form @xmath49 in the following , we make use of the renormalization group ( rg ) equation for the running of @xmath50 , @xmath51 where @xmath52 with @xmath53 , @xmath54 , and @xmath55 being colour factors and @xmath56 being the number of active quark flavours . using eq . ( [ running ] ) to perform a change of integration variable in eq . ( [ ansatz ] ) , we obtain @xmath57z(a_s(\mu_0 ^ 2 ) ) . \label{ansatz2}\ ] ] substituting then eq . ( [ zpertexp ] ) into eq . ( [ ansatz2 ] ) , differentiating it with respect to @xmath28 , and keeping only the first term in the @xmath28 expansion , we obtain the following condition for the @xmath58 matrix : @xmath59=\frac{p^{(1)od}}{\beta_0},\ ] ] where @xmath60 solving it , we find : @xmath61 at this point , an important comment is in order . in the conventional approach to solve eq.([ap ] ) , one expands about the diagonal lo matrix given in eq . ( [ pmbasis ] ) , while here we expand about the all - order diagonal part of the matrix given in eq . ( [ sfdec ] ) . the motivation for us to do this arises from the fact that the functional dependence of @xmath62 on @xmath28 is different after resummation . now reverting the change of basis specified in eq . ( [ changebasisin ] ) , we find the gluon and quark - singlet fragmentation functions to be given by @xmath63 as expected , this suggests to write the gluon and quark - singlet fragmentation functions in the following way : @xmath64 where @xmath65 evolves like a plus component and @xmath66 like a minus component . we now explicitly compute the functions @xmath67 appearing in eq . ( [ decomp ] ) . to this end , we first substitute eq . ( [ ansatz ] ) into eq . ( [ gensol ] ) . using eqs . ( [ diagpart ] ) and ( [ zmatrix ] ) , we then obtain @xmath68 where @xmath69 and @xmath70 . \label{rengroupexp}\ ] ] has a rg - type exponential form . finally , inserting eq . ( [ result1 ] ) into eq . ( [ inverbasis ] ) , we find by comparison with eq . ( [ decomp ] ) that @xmath71 where @xmath72 and @xmath73 are perturbative functions given by @xmath74 at @xmath75 , we have @xmath76 where @xmath77 is given by eq . ( [ zmatrix ] ) . as already mentioned in section [ sec : intro ] , reliable computations of average jet multiplicities require resummed analytic expressions for the splitting functions because one has to evaluate the first mellin moment ( corresponding to @xmath78 ) , which is a divergent quantity in the fixed - order perturbative approach . as is well known , resummation overcomes this problem , as demonstrated in the pioneering works by mueller @xcite and others @xcite . in particular , as we shall see in section [ multiplicities ] , resummed expressions for the first mellin moments of the timelike splitting functions in the plus - minus basis appearing in eq . ( [ pmbasis ] ) are required in our approach . up to the nnll level in the @xmath8 scheme , these may be extracted from the available literature @xcite in closed analytic form using the relations in eq . ( [ changebasis ] ) . note that the expressions are generally simpler in the plus - minus basis , while the corresponding results for the resummation of @xmath79 and @xmath80 can be highly nontrivial and complicated in higher orders of resummation . an analogous observation was made for the double - logarithm aymptotics in the kirschner - lipatov approach @xcite , where the corresponding amplitudes obey nontrivial equations , whose solutions are rather complicated special functions . for future considerations , we remind the reader of an assumpion already made in ref . @xcite according to which the splitting functions @xmath81 and @xmath82 are supposed to be free of singularities in the limit @xmath83 . in fact , this is expected to be true to all orders . this is certainly true at the ll and nll levels for the timelike splitting functions , as was verified in our previous work @xcite . this is also true at the nnll level , as may be explicitly checked by inserting the results of ref . @xcite in eq . ( [ changebasis ] ) . moreover , this is true through nlo in the spacelike case @xcite and holds for the lo and nlo singularities @xcite to all orders in the framework of the balitski - fadin - kuraev - lipatov ( bfkl ) dynamics @xcite , a fact that was exploited in various approaches ( see , e.g. , refs . @xcite and references cited therein ) . we also note that the timelike splitting functions share a number of simple properties with their spacelike counterparts . in particular , the lo splitting functions are the same , and the diagonal splitting functions grow like @xmath84 for @xmath85 at all orders . this suggests the conjecture that the double - logarithm resummation in the timelike case and the bfkl resummation in the spacelike case are only related via the plus components . the minus components are devoid of singularities as @xmath83 and thus are not resummed . now that this is known to be true for the first three orders of resummation , one has reason to expect this to remain true for all orders . using the relationships between the components of the splitting functions in the two bases given in eq . ( [ changebasis ] ) , we find that the absence of singularities for @xmath26 in @xmath86 and @xmath87 implies that the singular terms are related as @xmath88 where , through the nll level , @xmath89+\mathcal{o}(\omega^2 ) . \label{motivation}\ ] ] an explicit check of the applicability of the relationships in eqs . ( [ tolja3 ] ) and ( [ tolja3.1 ] ) for @xmath21 with @xmath90 themselves is performed in the appendix . of course , the relationships in eqs . ( [ tolja3 ] ) and ( [ tolja3.1 ] ) may be used to fix the singular terms of the off - diagonal timelike splitting functions @xmath91 and @xmath80 using known results for the diagonal timelike splitting functions @xmath92 and @xmath79 . since refs . @xcite became available during the preparation of ref . @xcite , the relations in eqs . ( [ tolja3 ] ) and ( [ tolja3.1 ] ) provided an important independent check rather than a prediction . we take here the opportunity to point out that eqs . ( [ evolsol ] ) and ( [ rlo ] ) together with eq . ( [ motivation ] ) support the motivations for the numerical effective approach that we used in ref . @xcite to study the average gluon - to - quark jet multiplicity ratio . in fact , according to the findings of ref . @xcite , substituting @xmath93 , where @xmath94 into eq . ( [ motivation ] ) exactly reproduces the result for the average gluon - to - quark jet multiplicity ratio @xmath95 obtained in ref . @xcite . in the next section , we shall obtain improved analytic formulae for the ratio @xmath95 and also for the average gluon and quark jet multiplicities . here we would also like to note that , at first sight , the substitution @xmath96 should induce a @xmath97 dependence in eq . ( [ elements ] ) , which should contribute to the diagonalization matrix . this is not the case , however , because to double - logarithmic accuracy the @xmath97 dependence of @xmath98 can be neglected , so that the factor @xmath99 does not recieve any @xmath97 dependence upon the substitution @xmath96 . this supports the possibility to use this substitution in our analysis and gives an explanation of the good agreement with other approaches , e.g. that of ref . @xcite . nevertheless , this substitution only carries a phenomenological meaning . it should only be done in the factor @xmath99 , but not in the rg exponents of eq . ( [ rengroupexp ] ) , where it would lead to a double - counting problem . in fact , the dangerous terms are already resummed in eq . ( [ rengroupexp ] ) . in order to be able to obtain the average jet multiplicities , we have to first evaluate the first mellin momoments of the timelike splitting functions in the plus - minus basis . according to eq . ( [ changebasis ] ) together with the results given in refs . @xcite , we have @xmath100 where @xmath101 , \\ k_2&=&\frac{1}{288 } \left[1193 - 576\zeta_2 -56\frac{n_f t_r}{c_a } \left(5 + 2\frac{c_f}{c_a}\right)\right ] + 16 \frac{n^2_f t^2_r}{c^2_a } \left(1 + 4\frac{c_f}{c_a}-12\frac{c^2_f}{c^2_a}\right),\quad\end{aligned}\ ] ] and @xmath102 where @xmath103 { \left(2c_a a_s^3\right)}^{1/2}. \label{nllseconda}\ ] ] for the @xmath104 component , we obtain @xmath105 finally , as for the @xmath106 component , we note that its lo expression produces a finite , nonvanishing term for @xmath26 that is of the same order in @xmath28 as the nll - resummed results in eq . ( [ nllfirst ] ) , which leads us to use the following expression for the @xmath106 component : @xmath107 at nnll accuracy . we can now perform the integration in eq . ( [ rengroupexp ] ) through the nnll level , which yields @xmath108\right\ } \left(a_s(q^2)\right)^{d_+ } , \\ t_{-}^\mathrm{nnll}(q^2)&=&t_{-}^\mathrm{nll}(q^2 ) = \left(a_s(q^2)\right)^{d_- } , \label{nnllresultm}\end{aligned}\ ] ] where @xmath109 in order to estimate the contribution to an observable of interest from orders of perturbation theory beyond our calculation , we may shift the argument of the strong - coupling constant as @xmath110 applying this shift to eqs . ( [ evolsolaa])([anomdim ] ) , there is only one change in the rg exponents , namely @xmath111\right\ } \left(a_s(\xi q^2)\right)^{d_+}. \label{shifta}\ ] ] according to eqs . ( [ rengroupexp ] ) and ( [ evolsol ] ) , the @xmath112 components are not involved in the @xmath97 evolution of average jet multiplicities , which is performed at @xmath113 using the resummed expressions for the plus and minus components given in eq . ( [ nllfirst ] ) and ( [ nllsecond ] ) , respectively . we are now ready to define the average gluon and quark jet multiplicities in our formalism , namely @xmath114 with @xmath19 , respectively . on the other hand , from eqs . ( [ evolsol ] ) and ( [ rlo ] ) , it follows that @xmath115 using these definitions and again eq . ( [ evolsol ] ) , we may write general expressions for the average gluon and quark jet multiplicities : @xmath116 at the lo in @xmath28 , the coefficients of the rg exponents are given by @xmath117 for @xmath19 . it would , of course , be desirable to include higher - order corrections in eqs . ( [ lonnll ] ) . however , this is highly nontrivial because the general perturbative structures of the functions @xmath118 and @xmath119 , which would allow us to resum those higher - order corrections , are presently unknown . fortunatly , some approximations can be made . on the one hand , it is well - known that the plus components by themselves represent the dominant contributions to both the average gluon and quark jet multiplicities ( see , e.g. , ref . @xcite for the gluon case and ref . @xcite for the quark case ) . on the other hand , eq . ( [ rmin ] ) tells us that @xmath120 is suppressed with respect to @xmath121 because @xmath122 . these two observations suggest that keeping @xmath123 also beyond lo should represent a good approximation . nevertheless , we shall explain below how to obtain the first nonvanishing contribution to @xmath124 . furthermore , we notice that higher - order corrections to @xmath125 and @xmath126 just represent redefinitions of @xmath127 by constant factors apart from running - coupling effects . therefore , we assume that these corrections can be neglected . note that the resummation of the @xmath128 components was performed similarly to eq . ( [ rengroupexp ] ) for the case of parton distribution functions in ref . such resummations are very important because they reduce the @xmath97 dependences of the considered results at fixed order in perturbation theory by properly taking into account terms that are potentially large in the limit @xmath83 @xcite . we anticipate similar properties in the considered case , too , which is in line with our approximations . some additional support for this may be obtained from @xmath129 super yang - mills theory , where the diagonalization can be performed exactly in any order of perturbation theory because the coupling constant and the corresponding martices for the diagonalization do not depended on @xmath97 . consequently , there are no @xmath130 terms , and only @xmath131 terms contribute to the integrand of the rg exponent . looking at the r.h.s . ( [ renfact ] ) and ( [ pertfun ] ) , we indeed observe that the corrections of @xmath132 would cancel each other if the coupling constant were scale independent . we now discuss higher - order corrections to @xmath133 . as already mentioned above , we introduced in ref . @xcite an effective approach to perform the resummation of the first mellin moment of the plus component of the anomalous dimension . in that approach , resummation is performed by taking the fixed - order plus component and substituting @xmath93 , where @xmath134 is given in eq . ( [ replacement ] ) . we now show that this approach is exact to @xmath135 . we indeed recover eq . ( [ llgamma0 ] ) by substituting @xmath93 in the leading singular term of the lo splitting function @xmath136 , @xmath137 we may then also substitute @xmath93 in eq . ( [ evolsola ] ) before taking the limit in @xmath26 . using also eq . ( [ motivation ] ) , we thus find @xmath138+\mathcal{o}(a_s ) , \label{rplusll}\ ] ] which coincides with the result obtained by mueller in ref . @xcite . for this reason and because , in ref . @xcite , the average gluon and quark jet multiplicities evolve with only one rg exponent , we inteprete the result in eq . ( 5 ) of ref . @xcite as higher - order corrections to eq . ( [ rplusll ] ) . complete analytic expressions for all the coefficients of the expansion through @xmath139 may be found in appendix 1 of ref . this interpretation is also explicitely confirmed in chapter 7 of ref . @xcite through @xmath132 . since we showed that our approach reproduces exact analytic results at @xmath135 , we may safely apply it to predict the first non - vanishing correction to @xmath124 defined in eq . ( [ rmin ] ) , which yields @xmath140 however , contributions beyond @xmath141 obtained in this way can not be trusted , and further investigation is required . therefore , we refrain from considering such contributions here . for the reader s convenience , we list here expressions with numerical coefficients for @xmath133 through @xmath139 and for @xmath124 through @xmath135 in qcd with @xmath142 : @xmath143 we denote the approximation in which eqs . ( [ nnllresult])([nnllresultm ] ) and ( [ lonnll ] ) are used as @xmath144 , the improved approximation in which the expression for @xmath133 in eq . ( [ lonnll ] ) is replaced by eq . ( [ dreminscaleplus ] ) , i.e. eq . ( 5 ) in ref . @xcite , as @xmath145 , and our best approximation in which , on top of that , the expression for @xmath124 in eq . ( [ lonnll ] ) is replaced by eq . ( [ dreminscaleminus ] ) as @xmath146 . we shall see in section [ analysis ] , where we compare with the experimental data and extract the strong - coupling constant , that the latter two approximations are actually very good and that the last one yields the best results , as expected . in all the approximations considered here , we may summarize our main theoretical results for the avarage gluon and quark jet multiplicities in the following way : @xmath147 where @xmath148 the average gluon - to - quark jet multiplicity ratio may thus be written as @xmath149 , \label{ratiogen}\ ] ] where @xmath150 it follows from the definition of @xmath151 in eq . ( [ nnllresult ] ) and from eq . ( [ n2 ] ) that , for @xmath152 , eqs . ( [ quarkgen ] ) and ( [ ratiogen ] ) become @xmath153 these represent the initial conditions for the @xmath97 evolution at an arbitrary initial scale @xmath17 . in fact , eq . ( [ quarkgen ] ) is independ of @xmath154 , as may be observed by noticing that @xmath155 for an arbitrary scale @xmath156 ( see also ref . @xcite for a detailed discussion of this point ) . in the approximations with @xmath123 @xcite , i.e. the @xmath144 and @xmath145 ones , our general results in eqs . ( [ quarkgen ] ) , and ( [ ratiogen ] ) collapse to @xmath157 \hat{t}_-^\mathrm{res}(0,q^2,q_0 ^ 2 ) , \nonumber\\ r(q^2 ) & = & \frac{r_{+}(q^2)}{\left[1 + \frac{r_{+}(q^2)}{r_{+}(q_0 ^ 2)}\left ( \frac{d_s(0,q^2_0)r_{+}(q_0 ^ 2)}{d_g(0,q^2_0 ) } -1 \right ) \frac{\hat{t}_{-}^\mathrm{res}(0,q^2,q^2_0)}{\hat{t}_{+}^\mathrm{res}(0,q^2,q^2_0)}\right ] } .\end{aligned}\ ] ] the nnll - resummed expressions for the average gluon and quark jet multiplicites given by eq . ( [ quarkgen ] ) only depend on two nonperturbative constants , namely @xmath158 and @xmath159 . these allow for a simple physical interpretation . in fact , according to eq . ( [ incond ] ) , they are the average gluon and quark jet multiplicities at the arbitrary scale @xmath17 . we should also mention that identifying the quantity @xmath133 with the one computed in ref . @xcite , we assume the scheme dependence to be negligible . this should be justified because of the scheme independence through nll established in ref . @xcite . we note that the @xmath97 dependence of our results is always generated via @xmath98 according to eq . ( [ running ] ) . this allows us to express eq . ( [ nnllresult ] ) entirely in terms of @xmath160 . in fact , substituting the qcd values for the color factors and choosing @xmath142 in the formulae given in ref . @xcite , we may write at nnll @xmath161^{d_1 } , \nonumber\\ \hat{t}_+^\mathrm{res}(q^2,q_0 ^ 2)&= & \exp\left[d_2\left(\frac{1}{\sqrt{\alpha_s(q^2 ) } } -\frac{1}{\sqrt{\alpha_s(q_0 ^ 2)}}\right ) + d_3\left(\sqrt{\alpha_s(q^2)}-\sqrt{\alpha_s(q_0 ^ 2)}\right)\right ] \nonumber\\ & & { } \times\left[\frac{\alpha_s(q^2)}{\alpha_s(q_0 ^ 2)}\right]^{d_4},\end{aligned}\ ] ] where @xmath162 we conclude this section by discussing the theoretical uncertainties in @xmath133 and @xmath124 due to unknown higher - order corrections . similarly to eq . ( [ shifta ] ) , we may estimate them by studying the scale dependence . performing the shift of eq . ( [ shift ] ) in eqs . ( [ dreminscaleplus ] ) and ( [ dreminscaleminus ] ) , we obtain @xmath163 we are now in a position to perform a global fit to the available experimental data of our formulas in eq . ( [ quarkgen ] ) in the @xmath144 , @xmath145 , and @xmath146 approximations , so as to extract the nonperturbative constants @xmath158 and @xmath159 . we have to make a choice for the scale @xmath17 , which , in principle , is arbitrary . we wish to choose it by optimizing the apparent convergence properties of the perturbative qcd expansion . to this end , we analyse in figs . [ fig : plotplus1 ] and [ fig : plotplus2 ] the dependence on the scaling parameter @xmath164 of @xmath165 governed by eq . ( [ shifta ] ) at different logarithmic accuracies for @xmath166 gev and @xmath167 gev , respectively . we put @xmath168 gev because this is in the center of the range where the majority of the available data located . we observe a strong reduction of the scale dependence as we pass from ll via nll to nnll , both for @xmath166 gev and @xmath167 gev . the perturbative series appears to be more rapidly converging at relatively large values of @xmath17 . therefore , we adopt @xmath167 gev in the following . another good reason for this choice is that , according to eq . ( [ incond ] ) , @xmath158 and @xmath159 represent the avarage gluon and quark jet multiplicities , respectively , at the scale @xmath17 , so that the fit results for our initial conditions may be directly compared with the experimental data at @xmath167 gev . in fig . [ fig : plotminus ] , we compare the scale dependence of @xmath169 , which is obtained by simply replacing @xmath97 with @xmath170 in eq . ( [ nnllresult ] ) , with the one of @xmath165 evaluated according to eq . ( [ shifta ] ) , for @xmath168 gev and @xmath167 gev . we observe from fig . [ fig : plotminus ] that the scale variation is very similar in both cases . in fig . [ fig : plotrplus ] , we study the scale dependence of @xmath133 evaluated at lo , nlo , nnlo , and n@xmath9lo according to eq . ( [ dreminscale ] ) . we observe that the scale dependence gradually increases as we pass from lo via nlo to nnlo , while it decreases in the step from nnlo to n@xmath9lo , and hence conclude that only the latter order may be trusted . prior to presenting our fits , we explain our definition of confidence level ( cl ) , which we adopt from ref . suppose a fit of the free parameters to @xmath171 experimental data points yields the minimum @xmath172 value @xmath173 . we then determine the 90% cl limits on a fit parameter by varying it so that the resulting @xmath172 values stay within the range @xmath174 where @xmath175 are defined such that @xmath176 with @xmath177 .fit results for @xmath178 and @xmath179 at @xmath167 gev with 90% cl errors and minimum values of @xmath180 achieved in the @xmath144 , @xmath145 , and @xmath146 approximations . [ cols="^,^,^,^",options="header " , ] in section [ analysis ] , we took @xmath181 to be a fixed input parameter for our fits . motivated by the excellent goodness of our @xmath145 and @xmath146 fits , we now include it among the fit parameters , the more so as the fits should be sufficiently sensitive to it in view of the wide @xmath97 range populated by the experimental data fitted to . we fit to the same experimental data as before and again put @xmath167 gev . the fit results are summarized in table [ tab : fit2 ] . we observe from table [ tab : fit2 ] that the results of the @xmath145 @xcite and @xmath146 fits for @xmath178 and @xmath179 are mutually consistent . they are also consistent with the respective fit results in table [ tab : fit ] . as expected , the values of @xmath180 are reduced by relasing @xmath181 in the fits , from 3.71 to 2.84 in the @xmath145 approximation and from 2.95 to 2.85 in the @xmath146 one . the three - parameter fits strongly confine @xmath181 , within an error of 3.7% at 90% cl in both approximations . the inclusion of the @xmath124 term has the beneficial effect of shifting @xmath181 closer to the world average , @xmath182 @xcite . in fact , our @xmath146 value , @xmath183 at 90% cl , which corresponds to @xmath184 at 68% cl , is in excellent agreement with the former . in order to illustrate the sensitivity of our @xmath145 and @xmath146 fits to @xmath181 , we show in fig . [ fig : as_parabola ] the values of @xmath172 obtained by varying @xmath181 while keeping @xmath178 and @xmath179 at their respective central values listed in table [ tab : fit2 ] . prior to our analysis in ref . @xcite , experimental data on the average gluon and quark jet multiplicities could not be simultaneously described in a satisfactory way mainly because the theoretical formalism failed to account for the difference in hadronic contents between gluon and quark jets , although the convergence of perturbation theory seemed to be well under control @xcite . this problem may be solved by including the minus components governed by @xmath185 in eqs . ( [ quarkgen ] ) and ( [ ratiogen ] ) . this was done for the first time in ref . @xcite , albeit in connection with the lo result @xmath123 . the quark - singlet minus component comes with an arbitrary normalization and has a slow @xmath97 dependence . consequently , its numerical contribution may be approximately mimicked by a constant introduced to the average quark jet multiplicity as in ref . @xcite . in the present paper , we improved the analysis of ref . @xcite in various ways . the most natural possible improvement consists in including higher - order correction to @xmath124 . here , we managed to obtain the nlo correction , of @xmath141 , using the effective approach introduced in ref . @xcite , which was shown to also exactly reproduce the @xmath141 correction to @xmath133 . our general result corresponding to eq . ( [ quarkgen ] ) depends on two parameters , @xmath186 and @xmath187 , which , according to eq . ( [ incond ] ) , represent the average gluon and quark jet multiplicities at an arbitrary reference scale @xmath17 and act as initial conditions for the @xmath97 evolution . looking at the perturbative behaviour of the expansion in @xmath10 and the distribution of the available experimental data , we argued that @xmath167 gev is a good choice . we fitted these two parameters to all available experimental data on the average gluon and quark jet multiplicities treating @xmath181 as an input parameter fixed to the world avarage @xcite . we worked in three different approximations , labeled @xmath144 , @xmath145 , and @xmath146 , in which the logarithms @xmath3 are resummed through the nnll level , @xmath133 is evaluated at lo or approximately at n@xmath9lo , and @xmath124 is evaluated at lo or nlo . including the nlo correction to @xmath124 , given in eq . ( [ frminus ] ) , significantly improved the quality of the fit , as is evident by comparing the values of @xmath180 for the @xmath145 and @xmath146 fits in table [ tab : fit ] . motivated by the goodness of our @xmath145 and @xmath146 fits with fixed value of @xmath181 in ref . @xcite and here , we then included @xmath181 among the fit parameters , which yielded a further reduction of @xmath180 . the fit results are listed in table [ tab : fit2 ] . also here , the inclusion of the nlo correction to @xmath124 is beneficial ; it shifts @xmath181 closer to the world average to become @xmath184 . a few comments are in order regarding the renormalization scheme and the counting of higher - order corrections in our analysis in order to allow for an appropriate classification of our determination of @xmath181 in the context of a global analysis yielding a world average . we worked in the @xmath8 renormalization scheme , which has become the standard choice in the literature . we reach beyond ordinary fixed - order analyses by resumming the logarithms @xmath3 through the nnll level . furthermore , our expressions are completely rg - improved in the sense that all @xmath97 dependence is accommodated in @xmath160 . unlike usual higher - order calculations in the qcd - improved parton model , the perturbation series of the coefficients @xmath188 are organized in powers of @xmath10 rather than @xmath189 . in the case of @xmath133 , which starts at @xmath190 , our exact knowledge reaches through @xmath191 , i.e. nnlo , while our @xmath192 term represents an educated guess in the sense that it was obtained using a procedure that , strictly speaking , was only tested through nnlo . in the case of @xmath124 , the @xmath190 term vanishes , and the @xmath193 term is listed in eq . ( [ frminus ] ) , i.e.we have control through nlo . however , the coefficients of @xmath124 in eq . ( [ quarkgen ] ) are numerically suppressed relative to those of @xmath133 , by approximately a factor of @xmath193 . in fact , the shift in @xmath194 ( @xmath195 ) induced by the @xmath193 term of @xmath124 is comparable to ( about a factor of three smaller than ) the one induced by the @xmath191 term of @xmath133 . we thus conclude that our determination of @xmath181 is effectively of nnlo . the next steps towards @xmath196 accuracy include an improved computation of the coefficient @xmath124 and an extended resummation of the plus and minus components of the splitting functions . at the lhc , jet multiplicity observables can be measured at unprecedented values of @xmath97 , which will allow for stringent tests of qcd and provide a strong lever arm for high - precision determinations of @xmath181 using the formalism elaborated in ref . @xcite and here . the work p.b . and of a.v.k . was supported in part by the heisenberg - landau program . the work of a.v.k . was supported in part by the russian foundation for basic research rfbr through grant no . 13 - 02 - 01005 . this work was supported by the german federal ministry for education and research bmbf through grant no . 05h12gue and by the german research foundation dfg through the collaborative research centre no . 676 _ particles , strings and the early universe the structure of matter and space time_. here we prove the relations given in eqs . ( [ tolja3 ] ) and ( [ tolja3.1 ] ) between the singular parts of the diagonal and nondiagonal splitting functions in mellin space @xmath197 with @xmath198 and show that they are _ approximately _ true also for the regular parts . following ref . @xcite , we introduce the notation @xmath199 in the following , the resummed functions @xmath197 are built up by their parts @xmath200 corresponding to the considered levels of resummation , with @xmath201 representing the ll , nll , and nnll levels , respectively . the results read : @xmath202 , \nonumber\\ p_{qq}^{(2)}(\omega , a_s)&= & c_f f_a a_s k^{qq } \nonumber\\ p_{gg}(\omega , a_s)&=&-p_{qq}(\omega , a_s ) + \tilde{p}_{gg}^{(0)}(\omega , a_s)+ \tilde{p}_{gg}^{(1)}(\omega , a_s ) + \tilde{p}_{qq}^{(2)}(\omega , a_s ) , \nonumber\\ \tilde{p}_{gg}^{(0)}(\omega , a_s ) & = & \frac{\omega}{4}(s-1 ) , \nonumber\\ \tilde{p}_{gg}^{(1)}(\omega , a_s ) & = & \frac{a_sc_a}{6 } \left[(11 + 2f_a(1 - 2c^f_a)\right]\left(1-s^{-1}\right ) , \nonumber\\ \tilde{p}_{gg}^{(2)}(\omega , a_s ) & = & c_a a_s \omega \left[k^{gg}_1 ( s-1 ) -k^{gg}_2 \left(1-s^{-1}\right ) - k^{gg}_3 \left(1-s^{-3}\right)\right ] , \label{qq+gg0 + 1 + 2}\end{aligned}\ ] ] where @xmath203 , @xmath204 , and @xmath205 \frac{l}{s } - \left[11 - 2f_a\left(3 - 10c^f_a\right)\right ] \left(1- \frac{s-1}{2\eta } \right)\right . \nonumber\\ & & { } -\left . \left[51 - 3f_a\left(1 - 4c^f_a\right)\right ] \frac{s-1}{2 } -20\frac{(s-1)l}{2\eta } - 2 \left[5 - 2f_a\left(1 - 3c^f_a\right)\right ] \frac{s-1}{2\eta } l^2 \right\ } , \nonumber\\ k^{gg}_1 & = & \frac{1193}{576}-\zeta_2 - \frac{7f_a}{144 } \left(5 + 2c^f_a\right ) + \frac{f_a^2}{144 } \left[1 + 4c^f_a\left(1 - 3c^f_a\right)\right ] , \nonumber\\ k^{gg}_2 & = & \frac{415}{288}-\zeta_2 + \frac{f_a}{36 } \left(5 + 2c^f_a\right ) - \frac{f_a^2}{72 } \left[1 - 4c^f_a\left(2 - 3c^f_a\right)\right ] , \nonumber\\ k^{gg}_3 & = & \frac{1}{576 } \left[1 + 2c^f_a\left(1 - 2c^f_a\right)\right]^2,\end{aligned}\ ] ] with @xmath206 . through nnll accuracy , the nondiagonal splitting functions may be represented as @xmath207 where @xmath208 , \nonumber\\ \overline{p}_{gq}^{(2)}(\omega , a_s ) & = & c_f a_s \omega \left\ { \frac{1}{9 } k^{gq}_1- k^{gq}_2 \left[1-\frac{s-1}{2\eta}(l+2)\right ] \right\ } , \label{qg2}\end{aligned}\ ] ] with @xmath209 we observe from eq . ( [ qg ] ) that the relation for the nnll - resummed parts of the splitting functions @xmath91 and @xmath92 in eq . ( [ tolja3.1 ] ) is not only correct for their terms singular as @xmath83 , but also for their regular ones . the situation is different for the relation between @xmath80 and @xmath79 in eq . ( [ tolja3 ] ) , which does not carry over to the regular terms , as is evident from eq . ( [ gq+1 + 2 ] ) . however , the additional terms in eq . ( [ qg2 ] ) have simple forms compared to the expression for @xmath79 in eq . 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ex/9708029 ] . g. abbiendi _ et al . _ [ opal collaboration ] , phys . rev . d * 69 * ( 2004 ) 032002 [ hep - ex/0310048 ] .	we develop a new formalism for computing and including both the perturbative and nonperturbative qcd contributions to the scale evolution of average gluon and quark jet multiplicities . the new method is motivated by recent progress in timelike small-@xmath0 resummation obtained in the @xmath1 factorization scheme . we obtain next - to - next - to - leading - logarithmic ( nnll ) resummed expressions , which represent generalizations of previous analytic results . our expressions depend on two nonperturbative parameters with clear and simple physical interpretations . a global fit of these two quantities to all available experimental data sets that are compatible with regard to the jet algorithms demonstrates by its goodness how our results solve a longstandig problem of qcd . we show that the statistical and theoretical uncertainties both do not exceed 5% for scales above 10 gev . we finally propose to use the jet multiplicity data as a new way to extract the strong - coupling constant . including all the available theoretical input within our approach , we obtain @xmath2 in the @xmath1 scheme in an approximation equivalent to next - to - next - to - leading order enhanced by the resummations of @xmath3 terms through the nnll level and of @xmath4 terms by the renormalization group , in excellent agreement with the present world average . pacs numbers : 12.38.cy , 12.39.st , 13.66.bc , 13.87.fh
the observational evidence for an acceleration of the expansion of the universe is now overwhelming , although the precise cause of this phenomenon is still unknown ( see , e.g. , @xcite for recent reviews ) . in this concern , and besides the need for more accurate estimates of cosmological parameters , the current state of affairs also brings to light some other important aspects regarding the physics of the mechanism behind cosmic acceleration . certainly , one of these aspects concerns the thermodynamical behavior of a dark energy - dominated universe , and questions such as `` what is the thermodynamic behavior of the dark energy in an expanding universe ? '' or , more precisely , `` what is its temperature evolution law ? '' must be answered in the context of this new conceptual set up . another interesting aspect in this discussion is whether thermodynamics in the accelerating universe can place constraints on the time evolution of the dark energy and can also reveal some physical properties of this energy component . the aim of this paper is twofold . first , to derive physical constraints on the dark energy from the second law of thermodynamics and to deduce the temperature evolution law for a dark component with a general equation - of - state ( eos ) parameter @xmath6 ; second , to perform a joint statistical analysis involving current observational data together with the thermodynamic bounds on @xmath6 . to do that , we assume the following generalized formula for the time evolution of @xmath6 @xcite @xmath7 which recovers some well - known eos parameterizations in the following limits : @xmath8 where @xmath9 and @xmath10 stand for the dark energy pressure and energy density , respectively ( see also @xcite for other eos parameterizations ) . the analyses are performed using one the most recent type ia supernovae ( sne ia ) observations , the nearby + sdss + essence + snls + hubble space telescope ( hst ) set of 288 sne ia discussed in ref . @xcite ( which we refer to as sdss compilation ) . we consider two sub - samples of this latter compilation that use salt2 @xcite and mlcs2k2 @xcite sn ia light - curve fitting method . along with the sne ia data , and to help break the degeneracy between the dark energy parameters we also use the baryonic acoustic oscillation ( bao ) peak at @xmath11 @xcite and the current estimate of the cmb shift parameter @xmath12 @xcite . we work in units where c = 1 . throughout this paper a subscript 0 stands for present - day quantities and a dot denotes time derivative . let us first consider a homogeneous , isotropic , spatially flat cosmologies described by the friedmann - robertson - walker ( frw ) flat line element , @xmath13 , where @xmath14 is the cosmological scalar factor . the matter content is assumed to be composed of baryons , cold dark matter and a dark energy component . in such a background , the thermodynamic states of a relativistic fluid are characterized by an energy momentum tensor ( perfect - type fluid ) @xmath15 a particle current , @xmath16 and an entropy current , @xmath17 where @xmath18 is the usual projector onto the local rest space of @xmath19 and @xmath20 and @xmath21 are the particle number density and the specific entropy ( per particle ) , respectively @xcite . the conservation laws for energy and particle number densities read @xmath22 @xmath23 where semi - colons mean covariant derivative , @xmath24 is the scalar of expansion and the quantities @xmath9 , @xmath10 , @xmath20 and @xmath21 are related to the temperature @xmath25 trough the gibbs law : @xmath26 . from the energy conservation equation above , it follows that the energy density for a general @xmath6 component can be written as @xmath27\;.\ ] ] following standard lines ( see , e.g. , @xcite ) , it is possible to show that the temperature evolution law is given by @xmath28 where we have split the dark energy pressure as @xmath29 where @xmath30 . by combining the above equations , we also find that @xmath31\;,\ ] ] where we have used that @xmath32 , as given by the conservation of particle number density [ eq . ( [ nalpha ] ) ] . for parameterization p@xmath33 , shown in eq . ( [ pbeta ] ) , eq . ( [ temp ] ) can be rewritten as @xmath34\;,\ ] ] which reduces to the generalized stefan - boltzmann law for @xmath35 @xcite ( see also @xcite ) . from the above expressions , we confirm the results of ref . @xcite ( for a constant eos parameter ) and find that dark energy becomes hotter in the course of the cosmological expansion since the its eos parameter must be a negative quantity . a possible physical explanation for this behavior is that thermodynamic work is being done on the system ( see , e.g. , fig . 1 of ref . in particular , for the vacuum state ( @xmath36 ) we obtain @xmath37 . to illustrate this behavior , fig . 1 shows the dark energy temperature as a function of the scale factor for @xmath38 ( p2 ) and @xmath39 ( p3 ) , with the dark energy density @xmath10 blowing up as @xmath40 at high-@xmath41 . we , therefore , do not consider this parameterization in our analyses . ] by assuming arbitrary values of @xmath4 , @xmath5 and @xmath42 , where @xmath43 k. from this analysis , it is clear that an important point for the thermodynamic fate of the universe is to know how long the dark energy temperature will take to become the dominant temperature of the universe . a basic difficulty in estimating such a time interval , however , is that the present - day dark energy temperature has not been measured , being completely unknown . 0.1 in by considering that the chemical potential for this @xmath6-fluid is null ( as occurs for @xmath44 ) , the euler s relation defines its specific entropy , i.e. , @xmath45 now , by combining the above equations , it is straightforward to show that the product @xmath46 , so that @xmath47 for a constant eos parameter , the above expression recovers some of the results of ref . note also that the vacuum entropy is zero ( @xmath36 ) whereas for phantom dark energy ( @xmath48 ) , which violates all the energy conditions @xcite , the entropy assumes negative values being , therefore , meaningless ( for a discussion on the behavior of a phanton fluid with nonzero chemical potential , see @xcite . see also @xcite for an alternative explanation in which the temperature of the phantom component takes negative values and @xcite for other thermodynamic analyses of dark energy ) . two cases of interest arise directly from eq . ( [ entropy ] ) . the case in which @xmath49 implies necessarily that @xmath50 for all the above parameterizations are imcompatible with the case @xmath51 . ] . the second case is more interesting , with the @xmath6-fluid mimicking a fluid with bulk viscosity where the viscosity term is identified with the varying part of the dark energy pressure @xmath52 . for this latter case , we note that the positiveness of @xmath53 implies that @xmath54 which clearly is not defined at @xmath55 , where @xmath56 . finally , by combining eqs . ( [ euler ] ) and ( [ entropy ] ) with the conservation of the particle number density shown earlier , we obtain from the second law of thermodynamics that @xmath57 or , equivalently , @xmath58 . in this section , we combine the above physical constraints ( [ c1 ] ) and ( [ c2 ] ) with current observational data in order to impose bounds on the dark energy parameters . we use one of the most recent sne ia data sets available , namely , the sdss compilation discussed in ref . @xcite . this compilation comprises 288 sne ia and uses both salt2 @xcite and mlcs2k2 @xcite light - curve fitters ( see also @xcite for a discussion on these light - curve fitters ) and is distributed in redshift interval @xmath60 . along with the sne ia data , and to help break the degeneracy between the dark energy parameters @xmath4 and @xmath61 we use the bao @xcite and shift parameters @xcite @xmath62 @xmath63 where @xmath64^{1/3}$ ] is the so - called dilation scale , defined in terms of the dimensionless comoving distance @xmath65 , @xmath11 and @xmath66 . in our analyses , we minimize the function @xmath67 , which takes into account all the data sets mentioned above and marginalize over the present values of the matter density @xmath68 and hubble parameters @xmath69 . figures 2 and 3 show the main results of our joint analyses . we plot contours of @xmath70 in the parametric space @xmath4 - @xmath5 for p2 and p3 , respectively . the light gray region displayed in the plots stands for the physical constraint ( [ c1 ] ) . since this inequality is a function of time , the region is plotted by assuring its validity from @xmath71 up to today at @xmath55 . the resulting parametric space , when all the observational data discussed above are combined with the constraints ( [ c1 ] ) and ( [ c2 ] ) , corresponds to the small hachured area right below the @xmath35 line . these results clearly illustrate the effect that the thermodynamic bounds discussed in the previous section may have on the determination of the dark energy eos parameters . in particular , we note that the resulting allowed regions are even tighter for the logarithmic parameterization p2 than for p3 ( cpl ) . since the salt2 compilation allows for more negative values of @xmath4 , the joint constraints involving this sne ia sub - sample are also more restrictive ( figs . 2b and 3b ) . for completeness , we display in table i the changes in the 2@xmath21 estimates of @xmath4 and @xmath5 due to the thermodynamic bounds ( [ c1 ] ) and ( [ c2 ] ) . [ n - table ] lccc + test & & @xmath4 & @xmath5 + + sne ia ( mlcs2k2 ) .................... & p2 & @xmath72 & @xmath73 + sne ia ( mlcs2k2)@xmath74 + t ] .......... & p2 & @xmath75 & @xmath76 + sne ia ( salt2)@xmath74 ........................ & p2 & @xmath77 & @xmath78 + sne ia ( salt2)@xmath74 + t@xmath79 ............... & p2 & @xmath80 & @xmath81 + sne ia ( mlcs2k2)@xmath74 .................... & p3 & @xmath82 & @xmath83 + sne ia ( mlcs2k2)@xmath74 + t@xmath79 .......... & p3 & @xmath84 & @xmath85 + sne ia ( salt2)@xmath74 ........................ & p3 & @xmath86 & @xmath87 + sne ia ( salt2)@xmath74 + t@xmath79 ............... & p3 & @xmath88 & @xmath89 + in spite of its fundamental importance for an actual understanding of the evolution of the universe , the relevant physical properties of the dominant dark energy component remain completely unknown . in this paper we have investigated some thermodynamic aspects of this energy component assuming that its constituents are massless quanta with a general time - dependent eos parameter @xmath6 . we have discussed its temperature evolution law and derived constraints from the second law of thermodynamics on the values of @xmath4 and @xmath5 for a family of @xmath6 parameterizations given by eq . ( [ pb ] ) . when combined with current data from sne ia , bao and cmb observations , we have shown that such constraints provide very restrictive limits on the parametric space @xmath4 - @xmath5 ( see figs . 2 and 3 ) . finally , it is also worth mentioning that in the present analysis we have assumed that the chemical potential @xmath90 for the @xmath6-fluid representing the dark energy is null . a more general analysis relaxing this condition ( @xmath91 ) is currently under preparation and will appear in a forthcoming communication . b. ratra & m. s. vogeley , pasp * 120 * , 235 ( 2008 ) ; r. r. caldwell & m. kamionkowski , ann . nucl . part . sci . * 59 * , 397 ( 2009 ) ; a. silvestri & m. trodden , rept . . phys . * 72 * , 09690 ( 2009 ) ; m. sami , curr . sci . * 97 * , 887 ( 2009 ) . y. wang and p. m. garnavich , astrophys . j. * 552 * , 445 ( 2001 ) ; c. r. watson and r. j. scherrer , phys . d * 68 * , 123524 ( 2003 ) ; p.s . corasaniti et al . , phys . d * 70 * , 083006 ( 2004 ) ; v. b. johri , astro - ph/0409161 ; y. wang and m. tegmark , phys . * 92 * , 241302 ( 2004 ) ; h. k. jassal , j. s. bagla , and t. padmanabhan , mon . not . . soc . * 356 * , l11 ( 2005 ) ; e. m. barboza jr . and j. s. alcaniz , phys . b * 666 * , 415 ( 2008 ) . m. m. phillips , astrophys . j. * 413 * , l105 ( 1993 ) ; a. g. riess , w. h. press , and r. p. kirshner , astrophys . j. * 438 * , l17 ( 1995 ) ; s. jha , a. g. riess , and r. p. kirshner , astrophys . j. * 659 * , 122 ( 2007 ) . g. izquierdo and d. pavon , phys . b * 633 * , 420 ( 2006 ) ; n. bilic , fortsch . phys . * 56 * , 363 ( 2008 ) ; y. s. myung , phys . b * 671 * , 216 ( 2009 ) ; e. n. saridakis , p. f. gonzalez - diaz and c. l. siguenza , class . . grav . * 26 * , 165003 ( 2009 ) .	a significant observational effort has been directed to unveil the nature of the so - called dark energy . however , given the large number of theoretical possibilities , it is possible that such a task can not be performed on the basis only of the observational data . in this article we discuss some thermodynamic properties of this energy component assuming a general time - dependent equation - of - state parameter @xmath0 , where @xmath1 and @xmath2 are constants and @xmath3 may assume different forms . we show that very restrictive bounds can be placed on the @xmath4 - @xmath5 space when current observational data are combined with the thermodynamic constraints derived .
given a compact metric space @xmath0 , a map @xmath8 is said to be _ minimal _ if the forward orbit @xmath9 is dense in @xmath0 , for every @xmath10 . in such a case we call @xmath0 minimal with respect to @xmath11 , or simply minimal if there is no confusion as to the map . in the present paper we contribute to the following two well known problems . we shall refer to a space @xmath0 as minimal if it admits a minimal map . * is minimality with respect to homeomorphisms preserved under cartesian product in the class of compact spaces ? * what spaces admit minimal noninvertible maps ? as with many other topological and dynamical properties , a fundamental question is to determine whether the property of being a minimal space is preserved under the cartesian product , and it seems quite surprising that it has not been settled until now . note that , for example for the fixed point property this question had been resolved in the negative already 50 years ago by lopez @xcite . in addition , basic examples of minimal spaces such as the circle , or the cantor set , are known to preserve minimality under the cartesian product , and there have not been even prospective candidates to provide a counterexample . although no product of a homeomorphism with itself @xmath12 is minimal , by the fact that it keeps the diagonal @xmath13 invariant , the cartesian product typically gives rise to new homeomorphisms that do not factor as conjugate homeomorphisms onto both coordinate spaces , and in this way minimal homeomorphisms of the product can often be obtained . in the present paper , however , we finally settle this problem in the negative , by providing a class of counterexamples . each space @xmath1 in the class is such that it admits a minimal homeomorphism but no cartesian power of it @xmath14 does . each such space also admits a monotone map onto a suspension of a minimal cantor system , and the minimal homeomorphisms it admits are extensions of minimal homeomorphisms of the suspension . in the late 1960s auslander raised the question concerning the existence of minimal noninvertible maps . in a joint work with yorke he showed that the cantor set admits such maps @xcite . since then more examples have been given , but as in the case of minimal homeomorphisms and flows , the classification of spaces admitting such maps is a difficult well known open question . in 1979 auslander and katznelson showed that the circle admits no such maps @xcite , despite supporting minimal homeomorphisms . in 2001 kolyada , snoha and trofimchuk constructed such maps for the 2-torus , proving that any minimal skew product homeomorphism of the 2-torus @xmath15 having an asymptotic pair of points has an almost one - to - one factor which is a noninvertible minimal map of @xmath15 @xcite . modifying this approach , in 2003 bruin , kolyada and snoha proved in @xcite that any minimal skew product homeomorphism of @xmath15 having an asymptotic pair of points has a factor which is a noninvertible minimal map of a 2-dimensional nonhomogeneous metric continuum @xmath0 , such that any homeomorphism of @xmath0 has a fixed point . tywoniuk showed in @xcite that solenoids , unlike the circle , admit noninvertible minimal maps . it had been a long - standing open question if the circle is the only nondegenerate continuum that admits a minimal homeomorphism but no minimal noninvertible map . it is a well known unresolved conjecture that the pseudo - circle is another such continuum , motivated by the fact that it admits minimal homeomorphisms @xcite . recently downarowicz , snoha and tywoniuk @xcite have answered the general question in the negative by providing a family of counterexamples that belong to the class of so - called slovak spaces . these spaces have their homeomorphism groups generated by a minimal homeomorphism , which in some cases has positive entropy , but they all admit no minimal noninvertible maps . in the present paper we show a new , very general class of compact spaces that admit minimal noninvertible maps . namely , any compact , finite dimensional metric space that admits a continuous , aperiodic minimal flow belongs to that class . in addition to some manifolds with zero euler characteristic , examples include peculiar minimal sets of flows on compact manifolds , such as the denjoy and kuperberg minimal sets . we shall start in section 2 with the construction of a new class of spaces admitting minimal noninvertible maps . the maps are obtained as perturbations of time @xmath16 homeomorphisms of aperiodic minimal continuous flows on metric spaces . our inverse limit approach is motivated by the method introduced by aarts & oversteegen , who in @xcite used it to construct a transitive homeomorphism of the sierpinski carpet . a novel variant of this method is then used in section 3 to modify minimal cantor systems suspensions to obtain minimal continua , none of which admits minimality in the cartesian product of finitely many copies of itself . these continua resemble some of the slovak spaces constructed by different methods by downarowicz , snoha and tywoniuk , where a dense orbit of a minimal homeomorphism was blown up to a null seqeunce of @xmath17-curves . our inverse limit approach allows us to introduce a null sequence of pseudo - arcs instead , which in the resulting spaces forces factorwise rigidity , as well as almost cyclicity of their homeomorphism groups , in the sense that they are isomorphic to either @xmath18 or @xmath19 . in the final section of the paper we illustrate how these techniques can be applied in a wider context to spaces that do not admit minimal flows but instead minimal maps that preserve a local product structure in a sufficiently smooth manner . in what follows , we identify the unit circle with @xmath20 and the 2-dimensional torus with @xmath21 . a continuous surjection @xmath22 is _ almost 1 - 1 _ if the set @xmath23 is dense in @xmath1 . a dynamical system @xmath24 is an _ almost 1 - 1 extension _ of @xmath25 if there exists an almost 1 - 1 factor @xmath26 . note that if @xmath24 is a minimal almost 1 - 1 extension of @xmath25 then @xmath27 is dense in @xmath0 . in fact both sets @xmath28 and @xmath27 are residual in that case . in @xcite kolyada , snoha , and trofimchuk proved that any minimal skew product homeomorphism of the 2-torus having an asymptotic pair of points has an almost 1 - 1 factor which is a noninvertible minimal map . by an inverse limit technique of aarts and oversteegen @xcite , as well as the application of lemma 3.1 in @xcite , we prove the following related result . [ generalnonivert ] let @xmath3 be a continuous , aperiodic minimal flow on the compact , finite dimensional metric space @xmath4 . then there is a generic choice of parameters @xmath5 , such that the homeomorphism @xmath6 admits a noninvertible minimal map @xmath7 as an almost 1 - 1 extension . by standard techniques , see e.g. @xcite , there exists a residual set @xmath29 of parameters @xmath16 such that the time@xmath16 map @xmath30 is minimal , and hence we may fix such a @xmath31 and denote @xmath32 by the theorem of bebutov and its consequences ( see , e.g. , @xcite ) there is for each @xmath33 and @xmath34 a local section @xmath35 with corresponding flow box neighbourhood @xmath36 and a homeomorphism onto its image @xmath37 that conjugates the time @xmath16-tmap @xmath38 of the flow restricted to @xmath36 with addition by @xmath16 in the first coordinate on the image . specifically , for some @xmath39 the @xmath40 ball centered at @xmath41 , @xmath42 contains a set @xmath35 so that for each @xmath43 } f_t\left(b(x,\delta)\right)=u(x , t)$ ] there is a unique @xmath44 such that @xmath45 and the homeomorphism @xmath46 maps @xmath47 into the set @xmath48 and the point @xmath49 as above to @xmath50 where @xmath51 the fact that we can choose a finite dimensional euclidean space @xmath52 follows from the standard fact that finite dimensional separable metric spaces can be embedded in a finite dimensional euclidean space . we now fix a particular @xmath53 and then construct for @xmath54 a nested sequence of local sections @xmath55 ( i.e. @xmath56 ) and corresponding flow box neighbourhoods @xmath57 , with corresponding homeomorphisms @xmath58 onto their images , constructed in such a way that @xmath59 when restricted to @xmath60 and @xmath61 for all @xmath62 for each @xmath63 denote @xmath64 . we put @xmath65 and will define spaces @xmath66 for @xmath67 inductively . using @xmath68 to denote the euclidean metric in @xmath69 , we define a new metric @xmath70 on @xmath71 by @xmath72 the completion of @xmath71 with respect to @xmath70 has remainder given by an interval that can naturally be identified with @xmath73 $ ] , corresponding to the limiting value of @xmath74 for the points in the remainder . by the compactness of @xmath73,$ ] any restriction of this completion to a compact subset of @xmath69 intersected with @xmath71 is compact . the level sets of @xmath74 , @xmath75 form cones that separate @xmath71 and intersect _ each _ line that is an image of a flow line in exactly one point . since the level sets of @xmath74 are closed under non zero scalar multiplication , any neighbourhood of @xmath76 in @xmath69 will intersect each level set . thus , under the hypotheses of aperiodicity and minimality , the completion of the images of @xmath77 in @xmath69 will also have intervals as remainders . denote by @xmath78 the compactification of @xmath79 by @xmath80 $ ] in the following way . we take @xmath81 so that the image of @xmath82 is at @xmath76 and we complete using the metric @xmath70 as described above . we then identify @xmath83 with the interval obtained as remainder in the completion of the metric @xmath84 in the same way , @xmath85 is obtained from @xmath78 by blowing up the point @xmath86 and recursively the space @xmath87 is obtained from @xmath66 by removing the point @xmath88 and compactifying the resulting hole by a closed interval @xmath89 identified with @xmath73 $ ] as with @xmath90 it is not hard to see that @xmath78 is homeomorphic to @xmath91 because @xmath92 and the compactification of @xmath93 in @xmath78 are homeomorphic by a homeomorphism which is the identity on the boundary of @xmath92 . by the same argument all @xmath66 are homeomorphic to @xmath65 . observe that compactification of @xmath94 by intervals using the above method gives raise to a space homeomorphic to @xmath66 . this is due to the fact that @xmath95 . denote by @xmath96 the inverse limit of the spaces @xmath66 with bonding maps @xmath97 defined by @xmath98 for @xmath99 and @xmath100 for @xmath101 . then by a result of brown ( * theorem 4 ) we obtain that @xmath96 is homeomorphic to @xmath4 because all the maps @xmath102 are monotone . now we shall define a minimal but noninvertible map @xmath103 on @xmath96 . fix any @xmath104 . if for every @xmath105 the coordinate @xmath106 then we put @xmath107 for every @xmath108 . now suppose that for some @xmath109 we have @xmath110 . then @xmath111 for all @xmath112 and @xmath113 for @xmath114 . if @xmath115 then we put @xmath116 for every @xmath108 . for the case @xmath117 , observe that since both @xmath89 and @xmath118 are identified with @xmath73 $ ] , we have a homeomorphism @xmath119 which is the identity after this identification . we put @xmath120 for @xmath121 and for @xmath114 we put inductively @xmath122 . this way @xmath103 is defined on @xmath96 and it is clear that it is surjective . we must show that it is continuous . if @xmath104 is such that @xmath106 for any @xmath67 then it is clear that @xmath103 is continuous at @xmath123 because in each @xmath124 there is a neighborhood of @xmath125 disjoint from @xmath126 . on the other hand if for some @xmath109 @xmath127 , @xmath88 and @xmath128 are contained in a flow box @xmath129 that was used in the construction of the compactification , which means that @xmath130 can be locally identified with a translation in @xmath69 that preserves the level sets of @xmath74 used in the definition of the compactifications . it easily follows that @xmath103 is continuous also at such @xmath123 . finally , we must show that @xmath103 is minimal . every minimal homeomorphism is forward minimal , hence for every @xmath131 its forward orbit @xmath132 is dense in @xmath4 . in particular , the forward orbit of every @xmath133 is dense in @xmath134 . furthermore , for every @xmath135 and every @xmath136 , the segment @xmath137)$ ] has empty interior in @xmath4 , because otherwise @xmath4 would be a single closed orbit . then it is clear that for every @xmath104 and every @xmath105 the set @xmath138 is dense in @xmath66 . but then the forward orbit of @xmath123 under @xmath103 is dense in @xmath96 . this shows that @xmath103 is minimal . it is also clear that @xmath103 is not invertible . by the same argument , the natural projection @xmath139 is one - to - one on a dense set , hence is almost 1 - 1 extension ( see also ( * ? ? ? * theorem 2.7 ) ) . the proof is completed . [ rem : r1 ] since our goal was to construct a noninvertible map , we have only compactified with intervals for points in the negative orbit of @xmath140 however , it is clear that we could have compactified with intervals for points in the complete orbit of @xmath141 to form an invertible map instead . a large but not exhaustive class of spaces admitting a minimal flow can be obtained by suspension as follows . let @xmath142 be a minimal homeomorphism of a compact metric space @xmath143 put @xmath144 , where @xmath145 is the equivalence relation given by : @xmath146 provided that @xmath147 and @xmath148 . then the _ suspension flow _ defined by @xmath46 is the continuous flow induced on @xmath0 given by @xmath149 . since the orbits of @xmath46 are dense , the flow orbits are dense in @xmath0 , and so @xmath0 is a continuum . as before , there exists a residual set @xmath29 of parameters @xmath16 such that @xmath150 is minimal . by a _ minimal suspension _ of @xmath46 we mean any homeomorphism @xmath151 for @xmath152 . and as long as @xmath153 is infinite , @xmath154 is aperiodic . then a direct application of theorem [ generalnonivert ] gives the following . [ r2:t2 ] suppose that @xmath154 is suspension flow defined by a minimal homeomorphism @xmath142 on an infinite , finite dimensional compact metric space @xmath153 and let @xmath0 be the phase space of @xmath154 . then @xmath0 is a continuum that admits a noninvertible minimal map . in the following section we will make use of this construction with @xmath153 the cantor set . let @xmath154 be the suspension flow of an adding machine and @xmath0 be its phase space . then @xmath0 is a solenoid . by corollary [ r2:t2 ] we obtain that @xmath0 admits a noninvertible dynamical system . such a system was constructed before in @xcite by a completely different , long and technical argument . let @xmath155 be a kuperberg minimal set of the smooth flow @xmath154 on @xmath156 without a closed orbit , first constructed in @xcite . the space @xmath155 is a continuum with unstable shape @xcite that admits a noninvertible minimal map @xmath157 . consider the standard denjoy minimal set for circle homeomorphism @xmath158 obtained by a blow - up of @xmath109 orbits in the irrational circle rotation by @xmath159 ( e.g. see @xcite ) . clearly this minimal set is the unique invariant cantor set @xmath153 for @xmath158 . let @xmath160 be the suspension flow of @xmath161 and @xmath162 be its phase space . then @xmath162 belongs to the class of so - called denjoy continua . this is a well - studied class of spaces , whose elements played a crucial role in schweitzer s counterexample to the seifert conjecture @xcite . by corollary [ r2:t2 ] we obtain that each @xmath162 admits a noninvertible minimal dynamical system . alternatively , we could realise these as minimal subsets of flows on the torus . it is well known that the only compact surfaces admitting a minimal system are the 2-dimensional torus and the klein bottle ( e.g. see ( * ? ? ? * theorem 3.17 ) ) . on the other hand , every flow on the klein bottle has a closed orbit ( e.g. see @xcite , cf . @xcite ) . hence , @xmath15 is the only compact surface admitting a minimal flow , and so this is the only closed surface to which theorem [ generalnonivert ] applies . it is not clear ( beyond tori and related examples ) which higher dimensional closed manifolds admit minimal flows even the case of @xmath163 remains unknown . in this section we adapt the technique of the previous section to replace the inserted arcs with pseudo arcs to create a space with special properties . recall that in @xcite downarowicz , snoha and tywoniuk introduced the notion of slovak spaces . we say that a compact @xmath0 is a _ slovak space _ if its homeomorphism group @xmath164 is cyclic and generated by a minimal homeomorphism . in the present section we shall appeal to a slightly broader class of spaces which we define as follows . we say that a compact space @xmath0 is an _ almost slovak space _ if its homeomorphism group @xmath165 with @xmath166 where @xmath167 is cyclic and generated by a minimal homeomorphism , @xmath168 and for every @xmath169 we have @xmath170 . let @xmath142 be a minimal homeomorphism of a cantor set @xmath153 . now we shall adapt the approach from the proof of theorem [ generalnonivert ] , to construct an almost slovak space @xmath1 . we shall later show that with the appropriate choice of @xmath46 @xmath14 is minimal only if @xmath115 . recall that the _ composant _ of the point @xmath41 of the space @xmath0 is the union of all proper subcontinua of @xmath0 containing @xmath171 [ thm : h ] let @xmath172 be a minimal homeomorphism of a cantor set @xmath153 . there exists a minimal suspension @xmath173 of @xmath172 , a continuum @xmath1 , a minimal homeomorphism @xmath174 and a factor map @xmath175 such that : a. @xmath176 is almost 1 - 1 , b. all non - singleton fibers @xmath177 are pseudo - arcs , c. there exists a composant @xmath178 such that if @xmath179 then @xmath180 . d. @xmath181 for all @xmath41 we may assume that @xmath182 $ ] and @xmath183 and @xmath184 for every @xmath185 . by the results of @xcite we may also assume that @xmath186 of the suspension flow of @xmath46 for some @xmath16 so that the construction from theorem [ generalnonivert ] can be carried out where instead of replacing the points of only the backward orbit of a particular point @xmath187 by arcs , we replace all points in the orbit of @xmath187 with a null sequence of arcs ( see remark [ rem : r1 ] and corollary [ r2:t2 ] ) . more specifically , fix any @xmath188 . by theorem [ generalnonivert ] there exists an almost 1 - 1 extension @xmath189 of @xmath173 and a sequence of maps @xmath190 , @xmath191 ( used to construct @xmath103 by an inverse limit technique ) such that @xmath192 is one - to - one except on the set @xmath193 , where each @xmath194 , @xmath195 is an interval . furthermore @xmath196 , where @xmath197 is one - to - one at every point except on two intervals @xmath198 , each of which is collapsed to a distinct point . note that for sufficiently small @xmath185 the map @xmath197 is invertible on the complement of the set @xmath199 . let @xmath200\to [ 0,1]$ ] be a map such that @xmath201 , @xmath202 and the inverse limit of @xmath203 $ ] with @xmath204 as the sole bonding map is the pseudo - arc ( e.g. see @xcite or @xcite ) . extend @xmath204 to a continuous surjection on @xmath205 $ ] by putting @xmath206 for all @xmath207 . put @xmath208 . let @xmath209 and let @xmath210 be a sufficiently small neighborhood of @xmath211 homeomorphic to @xmath212 , where @xmath153 is a cantor set and @xmath213\times \{0\}$ ] . there exists a nested sequence of clopen sets @xmath214 such that @xmath215 . now for each @xmath109 let @xmath216 be a map such that @xmath217 , @xmath218 for @xmath219 , @xmath220 , and @xmath221 otherwise . let @xmath222 . observe that for each @xmath223 there exists an @xmath224 , and an open neighborhood @xmath225 of @xmath226 , such that @xmath227 for all @xmath228 and @xmath229 . but since a finite number of the first coordinates in a sequences from an inverse limit does not affect the topological structure of @xmath78 , we see that if @xmath230 and @xmath231 for every @xmath109 , then a small neighborhood of @xmath41 is homeomorphic to @xmath232 . if , on the other hand , @xmath233 for some ( thus all ) @xmath109 then @xmath234 , which is a unique maximal pseudo - arc embedded in @xmath78 . we may view @xmath78 as @xmath0 with removed @xmath187 and the resulting `` hole '' compactified by a pseudo - arc . note that we have a natural projection @xmath235 given by @xmath236 , where we identify @xmath237 . by the same method we define @xmath66 and @xmath238 which is one - to - one except on the set @xmath239 where @xmath240 is a pseudo - arc for every @xmath195 . furthermore observe that @xmath197 induces a natural projection @xmath241 which collapses each of two pseudo - arcs to a point . let @xmath242 and let @xmath243 be the natural projection induced by the maps @xmath244 . observe that @xmath176 is one - to - one onto every point that does not belong to the orbit of @xmath187 , and @xmath245 is a pseudo - arc for every @xmath246 . furthermore , all composants of @xmath1 are topological one - to - one images of the real line , except the composant @xmath247 containing countably many pseudo - arcs connected by arcs ( note that in @xmath248,g)$ ] the constant sequences @xmath249 and @xmath250 belong both to the pseudo - arc @xmath251,g)$ ] and the arcs @xmath252,g)$ ] and @xmath253,g)$ ] , respectively ) . it remains to define a homeomorphism @xmath254 . for each @xmath255 such that @xmath256 we put @xmath257 . if @xmath258 then take any @xmath259 and denote @xmath260 and @xmath261 . by the method of our construction we have a linear map @xmath262 that extends to a continuous map in a neighborhood of @xmath263 in such a way that @xmath264 for all @xmath49 sufficiently close to @xmath265 . furthermore , for each @xmath259 we have @xmath266 for each @xmath267 . this way @xmath103 is defined also on @xmath245 for every @xmath268 , where it is a homeomorphism , since @xmath269 is a homeomorphism onto its image on the set @xmath270 . note that each pseudo - arc in @xmath66 has empty interior in @xmath66 , hence the set of points @xmath49 such that @xmath271 is residual in @xmath1 , thus demonstrating that @xmath176 is almost 1 - 1 . it is clear from the construction that all non - singleton fibers @xmath272 are pseudo - arcs and that all pseudo - arcs are contained in the same composant @xmath273 an almost 1 - 1 extension of a minimal system is always minimal , and so @xmath103 is minimal . the condition @xmath181 for all @xmath41 is a direct consequence of the construction . the proof is completed . there exists a compact connected metric space @xmath1 admitting a minimal homeomorphism such that @xmath274 does not admit a minimal homeomorphism . let @xmath46 be the homeomorphism induced on a minimal cantor set by a denjoy extension of an irrational rotation of the unit circle and let @xmath130 be a minimal suspension of @xmath275 let @xmath174 be a minimal dynamical system provided by an application of theorem [ thm : h ] . by the construction of @xmath1 , all composants of @xmath1 but one are one - to - one continuous images of @xmath276 there is also one special composant @xmath247 which consists of countably many pseudo - arcs connected by arcs . we shall show that for any homeomorphism @xmath277 there is an @xmath67 such that @xmath278 is of the form @xmath279 , with @xmath280 homeomorphisms . since composants are dense in a continuum , it is enough to consider the composant @xmath281 . , title="fig:",scaledwidth=60.0% ] [ fig : patiles ] this composant @xmath281 is tiled " with the following 4 types of `` squares '' : @xmath282 @xmath283 , @xmath284 @xmath285 , @xmath286 @xmath287 and @xmath288 @xmath289 , where @xmath290 is a ( maximal ) pseudo - arc in @xmath247 and @xmath29 is an arc that connects two such pseudo - arcs . note that any square of type @xmath288 is homeomorphic with any square of type @xmath291 [ pp ] no square of type @xmath284 contains a homeomorphic copy of a square of type @xmath282 or @xmath292 to prove the claim , suppose on the contrary that a square @xmath293 of type @xmath284 contains a copy of a square of type @xmath282 or @xmath292 then @xmath293 contains an arc @xmath294 . since the projection of @xmath294 onto one of the two coordinate spaces must be nondegenerate , it follows that a subcontinuum of @xmath290 is a continuous image of the arc . this contradicts the fact that @xmath290 does not contain any locally connected continuum , thereby establishing the claim . applying similar arguments we also obtain the following . no square of type @xmath288 contains a homeomorphic copy of a square of type @xmath295 for simplicity of notation , by convention we will write @xmath296 to denote that the image of a square of type @xmath284 is contained in a square of type @xmath297 it follows by the above claims that @xmath296 , @xmath298 , and @xmath299 or @xmath300 . we call a point @xmath41 of a square of type @xmath284 a _ corner point _ if it also belongs to a square of type @xmath282 . by the above discussion , @xmath301 must transform corner points to corner points . therefore a diagonal in the lattice of corners of the form @xmath302 must be mapped onto a diagonal of the same form , because squares of type @xmath282 may not `` cross '' squares of any other type . then there is an @xmath303 such that @xmath304 is a translation of diagonals , because if it translates one diagonal then , by the rigid structure of the diagonals and corner points , it must also translate all other diagonals in exactly the same way . denote by @xmath305 the set of all corner points in @xmath281 and observe that @xmath306 is dense in @xmath1 . by the above discussion there are homeomorphisms @xmath307 such that @xmath308 . but then @xmath309 can be extended continuously to @xmath274 in a unique way , so @xmath310 , as claimed . now , by way of contradiction suppose @xmath274 admits a minimal homeomorphism @xmath311 . by the above , we have @xmath310 for some @xmath312 since @xmath274 is a continuum , minimality of @xmath301 implies that @xmath304 is minimal as well . then for simplicity we may assume that @xmath115 . note that since each @xmath313 is a homeomorphism , following the same lines of argument as above , @xmath313 must preserve the structure @xmath314 of the special composant @xmath247 . since the pseudo - arc has the fixed point property @xcite , the homeomorphism @xmath313 can not permute a finite number of pseudo - arcs in @xmath247 . this shows that @xmath313 must preserve the ordering of pseudo - arcs in @xmath247 . more formally , if we enumerate consecutive pseudo - arcs in @xmath247 , say @xmath315 with @xmath316 , then there are @xmath317 such that @xmath318 and @xmath319 . by our construction we see that @xmath320 and hence the relation @xmath321 iff @xmath322 or @xmath323 for some @xmath324 is closed equivalence relation , and so @xmath325 induces a homeomorphism @xmath326 . but @xmath327 coincides with @xmath328 on a dense set and so @xmath329 . by the same argument @xmath330 is an extension of @xmath331 . since each suspension minimal system from a denjoy homeomorphism is an extension of a rotation @xmath294 by an @xmath332 on the unit circle , we obtain that @xmath301 is an extension of the map @xmath333 defined on the two dimensional torus . but the numbers @xmath334 and @xmath335 are rationally dependent , and hence @xmath333 is not minimal ( e.g. see @xcite ) . but any factor of a minimal homeomorphism has to be minimal , proving that @xmath301 is not minimal . it is clear from the proof that the cartesian product of @xmath109 copies of @xmath1 , is not minimal for any @xmath117 . does there exist a minimal space @xmath1 , such that @xmath336 is minimal for some @xmath337 , but @xmath338 is not minimal for some @xmath339 ? while we have only applied our technique to spaces supporting a flow , in fact the technique only relies on having a map that preserves a local product structure along with the `` angular '' relation . in the case of flows , this was accomplished by using the local translation structure , but this can also be achieved with certain classes of smooth maps . a large class of skew products provide examples of such maps as we illustrate below . in addition to the intrinsic interest of the extension of the construction , it allows us to produce an example of a minimal but noninvertible map of the klein bottle , which is known not to admit a minimal flow . this raises the possibility that the technique could lead to similar examples on a significantly larger class of manifolds than those that admit minimal flows . [ thm : skewt2 ] let the homeomorphism @xmath340 be a skew product defined by @xmath341 where @xmath342 is a differentiable function on @xmath343 . for every @xmath344 there exists a map @xmath345 and an almost 1 - 1 factor map @xmath346 such that @xmath347 is a singleton for every @xmath348 and which is an arc otherwise . furthermore @xmath349 . fix any @xmath350 and let @xmath210 be a small open neighborhood of @xmath351 first observe that if @xmath352 then @xmath353 which means that @xmath130 preserves vertical lines in @xmath210 . now assume that @xmath354 converges to @xmath355 on a line which is not vertical ; that is , @xmath356 and there is @xmath357 such that @xmath358 for every @xmath109 . then if we write a similar formula for the coordinates of @xmath359 and @xmath360 we obtain @xmath361 this implies that an arc obtained as the image of a radial line around @xmath355 in a small neighborhood has an asymptote at @xmath360 and the asymptote is different for different @xmath362 , and lines of all possible directions as obtained as such an asymptote . then we can perform the `` blow up '' procedure as in theorem [ generalnonivert ] applied to the entire orbit of @xmath355 , except for the compactification we use the circle of lines through the origin projected to an arc to compactify the pinched torus @xmath363 with an additional arc . this makes the corresponding map on the resulting inverse limit space ( which is still the torus ) continuous because the assignment of a line to the corresponding asymptote is continuous . by the construction using inverse limits , it is also clear that the condition @xmath349 is satisfied . define now the relation @xmath145 on @xmath15 by @xmath364 and @xmath365 . clearly @xmath145 is a closed equivalence relation , and it is well known that @xmath366 is the klein bottle @xmath367 and we use @xmath368 to denote the projection . let the homeomorphism @xmath340 be a skew product defined by @xmath369 for some @xmath370 . if @xmath371 then @xmath130 induces a homeomorphism @xmath301 on @xmath367 satisfying @xmath372 [ thm : skewt3 ] let the homeomorphism @xmath340 be a skew product defined by @xmath373 which preserves the relation @xmath145 , and let @xmath301 be the homeomorphism induced on the klein bottle @xmath367 . for every @xmath374 there exists a map @xmath375 and an almost 1 - 1 factor map @xmath376 such that @xmath347 is a singleton for every @xmath377 and an arc otherwise . furthermore @xmath378 . the proof is the same as in theorem [ thm : skewt2 ] . simply , if we aim to `` blow up '' the point in @xmath379 then , since @xmath380 is a local homeomorphism , we can pull @xmath355 back to @xmath381 in @xmath15 and blow up the orbits of both @xmath41 and @xmath49 as in theorem [ thm : skewt2 ] and then project this back to @xmath367 via the local homeomorphism @xmath382 we can then extend @xmath301 to the compactifying intervals using the same limits ( radial lines ) as for @xmath130 . the procedure described above works regardless of the special properties that @xmath130 might have . in @xcite parry presented a simple and elegant argument describing how to obtain a minimal homeomorphism of the klein bottle by defining a function @xmath342 satisfying @xmath383 and such that the skew product on @xmath15 given by @xmath373 is minimal . in fact , @xmath342 was defined by a convergent fourier series of a special type , allowing parry to obtain a family of minimal systems on @xmath367 in this way . motivated by this , sotola and trofimchuk showed by careful calculations in @xcite formulas to modify parry examples to non - minimal systems . they also mentioned ( see remark 2.9 in @xcite ) that their construction may lead to a minimal homeomorphism @xmath301 on @xmath367 with a continuum @xmath384 with @xmath385 which by results of @xcite may serve as an alternative proof . we see now that , based on the examples of parry , we can obtain similar examples in a direct way . recall that any almost 1 - 1 extension of a minimal homeomorphism is minimal . let @xmath373 be a minimal homeomorphism of @xmath15 constructed by parry , which induces a minimal homeomorphism @xmath301 on @xmath367 , with @xmath342 is differentiable and satisfying @xmath383 . then `` blowing up '' orbit of any point @xmath374 by theorem [ thm : skewt3 ] we obtain a minimal homeomorphism @xmath46 on @xmath367 which is almost 1 - 1 extension of @xmath301 and contains an arc @xmath70 such that @xmath386 . clearly , when constructing inverse limit in theorem [ thm : skewt3 ] we do not have to blow up forward orbit of @xmath355 , which allows us to construct noninvertible examples as described at the beginning of this section . let @xmath373 be a minimal homeomorphism of @xmath15 constructed by parry , which induces a minimal homeomorphism @xmath301 on @xmath367 , with @xmath342 is differentiable and satisfying @xmath383 . then `` blowing up '' the backward orbit of any point @xmath374 as in theorem [ thm : skewt3 ] we obtain a minimal but noninvertible map @xmath46 on @xmath367 which is almost 1 - 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( 2 ) * 100 * ( 1974 ) , 386400 . otola , j. , trofimchuk , s. , _ construction of minimal non - invertible skew - product maps on 2-manifolds_. proc . * 144 * ( 2016 ) , 723732 . tywoniuk , d. _ minimal non - invertible transformations of solenoids . _ colloq . math . * 127 * ( 2012 ) , no . 2 , 243252 . walters , p. _ an introduction to ergodic theory . _ graduate texts in mathematics , * 79*. springer - verlag , new york - berlin , 1982 .	the following well known open problem is answered in the negative : given two compact spaces @xmath0 and @xmath1 that admit minimal homeomorphisms , must the cartesian product @xmath2 admit a minimal homeomorphism as well ? a key element of our construction is an inverse limit approach inspired by combination of a technique of aarts & oversteegen and the construction of slovak spaces by downarowicz & snoha & tywoniuk . this approach allows us also to prove the following result . let @xmath3 be a continuous , aperiodic minimal flow on the compact , finite dimensional metric space @xmath4 . then there is a generic choice of parameters @xmath5 , such that the homeomorphism @xmath6 admits a noninvertible minimal map @xmath7 as an almost 1 - 1 extension .
the problem of the crossover from a bcs - like superconducting state to a bose condensate of local pairs @xcite has gained new interest in the context of high-@xmath1 superconductors . while there is still no microscopic theory of how superconductivity arises from doping the antiferromagnetic and insulating parent compounds @xcite , it is clear that the superconducting state can be described in terms of a generalized pairing picture . the many body ground state is thus a coherent superposition of two particle states built from spin singlets in a relative d - wave configuration @xcite . the short coherence length @xmath6 parallel to the basic @xmath7-planes which is of the same order than the average interparticle spacing @xmath8 , however indicates that neither the bcs picture of highly overlapping pairs nor a description in terms of composite bosons is applicable here . it is therefore of considerable interest to develop a theory , which is able to cover the whole regime between weak and strong coupling in a unified manner . on a phenomenological level such a description is provided by the ginzburg - landau ( gl ) theory . indeed , it is a nonvanishing expectation value of the complex order parameter @xmath9 which signals the breaking of gauge invariance as the basic characteristic of the superconducting state , irrespective of whether the pair size is much larger or smaller than the interparticle spacing . a gl description of the bcs to bose crossover was developed for the s - wave case by drechsler and one of the present authors @xcite in two dimensions and by s de melo , randeria and engelbrecht @xcite for the three - dimensional case . in our present work the theory for the two - dimensional case is reconsidered , including a discussion of the nelson - kosterlitz jump of the order parameter and the generalization to the experimentally relevant situation of d - wave superconductivity . moreover we also calculate the characteristic lengths @xmath2 and @xmath3 and compare our results with measured properties of high-@xmath1 compounds . the remarkable success with which the standard bcs model has been applied to conventional superconductors relies on the fact that in the weak coupling limit the details of the attractive interaction are irrelevant . by an appropriate rescaling of the parameters the properties of all weak coupling superconductors are therefore universal . it is one of our aims here to investigate to which extent such a simplifying description also exists in more strongly coupled superconductors . starting from a microscopic model with an instantaneous attractive interaction , we find that the resulting gl functional takes the standard form for arbitrary strength of the coupling . by adjusting a single dimensionless parameter to the measured upper critical field near @xmath1 , we obtain consistent values for both the dimensionless ratio @xmath10 between the coherence length and interparticle spacing as well as the observed value @xmath11 of the gl parameter in optimally doped high-@xmath1 compounds . therefore , in spite of the rather crude nature of the original microscopic model , our gl theory is quantitatively applicable to strongly coupled superconductors which are far from the standard weak coupling limit , although not yet in the crossover regime to bose - like behaviour . the plan of the paper is as follows : in section ii we introduce our microscopic model which has an attractive interaction in the singlet d - wave channel . from this we derive , via a hubbard - stratonovich transformation , a statistical gl theory which is valid near @xmath1 . the relevant coefficients of the gl functional are calculated for arbitrary interaction strength . in section iii we discuss the appropriate microscopic definition of the order parameter and the evolution from the bcs to the bose limit of the well known kosterlitz - thouless jump in the superfluid density of two - dimensional superconductors . ignoring the subtleties of the kosterlitz - thouless transition , in section iv we use a gaussian approximation to determine the critical temperature and the associated value of the chemical potential at the transition . finally in section v we determine the characteristic lengths @xmath2 and @xmath3 near the transition for arbitrary strength of the coupling . adjusting the coupling to the experimental values of the slope of @xmath12 near @xmath1 , we then determine the associated dimensionless ratios @xmath13 and @xmath14 . they agree rather well with the observed values in ybco and bscco . a brief conclusion and a discussion of open problems is given in section vi . as a general model describing fermions with an instaneous pairwise interaction @xmath15 in a translationally invariant system , we start from the hamiltonian ( @xmath16 is the volume of the system . ) @xmath17 with an arbitrary single particle energy @xmath18 which we will later replace by an effective mass approximation @xmath19 . in the two - dimensional case , which we consider throughout , the fourier transform @xmath20 of the interaction potential may be expanded in its relative angular momentum contributions @xmath21 by @xmath22 with @xmath23 the angle between @xmath24 and @xmath25 . in the following we are only interested in d - wave pairs with symmetry @xmath26 . we therefore omit all contributions @xmath27 and also neglect the dependence on the absolute values @xmath28 and @xmath29 of the momenta . assuming the interaction is separable , we thus approximate @xmath30 with @xmath31 and @xmath32 a negative constant characterizing the strength of the attractive interaction . finally the restriction to singlet pairing is incorporated trivially by considering only interactions between fermions with opposite spins @xmath33 . in this manner we obtain a gorkov - like reduced interaction hamiltonian @xmath34 ( note that the shift by @xmath35 in eq . ( [ ref3 ] ) is necesssary to guarantee that the interaction is symmetric with respect to @xmath36 . ) . for the derivation of a gl functional below , it is convenient to introduce pair operators @xmath37 via @xmath38 the contribution @xmath39 may then be written in the form @xmath40 of an attractive interaction between pairs of fermions with opposite spin and total momentum @xmath41 . in the following we want to derive a functional integral representation of the grand partition function @xmath42 which gives the standard gl theory as its mean field limit . since we are interested in a superconducting state with a nonzero anomalous average @xmath43 , it is convenient to formally linearize the interaction term @xmath39 by a hubbard - stratonovich transformation @xcite . the grand partition function is thus expressed in terms of a functional integral @xmath44 . here tr}\;\,{\rm t}\exp \bigg ( -\frac{1}{\hbar}\int\limits_0^{\beta\hbar}d\tau\ ; h_z(\tau ) \bigg)\ ] ] is a functional of the auxiliary field @xmath46 , which acts as a space- and time-dependent external potential on a noninteracting fermi system with hamiltonian @xmath47 the physical interpretation of the c - number field @xmath46 is obtained by noting that its expectation value @xmath48 is directly proportional to the anomalous average @xmath49 . thus up to some normalization constant , which will be determined below , the field @xmath46 is just the spatial fourier transform of the complex order parameter @xmath50 describing the superconducting state . it depends both on position and imaginary time @xmath51\/$ ] which is characteristic for a quantum gl functional . since the @xmath26-symmetry in a rotational invariant system is connected with a one dimensional irreducible representation @xcite , the order parameter is still a simple complex scalar , similar to the more familiar isotropic s - wave case . obviously the trace in the time ordered exponential in eq . ( [ ref10 ] ) can not be calculated exactly . however it is straightforward to evaluate @xmath52\/$ ] perturbatively in @xmath53 . the naive justification for this is that close to @xmath1 the order parameter is small . strictly speaking however , the functional integral in eq . ( [ ref9 ] ) requires to integrate over arbitrary realizations of @xmath46 . in order to obtain the standard form of the statistical gl functional , however , the expansion is truncated at fourth order in the exponent of @xmath52\/$ ] . in the language of field theory we are therefore calculating the bare coupling constants , which serve as the starting point for treating the behaviour at long wavelenghts . by a straightforward perturbative calculation @xcite up to fourth order in @xmath53 , the functional @xmath52\/$ ] turns out to be of the form @xmath54 = z_0\;\exp\bigg(\frac{1}{v\beta\hbar^2}\sum\limits_{\boldsymbol{q } } \sum\limits_{\omega_n } \tilde{a}(\boldsymbol{q},\omega_n)\ ; -\,\frac{1}{2v^3\beta^3\hbar^4}\sum\limits_{1\,2\,3 } b(1,2,3)\ ; z(1 ) z^{\star}(2 ) z(3 ) z^{\star}(1 - 2 + 3)\bigg ) \;.\end{gathered}\ ] ] here @xmath55 is the grand partition function of noninteracting fermions while @xmath56 is the fourier transform of the @xmath57-dependence of @xmath46 with bosonic matsubara frequencies @xmath58 , @xmath59 integer . in the quartic term we have used the short hand notation @xmath60 , etc . the functions @xmath61 and @xmath62 can be expressed in terms of the normal state green function ( @xmath63 , @xmath64 ) @xmath65 via @xmath66 and a similar expression with four factors @xmath67 for @xmath62 . in order to obtain the standard form of a quantum gl functional , the coefficients @xmath68 and @xmath69 have to be expanded for small @xmath41 and @xmath70 . to lowest order in the spatial and temporal gradients of the order parameter , it is sufficient to keep only the leading terms in @xmath71 and replace @xmath72 by its constant value at zero momentum and frequency . this expansion is valid , provided the contributions of order @xmath73 and @xmath74 in eq . ( [ ref17 ] ) are negligible . from an explicit calculation of these higher order terms it may be shown @xcite that the order parameter must vary slowly on length scales of order @xmath75 with @xmath76 the two particle binding energy introduced in eq . ( [ ref23 ] ) below . physically , the length @xmath77 is just the radius of a bound state with energy @xmath76 . in the weak coupling limit this length coincides with the standard bcs coherence length @xmath78 which , for the clean limit considered here , is identical with the gl coherence length @xmath79 as defined in ( [ ref49 ] ) . the standard form of the gl functional with a gradient term @xmath80 is therefore valid provided the order parameter varies on scales larger than @xmath81 . with increasing strength of the coupling @xmath76 , the pair radius @xmath77 decreases and thus the validity of the expansion ( 17 ) extends to variations on shorter length scales . regarding the dependence on @xmath57 , the requirement is that @xmath50 must vary slowly on time scales @xmath82 . for weak coupling this is a rather large scale of order @xmath83 ( we set @xmath84 ) . similar to the spatial dependence , however , the necessary scale for the @xmath57-dependence of the order parameter for which the leading terms kept in ( 17 ) are sufficient , decreases with increasing coupling . thus , in the bose limit , the description of the time dependence of the order parameter by a first order derivative like in the well known gross - pitaevskii equation @xcite becomes exact ( see also section vi ) . with these approximations our gl functional in @xmath85)\ ] ] finally takes the form @xmath86 = \frac{1}{\beta\hbar}\int\limits_{v}d^2 r \int\limits_{0}^{\beta\hbar}d\tau\bigg(a\,\|z(\boldsymbol{r},\tau)\|^2 + c\ , \frac{\hbar^2}{2m}\,\|\boldsymbol{\nabla}z(\boldsymbol{r},\tau)\|^2 \\ + \ , d\,\hbar\ , z^{\star}(\boldsymbol{r},\tau)\,\partial_{\tau}z(\boldsymbol{r},\tau ) + \frac{b}{2}\ , \|z(\boldsymbol{r},\tau)\|^4\bigg)\end{gathered}\ ] ] which reduces to the familiar expression if @xmath53 is independent of @xmath57 . the coefficients @xmath87 and @xmath62 are given by @xmath88 and @xmath89 now the sum over wavevectors @xmath24 in eq . ( [ ref21 ] ) diverges at large @xmath28 and thus the bare value of @xmath87 is undefined . in the weak coupling limit this divergence is usually eliminated by argueing that the interaction is finite only in a thin shell around the fermi surface . in the present case however , where the condensation in the strong coupling limit really affects the whole fermi sphere , such a procedure is no longer possible . instead , as was pointed out by randeria et al.@xcite , we have to connect the bare coupling constant to the low energy limit of the two - body scattering problem . in two dimensions this relation is of the form @xmath90 where @xmath91 is a high energy cutoff which precisely cancels the large @xmath28 divergence on the right hand side of eq . ( [ ref21 ] ) . the parameter @xmath92 is the binding energy of the two particle bound state in vacuum , which in fact must be finite in order to obtain a superconducting instability in two dimensions . in our present model , which neglects the dependence of @xmath20 on the absolute values of @xmath24 and @xmath25 , the existence of a bound state is indeed a necessary condition for superconductivity even in the case of d - wave pairing , although quite generally it only applies in the s - wave case @xcite . since in the effective mass approximation @xmath93 which we are using throughout , the free fermion density of states in two dimensions is constant , the coefficient @xmath87 can now be calculated analytically in terms of @xmath76 as @xmath94 ( @xmath95 is the euler constant . ) . the coefficient @xmath62 in ( 22 ) is finite without a cutoff and given by @xmath96 here @xmath97 and @xmath98 are the well known heaviside and sign functions . moreover we we have replaced @xmath99 and @xmath100 by their values at the critical point . similarly , the values of the two remaining coefficients @xmath101 and @xmath102 at the critical point are @xmath103 and @xmath104 all four gl coefficients can thus be expressed essentially in analytical form for arbitrary strength of the interaction . comparing these results with those for the s - wave case @xcite , it turns out that up to a geometrical factor @xmath105 in @xmath62 the coefficients are identical at given values of @xmath106 and @xmath107 , provided the two - particle binding energy @xmath76 is simply identified with the corresponding s - wave value . in order to relate the formal auxiliary field @xmath108 in the functional ( [ ref20 ] ) to the usual superconducting order parameter @xmath50 , the standard procedure in weak coupling is to take @xmath109 , which gives the conventional coefficient @xmath110 in front of @xmath111 . the gradient term is thus identical with the kinetic energy of a schrdinger field for a single quantum mechanical particle with mass @xmath112 , describing a pair built from constituents with mass @xmath113 . as pointed out by de gennes @xcite , however , the value of @xmath114 is arbitrary in principle , as long as one is considering the classical gl functional with @xmath9 independent of @xmath57 . indeed all measurable quantities obtained from the classical gl functional depend only on ratios like @xmath115 . an arbitrary choice for @xmath114 can therefore always be compensated by an appropriate rescaling of @xmath9 . this situation is changed , however , in a quantum mechanical treatment , where the order parameter also depends on @xmath57 , i.e. dynamics enters . in this case there is a different natural normalization of @xmath116 in which the coefficient of the @xmath117-contribution is just @xmath118 @xcite . with this choice of normalization , the order parameter @xmath50 is precisely the c - number field in a coherent state path integral @xcite associated with a genuine bose field operator @xmath119 with canonical commutation relations @xmath120 = \delta(\boldsymbol{r}-\boldsymbol{r'})\/$ ] . while this normalization is evidently the most appropriate one in the strong coupling bose limit - where it agrees with the standard choice as we will see - it can be used for arbitrary coupling , even in the bcs - limit . including the charge @xmath121 of a pair by generalizing the gradient to a covariant derivative in the standard way and adding the energy associated with the magnetic field @xmath122 , the resulting free energy functional reads ( @xmath123 is the velocity of light ) @xmath124 = \\ \frac{1}{\beta\hbar}\int\limits_{v}d^2r \int\limits_{0}^{\beta\hbar}d\tau\big(-\mu_{\star}\ , \frac{1}{2m_{\star}}\,\big\|\big(\frac{\hbar}{i}\boldsymbol{\nabla } + \frac{2e}{c_0}\boldsymbol{a}\big)\psi\big\|^2 \\ + \ , \hbar\ , \psi^{\star}\,\partial_{\tau}\psi + \frac{g_{\star}}{2}\ , \|\psi\|^4\big ) + \frac{\boldsymbol{h}^2}{8\pi}\;.\end{gathered}\ ] ] with this normalization , the three remaining independent coefficients now have a very direct physical interpretation @xcite : the coefficient @xmath125 is the effective chemical potential of the bosons , @xmath126 their effective mass and @xmath127 a measure of the repulsive interaction between the composite bosons . in the following we will concentrate on the effective mass @xmath114 ( at @xmath1 ) which , according to ( [ ref26],[ref27 ] ) is completely determined by the ratio @xmath128 . now in the weak coupling limit @xmath129 is equal to the fermi energy ( see section iv below ) and thus @xmath130 vanishes like @xmath131 in the case of a bcs - superconductor . by contrast , for strong coupling @xmath129 approaches @xmath132 , i.e. is large and negative . in the bose limit @xmath114 is therefore equal to @xmath133 as expected . relating @xmath128 to the dimensionless coupling strength @xmath134 by the gaussian approximation discussed in the following section , we obtain a monotonic increase of @xmath114 from exponentially small values to @xmath133 as a function of coupling , as is shown in fig . [ fig1 ] . as was pointed out above , the mass @xmath114 of a cooper pair defined in such a way can not be observed in any static measurement like the penetration depth . to discuss this , we consider the two dimensional current density @xmath135 } { \delta\boldsymbol{a}(\boldsymbol{r})}= -2e\|\psi\|^2\frac{\hbar}{m_{\star}}\big(\boldsymbol{\nabla}\phi + \frac{2e}{\hbar c_0}\boldsymbol{a}\big)\ ] ] which follows from ( [ ref28 ] ) for a @xmath57-independent order parameter @xmath136 . for a spatially constant magnitude @xmath137 of the order parameter , this leads immediately to the london equation @xmath138 as was noted above , it is only the ratio @xmath115 which enters here and thus static magnetic properties are independent of the choice for @xmath114 . specifically we consider a thin superconducting film with thickness @xmath139 . the in - plane penetration depth @xmath3 is then related to @xmath140 via @xcite @xmath141 here we have introduced a further length @xmath142 which is the effective magnetic penetration depth in a thin film . typically this length is of the order of one centimeter and thus for sample sizes which are smaller than that , magnetic screening may be neglected . in such a situation the difference between a charged and a neutral superfluid becomes irrelevant . a superconducting film thus exhibits a kosterlitz - thouless transition @xcite , in which the renormalized helicity modulus @xmath143 jumps from @xmath144 to zero at @xmath1 . using ( 31 ) this jump translates into one for the two dimensional screening length @xmath142 of size @xcite @xmath145 where @xmath146 is the standard flux quantum . consistent with our remarks above , this jump is completely universal and independent of @xmath114 , applying both to bcs- or bose - like superconductors , provided @xmath142 is larger than the sample size and the density of vortices is low @xcite . in order to define a proper superfluid density @xmath147 , we consider the relation between the order parameter @xmath9 and the microscopic anomalous average . from eq . ( [ expect ] ) we have @xmath148 . neglecting the internal d - wave structure of the order parameter and the logarithmic factor in ( [ ref23 ] ) , it is straightforward to see that @xmath149 with @xmath150 , @xmath151 the fermionic field operators ( the factor @xmath152 which is omitted in ( [ ordpar ] ) diverges as @xmath153 . this is a reflection of the fact that the product of two field operators at the same point can properly defined only with a cutoff . ) . since @xmath77 is the radius of a pair , the relation ( [ ordpar ] ) shows that @xmath154 is just the areal density of pairs . this remains true even in the bose limit where @xmath155 while the product @xmath156 eventually behaves like a true bose field operator @xmath157 . the standard definition @xmath158 of the superfluid density can thus be applied for arbitrary coupling . by contrast , the bose order parameter @xmath159 coincides with @xmath160 only in the strong coupling limit . for weak coupling it is given by an expression like ( [ ordpar ] ) but with the interparticle spacing @xmath8 instead of @xmath77 as the prefactor . thus @xmath161 is essentially the probability density for two fermions with opposite spin at the same point . in the bcs limit this density is exponentially small due to the large size of a pair . the superfluid density in turn is still of order one even in weak coupling and indeed at zero temperature @xmath147 must be equal to the full density @xmath59 for any superfluid ground state in a translational system as considered here @xcite . using @xmath162 , the relation ( [ ref32 ] ) can be rewritten in terms of a jump @xmath163 of the renormalized superfluid density . the superfluid fraction @xmath164 therefore has a jump of order @xmath165 . since this ratio approaches zero in weak coupling , there is a smooth crossover between the universal jump of the superfluid density in a bose superfluid @xcite and the behaviour in a strict bcs model where @xmath166 vanishes _ continuously _ near @xmath1 even in two dimensions . indeed the bcs - hamiltonian is equivalent to a model with an infinite range interaction of strength @xmath167 for which mean field theory is exact @xcite . in the following we will neglect the subtleties associated with the kosterlitz - thouless nature of the transition , which is anyway masked by the coupling between different cuo@xmath168-planes in real high-@xmath1 superconductors , giving a three dimensional critical behaviour near @xmath1 @xcite . in order to calculate directly observable quantities from our gl functional ( [ ref28 ] ) , we have to determine both the critical temperature @xmath1 and the corresponding chemical potential @xmath129 in terms of the binding energy @xmath76 . now it is obvious that an exact evaluation of the functional integral over @xmath50 is impossible . we will therefore use the gaussian approximation above @xmath1 , which is obtained by simply omitting the @xmath169-term . with this approximation our complete grand canonical potential @xmath170 per volume takes the form @xmath171 the critical temperature and chemical potential then follow from the standard condition @xmath172 for a bifurcation to a nonzero order parameter , and the particle number equation @xmath173 - 1}\;.\end{gathered}\ ] ] here @xmath174 is the number density of a free fermion gas in two dimensions . ( [ ref35 ] ) is identical with the thouless criterion @xcite for a superconducting instability , which is equivalent to the condition that the ladder approximation to the exact pair field susceptibility @xmath175 diverges @xcite . it is a straightforward generalization of the usual gap equation to arbitrary coupling . the number equation ( [ ref36 ] ) deserves some more comments . since we have @xmath176 quite generally , it is easy to see that @xmath177 at @xmath178 and @xmath179 . therefore eq . ( [ ref36 ] ) has the simple intuitive interpretation that the total number of particles is split into the number of free fermions still present at @xmath180 plus the number of fermions already bound together in pairs , whose mean occupation number is just the bose distribution . now formally this distribution function arises from the summation over the matsubara frequencies @xmath70 in eq . ( [ ref34b ] ) precisely because our coefficient @xmath181 has been expanded only to linear order in @xmath70 . the omission of the higher order terms in this expansion is therefore connected with neglecting scattering state contributions , which would give an additional term in eq . ( [ ref36 ] ) beyond the completely free and fully bound number of fermions . such a contribution is important in the three - dimensional case where true bound states exist only beyond a critical strength of the coupling @xcite . for our present discussion of the problem in two dimensions , however , there are only free fermions or true bound states . therefore there is no contribution in eq . ( [ ref36 ] ) from scattering states and one expects that the expansion of @xmath181 to linear order in @xmath70 is reliable at arbitrary strength of interaction . there is however a rather different problem which appears in the two - dimensional case . as was discussed above , a superconducting transition exists only in the kosterlitz - thouless sense . this problem shows up in our gaussian approximation , since the bose integral in eq . ( [ ref36 ] ) diverges at @xmath182 . now at this level of approximation this is just a reflection of the fact that @xmath183 for an ideal bose gas in two dimensions , because - as pointed out above - the omission of the @xmath169-term corresponds to neglecting the repulsive interaction between the bosons . from our analytical results ( [ ref25 ] ) and ( [ ref27 ] ) for @xmath62 and @xmath102 , which do not contain @xmath76 , it is however straightforward to calculate the effective interaction @xmath184 for arbitrary coupling . it turns out that @xmath184 is a monotonically decreasing function of the coupling . in the limits @xmath185 of a bcs - like or @xmath186 of a bose - like system , we find @xmath187 thus , in two dimensions , there is always a finite repulsive interaction between the pairs , which is of purely statistical origin @xcite . in particular @xmath184 remains finite in the bose limit , where it arises from processes with a virtual exchange of one of the constituent fermions in a bose - bose scattering process @xcite . the fact that @xmath184 is very large in the weak coupling limit is simply a consequence of the large pair size @xmath188 ( note that @xmath189 in the weak coupling limit ) , but does not imply that the @xmath169-term is particularly relevant in this regime . on the contrary , using the gaussian approximation , it is straightforward to show @xcite that in this limit the product @xmath190 which effectively renormalizes the boson chemical potential @xmath191 , is of order @xmath192 which is roughly @xmath1 in the weak coupling limit . for bcs - like superconductors the @xmath169-contribution is therefore irrelevant except very close to @xmath1 , a fact which is well known from the standard theory of conventional superconductors . now the finite value of @xmath184 guarantees that even in two dimensions there is a finite critical temperature below which the superfluid density is nonvanishing . unfortunately it is not possible to incorporate the kosterlitz - thouless nature of the transition in an approximate treatment of the gl functional . however considering the effectively three - dimensional structure of high-@xmath1 superconductors , this problem may be circumvented by including the motion in the direction perpendicular to the planes . a very simple method to incorporate this is provided by adding a transverse contribution @xmath193 to the kinetic energy by @xcite @xmath194 whose average is equal to the thermal energy @xmath195 . replacing the integral in eq . ( [ ref36 ] ) by @xmath196 with @xmath197 we obtain an effectively three - dimensional system . this becomes evident by writing the contribution of the bound pairs in ( [ ref36 ] ) in the form @xmath198 - 1}\;.\ ] ] the density of states is thus proportional to @xmath199 as in three dimensions , making @xmath200 finite at @xmath182 . although rather crude , this approximation gives a value for @xmath1 in the bose limit , which is very close to the kosterlitz - thouless value for the transition temperature @xcite @xmath201 of a dilute hard core bose gas on a lattice@xcite with boson mass @xmath133 and number density @xmath202 . using our results for the gl coefficients @xmath87 , @xmath101 and @xmath102 and the replacement ( [ ref43 ] ) in the number equation , we can now determine both @xmath1 and @xmath129 for arbitrary coupling from @xmath203 and eq . ( [ ref36 ] ) . the corresponding results are shown in figs . 2 and 3 in units of the characteristic energy scale @xmath204 and as a function of the dimensionless effective coupling @xmath205 . for weak coupling @xmath206 , the critical temperature is monotonic in @xmath76 , behaving like @xmath207 . for intermediate coupling , it exhibits a maximum @xcite . a similar but less pronounced behaviour is found in three dimensions , again in the gaussian approximation @xcite . in a more refined self - consistent treatment @xcite , however , the critical temperature is a monotonically increasing function of the coupling . it is likely that the same situation also applies in two dimensions , but unfortunately there is at present no quantitative theory taking into account the repulsive interaction between the pairs in this case . in the bose limit the transition temperature becomes independent of the original attractive interaction and is completely determined by the boson density @xmath202 and the effective mass @xmath208 . it approaches a value of about @xmath209 of the fermi energy , which is likely to be an upper limit for @xmath1 in the present problem . the chemical potential @xmath129 at the transition decreases monotonically from its weak coupling value @xmath204 to @xmath132 in the bose limit . it changes sign at @xmath210 , where the behaviour crosses over from bcs- to bose - like ( in fig [ fig3 ] we have supressed a small dip in @xmath129 around these couplings which is an artefact of the pronounced maximum in @xmath1 ) . apart from the chemical potential @xmath129 , the nature of the transition can also be inferred from evaluating the number of preformed bosons at @xmath1 . this quantity , which is just half of the contribution ( [ ref43 ] ) to the number equation is shown in fig . it is obvious that the nature of the phase transition changes rather quickly in a range of couplings between @xmath211 and @xmath212 . for smaller couplings , the density of preformed pairs near @xmath1 is essentially negligible and binding occurs simultaneously with condensation . on the other hand , for @xmath213 basically all pairs are already present above @xmath1 and the transition to superconductivity is that of a true bose system . in the following we want to determine both the coherence length @xmath2 and the penetration depth @xmath3 as a function of the coupling . the former is defined both above and below @xmath1 and may be obtained from the gl coefficients simply via @xmath214 the definition of a penetration depth in a two - dimensional superconductor has been discussed in section iii . within the gaussian approximation we may replace @xmath140 in ( 31 ) by @xmath215 which leads to @xmath216 here we have introduced the bare value @xmath217 of the london penetration depth defined by @xmath218 with @xmath219 the nominal three - dimensional carrier density . note that both @xmath2 and @xmath3 have been written in terms of the original static gl - coefficients @xmath220 and @xmath101 , in order to stress that the characteristic lengths are independent of the normalization of @xmath9 , i.e. the kinetic coefficient @xmath102 does not enter here . since the coefficient @xmath87 vanishes at the transition , both @xmath2 and @xmath3 diverge . now in a full treatment of the gl - functional , including the @xmath221-term , the behaviour very close to @xmath1 in a single layer would be of the kosterlitz - thouless type . the correlation length would thus diverge like @xcite @xmath222 while @xmath223 would jump from zero to a finite value below @xmath1 ( see ( 32 ) ) . in the three - dimensional case , the behaviour very close to @xmath1 is that of a 3d xy - model with nontrivial but well known critical exponents @xcite . in our gaussian approximation this complex structure is replaced by a simple mean field behaviour . however there is a subtle point even at this level of approximation . indeed the coefficient @xmath87 depends both on temperature and chemical potential and it is only in the strict bcs - limit , where the latter is a fixed constant equal to @xmath204 . with increasing coupling , however , the chemical potential changes and thus the relevant limit close to @xmath1 is to consider @xmath224 as @xmath225 . now by using the exact relation ( [ ref39 ] ) , it is straightforward to show @xcite that @xmath226 vanishes quadratically near @xmath1 . the resulting critical exponent for the correlation length would thus be @xmath227 . indeed this is the exponent expected for an ideal bose gas in three dimensions , to which our gaussian approximation is effectively equivalent . now in order to allow a comparison of our results with measured values of @xmath2 and @xmath3 , which are found to obey a mean field behaviour with @xmath228 except very close to @xmath1 @xcite , we neglect the temperature dependence of the chemical potential near @xmath1 . as a result @xmath229 vanishes linearly near @xmath1 , giving the standard mean field divergence of @xmath2 and @xmath3 . obviously this approximation is only reliable on the weak coupling side of the transition . as we will see below , however , this is indeed the relevant regime even in high-@xmath1 superconductors . we thus expect that our results are at least qualitatively reliable for these systems . for strong coupling , the derivative @xmath230 of @xmath231 near @xmath1 vanishes like @xmath232 . as a result , the characteristic lengths @xmath2 and @xmath3 would increase exponentially in the bose limit which is unphysical however . we have therefore restricted our calculation of the gl coherence length @xmath79 defined by @xmath233 to coupling strengths @xmath234 smaller than about @xmath212 , where the system starts to cross over to bose like behaviour . the corresponding result for @xmath79 in units of @xmath235 is shown in fig . [ fig5 ] . it exhibits the expected decrease of the coherence length from its weak coupling limit @xmath236 to values of order one near the crossover regime , before it starts to rise again . as was noted above our approximations in determining @xmath79 are no longer reliable in this regime . while in three dimensions @xmath79 is expected to increase like @xmath237 @xcite with @xmath238 the relevant bose - bose scattering length @xcite , the actual behaviour in two dimensions is unknown . fortunately , however , this problem does not arise in determining the gl - parameter @xmath239 which , in two dimensions , can be expressed as @xmath240 here we have introduced the equivalent of the classical electron radius @xmath241 where @xmath113 is the band mass . since the problematic coefficient @xmath87 has dropped out in @xmath242 , we can use ( [ ref52 ] ) to determine the gl - parameter in the whole regime between weak and strong coupling . in the bcs - limit @xmath242 is exponentially small , behaving like @xmath243 . the associated penetration depth @xmath244 is thus essentially equal to the london value @xmath217 . by contrast , for the bose case @xmath242 approaches a constant value @xmath245 which is large compared to one , since @xmath246 for realistic values of the sheet thickness @xmath139 . the complete dependence of @xmath242 in the crossover regime is shown in fig . [ fig6 ] . it is a monotonic function of the binding energy . thus with increasing coupling there is always a transition from type i to type ii behaviour in two dimensions , even for a clean superconductor with no impurities as considered here . with these results , we are now in a position to compare our simple model with experimental values for optimally doped high-@xmath1 superconductors . since the critical temperature is exponentially sensitive to the coupling strength and also is likely to be considerably reduced compared to our result in fig . 2 by fluctuation effects in the crossover regime , we have refrained from taking @xmath1 as a reliable parameter for adjustment . instead we have used the measured values of the slope of the upper critical field @xmath0 near @xmath1 , which determines the gl coherence length @xmath79 via @xmath247 in order to fix the dimensionless coupling strength @xmath205 from the measured values @xcite @xmath248 for optimally doped ybco ( @xmath249 ) and @xcite @xmath250 for the corresponding compound bscco ( @xmath251 ) , we need the in plane carrier densities @xmath59 which determine an effective value of @xmath252 ( it should be pointed out that @xmath253 is introduced here only as a measure of the carrier density . in fact due to the strong attractive interaction , the actual momentum distribution of the fermions just above @xmath1 will be far from the standard fermi distribution , except in the bcs limit . ) . the three - dimensional carrier densities in optimally doped ybco and bscco are @xcite @xmath254 and @xcite @xmath255 respectively . with the corresponding values @xcite @xmath256 and @xmath257 of the effective sheet thickness , the resulting fermi momenta are @xmath258 for ybco and @xmath259 for bscco . the dimensionless ratio @xmath260 and @xmath261 between the coherence length and the average interparticle spacing then allows us to determine the effective coupling strength . from fig . [ fig5 ] we find that @xmath262 is equal to @xmath263 for ybco and @xmath264 for bscco . as figs . [ fig3 ] and [ fig4 ] show , these coupling strengths describe superconductors which are still on the weak coupling side of the crossover from bcs to bose behaviour . for instance the density of preformed bosons at @xmath1 is less then 1% in both cases ( see fig . [ fig4 ] ) . in order to check whether our description is consistent , we determine the associated values of the gl parameter @xmath242 near @xmath1 . from the above values of @xmath139 and the band masses @xcite @xmath265 for ybco and @xcite @xmath266 for bscco ( @xmath267 is the free - electron mass . ) we find that @xmath268 is equal to @xmath269 and @xmath270 respectively for the two compounds considered here . from fig . [ fig6 ] we thus obtain @xmath271 and @xmath272 for optimally doped ybco and bscco . these numbers agree very well with the experimentally determined values of @xmath242 , which are @xcite @xmath273 and @xcite @xmath274 . our simple one parameter model therefore gives a consistent quantitative description of the characteristic lengths @xmath2 and @xmath3 . to summarize , we have studied the crossover in the superconducting transition between bcs- and bose - like behaviour within a gl description . it has been found that optimally doped high-@xmath1 superconductors are still on the weak coupling side of this crossover , although they are certainly far away from the bcs - limit . our microscopic model is characterized by a single dimensionless parameter , similar to the familiar bcs - hamiltonian . while the gl functional has the standard form for arbitrary coupling its coefficients depend strongly on the coupling strength . the crossover , at least in two dimensions , is essentially identical for the s- or d - wave case . for static properties , only two of the relevant coefficients @xmath87 , @xmath62 and @xmath101 are independent , since the normalization of the order parameter is arbitrary . fixing @xmath79 from experiment therefore leaves only one further parameter for instance @xmath242 as an independent predicted quantity . the good agreement of @xmath242 with measured values supports our conclusion that the optimally high-@xmath1 compounds are intermediate between bcs and bose behaviour . since the crossover regime is rather narrow , however , weak coupling theories are still a reasonable approximation for the relevant coupling strengths . this is consistent with the empirical fact that a weak coupling approach apparently works well in many cases . evidently there are a number of important open questions . they may be divided into two classes : the first one concerns the problem of a better and more complete treatment of our model itself . the second class is related to the problem , to which extent this model is applicable to high-@xmath1 compounds and what are the necessary ingredients for a more realistic description . regarding our microscopic hamiltonian as a given model , it is obvious that our treatment of the associated gl phenomenology is still incomplete . in particular the gaussian approximation is obviously not sufficient to give a quantitatively reliable result for @xmath1 at intermediate coupling . moreover , the behaviour of the characteristic lengths @xmath2 and @xmath3 in the strong coupling limit is completely unknown . in order to go beyond the gaussian approximation , it is necessary to include the pair interaction ( i. e. the @xmath169-term ) properly . progress in this direction has been made in the three - dimensional case by haussmann @xcite and very recently by pistolesi and strinati @xcite . using a self - consistent and conserving approximation for the green- and vertex functions , haussmann obtained a smooth increase of @xmath1 with coupling , thus eliminating the unphysical maximum in the crossover regime found in the gaussian approximation . this approach is rather different from our present one and requires extensive numerical work . pistolesi and strinati have performed an essentially analytical calculation of the correlation length @xmath275 at zero temperature , using a loop expansion in a functional approach similar to our present one . they have shown that the pair radius @xmath276 coincides with @xmath275 not only in the bcs - limit , but down to values around @xmath277 . similar to our results for the gl coherence length @xmath79 in fig . [ fig5 ] , @xmath278 reaches a minimum of order one in the crossover regime before it starts to rise again . since in three dimensions the behaviour at strong coupling is that of a weakly interacting bose gas with scattering length @xmath279 @xcite , @xmath275 eventually increases like @xmath280 while @xmath281 approaches zero @xcite . unfortunately for the two - dimensional case , where the boson interaction @xmath184 is finite even at very strong coupling @xcite , the kosterlitz - thouless nature of the transition makes it very difficult to improve upon the simple approximations used here . a first step in this direction was taken by traven@xcite . he showed that interactions between the pair fluctuations guarantee a nonvanishing superfluid density at finite temperature , in agreement with our arguments below eq . ( [ ref40 ] ) . however there seems to be no quantitative calculation of the coherence length in a two - dimensional bose - like regime even near zero temperature . a different problem we want to mention here is that of the proper time dependent gl theory . since our quantum gl functional is derived from a microscopic hamiltonian , in principle it contains the complete information about the order parameter dynamics , at least as far as intrinsic effects are concerned . neglecting higher order terms in the expansion in @xmath70 , the resulting equation of motion for the order parameter in real time @xmath282 is @xcite @xmath283}{\delta\psi^{\star}(\boldsymbol{r},t)}\;.\ ] ] due to the analytic continuation , the coefficient @xmath284 has now acquired a finite imaginary part @xmath285 which describes irreversible relaxation . for a better comparison with the standard literature , it is convenient to choose the conventional order parameter @xmath286 , where the prefactor of the gradient term is @xmath110 . with this choice of normalization , the kinetic coefficient @xmath287 is identical with the effective mass discussed in section iii . it is then evident from fig . [ fig1 ] that a gross - pitaevskii - like dynamics where @xmath288 , is only valid in the bose limit , while @xmath289 is exponentially small for weak coupling . indeed for bcs - like systems it is @xmath290 which is dominant , being of order @xmath291 in the three - dimensional case @xcite . this result reflects the fact that for weak coupling superconductors the order parameter dynamics is purely relaxing . going beyond the bcs - limit , the associated kinetic coefficient @xmath290 has only been evaluated in three dimensions @xcite , where it exhibits a maximum at intermediate coupling . its behaviour in the bose - limit and in two dimensions in general , however , is completely unknown . since the incorporation of scattering states in the three - dimensional case requires to go beyond the linear expansion in @xmath70 , it is likely however , that a simple first order equation like eq . ( [ time ] ) is in fact not appropriate for describing the dynamics at intermediate coupling . it is only at very low temperatures where the situation becomes simple again . indeed from quite general arguments @xcite , the dynamics as @xmath292 is expected to be of the gross - pitaevskii form , irrespective of the strength of the coupling . finally we mention that a proper microscopic calculation of the complex coefficient @xmath102 is relevant for understanding the hall effect in high-@xmath1 compounds @xcite . concerning the question to which extent our model is really applicable to high-@xmath1 superconductors , it is obvious that most of the complexity of these systems is neglected here . in particular we have assumed that the normal state is characterized by a given density of effective mass fermions with some instantaneous attractive interaction @xcite . such a system will certainly be very different from a conventional fermi liquid for intermediate or strong coupling . our conclusion that we are still on the weak coupling side of the crossover , with a negligible density of preformed pairs , is consistent with the phenomenology of optimally ( and perhaps overdoped ) high-@xmath1 superconductors , however it is certainly inappropriate for the underdoped cuprates . indeed these systems exhibit a gap far above @xmath1 which may be interpreted in terms of preformed bosons . a gl description of underdoped compounds was very recently developed by geshkenbein , ioffe and larkin @xcite . assuming that bosons form far above @xmath1 only in parts of the fermi surface and coexist with unpaired fermions through @xmath1 , they obtain a reasonable description of the phenomenology of certain underdoped materials . this behaviour is quite different from that obtained in our model , which completely neglects any effects of the coulomb repulsion and band structure . it is obvious that for a quantitative description of high-@xmath1 superconductors both coulomb correlations and band structure effects have to be included , which requires to use lattice models . they provide a quantitative description of these complex systems at least in the normal state and allow the calculation of microscopic properties like spectral functions , etc @xcite . unfortunately with these models it still seems impossible to really explain how d - wave superconductivity arises from the strongly spin and charge correlated normal state . as a result , effective models like the negative - u hubbard model are often used to discuss microscopic properties of high-@xmath1 compounds @xcite . our present approach is more phenomenological , starting from a model in which all microscopic details are neglected except for the fact that we have a strong pairing interaction in a system of fermions with given density . the advantage of such an approach is that it allows a simple calculation of the relevant lengths @xmath2 and @xmath3 and quantities following from that like the critical fields . the fact that the resulting gl theory gives a consistent description of optimally doped cuprates indicates that at least at this level the microscopic details are not relevant . certainly our results supporting this view are quite limited so far and it is necessary to investigate this further . since the coefficients of the gl functional near @xmath1 are quite generally determined by the properties in the _ normal _ state , an interesting future direction would be to see whether superconducting properties below @xmath1 can quantitatively be obtained from the gl functional by properly incorporating the anomalous behaviour in the normal state . one of the authors ( w. z. ) would like to thank a. j. leggett for his hospitality at the university of illinois where this work was completed and for useful discussions . part of this work was supported by a grant from the deutsche forschungsgemeinschaft ( s. s. ) . a. j. leggett , in _ modern trends in the theory of condensed matter _ , edited by a. pekalski und j. przystawa ( springer , berlin , 1980 ) . p. nozires and s. schmitt - 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wave channel of arbitrary strength . by means of a hubbard - stratonovich transformation a statistical ginzburg - landau theory is derived , which describes the smooth crossover from a weak - coupling bcs superconductor to a condensate of composite bosons . adjusting the interaction strength to the observed slope of @xmath0 at @xmath1 in the optimally doped high-@xmath1 compounds ybco and bscco , we determine the associated values of the ginzburg - landau correlation length @xmath2 and the london penetration depth @xmath3 . the resulting dimensionless ratio @xmath4 and the ginzburg - landau parameter @xmath5 agree well with the experimentally observed values . these parameters indicate that the optimally doped materials are still on the weak coupling side of the crossover to a bose regime .
let the real function @xmath2 satisfy an initial - boundary value problem for the sine - gordon equation on the half - line , @xmath20 , where @xmath21 is a positive constant . the function @xmath2 can be constructed as follows [ 1],[2 ] : * given initial conditions construct the spectral functions @xmath5 and @xmath6 . these functions are defined in terms of @xmath22 , where the vector @xmath23 is an appropriate solution of the @xmath24-part of the associated lax pair evaluated at @xmath25 . thus @xmath23 is defined in terms of the initial conditions @xmath8 and @xmath9 . * from the given boundary condition , characterize the unknown boundary value at @xmath26 by the requirement that the spectral functions @xmath27 satisfy the global relation @xmath28 where @xmath29 is analytic for @xmath30 and is of @xmath31 as @xmath32 . the functions @xmath10 and @xmath11 are defined in terms of @xmath33 , where the vector @xmath34 is an appropriate solution of the @xmath35-part of the associated lax pair evaluated at @xmath26 . thus @xmath34 is defined in terms of @xmath36 and @xmath13 . * given @xmath37 , construct @xmath2 through the solution of a @xmath3 matrix riemann - hilbert problem . the function @xmath2 satisfies the sine - gordon equation as well as the given initial and boundary conditions . the most complicated step in the above construction is the characterization of the missing boundary value . for example , for the dirichlet problem where the function @xmath36 is prescribed as the boundary condition , it is shown in [ 2 ] that the unknown boundary value @xmath13 can be obtained through the solution of a system of four _ nonlinear _ odes . it was shown in [ 1 ] and [ 3 ] that for some particular boundary conditions , which we refer to as _ linearizable _ boundary conditions , it is possible to bypass the above system of nonlinear odes and to construct @xmath7 using only algebraic manipulations . in particular , it was shown in [ 1 ] that this is the case for the boundary condition @xmath12 , @xmath38 constant . in this paper we show that there exists another linearizable boundary condition which involves two constants @xmath16 and @xmath17 . for completeness we also include the case @xmath12 . [ [ theorem-1.1 ] ] theorem 1.1 + + + + + + + + + + + let the real function @xmath2 satisfy the sine - gordon equation @xmath39 ( where @xmath21 is a given constant ) the initial conditions @xmath40 and either of the following two boundary conditions , @xmath41 or @xmath42 where @xmath43 are real constants . assume that @xmath44 and @xmath45 are schwartz functions for @xmath46 integer , and that the initial conditions are compatible with the boundary conditions at @xmath47 , i.e. assume that for ( 1.4a ) and ( 1.4b ) the following conditions are valid respectively @xmath48 the above initial - boundary value problems have a unique global solution given by @xmath49 where @xmath50 and @xmath51 denote the ( 12 ) and ( 22 ) entries of the @xmath3 matrix @xmath52 which satisfies the following riemann - hilbert problem . \(i ) @xmath53 is meromorphic in @xmath4 for @xmath54 , where @xmath55 , which is depicted in figure [ fig1.1 ] , is defined by @xmath56 and the domains @xmath57 .2 in ( ii ) let the domains @xmath58 which are depicted in figure [ fig1.1 ] , be defined by @xmath59 @xmath60 the matrix @xmath53 satisfies the jump condition @xmath61 where @xmath53 is @xmath62 for @xmath63 , @xmath53 is @xmath64 for @xmath65 , and the @xmath3 matrix @xmath66 is defined in terms of the spectral functions @xmath67 and the explicit function @xmath68 by the following formulae : @xmath69 ] .2 in @xmath70 \gamma(k)e^{2i\theta } & 1 \end{array } \right ) } , \quad j_3 = { \left ( \begin{array}{cc } 1 & \overline{\gamma(\bar k)}e^{-2i\theta } \\[2ex ] 0 & 1 \end{array } \right)},\ ] ] @xmath71 -\bar \gamma(k)e^{2i\theta } & 1 + \|\gamma(k)\|^2 \end{array } \right ) } , \quad j_2 = j_3j_4^{-1}j_1,\ ] ] @xmath72 } , \quad k \in d_2 ; \eqno ( 1.7)\ ] ] @xmath73 the functions @xmath5 and @xmath6 are defined in terms of @xmath74 and @xmath45 by @xmath75 where the vector @xmath22 with component @xmath76 and @xmath77 satisfies @xmath78 -i ( \dot q_0(x ) + q_1(x ) ) + \frac{\sin q_0(x)}{k } & - i ( k+ \frac{\cos q_0(x)}{k } ) \end{array } \right ) } \phi,\ ] ] @xmath79 @xmath80 and @xmath81 diag @xmath82 . for @xmath83 the ratio @xmath7 equals @xmath84 . for @xmath85 , the ratio @xmath7 for the cases ( 1.4a ) and ( 1.4b ) is given respectively by @xmath86 and @xmath87 + a(\frac{1}{k } ) [ \overline{\alpha(\bar{k } ) } - \alpha(k ) - 2\sin{(\frac{q_{0}(0)}{2 } ) } ] } { a(\frac{1}{k } ) [ \alpha(k ) + \overline{\alpha(\bar{k } ) } - 2i\cos{(\frac{q_{0}(0)}{2 } ) } ] - b(\frac{1}{k } ) [ \alpha(k ) - \overline{\alpha(\bar{k } ) } - 2\sin{(\frac{q_{0}(0)}{2 } ) } ] } \eqno ( 1.10b)\ ] ] where @xmath88 \(iii ) define the function @xmath89 by @xmath90 where @xmath91 and @xmath92 denote the denominator and the numerator of the rhs of equations ( 1.10 ) . if the function @xmath5 has zeros for i m @xmath93 , and/or the function @xmath89 has zeros in @xmath94 , then @xmath52 satisfies appropriate residue conditions , see section 4 . [ [ organization - of - the - paper ] ] organization of the paper + + + + + + + + + + + + + + + + + + + + + + + + + in section 2 we summarize the methodology of [ 1 ] , [ 3 ] for identifying and analyzing linearizable boundary conditions . if a given pde admits different lax pair [ 4 ] formulations , it is possible to search for linearizable boundary conditions for each of these different formulations . the lax pair analyzed in [ 1 ] , see equations ( 2.9 ) , gives rise to the case ( 1.4a ) ; the sg also admits an alternative lax pair [ 5 ] , see ( 2.11 ) , which gives rise to the linearizable case ( 1.4b ) . since the basic rh problem presented in theorem 1.1 is associated with the lax pair analyzed in [ 1 ] , we present in section 3 a general formalism which connects the spectral functions @xmath95 associated with two different lax pairs . as an application of this formalism we use the alternative lax pair ( 2.11 ) to identify ( 1.4b ) , but we solve the sg , for both boundary conditions ( 1.4a ) and ( 1.4b ) , using the lax pair of [ 1 ] . in section 4 we derive the residue conditions and in section 5 we discuss further these results . we first recall the definition of the spectral functions @xmath10 and @xmath11 [ 1 ] . [ [ definition-2.1 ] ] definition 2.1 + + + + + + + + + + + + + + suppose that the @xmath35-part of the lax pair of a given integrable pde is @xmath96 where @xmath97 @xmath98 is a @xmath3 matrix , the scalar @xmath99 is an analytic function of @xmath4 , and the @xmath3 matrix @xmath100 is an analytic function of @xmath4 , of @xmath2 , of @xmath101 , and of the derivatives of these functions . let @xmath102 be the unique solution of @xmath103 @xmath104 where @xmath21 is a positive constant . assume that @xmath105 is such that @xmath102 has the form @xmath106 \rho\overline{\phi_1(t,\bar k ) } & \phi_2(t , k ) \end{array } \right ) } , \quad \rho^2=1 . \eqno ( 2.4)\ ] ] the spectral functions @xmath10 and @xmath11 are defined by @xmath107 [ [ linearizable - boundary - conditions ] ] linearizable boundary conditions + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + let the transformation @xmath108 be defined by the requirement that it leaves @xmath99 invariant , i.e. @xmath109 suppose that there exists a nonsingular @xmath3 matrix @xmath92 such that @xmath110 n(k ) = if_2(k)\sigma_3-\tilde q(0,t , k ) . \eqno ( 2.7)\ ] ] then the solution @xmath102 of equations ( 2.3 ) satisfies @xmath111 this equation and the definitions ( 2.5 ) , imply a relation between @xmath112 and @xmath113 . using this relation , it is possible to compute @xmath7 using only the algebraic manipulation of the global relation [ 1 ] . [ [ two - different - lax - pairs - for - the - sine - gordon ] ] two different lax pairs for the sine - gordon + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the sg possesses the lax pair [ 6 ] @xmath114 @xmath115 where @xmath98 is a @xmath3 matrix , @xmath116 , and the @xmath3 matrix @xmath117 is defined by @xmath118 -i ( q_x + q_t ) + \frac{\sin q}{k } & - \frac{i}{k}(-1 + \cos q ) \end{array } \right ) } \eqno ( 2.10)\ ] ] with @xmath119 denoting @xmath2 . the sg also possesses the alternative lax pair [ 5 ] @xmath120 where @xmath98 is a @xmath3 matrix and the @xmath3 matrices @xmath121 and @xmath122 are defined by @xmath123 [ [ linearizable - case - of - the - second - lax - pair ] ] linearizable case of the second lax pair + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we now show that by analyzing the @xmath35-part of the second lax pair ( 2.11 ) evaluated at @xmath26 , it is possible to identify linearizable boundary conditions using the simple formulation reviewed earlier , see equations ( 2.6)-(2.8 ) . [ [ proposition-2.1 ] ] proposition 2.1 + + + + + + + + + + + + + + + let the @xmath124 matrix @xmath125 be the following solution of the @xmath35-part of the lax pair ( 2.11 ) evaluated at @xmath26 : @xmath126 @xmath127 where @xmath128 and @xmath122 is defined by equation ( 2.12b ) . then @xmath129 satisfies the `` symmetry '' relation @xmath130 where for equations ( 1.4a ) and ( 1.4b ) , @xmath92 is given respectively by @xmath131 0 & \frac { e^{-\frac{i\chi}{2 } } + k^2e^{\frac{i\chi}{2 } } } { e^{\frac{i\chi}{2 } } + k^2e^{- \frac{i\chi}{2 } } } \end{array } \right ) } \eqno ( 2.15)\ ] ] and @xmath132 -1 & \frac{\chi_2}{k-\frac{1}{k } } - \frac{i\chi_1}{k+\frac{1}{k } } \end{array } \right ) } . \eqno ( 2.16)\ ] ] _ proof _ in this case @xmath133 , thus equation ( 2.6 ) implies @xmath134 . let us introduce the following notations for the matrix @xmath92 , @xmath135 in this case equation ( 2.8 ) is @xmath136 if @xmath137 , then the ( 11 ) and ( 22 ) entries of this equation are identically satisfied , while both the ( 12 ) and ( 21 ) entries of equation ( 2.18 ) yield the equation @xmath138 we distinguish two cases \(i ) @xmath139 if @xmath140 equals the ( 22 ) entry of the matrix @xmath141 defined in ( 2.15 ) then equation ( 2.19 ) becomes the boundary condition ( 1.4a ) . \(ii ) @xmath142 , @xmath143 , @xmath144 . then @xmath145 letting @xmath146 i.e. @xmath147 equation ( 2.20 ) becomes the boundary condition ( 1.4b ) , and @xmath92 is given by equation ( 2.16 ) . equation ( 2.14 ) expressed a `` symmetry '' relation for the eigenfunction @xmath125 associated with the second lax pair ( 2.11 ) . the spectral functions @xmath10 and @xmath11 are defined in terms of the eigenfunction @xmath102 associated with the first lax pair ( 2.9 ) . in what follows we present a general result which , starting with the `` symmetry '' relation satisfied by @xmath125 , yields the `` symmetry '' relations satisfied by @xmath113 . .2 in * proposition 3.1 * let @xmath148 , @xmath149 , @xmath150 . let @xmath102 be the unique solution of the @xmath3 matrix equations @xmath151 @xmath152 where @xmath153 are as in definition 2.1 ( see equation ( 2.1 ) ) . assume that @xmath102 has the form ( 2.4 ) . define the spectral functions @xmath10 and @xmath11 by equations ( 2.5 ) where @xmath99 satisfies the relation ( 2.6 ) . let @xmath125 be the unique solution of the @xmath154 matrix equations @xmath155 @xmath156 suppose that there exists a non - singular @xmath3 matrix @xmath157 such that @xmath158 assume that @xmath125 satisfies the `` symmetry '' relation @xmath159 where @xmath92 is a @xmath3 nonsingular matrix . then @xmath102 satisfies the symmetry relation @xmath160 furthermore , the spectral functions @xmath10 and @xmath11 satisfy the symmetry relation @xmath161 - \rho \overline{b(\nu(\bar k)}e^{\overline{if_2(\bar k)}t } & \overline{a(\nu(\bar k)}e^{\overline{if_2(\bar k)}t } \end{array } \right ) } = \ ] ] @xmath162 - \rho \overline{b(\bar k)}e^{\overline{if_2(\bar k)}t } & \overline{a(\bar k)}e^{\overline{if_2(\bar k)}t } \end{array } \right ) } h(0)^{-1 } n(k)^{-1}h(0 ) . \eqno ( 3.6)\ ] ] _ proof . _ if the @xmath3 matrices @xmath163 and @xmath164 are related by equation ( 3.3 ) , then the @xmath3 matrices @xmath125 and @xmath102 are related by the equation @xmath165 indeed , equation ( 3.7 ) is identically satisfied at @xmath25 . furthermore replacing in equation ( 3.2a ) @xmath125 by the rhs of equation ( 3.7 ) we find @xmath166 replacing in this equation @xmath167 by @xmath168 , and using equation ( 3.3 ) we find an identity . since @xmath125 satisfies the `` symmetry '' relation ( 3.4 ) , and @xmath102 is related with @xmath125 through equation ( 3.7 ) , it is easy to show that @xmath102 satisfies the symmetry relation ( 3.5 ) . indeed , this latter equation follows from ( 3.7 ) by replacing @xmath4 with @xmath169 and then using equations ( 3.4 ) and ( 3.7 ) . having established the `` symmetry '' relation ( 3.5 ) it is straightforward to obtain a symmetry relation for the spectral functions @xmath170 . indeed , evaluating equation ( 3.5 ) at @xmath171 , and expressing @xmath172 in terms of the spectral functions ( see equations ( 2.4 ) , ( 2.5 ) ) , equation ( 3.5 ) yields equation ( 3.6 ) . [ [ the - sg - case ] ] the sg case + + + + + + + + + + + for the sg equation @xmath173 @xmath174 where @xmath117 and @xmath122 are defined by equations ( 2.10 ) and ( 2.12b ) respectively . it can be verified that in this case @xmath102 has the form ( 2.4 ) with @xmath175 . furthermore , equation ( 3.3 ) is valid with the @xmath3 nonsingular matrix @xmath157 given by @xmath176 thus for the boundary condition ( 1.4b ) , the spectral functions @xmath10 and @xmath11 satisfy equation ( 3.6 ) with @xmath99 , @xmath157 , and @xmath92 given by equations ( 3.8a ) , ( 3.9 ) , and ( 2.16 ) respectively . [ [ proof - of - theorem-1.1 ] ] proof of theorem 1.1 + + + + + + + + + + + + + + + + + + + + let @xmath2 be defined by equation ( 1.5 ) in terms of the solution @xmath52 of the @xmath3 matrix rh problem with the jump condition ( 1.6 ) , where the jump matrix @xmath66 is defined by equations ( 1.7 ) in terms of @xmath177 . let @xmath5 and @xmath6 be defined by equations ( 1.8 ) and ( 1.9 ) . let @xmath10 and @xmath11 be defined by equations ( 2.3)(2.5 ) where @xmath105 in equation ( 2.3 ) equals @xmath178 which is defined by equation ( 2.10 ) with @xmath36 and @xmath13 replaced by @xmath179 and @xmath180 . it was shown in [ 1 ] that if @xmath179 and @xmath180 are such that the global relation ( 1.1 ) is valid , then @xmath2 satisfies the sg , and also @xmath181 , @xmath182 , @xmath183 , @xmath184 . furthermore , it was shown in [ 1 ] that if @xmath185 then the definition of @xmath186 and the global relation ( 1.1 ) imply ( 1.10a ) . thus it only remains to be shown that if the boundary condition ( 1.4b ) is valid then @xmath187 satisfies equation ( 1.10b ) . in this respect we note that the definition of @xmath113 implies the symmetry relation ( 3.6 ) . in what follows we show that this relation together with the global relation ( 1.1 ) imply ( 1.10b ) . for this purpose it is convenient to assume that @xmath188 as @xmath189 and to let @xmath190 . in this case @xmath10 and @xmath11 are _ not _ entire functions but are analytic functions for @xmath191 . also , the global relation ( 1.1 ) becomes @xmath192 note that @xmath193 are defined for @xmath191 , while @xmath194 are defined for @xmath195 , thus each term of equation ( 3.10 ) is well defined for @xmath196 . the assumption @xmath197 as @xmath189 , and the definitions of @xmath157 and @xmath92 , ie . equations ( 3.9 ) and ( 2.16 ) , imply @xmath198 where @xmath140 is defined by equation ( 2.21a ) . similarly @xmath199 using equations ( 3.11 ) and ( 3.12 ) in equation ( 3.6 ) ( with @xmath200 ) , solving for @xmath201 , and letting @xmath190 , we find @xmath202a(k ) - [ \bar{\alpha } - \alpha + 2\sin{(\frac{q_{0}(0)}{2})}]b(k ) \big\ } , \\ b(\frac{1}{k } ) & = \frac{\alpha + \bar{\alpha } - 2i}{4(1+\alpha\bar{\alpha } ) } \big\ { [ \alpha + \bar{\alpha } - 2i\cos{(\frac{q_{0}(0)}{2})}]b(k ) - [ \bar{\alpha } - \alpha - 2\sin{(\frac{q_{0}(0)}{2})}]a(k ) \big\ } , \end{split } \tag{3.13}\ ] ] where @xmath191 . the `` symmetry '' relations ( 3.13 ) together with the global relation ( 3.10 ) , yield @xmath7 in terms of @xmath203 . indeed , for @xmath204 , @xmath205 replacing @xmath4 by @xmath206 in this equation , using equations ( 3.13 ) , and solving the resulting equation for @xmath7 we find equation ( 1.10b ) . it was shown in [ 1 ] that the solution of the basic rh problem is independent of @xmath21 . thus although the basic formula ( 1.10b ) was derived under the assumption that @xmath207 as @xmath189 , this formula is valid even without this assumption . we _ assume _ that @xmath5 has @xmath208 simple zeros @xmath209 , @xmath210 of which are in @xmath211 and the remaining @xmath212 of which are in @xmath213 . we denote by @xmath214 the derivative of @xmath5 with respect to @xmath4 , and we denote by @xmath215_{1}$ ] and @xmath215_{2}$ ] the first and second columns of the @xmath216 matrix @xmath53 . let @xmath217 denote the zeros of @xmath89 for @xmath218 , where @xmath89 is defined by equation ( 1.12 ) . the following residue conditions are valid : @xmath219_{1 } & = \frac{e^{2i\theta(k_{j})}}{\dot{a}(k_{j})b(k_{j } ) } [ \mu(x , t , k_{j})]_{2 } , \quad j = 1,\dots , n_{1 } , \quad k_{j } \in d_{1 } , \\ \operatorname{res}_{\bar{k}_{j } } [ \mu(x , t , k)]_{2 } & = -\frac{e^{-2i\theta(\bar{k}_{j})}}{\overline{\dot{a}(k_{j})}\ , \overline{b(k_{j } ) } } [ \mu(x , t,\bar{k}_{j})]_{1 } , \quad j = 1,\dots , n_{1 } , \quad \bar{k}_{j } \in d_{4 } , \\ \operatorname{res}_{\lambda_{j } } [ \mu(x , t , k)]_{1 } & = -\frac{\overline{n(\bar{\lambda}_{j } ) } e^{2i\theta(\lambda_{j})}}{\dot{a}(\lambda_{j } ) \dot{\delta}(\lambda_{j } ) } [ \mu(x , t,\lambda_{j})]_{2 } , \quad j = 1,\dots,\lambda , \quad \lambda_{j } \in d_{2 } , \\ \operatorname{res}_{\bar{\lambda}_{j } } [ \mu(x , t , k)]_{2 } & = -\frac{n(\lambda_{j } ) e^{-2i\theta(\bar{\lambda}_{j } ) } } { \overline{\dot{a}(\lambda_{j})}\ , \overline{\dot{\delta}(\lambda_{j } ) } } [ \mu(x , t,\bar{\lambda}_{j})]_{1 } , \quad j = 1,\dots,\lambda , \quad \bar{\lambda}_{j } \in d_{3 } , \tag{4.1}\end{aligned}\ ] ] where @xmath92 denotes the numerators of equations ( 1.10 ) , and @xmath220 are defined in theorem 1.1 . in order to derive these equations we first recall that the matrix @xmath52 appearing in the rh problem of theorem 1.1 is constructed from three appropriate matrix solutions , @xmath221 , of the lax pair ( 2.9 ) . these matrices can be written in the column vector form @xmath222 where superscripts denote the domains that these column vectors are bounded and analytic ( @xmath223 in @xmath213 , @xmath224 in @xmath225 etc . ) . it is shown in [ 1 ] that the matrix @xmath53 has the following form : @xmath226 where @xmath227 in order to derive equation ( 4.1a ) we condsider the equation @xmath228 , where @xmath229 is defined in theorem 1.1 and @xmath230 are given by the first and fourth equations in ( 4.3 ) . the second column of this equation yields @xmath231 evaluating this equation at @xmath232 we find @xmath233 where for simplicity of notation we suppress the @xmath234 dependence of @xmath235 and @xmath236 . hence @xmath237_{1 } = \frac{\mu_{2}^{(1)}(k_{j})}{\dot{a}(k_{j } ) } = \frac{e^{2i\theta(k_{j})}}{\dot{a}(k_{j})b(k_{j } ) } \mu_{3}^{(12)}(k_{j}),\ ] ] and since @xmath238_{2}$ ] , equation ( 4.1a ) follows . the derivation of equation ( 4.1b ) is similar . in order to derive equation ( 4.1c ) we consider the equation @xmath239 , where @xmath240 is defined in theorem 1.1 and @xmath241 are given by the first and second equations in ( 4.3 ) . the first column of this equation yields @xmath242 } \mu_{3}^{(12)}. \eqno ( 4.5)\ ] ] following arguments similar to those used in [ 3 ] it can be shown that the zeros of @xmath243 in @xmath213 coincide with the zeros of @xmath89 in @xmath213 , thus equation ( 4.5 ) yields @xmath244 hence @xmath245_{1 } = \frac{\mu_{1}^{(2)}}{\dot{d}(\lambda_{j } ) } = - \frac{\overline{n(\bar{\lambda}_{j } ) } e^{2i\theta(\lambda_{j})}}{a(\lambda_{j } ) \dot{\delta}(\lambda_{j } ) } \mu_{3}^{(12)}(\lambda_{j}),\ ] ] and since @xmath238_{2}$ ] , equation ( 4.1c ) follows . the derivation of ( 4.1d ) is similar . it was shown in [ 1 ] and [ 3 ] that there exists a particular class of boundary conditions for which initial - boundary value problems on the half - line can be solved with the same level of efficiency as the classical initial value problem on the line . these `` linearizable '' boundary conditions were identified by the requirement that the eigenfunction @xmath102 , which satisfies the @xmath35-part of the associated lax pair evaluated at @xmath26 ( as well as the condition that @xmath246 equals the identity matrix ) , satisfies the `` symmetry '' relation @xmath247 in this equation the map @xmath108 is the map which leaves the dispersion relation of the linearized version of the given nonlinear pde invariant ( for the sg , @xmath248 ) and @xmath92 is a non - singular matrix . in this paper we have generalized equation ( 5.1 ) to the equation @xmath249 where @xmath250 is a nonsingular matrix . furthermore we have given an algorithmic way of constructing @xmath251 by making use of the existence of different lax pair formulations for the same nonlinear pde . we expect that the systematic use of bcklund transformations will provide an approach to computing @xmath250 for any linearizable boundary condition . the main advantage of our method is _ not _ that it identifies linearizable boundary conditions , but that for such boundary conditions it expresses the solution @xmath2 in terms of a simple riemann - hilbert ( rh ) problem . the basic features of this problem are similar with the basic features of the rh problem characterizing the solution of the classical initial - value problem , namely these two rh problems : ( a ) have the _ same _ explicit @xmath234 dependence . ( b ) their jump matrices involve the functions @xmath5 and @xmath6 which are constructed from the initial conditions in a similar manner . regarding differences between these two rh problems , we note that the rh problem associated with initial - boundary value problems has the novelty that it is formulated on a more complicated contour , and it also involves some additional jump functions which however can be _ explicitly _ written in terms of @xmath5 and @xmath6 ( see equations ( 1.10 ) ) . the existence of a more complicated contour does _ not _ add any significant complexity to the analysis of the rh problem . also in some cases it is possible to map this contour to the usual contour which is the real @xmath4-axis . this is actually the case for the sg and the nonlinear schrdinger [ 7 ] ( but _ not _ for the korteweg - devries and the modified korteweg - devries ) . the simplicity of the basic rh problem has important implications for the analysis of the long time asymptotics . indeed , in the case that the boundary conditions decay as @xmath252 , it is possible using the deift - zhou approach [ 8 ] to obtain a complete characterization of the solution as @xmath253 and @xmath254 . the general structure of the asymptotics is given in [ 9 ] , where it is shown that the solution is dominated by solitons . a detailed investigation of these solitons for the boundary condition ( 1.4b ) , as well as for the boundary condition @xmath255 will be presented elsewhere . we also note that the simplicity of the basic rh problem makes it possible , using the deift - zhou - venakides approach [ 10 ] , to study the zero dispersion limit of initial - boundary value problems [ 7 ] , [ 11 ] . several authors have identified linearisable boundary conditions using the existence of symmetries and conservation laws , see for example [ 12][13 ] . the analysis of such boundary conditions using several formal rh problems was presented in [ 14 ] . the particular cases of either @xmath256 or @xmath257 are discussed in [ 15 ] using an extension of the problem from the half line to the infinite line ( such an extension is _ not _ possible for pdes with a third order derivative such as the kdv and the modified kdv equations ) . a discussion of the physical significance of the sg with the boundary condition ( 1.4b ) as well as several approaches for the analysis of this problem can be found in [ 16][21 ] . in particular in [ 18 ] the question of the integrability of both the classical and quantum sine - gordon theory involving @xmath258 with the boundary condition @xmath259 was investigated . by demanding that a modification of the first non - trivial integral of motion of the usual theory remain conserved it was found that in general @xmath260 where @xmath261 and @xmath262 are arbitrary real constants . in [ 18 ] the existence of an infinite number of integrals of motion for this system was assumed and the associated quantum field theory was studied . in [ 17 ] the question of integrability of the classical system with the boundary condition ( 5.3 ) was addressed . the results of [ 16 ] were used to prove that an infinite number of integrals of motion do exist , but only for certain @xmath263 . it was found that ( 5.4 ) is the most general boundary term compatible with the existence of infinitely many conserved quantities . this result thus agrees with [ 18 ] . the boundary condition ( 5.4 ) with @xmath264 or @xmath265 has appeared in classical considerations of sine - gordon theory . it was suggested in [ 18 ] that the scattering theory can depend on the extra parameter @xmath262 ; a similar question was investigated in [ 20 ] for real coupling affine toda field theory . i am deeply grateful to e. corrigan for suggesting this problem to me , and to a.r . its for some important suggestions and in particular for showing me the relation between the two different lax pair formulations of the sg equation . faddeev , l.a . takhtajan and v.e . zakharov , a complete description of the solutions of the sine - gordon equation , dan ussr , 219 * , 13341337 ( 1974 ) ; l.d . faddeev and l.a . takhtajan , _ hamiltonian methods in the theory of solitons _ , springer - verlag , 1987 . deift and x. zhou , a steepest descent method for oscillatory riemann - hilbert problems , bull . * 20 * , 119123 ( 1992 ) ; p.a . deift and x. zhou , a steepest descent method for oscillatory riemann - hilbert problems . asymptotics for the mkdv equation , ann . math . * 137 * , 295368 ( 1993 ) . s. ghoshal , a.b . zamolodchikov , boundary @xmath266 matrix and boundary state in two - dimensional integrable quantum field theory , int . j. mod a , * 9 * , 3841 - 3886 ( 1994 ) , erratum - ibid , a , * 9 * , 4353 , ( 1994 ) .	a rigorous methodology for the analysis of initial boundary value problems on the half - line , @xmath0 , @xmath1 , for integrable nonlinear evolution pdes has recently appeared in the literature . as an application of this methodology the solution @xmath2 of the sine - gordon equation can be obtained in terms of the solution of a @xmath3 matrix riemann - hilbert problem . this problem is formulated in the complex @xmath4-plane and is uniquely defined in terms of the so called spectral functions @xmath5 , @xmath6 , and @xmath7 . the functions @xmath5 and @xmath6 can be constructed in terms of the given initial conditions @xmath8 and @xmath9 via the solution of a system of two _ linear _ ode s , while for _ arbitrary _ boundary conditions the functions @xmath10 and @xmath11 can be constructed in terms of the given boundary condition via the solution of a system of four _ nonlinear _ odes . in this paper we analyse two _ particular _ boundary conditions : the case of constant dirichlet data , @xmath12 , as well as the case that @xmath13 , @xmath14 , and @xmath15 are linearly related by two constants @xmath16 and @xmath17 . we show that for these particular cases , the system of the above nonlinear odes can be avoided , and @xmath7 can be computed explicitly in terms of @xmath18 and @xmath19 respectively . thus these `` linearizable '' initial - boundary value problems can be solved with absolutely the same level of efficiency as the classical initial value problem of the line .
sz her ( @xmath13 , gsc 2610 - 1209 , hip 86430 , tyc 2610 - 1209 - 1 ) is an algol - type system with an orbital period of 0.818 d and was announced to be a variable by ceraski ( 1908 ) and also dunr et al . ( 1909 ) . although the first observations of the system date back to 1902 ( shapley 1913 ; russell & shapley 1914 ; dugan 1923 ) , its properties are poorly known compared to those of other short - period algols . to date in the published literature , only one light - curve analysis has been published and it was presented by giuricin & mardirossian ( 1981 ) . they analyzed the two - color photoelectric light curves of broglia et al . ( 1955 ) using the wink model ( wood 1972 ) and concluded that the system is a semi - detached algol - type binary with a mass ratio of @xmath0=0.4 , an orbital inclination of @[email protected] , and a temperature difference between the components of @xmath14=2,600 k. recently , szkely ( 2003 ) and dvorak ( 2009 ) performed ccd observations in order to locate @xmath15 scuti - type pulsations but failed to detect them . although the orbital period of sz her has been examined several times ( kreiner 1971 ; mallama 1980 ; zavala et al . 2002 ) , a detailed study of its orbital period was made by szkely ( 2003 ) and soydugan ( 2008 ) . they reported that the period change can be described using either a sine curve or a single light - time effect ( lite ) ephemeris due to a third body with implied periods of 66 and 71 yr , respectively . soydugan ( 2008 ) also suggested that the timing residuals from the lite fit indicate an additional short - term oscillation with a period within about 20 yr . more than one thousand eclipse timings , spanning @xmath16 110 yr , should be sufficient to resolve the confusion regarding the orbital behavior of sz her . nonetheless , the period variation of this system has not yet been studied as conclusively as required . in this article , a new photometric study of sz her based on modern observations and analyses is presented , and it is demonstrated that the sz her system is likely a quadruple one containing two low - mass companions . we performed new ccd photometry of sz her on 13 nights from 2008 february 28 through may 17 . the observations were taken with a site 2k ccd camera and a @xmath17 filter set attached to the 61-cm reflector at sobaeksan optical astronomy observatory ( soao ) in korea . the instrument and reduction method used were the same as those described by lee et al . ( 2007 , 2010b ) . gsc 2610 - 1116 ( @xmath18 , tyc 2610 - 1116 - 1 ) and gsc 2610 - 0821 , imaged on the chip at the same time as the program target , were selected as comparison and check stars , respectively . the 1@xmath19-values of the dispersions of the magnitude differences between these stars are within @xmath200.01 mag for all bandpasses . the coordinates and tycho magnitudes for the three stars of interest are given in table 1 . a total of 1,728 individual observations were obtained among the four bandpasses ( 435 in @xmath21 , 437 in @xmath22 , 439 in @xmath23 , and 417 in @xmath24 ) and a sample of them is listed in table 2 . the light curves of sz her defined by the soao observations are plotted in figure 1 as the ( v@xmath25c ) differential magnitudes _ versus _ orbital phase , which was computed according to the ephemeris for our binary model determined later in this article with the wilson - devinney synthesis code ( wilson & devinney 1971 , hereafter w - d ) . in addition to these complete light curves , two eclipse timings were observed in 2004 june and 2011 may using the same telescope . the 2004 data were collected using the site 2k ccd camera and @xmath21 filter , and the 2011 ones using an fli img4301e ccd camera and @xmath26 filters . gsc 2610 - 1116 also served as the comparison star for these data collections . details of the new detector have been given previously by lee et al . the shape of the light curve of sz her is very similar to that of algol type . as shown in figure 1 , the light curve was completely covered and the depth differences between the primary and secondary eclipses indicate a significant temperature difference between the two components . in order to understand the physical properties of the system , the @xmath17 light curves in this study were analyzed simultaneously in a manner almost identical to those for xx cep ( lee et al . 2007 ) and cl aur ( lee et al . 2010a ) using the 2003 version of the w - d code and the so - called @xmath0-search procedure ( cf . lee et al . in this paper , the subscripts 1 and 2 refer to the primary and secondary stars being eclipsed at min i ( at phase 0.0 ) and min ii , respectively . the surface temperature of the hotter , and presumably more massive , primary star was assumed to be @xmath4=7,270 k from flower s ( 1996 ) table , according to ( @xmath27)[email protected]@xmath200.041 in the tycho-2 catalog ( hg et al . 2000 ) and @xmath29(@xmath27)[email protected] calculated following schlegel et al . the gravity - darkening exponents were initialized at standard values of @xmath30=1.0 and @xmath31=0.32 and the bolometric albedos at @xmath32=1.0 and @xmath33=0.5 , as surmised from the components temperatures . linear bolometric and monochromatic limb - darkening coefficients were interpolated from the values of van hamme ( 1993 ) in concert with the model atmosphere option . furthermore , a synchronous rotation for both components and a circular orbit were adopted and the detailed reflection effect was considered . the only photometric solution of sz her was reported by giuricin & mardirossian ( 1981 ) 30 years ago and a spectroscopic orbit has not yet been established . thus , an extensive @xmath0-search procedure was conducted for a series of models with varying @xmath0 values . in this process , we first considered the orbital inclination ( @xmath1 ) , effective temperature ( @xmath34 ) , dimensionless surface potential ( @xmath35 ) , and luminosity ( @xmath36 ) . this procedure showed acceptable photometric solutions for mode 5 only , which are semi - detached systems for which the less massive secondary stars accurately fill their inner roche lobes . as displayed in figure 2 , the @xmath0-search results indicate that the minimum value of the weighted sum of the squared residuals ( @xmath37 ) is approximately @xmath0=0.49 . then , we treated this @xmath0 value as an adjustable parameters and included limb - darkening coefficients , albedos , and gravity darkening exponents as additional free variables . the final values are given in table 3 and are plotted in figure 1 as solid curves . in the figure , the model light curves describe the soao multiband data satisfactorily . our light - curve synthesis demonstrates that sz her is an algol - type semi - detached system in which the primary component fills its limiting lobe by approximately 77% and is slightly larger than the lobe - filling secondary component . the gravity darkening exponent of the secondary component is consistent with the standard convective @xmath38 value , while its albedo is close to the standard radiative @xmath39 value . in these analyses , we searched for a possible third light source but found that the parameter remained zero within its error . the dereddened color ( @xmath27)@[email protected] and temperature of the primary component correspond to a normal main - sequence star with a spectral type of about a9 . we estimated the absolute dimensions for the binary system from our photometric solution and from harmanec s ( 1988 ) relation between the spectral type and stellar mass . these are given in table 4 , where the luminosity ( @xmath41 ) and bolometric magnitudes ( @xmath42 ) were computed by adopting @xmath43@xmath11=5,780 k and @xmath42@xmath11=+4.73 for solar values . for the absolute visual magnitudes ( @xmath44 ) , we used the bolometric corrections ( bcs ) appropriate for the temperature of each component from the expression between @xmath45 and bc given by torres ( 2010 ) . with an apparent visual magnitude of @xmath22=+10.06 and the interstellar absorption of @xmath46=0.11 , we calculated the distance of the system to be 294 pc . this result is consistent with 306 pc taken from the trigonometric parallax ( [email protected] mas ; perryman et al . 1997 ) . a comparison of the sz her parameters with the mass - radius , mass - luminosity , and hertzsprung - russell ( hr ) diagrams ( . ibanolu et al . 2006 ) clearly demonstrates that the primary component lies in the main - sequence band , while the secondary is slightly beyond the terminal - age main sequence and its radius and luminosity are about two times oversized and more than four times overluminous compared with dwarf stars of the same mass . in these diagrams , the locations of the two components conform to the general pattern of classical algols . the mass and temperature of the secondary star correspond to a spectral type of approximately k2 to k3 . from the current observations , six new times of minimum light and their errors were determined using the method of kwee & van woerden ( 1956 ) and with the weighted mean for the values in each filter . these are listed in table 5 , wherein 73 additional eclipses were obtained using the data from the wasp ( wide angle search for planets ) public archive ( butters et al . 2010 ) . for ephemeris computations , we have collected a total of 1050 timings ( 949 visual , 20 photographic , 16 photoelectric and 65 ccd ) from the literature ( kreiner et al . 2001 ; baldwin & samolyk 2002 , 2004 ; locher 2002a , 2002b ; baki et al . 2003 ; szkely 2003 ; diethelm 2003 , 2004 ; nelson 2005 ; cook et al . 2005 ; kim et al . 2006 ; nagai 2004 , 2006 ; hbscher , et al . 2006 , 2009 ; senavci et al . 2007 ; samolyk 2008a , 2008b ; brt et al . 2008 ; liakos & niarchos 2009 ; doru et al . 2009 , 2011 ; dvorak 2010 ; erkan et al . 2010 ; hbscher & monninger 2011 ) to add to the current measurements . most earlier timings were extracted from the database published by kreiner et al . the secondary minima are much shallower than the primary ones and the @xmath47@xmath48 residuals from the two eclipse types are in phase with each other . thus , we did not use all secondary eclipses in the subsequent analysis . because many timings have been published without errors , the following standard deviations were assigned to the timing residuals based on an observational technique : @xmath200.0036 d for visual , @xmath200.0020 d for photographic , and @xmath200.0013 d for photoelectric and ccd minima . relative weights were then scaled from the inverse squares of these values ( lee et al . 2007 ) . previous researchers ( szkely 2003 ; soydugan 2008 ) have suggested that the period variations of sz her can be represented using an lite caused by the presence of a third body in the system . first of all , we fitted the minimum epochs to the single lite ephemeris as follows : @xmath49 where @xmath50 is the lite due to a hypothetical distant companion to the eclipsing close pair ( irwin 1952 , 1959 ) and includes five parameters ( @xmath51 , @xmath52 , @xmath53 , @xmath54 , @xmath34 ) . the levenberg@xmath25marquardt ( lm ) technique ( press et al . 1992 ) was used to evaluate the seven parameters of the ephemeris . the results are summarized in column ( 2 ) of table 6 , together with their related quantities . as displayed in figure 3 , the single lite ephemeris failed to provide a satisfactory result . because the timing residuals in the lower panel of the figure indicate the existence of further effects , some combination of long- and short - term period variations appears possible . using the period04 program ( lenz & breger 2005 ) , which can extract individual frequencies from the multi - periodic content of an astronomical time series containing gaps , we looked to see if the @xmath47@xmath55 residuals from the linear terms represent multi - periodic variations . as can be seen from figure 4 , two frequencies of @xmath56=0.0000353 cycle d@xmath57 and @xmath58=0.0000706 cycle d@xmath57 were detected corresponding to 77.6 yr and 38.8 yr , respectively . therefore , after considering the two periods and testing several other forms , such as a quadratic _ plus _ single - lite ephemeris , a two - lite ephemeris and a quadratic _ plus _ two - lite ephemeris , we found that the times of minimum light are best fitted using the following ephemeris : @xmath59 where @xmath50 and @xmath60 are the light times due to a third and fourth body , respectively . the lm method was applied again in order to simultaneously locate the lite parameters of the third and newly assumed fourth bodies . the calculations converged quickly to yield the entries listed in columns ( 3)(4 ) of table 6 . the @xmath47@xmath61 residuals from the linear terms are plotted in the top panel of figure 5 . the second and third panels display the @xmath62 and @xmath63 orbits , respectively , and the bottom panel represents the residuals from the full ephemeris . the long - term orbit ( @xmath50 ) are currently preliminary because about 1.3 cycles of the 86-yr period have been covered , while the short - term orbit ( @xmath60 ) has a relatively high determinacy because the observations have already covered about 2.6 cycles . on the other hand , it is alternatively possible that the @xmath47@xmath48 diagrams may be described by abrupt period changes instead of continuous period variations . as displayed in figures 3 and 5 , the orbital period of sz her seemed to experience period jumps around years 1920 , 1960 , 1978 , 1987 , 2002 , and 2008 . they could possibly have been produced either by episodic mass transfer events or by impulsive mass ejections from one ( or both ) component(s ) . assuming constant periods before and after the years , we applied linear least - squares fits separately to the seven sections . the results are plotted as the thick lines in the top panel of figure 5 . a combination of the straight lines resulted in a larger @xmath64 = 1.031 than the two - lite ephemeris . as seen in the figure , the sudden period changes seem to have alternated cyclically in algebraic sign . then , the alternating sign changes would require some preferred reciprocating mechanism . however , in view of the semi - detached nature of the binary , no pair of unstable locales is obvious and then it is difficult for the jumps to produce perfectly smooth and tilted periodic components in the @xmath47@xmath48 residuals . these might be an indication of sinusoidal variations rather than abrupt period changes . for the eclipsing binary sz her we obtained six times of minimum light from the eclipse light curves using the 61cm - reflector at soao . this data set was further augmented with additional photometric data provided by the wasp public archive . these two data sets were added to previous measurements of minimum light from earlier epochs . we then carried out a detailed analysis of the resulting @xmath47@xmath48 diagram by fitting keplerian lite models to the data set . the best fit to the data suggests the existence of two companions with the orbital parameters listed in table 6 . the period ratio of @xmath65/@[email protected] would suggest that the two companions are in a 2:1 mean motion orbital resonance . we think that a long - term gravitational interaction between the two objects would result in capture into the 2:1 resonant configuration ( cf . kley et al . 2004 ) . to our knowledge , this would be the fourth case when two circumbinary companions would be in or close to any kind of resonance . lee et al . ( 2009 ) discovered two substellar companions revolving around the sdb+m eclipsing system hw vir in nearly 5:3 or 2:1 resonant captures and beuermann et al . ( 2010 ) announced the existence of two planets in a 2:1 ( or possibly 5:2 ) mean motion orbiting the post - common envelope binary nn ser . another interesting case is the w uma - type binary star wz cep : jeong & kim ( 2011 ) suggested that two periodicities of 41.3 yr and 11.8 yr exist in the @xmath47@xmath48 residuals and indicate lites due to two circumbinary companions . the periods are exactly in a commensurable 7:2 relation between their mean motions . if the two circumbinary objects are on the main sequence and in the orbital plane ( @[email protected] ) of the eclipsing pair sz her , the masses of the third and fourth bodies become @xmath10=0.22 m@xmath11 and @xmath12=0.19 m@xmath11 , respectively . following the empirical relations presented by southworth ( 2009 ) , the radii and temperatures are calculated to be @xmath67=0.23 r@xmath11 and @xmath68=3018 k , and @xmath69=0.20 r@xmath11 and @xmath70=3008 k for the third and fourth bodies , respectively . these values correspond to a spectral type of about m67 for both bodies and contribute only 0.1% to the total bolometric luminosity of the supposed quadruple system . the semi - amplitudes of the systemic radial velocity variation of the eclipsing pair due to the additional objects are approximately 1 km s@xmath57 . the two limits indicate that it will be difficult to detect these companions orbiting the eclipsing binary independently from spectroscopic data . this difficulty is further substantiated due to the large orbital periods suggested by the derived lite models . however , the semi - major axes of the third and fourth companions relative to the binary center of mass are about 26.6 au and 16.5 au , respectively , corresponding to the angular sizes of 0.09 arcsec and 0.05 arcsec . the ( @xmath71 ) color index for such m - type stars is about @xmath287.3 mag , so the objects can be as bright as @xmath72 13 mag . hence , they might be detected by careful observations with infrared photometry and direct speckle imaging interferometry . in classical algols , another possible mechanism for the period modulations is a magnetic activity cycle for systems with a secondary spectral type later than f5 ( hall 1989 ; applegate 1992 ) . according to this mechanism , the variable rotational oblateness of a magnetically active star produces a change in its gravitational quadratic moment , hence forcing a change in the orbital period . with the periods and amplitudes for the two - lite listed in table 6 , the model parameters were calculated for the secondary components from the applegate formulae . the parameters are listed in table 7 , where the rms luminosity changes ( @xmath73 ) converted to magnitude scale were obtained using equation ( 4 ) in the paper of kim et al . the variations of the gravitational quadrupole moment ( @xmath74 ) are two orders of magnitude smaller than the typical values of @xmath75 for close binaries ( lanza & rodono 1999 ) . recently , lanza ( 2006 ) noted that the applegate mechanism is not sufficiently adequate to explain the period modulation of close binaries with a late - type secondary . these suggest that this kind of mechanism can not explain the observed period variations of sz her . because sz her is in a semi - detached configuration with the less massive secondary component filling its inner roche lobe , from both theoretical and intuitive viewpoints , a period increase could be produced through mass transfer from the secondary to the primary star . this implies that a long - term secular variation may be hidden in the @xmath47@xmath48 data set and the algol system may be in a weak phase of mass transfer . as listed in columns ( 5)(6 ) of table 6 , fitting the eclipse timings to a quadratic _ plus _ two - lite ephemeris indicates that the quadratic term ( @xmath76 ) represents a continuous period increase with a rate of d@xmath77/d@xmath78 = @xmath282.5@xmath7910@xmath80 d yr@xmath57 . from these fits , it was found that this contribution is not significant and a secular term does not adequately describe the timing data , showing a larger @xmath64 value . furthermore , this value corresponds to a mass transfer rate of 1.4@xmath7910@xmath80 m@xmath11 yr@xmath57 , which is found to be the smallest rate amongst the classical semi - detached algol - type systems . no difference in the final fitted parameters is observed when comparing the two - lite ( @xmath64 = 1.013 ) and quadratic _ plus _ two - lite ( @xmath64 = 1.014 ) timing ephemeris . we are therefore left with the two - lite ephemeris as a candidate that best explains the compiled timing data set of sz her with a possible mass transfer being negligible in this description . if the existence of the third and fourth components in sz her is true , they may have played an important role in the formation and evolution of the semi - detached eclipsing system , which may ultimately evolve into a contact configuration by re - distributing most of its angular momentum to the outer circumbinary companions . when more systematic and continuous observations ( e.g. , eclipse timings and spectroscopy ) are undertaken , all of this is understood better than now and the absolute dimensions and evolutionary status of this system will be advanced greatly . the authors thank professor chun - hwey kim for his help using the @xmath47@xmath48 database of eclipsing binaries and the staff of the sobaeksan optical astronomy observatory for assistance with our observations . we appreciate the careful reading and valuable comments of the anonymous referee and dr . tobias c. hinse . this research has made use of the simbad database maintained at cds , strasbourg , france . we have used data from the wasp public archive in this research . the wasp consortium comprises of the university of cambridge , keele university , university of leicester , the open university , the queen s university belfast , st . andrews university and the isaac newton group . funding 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upper panel , constructed with the linear terms of equation ( 1 ) , the continuous curve represents the lite orbit . the residuals from this lite ephemeris are plotted in the lower panel where an additional short - term oscillation appears to exist . cc , pe , pg , and vi denote ccd , photoelectric , photographic , and visual minima , respectively . ] @xmath55 residuals . as a result of the successive pre - whitening procedures , two frequencies of @xmath56=0.0000353 cycle d@xmath57 and @xmath58=0.0000706 cycle d@xmath57 are detected and these become periods of 77.6 and 38.8 yr . ] @xmath48 diagram of sz her with respect to equation ( 2 ) . in the top panel , the two - lite ephemeris is drawn as the continuous curve and the straight lines represent the linear fits to the seven sections separated by period jumps . the second and third panels display the long- and short - term lite orbits , respectively . the bottom panel shows the residuals from the complete ephemeris , which demonstrates much better performance than that in figure 3 . ] lccccc sz herculis & 2610 - 0129 & 17@xmath8139@xmath8236@xmath83 & + 32@xmath8456@xmath8546@xmath867 & @xmath2810.47 & @xmath280.38 + comparison & 2610 - 1116 & 17@xmath8139@xmath8200@xmath87 & + 32@xmath8454@xmath8536@xmath860 & @xmath2811.19 & @xmath281.28 + check & 2610 - 0821 & 17@xmath8139@xmath8203@xmath88 & + 32@xmath8458@xmath8556@xmath867 & & + crcrcrcr 2,454,525.24638 & -0.389 & 2,454,525.24769 & 0.183 & 2,454,525.24881 & 0.519 & 2,454,525.25950 & 0.756 + 2,454,525.25125 & -0.428 & 2,454,525.25268 & 0.152 & 2,454,525.25380 & 0.504 & 2,454,525.26414 & 0.759 + 2,454,525.25615 & -0.456 & 2,454,525.25753 & 0.128 & 2,454,525.25859 & 0.478 & 2,454,525.26878 & 0.754 + 2,454,525.26079 & -0.469 & 2,454,525.26217 & 0.117 & 2,454,525.26323 & 0.473 & 2,454,525.27343 & 0.742 + 2,454,525.26543 & -0.469 & 2,454,525.26681 & 0.120 & 2,454,525.26787 & 0.457 & 2,454,525.27795 & 0.743 + 2,454,525.27007 & -0.468 & 2,454,525.27145 & 0.117 & 2,454,525.27251 & 0.457 & 2,454,525.28251 & 0.748 + 2,454,525.27472 & -0.479 & 2,454,525.27606 & 0.101 & 2,454,525.27705 & 0.445 & 2,454,525.28704 & 0.753 + 2,454,525.27926 & -0.488 & 2,454,525.28061 & 0.111 & 2,454,525.28160 & 0.449 & 2,454,525.29158 & 0.741 + 2,454,525.28381 & -0.486 & 2,454,525.28516 & 0.103 & 2,454,525.28615 & 0.456 & 2,454,525.29601 & 0.751 + 2,454,525.28835 & -0.498 & 2,454,525.28970 & 0.100 & 2,454,525.29069 & 0.448 & 2,454,525.30726 & 0.741 + lcc @xmath89 ( hjd ) & + @xmath77 ( d ) & + @xmath0 & + @xmath1 ( @xmath2 ) & + @xmath34 ( k ) & 7,262@xmath2078 & 4,881@xmath2033 + @xmath35 & [email protected] & 2.8207 + @xmath39 & 1.0 & [email protected] + @xmath38 & 1.0 & [email protected] + @xmath90 & 0.468 & 0.534 + @xmath91 & [email protected] & [email protected] + @xmath92 & [email protected] & [email protected] + @xmath93 & [email protected] & [email protected] + @xmath94 & [email protected] & [email protected] + @xmath95 & [email protected] & 0.0832 + @xmath96 & [email protected] & 0.1262 + @xmath97 & [email protected] & 0.1684 + @xmath98 & [email protected] & 0.2139 + @xmath99 ( pole ) & [email protected] & [email protected] + @xmath99 ( point ) & [email protected] & [email protected] + @xmath99 ( side ) & [email protected] & [email protected] + @xmath99 ( back ) & [email protected] & [email protected] + @xmath99 ( volume)@xmath100 & 0.3180 & 0.3159 lcc @xmath101/m@xmath11 & 1.58 & 0.75 + @xmath23/r@xmath11 & 1.55 & 1.54 + @xmath102 @xmath38 ( cgs ) & 4.26 & 3.94 + @xmath102 @xmath103/@xmath104 & @xmath250.37 & @xmath250.69 + @xmath34 ( k ) & 7,262 & 4,881 + @xmath41/l@xmath11 & 5.98 & 1.20 + @xmath42 ( mag ) & @xmath282.79 & @xmath284.53 + bc ( mag ) & @xmath280.03 & @xmath250.36 + @xmath105 ( mag ) & @xmath282.76 & @xmath284.89 + @xmath106 ( mag ) & + distance ( pc ) & + llccllc 3,128.67869 & @xmath200.00006 & i & & 3,237.48623 & @xmath200.00014 & i + 3,137.67757 & @xmath200.00007 & i & & 3,242.39511 & @xmath200.00007 & i + 3,142.58640 & @xmath200.00004 & i & & 3,246.48534 & @xmath200.00015 & i + 3,144.63307 & @xmath200.00040 & ii & & 3,253.43933 & @xmath200.00048 & ii + 3,151.58572 & @xmath200.00006 & i & & 3,262.43879 & @xmath200.00066 & ii + 3,153.63084 & @xmath200.00022 & ii & & 3,278.39164 & @xmath200.00046 & i + 3,155.67585 & @xmath200.00011 & i & & 3,855.55946 & @xmath200.00061 & ii + 3,156.49399 & @xmath200.00006 & i & & 3,882.55882 & @xmath200.00027 & ii + 3,158.54003 & @xmath200.00035 & ii & & 3,907.51075 & @xmath200.00005 & i + 3,160.58427 & @xmath200.00015 & i & & 3,921.41866 & @xmath200.00008 & i + 3,162.62941 & @xmath200.00036 & ii & & 3,923.46831 & @xmath200.00040 & ii + 3,164.67494 & @xmath200.00003 & i & & 3,943.50694 & @xmath200.00004 & i + 3,165.49276 & @xmath200.00006 & i & & 3,948.41539 & @xmath200.00016 & i + 3,167.53872 & @xmath200.00022 & ii & & 4,298.56066 & @xmath200.00014 & i + 3,169.58367 & @xmath200.00005 & i & & 4,303.46897 & @xmath200.00007 & i + 3,171.63073 & @xmath200.00033 & ii & & 4,307.55961 & @xmath200.00022 & i + 3,173.67431 & @xmath200.00002 & i & & 4,312.46786 & @xmath200.00018 & i + 3,174.49230 & @xmath200.00005 & i & & 4,328.42428 & @xmath200.00054 & i + 3,176.53844 & @xmath200.00051 & ii & & 4,330.46611 & @xmath200.00014 & i + 3,178.58273 & @xmath200.00004 & i & & 4,581.21434@xmath100 & @xmath200.00037 & ii + 3,180.62827 & @xmath200.00050 & ii & & 4,581.62260 & @xmath200.00012 & i + 3,183.08448@xmath100 & @xmath200.00063 & ii & & 4,588.16795@xmath100 & @xmath200.00006 & i + 3,183.49127 & @xmath200.00003 & i & & 4,592.25831@xmath100 & @xmath200.00006 & i + 3,185.53634 & @xmath200.00043 & ii & & 4,604.12114@xmath100 & @xmath200.00042 & ii + 3,190.44593 & @xmath200.00043 & ii & & 4,631.52689 & @xmath200.00004 & i + 3,192.49023 & @xmath200.00007 & i & & 4,638.48360 & @xmath200.00050 & ii + 3,194.53696 & @xmath200.00043 & ii & & 4,640.52602 & @xmath200.00009 & i + 3,196.58119 & @xmath200.00005 & i & & 4,644.61676 & @xmath200.00007 & i + 3,199.44569 & @xmath200.00031 & ii & & 4,645.43473 & @xmath200.00003 & i + 3,201.48976 & @xmath200.00004 & i & & 4,647.48195 & @xmath200.00048 & ii + 3,203.53628 & @xmath200.00047 & ii & & 4,651.57417 & @xmath200.00057 & ii + 3,205.58026 & @xmath200.00005 & i & & 4,663.43290 & @xmath200.00010 & i + 3,208.44389 & @xmath200.00057 & ii & & 4,669.57011 & @xmath200.00097 & ii + 3,219.48818 & @xmath200.00008 & i & & 4,672.43205 & @xmath200.00006 & i + 3,224.39681 & @xmath200.00011 & i & & 4,674.48004 & @xmath200.00052 & ii + 3,226.44262 & @xmath200.00034 & ii & & 4,681.43086 & @xmath200.00005 & i + 3,228.48723 & @xmath200.00007 & i & & 4,683.47803 & @xmath200.00027 & ii + 3,230.53328 & @xmath200.00062 & ii & & 4,685.52134 & @xmath200.00008 & i + 3,233.39568 & @xmath200.00031 & i & & 5,685.23465@xmath100 & @xmath200.00006 & i + 3,235.44175 & @xmath200.00042 & ii & & & & + lcccccccc @xmath89 & 2,434,987.3975(15 ) & & & & & hjd + @xmath77 & 0.818096071(92 ) & & & & & d + @xmath107 & 3.23(66 ) & & 2.31(18 ) & 1.24(20 ) & & 2.30(17 ) & 1.24(19 ) & au + @xmath53 & 187.3(5.7 ) & & 88.6(7.5 ) & 285(10 ) & & 88.4(7.2 ) & 285(9 ) & deg + @xmath52 & 0.75(10 ) & & 0.718(90 ) & 0.48(17 ) & & 0.720(87 ) & 0.48(16 ) & + @xmath54 & 0.00809(78 ) & & 0.01148(13 ) & 0.02320(17 ) & & 0.01148(12 ) & 0.02319(17 ) & deg d@xmath57 + @xmath34 & 2,431,348(1271 ) & & 2,422,631(312 ) & 2,406,158(320 ) & & 2,422,629(299 ) & 2,406,141(320 ) & hjd + @xmath108 & 122(12 ) & & 85.8(1.0 ) & 42.5(1.1 ) & & 85.8(0.9 ) & 42.5(0.3 ) & yr + @xmath109 & 0.0124(26 ) & & 0.0133(10 ) & 0.0071(11 ) & & 0.0133(10 ) & 0.0071(11 ) & d + @xmath110 & 0.00226(51 ) & & 0.00167(13 ) & 0.00106(17 ) & & 0.00166(12 ) & 0.00106(16 ) & m@xmath11 + @xmath111 & 0.247(29 ) & & 0.222(9 ) & 0.189(15 ) & & 0.221(8 ) & 0.189(14 ) & m@xmath11 + @xmath76 & & & & & & & d + @xmath112/@xmath113 & & & & & & & d yr@xmath57 + reduced @xmath64 & 1.663 & & & & & + lccc @xmath114 & 0.1885 & 0.2031 & s + @xmath115 & @xmath116 & @xmath117 & + @xmath74 & @xmath118 & @xmath119 & g @xmath120 + @xmath121 & @xmath122 & @xmath123 & g @xmath124 s@xmath57 + @xmath125 & @xmath126 & @xmath126 & g @xmath124 + @xmath127 & @xmath128 & @xmath129 & s@xmath57 + @xmath130 & @xmath131 & @xmath132 & + @xmath133 & @xmath134 & @xmath135 & erg + @xmath136 & @xmath137 & @xmath138 & erg s@xmath57 + & 0.0100 & 0.0235 & l@xmath11 + & 0.0083 & 0.0195 & @xmath139 + @xmath73 & @xmath200.0015 & @xmath200.0035 & mag + @xmath21 & 2.9 & 4.2 & kg +	multiband ccd photometric observations of sz her were obtained between 2008 february and may . the light curve was completely covered and indicated a significant temperature difference between both components . the light - curve synthesis presented in this paper indicates that the eclipsing binary is a classical algol - type system with parameters of @xmath0=0.472 , @[email protected] , and @xmath3(@xmath4@xmath5)=2,381 k ; the primary component fills approximately 77% of its limiting lobe and is slightly larger than the lobe - filling secondary . more than 1,100 times of minimum light spanning more than one century were used to study an orbital behavior of the binary system . it was found that the orbital period of sz her has varied due to a combination of two periodic variations with cycle lengths of @xmath6=85.8 yr and @xmath7=42.5 yr and semi - amplitudes of @xmath8=0.013 d and @xmath9=0.007 d , respectively . the most reasonable explanation for them is a pair of light - time effects ( lites ) driven by the possible existence of two m - type companions with minimum masses of @xmath10=0.22 m@xmath11 and @xmath12=0.19 m@xmath11 , that are located close to the 2:1 mean motion resonance . if two additional bodies exist , then the overall dynamics of the multiple system may provide a significant clue to the formation and evolution of the eclipsing pair .
near the glass transition , molecular motion is greatly slowed , yet the motion of molecules is difficult to directly observe and the character of the molecular slowing is hard to determine @xcite . thus , colloidal suspensions of micrometer - sized spheres are a useful model system for investigating the nature of the glass transition , as they can be directly observed using optical microscopy @xcite . as the volume fraction of a colloidal suspension is increased , the motion of the particles becomes restricted due to the confining effect of the other particles : each particle is temporarily trapped in a `` cage '' formed by its neighbors @xcite . for liquid - like samples , these cages eventually deform and allow particles to diffuse through the sample @xcite . above a critical volume fraction ( @xmath1 for hard sphere colloids ) , the particles diffusion constant becomes zero and the macroscopic viscosity increases dramatically @xcite . a system at such state is considered a colloidal glass and @xmath0 is identified as the colloidal glass transition point . to gain insight into the nature of the colloidal glass transition , past work used optical microscopy to study the motion of individual particles as a function of volume fraction @xcite . these experiments discovered that particle motion in equilibrated samples is both spatially heterogeneous and temporally intermittent . however , only a few studies @xcite have focused on nonequilibrium , driven motion . in this letter , we study experimentally the motion of a magnetic bead pulled through a dense colloidal suspension . similar to previous work with magnetic probes , we are able to probe the local rheological properties of the suspension @xcite , although unlike those studies , our samples are heterogeneous on the scale of the magnetic probe . in our experiments , as the colloidal glass transition is approached the motion of the magnetic bead becomes complex . below a threshold force , the magnetic bead exhibits only localized caged motion . when the external force is above this threshold , the magnetic bead moves significantly faster , with a fluctuating velocity . closer to @xmath0 , the threshold force rises , and the velocity - force relationship becomes increasingly nonlinear . our measurements reveal dramatic changes in the drag force acting on the magnetic bead as the glass transition is approached . the colloidal suspensions are made of poly-(methylmethacrylate ) particles , sterically stabilized by a thin layer of poly-12-hydroxystearic acid @xcite . the particles have a radius @xmath2 @xmath3 m , a polydispersity of @xmath45% , and are dyed with rhodamine . the particles are slightly charged , shifting the phase transition boundaries from those of hard spheres . we observe the freezing transition at @xmath5 0.38 , the melting transition at @xmath6 0.42 , and the glass transition at @xmath7 0.58 . the glass transition is characterized by a vanishing diffusion constant for the particles motion , measured by confocal microscopy . the colloidal particles are suspended in a mixture of cyclohexylbromide/_cis_- and _ trans_- decalin which nearly matches both the density and the index of refraction of the colloidal particles . we add superparamagnetic beads with a radius of @xmath8 @xmath3 m ( m450 , coated with glycidyl ether reactive groups , dynal ) , at a volume fraction @xmath9 , so that the magnetic beads are well separated . we do not observe attraction or repulsion between the colloidal particles and the magnetic beads , in either dilute or concentrated samples . before beginning experiments , we mix the sample with an air bubble which breaks up any pre - existing crystalline regions , then we wait 20 minutes before taking data , to allow the air bubble to stop moving . the colloidal suspension is not observed to flow during the acquisition of the data . with a confocal microscope @xcite , we rapidly acquire images ( 1.88 image / s ) of area 80 @xmath3 m @xmath10 80 @xmath3 m , containing several hundred particles . the magnetic beads are not fluorescent and thus appear black on the background of dyed colloidal particles . to exert a force on these beads , a neodymium magnet is mounted on a micrometer held just above the sample . the force is calibrated by determining the velocity of the magnetic beads in glycerol for a given magnet position , and inferring the drag force from stokes law ( @xmath11 , where @xmath12 is the drag force , @xmath13 is the viscosity , and @xmath14 is the observed velocity ) @xcite . the imperfect reproducibility of the magnet position causes a 5@xmath15 uncertainty of the force . additionally , the variability in magnetic bead composition results in an uncertainty in the magnetic force on different beads , which we measure to be less than 10@xmath15 . also , the magnetic beads are not density matched , and their effective weight is 0.1 pn . this is nearly negligible compared to the applied horizontal forces in our experiments , and so in our data below we consider only the applied forces and the measured velocities in the @xmath16 direction . we observe very little vertical motion of the magnetic beads , typically less than 5% of the horizontal motion . this is less than our other measurement errors , discussed below . we study isolated magnetic beads at least 35 @xmath3 m and more typically @xmath1750 @xmath3 m from the sample chamber boundary and from other magnetic beads . our experiments are limited by crystallization as the average velocity of the magnetic bead decreases significantly in the crystallized regions . thus , we make sure that the data are collected well before crystallization appears . repeated measurements are reproducible before the onset of crystallization . we study the motion of the magnetic beads for a range of forces . within each image , we locate the center of the magnetic bead with an accuracy of at least 0.5 @xmath3 m @xcite . to show the behavior of the magnetic bead , we plot the average magnetic bead velocity against the applied force on a log - log plot in fig . [ fig1 ] for a range of volume fractions . the average velocity of the magnetic bead for a given @xmath12 is given by @xmath18 for the entire course of the measurement ; the error bars in fig . [ fig1 ] are from the standard deviation of repeated measurements of @xmath19 for the same magnetic bead at the same @xmath12 . for a dilute system ( @xmath20 , asterisks ) the data fall on a line with a slope of 1 , indicating that the velocity is linearly proportional to the applied force and stokes law applies ( fig . [ fig1 ] inset ) . due to the nonzero volume fraction , the effective viscosity is modified from that of the solvent ( @xmath21 mpa@xmath22s ) to the larger value of @xmath23 mpa@xmath22s , in reasonable agreement with past macroscopic measurements @xcite . at higher volume fractions the velocity - force relationship becomes more complex ( fig . [ fig1 ] ) . for data at large forces , the average velocity seems to grow as a power law with force , @xmath24 , with exponent @xmath25 between 1.5 and 3 . from left to right the curves increase in volume fraction , and the slope at the highest forces for each curve increases nearly monotonically , reflecting the approach of the glass transition . for all samples @xmath26 , we took data at extremely low forces , and found that the magnetic bead did not move more than our resolution limit within four hours ; instead , it exhibits random caged motion . this observation time is limited by the onset of crystallization of the sample ; in other words , the velocity is smaller than @xmath27 @xmath3 m/s . since the colloidal particles move due to brownian motion one would expect rearrangements to `` allow '' for the magnetic bead to move , even if this motion would be primarily random with a small bias from the external force . however , within our observation time scale we can not see this behavior , and conclude that the velocity of such a random walk is quite small . for @xmath28 and @xmath29 , for all forces checked , motion of the magnetic bead was observed . ( forces below 0.1 pn are unobtainable , as this is the effective weight of the magnetic bead . ) to determine how the velocity / force relationship changes as the volume fraction is increased , we fit the data to the following equation : @xmath30 where @xmath31 is a threshold force and @xmath25 is the exponent characterizing the growth of the velocity at large forces . @xmath32 is the velocity the magnetic bead would have if the applied force @xmath12 was equal to @xmath33 . this functional form was also used to fit data from a two - dimensional simulation similar to our experiments @xcite . the solid lines in fig . [ fig1 ] are the results of fitting the data with eq . ( 1 ) , with the exception of the @xmath20 data ( asterisks ) where stokes law applies . while the model seems a good fit to our data , our data are not strong enough to verify the model . however , the model provides a useful way to quantify the main features of the data . as the colloidal glass transition is approached the system becomes more viscous , and this is reflected in the changing fit parameters from eq . ( 1 ) shown in fig . [ fig2 ] . for low volume fractions , the three parameters @xmath32 , @xmath34 , and @xmath25 are all small , and stokes law may be the simplest explanation for the force / velocity data . the fit parameters change as the volume fraction @xmath35 . @xmath36 initially grows with increasing @xmath37 and then drops [ fig . [ fig2](a ) ] . the initial increase presumably reflects the change from the @xmath38 limit at low volume fractions . the decrease as the glass transition is approached may be due to the sharply increasing effective viscosity acting on the magnetic bead . @xmath32 is also sensitive to variations in the other fitting parameters , as for large @xmath12 , @xmath39 . the threshold force @xmath34 grows as @xmath37 increases , reaching a plateau value for large @xmath37 . we expect that at @xmath0 the threshold force will be finite , both from our data , and also as it seems more likely that @xmath34 diverges at @xmath40 , the random close - packing volume fraction @xcite . the value of the threshold force near @xmath0 , @xmath41 pn , is similar to that predicted by the work of schweizer and saltzman _ et al . they determined an effective free energy characterizing caged particle motion , and the derivative of this energy can be used to find an effective force to break out of a cage , resulting in a value of @xmath42 pn for @xmath43 @xcite . this prediction considers the caged particle to be the same size as the surrounding particles , and for our larger magnetic beads the stall force would likely be larger , closer to our experimental values . thus , our measured threshold forces @xmath34 may be related to the strength of colloidal cages ; previously only the sizes of these cages have been measured @xcite . the exponent @xmath25 changes dramatically as the glass transition is approached , as seen in fig . [ fig2](c ) , where it rises to almost 3 near @xmath0 . our results are in contrast to ref . @xcite ; while the model fits their data as well , they found @xmath44 for glassy systems . the difference may be due to the much different particle interactions : they studied unscreened point charges in a 2d system , qualitatively different from our colloidal particles . in our experiments , it is unclear if @xmath25 diverges at @xmath0 or continues to grow as @xmath45 is approached . experiments at @xmath46 are difficult to interpret due to the aging of the sample @xcite : for a given force the magnetic bead response depends on the age of the sample . the motion of the magnetic bead is not smooth , as is seen by a typical plot of its position as a function of time [ fig . [ fig3](a , b ) ] . while the largest `` jumps '' seen in fig . [ fig3](a ) are roughly the diameter of the surrounding colloidal particles ( @xmath47 @xmath3 m ) , at other times the magnetic bead takes much smaller steps . when the magnetic bead moves faster the fluctuations become less significant and the motion of the magnetic bead is smoother , shown in fig . [ fig3](b ) . we arbitrarily choose @xmath48 as a time scale and define the instantaneous velocity as @xmath49 / \delta t$ ] . this time scale is indicated with the scale bars shown in fig . [ fig3](a , b ) , and the results that follow are not sensitive to this choice ( and in particular do not change using @xmath50 rather than @xmath51 ) . the instantaneous velocity is shown in fig . [ fig3](c ) . the magnetic bead velocity has much larger fluctuations along the direction of motion [ @xmath52 , top trace in fig . [ fig3](c ) ] , while fluctuations in the transverse direction are smaller . the distributions of @xmath53 and @xmath54 , shown on fig . [ fig3](d ) , are gaussian ( solid fit lines ) . the fluctuations in @xmath16 and @xmath55 appear to be only weakly correlated . the velocity fluctuations reflect the fact that the magnetic bead is of similar size to the colloidal particles : its motion is sensitive to colloidal particle configurations . furthermore , dense colloidal suspensions are spatially heterogeneous systems @xcite and some regions may be `` glassier '' and thus harder to move through . as the glass transition is approached , the spatial heterogeneity increases @xcite , and one might expect that the velocity fluctuations would become more significant . however , this is not the case , as is seen in fig . [ fig4 ] which shows the standard deviation of the instantaneous velocity @xmath56 plotted against the average velocity @xmath19 . different symbols indicate different volume fractions , showing that it is not the volume fraction which determines the standard deviation but the average velocity : the fluctuations are larger when the average velocity is larger . denser suspensions have slower velocities for the same force ( as seen in fig . [ fig1 ] ) but also correspondingly smaller velocity fluctuations . in fig . [ fig4 ] , the data are more scattered at low velocities and for any given sample the data scatter on both sides of the fit line . for different choices of @xmath57 , we find that @xmath58 with the exponent @xmath59 . the smaller the @xmath57 the wider distribution of @xmath56 . however , the general dependence of @xmath56 on @xmath60 does not depend on @xmath57 . we find that @xmath61 and thus a similar trend holds for the transverse fluctuations . the changing character of the motion as the glass transition is approached can be understood by calculating the modified peclet number . this is the ratio of two time scales , @xmath62 characterizing the unforced motion of the colloidal particles , and @xmath63 characterizing the motion of the magnetic bead . the brownian time @xmath64 is the time it takes for colloidal particles to diffuse a distance equal to their radius @xmath50 and is based on their asymptotic diffusion constant @xmath65 . @xmath65 becomes small near the glass transition and becomes zero for a colloidal glass . the magnetic bead time scale @xmath66 is the time scale for the magnetic bead to move the distance @xmath50 and is based on the magnetic bead s average velocity @xmath19 . the modified peclet number is @xmath67 and is larger than 1 for all symbols shown in fig . [ fig1 ] @xcite . for the largest forces , @xmath68 . thus , the forced motion of the magnetic bead is much more significant than the brownian motion of the surrounding colloidal particles : the magnetic bead pushes these particles out of the way , plastically rearranging the sample . this results in a lowering of the effective viscosity acting on the magnetic bead at higher velocities . on the macroscopic scale , this may correspond to traditional rheological measurements at high strains . past work found for similar colloidal samples that the viscoelastic moduli decreased as the maximum strain increased @xcite . for extremely low forces , @xmath69 and the velocity should be linearly related to the force . however , the velocities predicted by linear response theory are well below what we observe ( and for low forces , below what we can measure ) . this is not surprising , as the lowest forces we can apply are fairly large ; a nondimensional way to characterize this is the ratio @xmath70 which is @xmath71 for @xmath12=0.1 pn . since @xmath72 in our experiments , the brownian motion should be unimportant , similar to a granular system . moreover , conjectures of the existence of a `` jamming transition '' speculate that the colloidal glass transition is similar to jamming in granular media @xcite , and so it is interesting to compare our results with studies performed in a granular media . _ @xcite immersed a cylinder in granular particles and measured the force exerted on the cylinder when it moved with constant speed relative to the particles . they found that the drag force on the cylinder is independent of the velocity , quite different from our behavior ( fig . [ fig1 ] ) . in our experiments , the solvent viscosity may be important , causing the velocity dependent drag force . in conclusion , we find that near the glass transition the motion of a microsphere through a colloidal suspension changes dramatically . a threshold force appears , below which the motion of the sphere is localized , and this threshold force increases as the glass transition is approached . the existence of the threshold force hints that the system may locally `` jam '' even when the colloidal suspension is globally still a liquid . above the threshold force the velocity is related to the force by a power law and the sphere moves with a fluctuating velocity , locally deforming the sample . at high forces , the velocity is related to the force by a power law . this can be inverted to obtain an effective drag force on the sphere , growing weakly with velocity , @xmath73 with @xmath25 growing from 1 far from the glass transition to nearly 3 close to the transition . these results indicate that the approach of the colloidal glass transition is signaled by a growing nonlinearity of the drag force . this may be a new way to characterize the glass transition and should serve as a useful test for some glass transition theories @xcite . we thank r. e. courtland , m. b. hastings , s. a. koehler , d. nelson , k. s. schweizer , t. squires , t. a. witten , and s. wu for helpful discussions . we thank a. schofield for providing our colloidal samples . this work was supported by nasa ( nag3 - 2284 ) .	we use confocal microscopy to study the motion of a magnetic bead in a dense colloidal suspension , near the colloidal glass transition volume fraction @xmath0 . for dense liquid - like samples near @xmath0 , below a threshold force the magnetic bead exhibits only localized caged motion . above this force , the bead is pulled with a fluctuating velocity . the relationship between force and velocity becomes increasingly nonlinear as @xmath0 is approached . the threshold force and nonlinear drag force vary strongly with the volume fraction , while the velocity fluctuations do not change near the transition .
transition from spatially uniform state to a self - organized or ordered structures is a universal feature in far from equilibrium systems and has been observed in many natural systems ( see overviews on this issue @xcite ) . this transition is triggered by a few number of modes often called turing modes that leads to the formation of spatially stationary structures that can be either periodic or localized in space . on the order hand , a transition to a time oscillation or a self - pulsing state through the andronov - hopf ( termed hopf in the following ) bifurcation has also been observed . these transitions are responsible for the symmetry breaking in space ( turing ) and in time ( hopf ) . the interaction between turing and hopf bifurcations concerns almost all fields of nonlinear science such as biology , chemistry , physics , fluid mechanics , and optics . when both instabilities are close one to another , the space - time dynamics of various spatially extended systems may be significantly impacted . in particular , it has been shown that in this regime either a bistable behavior between turing patterns and self - pulsing states or front waves between hopf and turing - type domains may occur @xcite . moreover , mixed - mode solutions resulting from the interplay between andronov - hopf and turing modes may dominate the spatio - temporal dynamics in optical frequency conversion systems @xcite . this interaction can also lead to a spatio - temporal chaos @xcite . in this paper , we consider a ring cavity filled with a non instantaneous and a nonlocal kerr medium , and driven by a coherent injected beam . we show that the threshold associated with turing and hopf bifurcations can occur arbitrarily close one to another leading to codimension - two point where both bifurcations coincide . in the monostable regime , we show that the turing pattern becomes unstable and leads to the formation of a turing - hopf mixed - mode . more importantly , we show that localized mixed - mode solutions can be stabilized when the system exhibits a coexistence between an homogeneous steady state and an extended mixed - mode solution . the slowly varying envelope of the electric field @xmath0 circulating inside the cavity is described by @xmath1 coupled with an additional equation for the material photoexcitation @xmath2 : @xmath3 with @xmath4 is the longitudinal coordinate , and the parameter @xmath5 is proportional to the ratio between the diffusion length and diffraction coefficients . this parameter describes the degree of nonlocality . indeed , if @xmath6 , the above propagation equations have a local nonlinear response . the strong nonlocal response is characterized by a large @xmath5 . the parameter @xmath7 is proportional to the ratio between the characteristics decay times associated with the electric field and the refractive index @xmath8 , respectively . at each round trip , the light inside the kerr media is coherently superimposed with the input beam . this can be described by the cavity boundary conditions @xmath9 and @xmath10 , with @xmath11 are linear phase shifts , @xmath12 is the ampliude of the injected field , and @xmath13 denotes the cavity length . the letters `` @xmath14 '' and `` @xmath15 '' indicate the transmission and the reflexion coefficients . we adopt the well know mean field approach proposed by lugiato and lefever @xcite a detailled calculations of the derivation of the mean field model can be found in the appendix of the paper @xcite . this approach is valid under the following approximations : the cavity possesses a high fresnel number and we assume that the cavity length is much shorter than the diffraction , diffusion and the nonlinearity spatial scales . a single longitudinal mode operation is also assumed . the mean field approach applied to our system leads to the adimensional coupled partial differential equations @xmath16 where @xmath17 is the frequency detuning between the injected light and the cavity resonance . @xmath12 is the injected field amplitude considered as being real without loss of generality . diffraction of the electric field in the transverse @xmath18 coordinate is modeled by @xmath19 . the density of material photoexcitation is denoted by @xmath8 that diffuses according to @xmath20 . in the limit of @xmath21 , the model has been derived for a kerr medium but in a single feedback mirror configuration @xcite . for a ring cavities , and in the particular limit where @xmath22 and @xmath23 , one obtains from eq . ( [ eq : dndt ] ) @xmath24 . by replacing @xmath8 by @xmath25 in eqs . ( [ eq : dedt ] ) , we recover the well known lugiato - lefever ( ll ) equation @xcite . the model eqs . ( [ eq : dedt ] ) and [ eq : dndt ] ) may be viewed as an extension of ll equation to include the non instantaneous response of the medium and the nonlocal effects . homogeneous steady state ( hss ) solutions of eqs . ( [ eq : dedt ] ) and ( [ eq : dndt ] ) are found by setting the time and space derivatives equal to zero : @xmath26 and @xmath27 $ ] with @xmath28 . the characteristic @xmath29 as a function of the input intensity @xmath30 is monstable when @xmath31 . the hss s undergo a bistable behavior when @xmath32 . the linear stability analysis of the hss with respect to a finite wavelength perturbation of the form @xmath33 leads to the third - order polynomial characteristic equation : @xmath34 , with @xmath35 , @xmath36 , @xmath37 and @xmath38 , with @xmath39 . the turing type of bifurcation occurs when @xmath40 and @xmath41 . the threshold associated with this instability and the critical wavelength at the turing bifurcation are obtained from the expression @xmath42 . this instability is characterized by an intrinsic wavelength which is determined by the dynamical parameters and not by the external geometrical constraints such as boundary conditions . a hopf bifurcation occurs if a pair of complex - conjugate roots has a vanishing real part and a nonzero imaginary part . in the limit @xmath43 and @xmath44 , only one hopf bifurcation point is possible . the associated critical intensity is explicitly given by @xmath45 . at this bifurcation point the real part vanishes at nonzero wavenumber @xmath46 . note that the hss s may undergo a homogeneous hopf bifurcation where the real part of the eigenvalues vanishes for @xmath47 . we are interested in the regime where turing and hopf instabilities are close one to another . for this purpose , we fix @xmath43 and @xmath48 , and we vary the coefficient @xmath5 and the input field intensity . an important and interesting feature is that the parameter @xmath5 controls the relative position of the thresholds associated with both turing and hopf instabilities . typical marginal stability curves are shown in fig . 1 . by increasing the value of the parameter @xmath5 , the first bifurcation is of turing type , and the hopf bifurcation appears as secondary instability as shown in fig . there exists a critical value of the parameter @xmath5 for which both bifurcations coincide , i.e. , @xmath49 as shown in fig . at this co - dimensional two point , one of the real roots of the characteristic equation vanishes @xmath40 with a non zero finite intrinsic wave length @xmath50 and two other roots are purely imaginary @xmath51 . note that the critical mode @xmath52 is also@xmath5-dependent . when further increasing @xmath5 , the first instability is the hopf bifurcation followed by the turing instability as shown in fig . , @xmath48 with @xmath53 and @xmath54 are hopf and turing instability thresholds , respectively . ( a ) @xmath55 , above the upper curve hss is hopf unstable . inside the lower curve , hss is turing unstable . ( b ) @xmath56 , note the co - dimension two instability where turing and hopf thresholds coincide . ( c ) @xmath57 , the situation is inverted with respect to ( a).,width=453 ] , @xmath48 , @xmath58 and @xmath55.,width=226 ] , @xmath48 , and @xmath55.,width=226 ] , @xmath59 , @xmath60 , and @xmath61.,width=302 ] , @xmath59 , @xmath62 , and @xmath61 . ( b ) bifurcation diagram showing the evolution of the intracavity field intensity as a function of the input field amplitude . stable ( unstable ) homogeneous steady states are indicated by solid ( dotted ) line . filled squares denote the maximum intensity of the stationary localized structures . empty squares indicate the maximum intensity associated with localized mixed - mode solutions . same parameters as in ( a ) with varying @xmath12.,width=377 ] in what follows we focus on numerical investigations of the model eqs . ( 1 ) and ( 2 ) by using an adaptive step - size bulirsch - stoer method @xcite . we choose parameters values such that the homogeneous steady state is destabilized first by turing instability . from this bifurcation point , a branch of stationary spatially periodic solutions appears supercitically with a well defined wavelength . however , when increasing further the injected field intensity a stable mixed - mode solution is spontaneously generated in the system . these solutions correspond to oscillations both in time and in space . a typical example of such a behavior is plotted in fig . 2 . to characterize the transition from stationary periodic patterns to a branch of mixed - mode turing - hopf structures , we plot in fig . 3 the maxima of both turing structures and mixed - mode solutions together with the homogeneous steady state . as we increase the amplitude of the injected field , the hss becomes unstable , and a spatially periodic structure emerges from the turing bifurcation point @xmath63 . these structures are stable in the parameter range @xmath64 . for @xmath65 , the turing structure becomes unstable and bifurcates to a stable mixed - mode solution . the extended solutions either turing structures or mixed - mode solutions are obtained in the supercritical regime . besides these extended structures there exists another type of solutions which are aperiodic but localized in space ( ls s ) . the latter are found in the subcritical regime associated with the turing instability . it is well known that the emergence of these solutions does not necessarily requires a bistable regime @xcite . the prerequisite condition for the generation of ls s is the coexistence between a single hss and the spatially periodic solutions . this solutions have been predicted theoretically @xcite and experimentally evidenced in an instantaneous and local optical kerr medium @xcite . notice that temporal localized solutions known also as temporal solitons have also been experimentally observed in a dispersive kerr cavity in @xcite . here we show that in the case of a non - instantaneous and a nonlocal kerr medium , modeled by the eqs . ( 1 ) and ( 2 ) , support localized structures . an example of such a behavior is shown in fig . 4 . we show in this figure only localized structures with one , two and three peaks . the number of localized peaks and their spatial distributions are determined by the initial conditions @xcite . there exists an infinite number of spatially localized solutions if the size of the system is infinite . when increasing the injected field intensity , localized structures exhibit a pulsing phenomenon , i.e. , time oscillations in a wide range of parameters leading to the formation of hopf - turing mixed - mode solutions . a typical example of such a behavior is shown in fig . the maxima of intensity associated with localized structures and localized mixed - mode solutions are plotted together with the homogeneous steady states in fig . when increasing the input intensity , a single peak stationary localized structure is formed in the range @xmath66 . for @xmath65 , stable localized mixed - mode solutions are generated . their maximum intensity are shown in fig . the width of these time dependent structures varies in the course of time . we have shown that extended mixed - mode solutions can be generated in the output of a driven ring cavity filled with a non - instantaneous and a nonlocal kerr medium . our investigations are focused on the regime where turing and hopf instabilities interact strongly . in the monostable regime , the mixed - mode solutions appear as a transition from spatially periodic structures . we have drawn the bifurcation diagram showing their stability domain . in the bistable regime , we have shown occurrence of stationary localized structures that exhibit multistability in a finite range of parameters . finally , we have established evidence of localized mixed - mode solutions that emerge from the branch of a single peak localized structures . we have constructed a bifurcation diagram associated with these localized structures . we are very grateful to the support from the `` centre national de la recherche scientifique ( cnrs ) '' , france . this research was partially supported by the interuniversity attraction poles program of the belgian science policy office , under grant iap p7 - 35 _ photonics@be_. the support of ministry of higher education and research as well as by the agence nationale de la recherche through the labex cempi project ( anr-11-labx-0007 ) is also acknowledge . m. t. received support from the fonds national de la recherche scientifique ( belgium ) . n. akhmediev and a. ankiewicz ( eds . ) , _ dissipative solitons : from optics to biology and medicine _ ( lecture notes in physics , volume 751 , 2008 ) . h. leblond and d. mihalache , `` models of few optical cycle solitons beyond the slowly varying envelope approximatio , '' phy . rep . * 523 * , 61 ( 2013 ) . m. tlidi , k. staliunas , k. panajotov , a.g . vladimiorv , and m. clerc , `` localized structures in dissipative media : from optics to plant ecology , '' phil . a * 372 * , 20140101 ( 2014 ) . m. tlidi , m. taki , m. le berre , e. reyssayre , a. tallet , and l. di menza , `` moving localized structures and spatial patterns in quadratic media with a saturable absorber , '' j. opt . b : quantum and semiclassical optics * 6 * , s421 ( 2004 ) . m. tlidi , p. mandel , and m. haelterman , `` spatiotemporal patterns and localized structures in nonlinear optics , '' phys . e * 56 * , 6524 ( 1997 ) . a. de wit , d. lima , g. dewel , and p. borckmans , `` spatiotemporal dynamics near a codimension - two point , '' phys . e * 54 * , 261 ( 1996 ) . firth , `` spatial instabilities in a kerr medium with single feedback mirror , '' j. mod . opt . * 37 * , 151 ( 1990 ) . l. a. lugiato and r. lefever , `` spatial dissipative structures in passive optical systems , '' phys . lett . * 58 * , 2209 ( 1987 ) . p. kockaert , p. tassin , g. van der sande , i. veretennicoff , and m. tlidi , `` negative diffraction pattern dynamics in nonlinear cavities with left - handed materials , '' phy . a * 48 * , 4605 ( 2006 ) . e. hairer , s.p . nrsett , and g. wanner , _ solving ordinary differential equations i : nonstiff problem _ ( springer - verlag , 1993 ) . m. tlidi , p. mandel , and r. lefever , `` localized structures and localized patterns in optical bistability , '' phys . 73 * , 640 ( 1994 ) . scorggie , w.j . firth , g.s . mcdonald , m. tlidi , r. lefever , and l.a . lugiato , `` pattern formation in a passive kerr cavity , '' chaos , solitons & fractals , * 4 * , 1323 ( 1994 ) . v. odent , m. taki , and e. louvergneaux , `` experimental evidence of dissipative spatial solitons in an optical passive kerr cavity , '' new j. phys . * 13 * , 113026 ( 2011 ) . f. lo , l. gelens , ph . emplit , m. haelterman , and s. coen , `` dynamics of one - dimensional kerr cavity solitons , '' opt . express * 21 * , 9180 ( 2013 ) .	we investigate the spatio - temporal dynamics of a ring cavity filled with a non - instantaneous kerr medium and driven by a coherent injected beam . we show the existence of a stable mixed - mode solution that can be either extended or localized in space . the mixed - mode solutions are obtained in a regime where turing instability ( often called modulational instability ) interacts with self - pulsing phenomenon ( andronov - hopf bifurcation ) . we numerically describe the transition from stationary inhomogeneous solutions to a branch of mixed - mode solutions . we characterize this transition by constructing the bifurcation diagram associated with these solutions . finally , we show stable localized mixed - mode solutions , which consist of time - periodic oscillations that are localized in space .
granular systems have been extensively studied due both to the theoretical challenges they present ( for a recent review see @xcite ) and to the applications of industrial importance that spring from the rich phenomena they exhibit ( see @xcite and references therein ) . these systems are characterized by an energy loss in collisions . this loss is at the base of many interesting phenomena , such as inelastic collapse@xcite , where the particles collide infinitely often in finite time , and clustering ( for a sample of theoretical , simulational , and experimental approaches see @xcite ) . different methods for keeping the system from collapsing have been devised , such as subjecting the particles to brownian forces@xcite , and forcing through the boundaries by putting the system in a box with one or more thermal - like walls ( see for example @xcite ) . this work focuses on the latter method . being one of the simplest types of forcing , several authors@xcite have studied a one - dimensional system in a box with one or two heated ( stochastic ) walls . of these , @xcite study cluster formation , although @xcite are not strictly one - dimensional . this article studies a quasielastic one - dimensional system not subject to gravity between two thermalizing walls . we focus on two control parameters : the total inelasticity parameter @xmath4 , where @xmath5 is the number of particles and @xmath6 is the restitution coefficient , and the externally imposed temperature gradient . the parameter @xmath7 has been shown to be relevant for the quasielastic system@xcite . by varying these parameters we determine the region in parameter space where clustering is fully inhibited , obtaining a fluidized state . we present a singular feature of the distribution function for the clustering regime , and then study how this feature is modified for the fluidized state . in @xcite the authors study a one - dimensional system of point - like particles between an elastic and a heated wall . they emphasize that a cluster inevitably forms away from the heated wall , regardless of how elastic the system is ( as long as it is not perfectly elastic ) . they also study the same system , but with both walls expelling the particles with a fixed velocity . in this case they find that the cluster forms away from the walls and roams slowly about the system , with two groups of fast particles connecting the cluster with the `` heated '' walls . in @xcite the same system is studied for different types of boundary conditions at the heated wall . the stochastic boundary condition studied has the form of a power of the velocity times the `` thermal '' condition ( the one that produces a maxwell - boltzmann distribution in the elastic case ) . the authors show that when the power that multiplies the thermal condition is positive the test - particle equation ( derived from the boltzmann equation ) has a steady - state solution . thus the thermal case does not have a steady state and develops a cluster away from the heated wall . the mechanism for the growth of the cluster is explained and verified numerically . in @xcite a similar system is studied : a long thin pipe of inelastic hard disks with heated walls ( at the same temperature ) at the ends of the pipe and periodic side walls . the pipe is thin enough for the particle order to be preserved . the probability distribution for the distance between the central particles is studied . this distribution gives a markedly denser system near the center than in the elastic case , although the limit to the elastic case is smooth , unlike the strictly one - dimensional case of @xcite . in @xcite the same author studies the velocity correlations that this system develops as inelasticity is increased , showing that a consistent description must take these correlations into account . in this paper we revisit the one - dimensional system of @xmath5 point - like particles interacting via collisions that conserve momentum but dissipate kinetic energy . to fix notation , the particle velocities after a collision are given by @xmath8 where @xmath9 is the velocity of particle @xmath10 before a collision , and @xmath11 , being @xmath6 the restitution coefficient . for the elastic case ( @xmath12 ) the particles simply exchange velocities . since the particles are point - like , the system is then indistinguishable from a system in which the particles do not interact . this one - dimensional system is interesting because dissipation is the first order correction to a free gas . besides , results for the one - dimensional system have been found to have unexpected relevance for higher - dimensional problems . for example , in two - dimensions the particles involved in inelastic collapse lie roughly on a line@xcite . also , the dissipation - induced temperature gradients calculated in @xcite for the one - dimensional case inspired the authors to look for dissipation - induced rayleigh - bnard - like convection for a two - dimensional system without an externally imposed temperature gradient@xcite . for a system with one thermal wall and open on the other side , under the influence of gravity , the quasielastic system may be kept fluidized@xcite : any cluster that starts to form is forced against the thermal wall , where it evaporates . in @xcite the test - particle equation@xcite which is the 1d boltzmann equation where the limit @xmath13 is taken , but keeping @xmath7 fixed is successfully applied , with close matching of theory and simulations even at the level of the distribution function . particles between two walls at the same temperature for @xmath14 . note the plateau after each fall . subfigure ( b ) shows the trajectory of the cluster for the same time interval . the walls are placed at @xmath15 and @xmath16 . the temperature at the walls is unity , and thus a unit of time measures how long it takes for a particle with the thermal speed to cross the system.,scaledwidth=45.0% ] . the distribution for particles reaching the wall ( @xmath17 ) exhibits a sharp peak . subfigure ( b ) shows the same distribution multiplied by @xmath18 to show that the peak behaves like @xmath19 . the microscopic velocity @xmath20 is measured in units of the thermal speed.,scaledwidth=45.0% ] the one - dimensional system under study is left to evolve between two thermal walls at temperatures @xmath0 and @xmath1 , with @xmath21 . we define the parameter @xmath22 to quantify how far from the symmetrical case is the system . under the same conditions , an elastic system has a perfectly bimodal velocity distribution with global homogeneous temperature equal to @xmath23 . for the sake of comparison , we simulate systems with @xmath24 . here @xmath25 represents a symmetrical setting , while @xmath26 represents an infinitely strong temperature gradient . as in @xcite , for @xmath25 a cluster unavoidably forms away from the thermal walls . after forming , the cluster performs an apparently random walk about the center , growing in size ( and therefore mass ) while the rest of the system grows more rarefied . with the decrease in density of the surrounding gas and the increased inertia of the cluster , an eventual collision with one of the walls is to be expected . when this happens part of the cluster evaporates , and what is left of it is expelled from the wall ( see fig . [ fg : cluster_t0t0_qn001_n1000 ] ) , thus restarting the growth process . thus not only is the system highly clumped , but also in a non - steady state . nevertheless , the gas that is far from the random - walk zone has a well - defined time average for the distribution function , as is seen in fig . [ fg : t0t0_distrib_pared_qn001 ] . a noteworthy feature of this distribution is that it exhibits apparently singular behavior for slow velocities . setting @xmath27 the symmetry of the system is broken . for @xmath28 the cluster performs a slightly asymmetric random walk , spending more time near the colder wall , and therefore colliding more often with it . thus the cluster can not grow as much as it did in the symmetric case before colliding with a wall . by increasing @xmath3 the cluster - wall collision frequency grows , even obtaining short `` windows '' in which the cluster completely evaporates . by further increasing @xmath3 these windows grow larger until a point is reached where no cluster forms . in this fashion a totally fluidized state is achieved , which may be tractable with the dissipative boltzmann equation . however , the distribution function obtained from the molecular dynamics ( md ) simulations exhibits a peculiar non - gaussian feature for slow velocities . this feature is a smoothed version of the apparent singularity of the symmetric case . to discern whether this feature is due to correlations or is present before they settle in , we compared newtonian molecular dynamics results with those obtained through direct - simulation monte carlo ( dsmc ) , which neglects correlations . the results agree very well , except when the system approaches the clustering regime . we simulate the system through event - driven molecular dynamics@xcite and through direct - simulation monte carlo@xcite . the direct - simulation monte carlo procedures use the null - collision technique@xcite where , overestimating the collision frequency ( using the maximum relative velocity within a cell ) , the number of collisions to be _ attempted _ is calculated through a poisson process . in the next step the collisions are attempted , choosing at random two particles within the cell , and making them collide with a probability proportional to their relative velocity . most of the molecular dynamics and dsmc simulations were done with @xmath29 particles . in the md simulations we detect clusters using a geometric criterion : we consider chains of particles that are nearer than a critical distance ( in our case @xmath30 and @xmath31 , to be certain that the conclusions are independent of the choice ) . the system length is one , and with a thousand particles the mean distance between neighbors for a homogeneous system is @xmath32 . thus we detect particles that are uncommonly near by three orders of magnitude . we discard chains of length three or less , since they may be random encounters . measuring the _ total _ length of the cluster we have found that on average it is of the order of @xmath31 ; thus the choice of @xmath31 as link - link distance is much larger than the true distance between them . as already stated , the boundary conditions are such that the ( homogeneous ) temperature of the corresponding elastic system ( equal to @xmath23 ) is one . as a function of @xmath7 for @xmath29 . @xmath24 throughout . the solid curve shows the lowest possible value @xmath3 can take without detecting clusters . the dashed curve shows the largest possible value @xmath3 can take with easy cluster detection . the small region between the curves represents a zone where clusters appear erratically.,scaledwidth=45.0% ] , @xmath33 , and @xmath34 ( @xmath35 ) . the solid line represents results from a molecular dynamics simulation , while the dots represent results from a monte carlo simulation . the density and temperature are related by @xmath36 and , since momentum is conserved and the system is stationary , the pressure is constant throughout the system . the density and temperature profiles are almost symmetrical because the normalized density is very close to unity.,scaledwidth=45.0% ] . the solid line represents results from a molecular dynamics simulation , while the dashed line represents results from a monte carlo simulation.,scaledwidth=45.0% ] figure [ fg : cluster_t0t0_qn001_n1000 ] shows the non - steady state of a granular system between two walls at the same temperature for @xmath14 . a cluster forms away from the walls , performing a random walk of varying amplitude . when the cluster reaches a wall , part of it evaporates , and the growth process begins anew . as is usual for the quasielastic case , we relabel the particles when they collide . this enables us to visualize this system as a group of barely interacting particles passing through each other . the picture for cluster evolution , as explained before , is the following : the cluster grows because the slowest particles , due to the asymmetry of the distribution function , drift towards the cluster@xcite . as it grows , the density of the gas surrounding it decreases , with the consequent saturation in growth . thus we have a `` brownian particle '' of increasing mass moving in an increasingly rarefied medium . this `` particle '' will be increasingly less affected by the surrounding medium , until it can no longer be kept away from the walls . the cluster moves several orders of magnitude slower than the thermal speed ( three orders of magnitude in fig . [ fg : cluster_t0t0_qn001_n1000 ] ) . upon reaching a wall , the front liners strike the wall and are expelled by it much faster than the other cluster members . these particles pass through the cluster , transferring momentum to it , as described in @xcite . thus these fast particles push the cluster away from the wall , where it can absorb particles again . the fast particles , however , no longer belong to the cluster . since the slowest particles in the gas are the ones that will be absorbed by the cluster@xcite , it is the number of slow particles in the gas that will determine the cluster s growth rate . after the cluster strikes a wall , the expelled particles will be fast particles , and they will not contribute to the growth of the cluster during the time it takes for the gas to cool down again : the only particles available for absorption are the ones that were available before the cluster - wall collision . this explains why the cluster keeps growing at approximately the same rate it did before the collision . after the gas has cooled down , the growth rate returns to its normal value . this is the end of the plateau seen in fig . [ fg : cluster_t0t0_qn001_n1000 ] after each cluster - wall collision . to quantify the evaporation process we proceed as in @xcite : as soon as the first particle belonging to the cluster reaches a wall , it is expelled with a speed much higher than the cluster velocity ; thus we may consider the ideal situation of a cluster of @xmath37 particles at rest being stricken by a fast particle with velocity @xmath38 ( in this case @xmath39 ) . after colliding with the first particle in the cluster , the new fast particle s speed will be @xmath40 . thus , after traversing the cluster , the fast particle s velocity will be @xmath41 . since momentum is conserved in collisions , the center of mass of the cluster will have acquired a speed of @xmath42 considering the case @xmath43 with fixed @xmath7 , as in @xcite , we may simplify this expression to @xmath44 by further considering the case @xmath45 we get @xmath46 . in this limit , if the cluster reaches the wall with velocity @xmath47 and is expelled from it with velocity @xmath48 , the number @xmath49 of particles evaporated will satisfy @xmath50 . for the situation shown in fig . [ fg : cluster_t0t0_qn001_n1000 ] , @xmath47 and @xmath48 are typically of the order of @xmath32 and @xmath51 , hence the number of particles evaporated will be of the order of @xmath52 . even if the system is in a non - steady state , the gas at the walls ( far from the random - walk zone ) has a well - defined time average for the distribution function . the distribution function at the left wall is shown in fig . [ fg : t0t0_distrib_pared_qn001 ] . there is an apparent singularity for slow velocities . the distribution is asymmetric as it should , since the particles leaving the wall ( @xmath53 ) follow a gaussian distribution . figure [ fg : t0t0_distrib_pared_qn001 ] also shows the distribution multiplied by @xmath18 . since the limit of @xmath54 for @xmath55 is finite and nonzero , we conclude that the distribution function exhibits a singularity that behaves like @xmath19 for slow velocities . as shown in @xcite , when @xmath25 ( the symmetric case ) the distribution shown in fig . [ fg : t0t0_distrib_pared_qn001 ] is not a solution of the steady - state test - particle equation : @xmath56 , \label{eq : steadystatetpe}\ ] ] where @xmath57 to establish this , let us study the behavior for small @xmath20 of a solution of this equation . assume that , for small @xmath20 , @xmath58 , with @xmath59 in order to have a finite density in the vicinity of @xmath60 . furthermore , assume that @xmath61 . inserting this behavior in eq . ( [ eq : steadystatetpe ] ) we obtain @xmath62 thus , in order to keep @xmath63 ( the amplitude of the singularity ) finite , we must have either @xmath64 or @xmath65 . integrating the distribution of fig . [ fg : t0t0_distrib_pared_qn001 ] we obtain @xmath66 . since @xmath67 , we must have @xmath68 . but this corresponds to a nonintegrable distribution , and therefore the distribution can not be steady . figure [ fg : clustering_threshold ] shows the regions in @xmath69-space where clustering is inhibited for @xmath29 . as is to be expected , as the inelasticity increases , a stronger temperature gradient is necessary to inhibit cluster formation . to discern whether the non - gaussian features of the velocity distribution function are derived from correlations in the system we compared results from md simulations ( full newtonian dynamics ) with results from dsmc simulations ( no velocity correlations assumed ) . figures [ fg : qn01ti05_n_t ] and [ fg : qn01ti05_q_dist ] show this comparison for a case far from the clustering threshold ( @xmath70 and @xmath33 ) . the temperature of the left and right walls are chosen so that the global temperature for the elastic case ( equal to @xmath23 ) is one . the curves match almost exactly . at the level of the distribution function , the results also match closely . the peculiar non - gaussian feature of the distribution function is clearly seen in fig . [ fg : qn01ti05_q_dist ] . there is some slight mismatch near the peak . for the fluidized case , since momentum is conserved and the system is stationary , the pressure is constant throughout the system . the number density and the granular temperature calculated here are related by @xmath36 ( @xmath71 in energy units ) . thus when the normalized density @xmath72 varies little throughout the system ( @xmath73 ) , the normalized temperature is @xmath74 thus obtaining the nearly symmetric profiles seen in figs . [ fg : qn01ti05_n_t ] and [ fg : qn01ti084_n_t ] . , @xmath75 , and @xmath76 ( @xmath77 ) . the solid line represents results from a molecular dynamics simulation , while the dots represent results from a monte carlo simulation.,scaledwidth=45.0% ] figures [ fg : qn01ti084_n_t ] , [ fg : qn05ti001_n_t ] , and [ fg : qn05ti001_q_dist ] compare the md and dsmc results for cases near cluster formation . the non - gaussian feature of the distribution function shows a systematic deviation for dsmc simulations : there is overpopulation for slow velocities . this is explained by considering that the dsmc method , like the boltzmann equation , neglects correlations . when the system approaches the clustering regime , increased dissipation induces correlations which tend to make the particles collide less@xcite . in dsmc these correlations are neglected , with the corresponding systematic overestimation in the collision frequency . this overestimation results in a lower temperature of the system about the density peak . , @xmath78 , and @xmath79 ( @xmath80 ) . the solid line represents results from a molecular dynamics simulation , while the dots represent results from a monte carlo simulation.,scaledwidth=45.0% ] , while subfigure ( b ) corresponds to the system of fig . [ fg : qn05ti001_n_t ] . the solid line represents results from a molecular dynamics ( md ) simulation , while the dashed line represents results from a monte carlo ( dsmc ) simulation . the md results have been rescaled so that the area under the md and dsmc curves is the same . there is a systematic overpopulation of slow particles in the distributions obtained from monte carlo simulations.,scaledwidth=45.0% ] we have shown that a system not subject to gravity between thermal walls unavoidably reaches a non - steady state when the walls are at the same temperature . a cluster forms in the bulk , slowly roaming about the system while absorbing particles . as it grows , the amplitude of the random walk increases , until at last the surrounding gas can not keep the cluster away from the walls . when the cluster reaches a wall , a part of it is ejected by the wall _ through the cluster _ ( relabeling the particles on collisions ) , effectively pushing the cluster away from the wall , and leaving it to grow again . most of the time the cluster is far from the walls . thus measuring the distribution function at a wall is measuring the distribution function of the gas that surrounds the cluster . this distribution function has a well - defined time average , and exhibits apparently singular behavior for slow particles , diverging like @xmath19 . imposing an external temperature gradient forces the cluster against the colder wall , inhibiting its growth . increasing the temperature difference leads to a system in which the cluster never forms : the system is completely fluidized . the distribution function of the gas exhibits peculiar non - gaussian features : a smooth version of the aforementioned singularity . therefore , any attempt at solving the boltzmann equation through moment methods must consider this feature in the initial _ ansatz _ , as is done for the problem of an infinitely strong shock wave in @xcite and @xcite . in fact , a solution for this problem was attempted using the four moment method of @xcite . as mentioned in @xcite , the fourth balance equation could not be freely chosen when the boundary conditions were symmetric : some choices gave undefined results . as is natural , by not including the non - gaussian feature in the _ ansatz _ for this calculation we obtained absurd results , such as higher temperature in the middle of the system than near the walls . we compared molecular dynamics with direct - simulation monte carlo . agreement between these two methods shows that the non - gaussian feature of the distribution function may be predicted by the dissipative boltzmann equation . as the system approaches cluster formation , correlations settle in . these correlations reduce the collision frequency among particles . dsmc neglects these correlations , and thus overestimates the number of collisions . this exaggerates the effects of dissipation , producing steeper profiles . we thank rodrigo soto , aldo frezzotti , and rosa ramrez for helpful discussions . this work has been partially funded by _ fundacin andes _ through a doctoral scholarship , _ fondecyt _ through grants 2990108 and 1000884 , and by _ fondap _ through grant 11980002 .	we study a 1d granular gas of point - like particles not subject to gravity between two walls at temperatures @xmath0 and @xmath1 . the system exhibits two distinct regimes , depending on the normalized temperature difference @xmath2 : one completely fluidized and one in which a cluster coexists with the fluidized gas . when @xmath3 is above a certain threshold , cluster formation is fully inhibited , obtaining a completely fluidized state . the mechanism that produces these two phases is explained . in the fluidized state the velocity distribution function exhibits peculiar non - gaussian features . for this state , comparison between integration of the boltzmann equation using the direct - simulation monte carlo method and results stemming from microscopic newtonian molecular dynamics gives good coincidence , establishing that the non - gaussian features observed do not arise from the onset of correlations .
majorana bound states ( mbs)@xcite are recently attracting increasing interest both theoretically and experimentally . these have been predicted to exist in artificial structures , such as nanowires with spin - orbit coupling ( soc ) in proximity to a superconductor,@xcite ferromagnetic atom chains on top of a superconductor,@xcite topological insulator / superconductor hybrid structures,@xcite and superconducting circuits.@xcite recently , possible signatures of mbs have been reported in nanowires,@xcite atom chains,@xcite and topological insulator / superconductor structures.@xcite the mbs attract considerable attention partly due to their hypothetical non - abelian anyonic statistics , which might allow the realization of topologically - protected quantum information manipulation.@xcite in parallel to the ongoing search of some unambiguous confirmation of mbs,@xcite there are also numerous theoretical studies on how to efficiently exploit these mbs . one promising application of mbs is to construct majorana qubits.@xcite it has been suggested that majorana qubits might be robust against local perturbations and are hence promising to store quantum information.@xcite ( note that majorana qubits are not totally protected from decoherence , as studied in , e.g. , refs . . ) furthermore , majorana qubits could be rotated by topologically - protected braiding operations.@xcite therefore , among various realizations of qubits,@xcite majorana qubits are considered to be promising candidates for building blocks of quantum information processors . the braiding operations alone are insufficient to realize a universal quantum gate based on a majorana qubit.@xcite for the implementation of arbitrary qubit rotations , other non - topological operations are required . several schemes of such non - topological operations assisted by , e.g. , phase gates,@xcite quantum dots,@xcite flux qubits,@xcite or microwave cavities,@xcite have been proposed in the literature . nanomechanical resonators@xcite could also be used to study non - topological operations of a majorana qubit . for example , quite recently , kovalev _ et al._@xcite have proposed to rotate a majorana qubit by a magnetic cantilever . indeed , nanomechanical resonators have been utilized to couple to a wide range of quantum systems , including electric circuits,@xcite optomechanical devices,@xcite atoms,@xcite cooper - pair boxes,@xcite spin qubits,@xcite or microwave cavities.@xcite with the assistance of nanomechanical resonators , it is possible to perform important applications such as quantum manipulations , quantum measurements , as well as efficient sensing . these applications exploit the advantages of nanomechanical resonators , e.g. , their large quality factors ( @xmath0-@xmath1 ) , high natural frequencies ( mhz - ghz ) , as well as the feasibility of reaching the quantum ground states by cooling methods.@xcite recently , nanomechanical resonators have also been exploited to measure or manipulate the mbs.@xcite nevertheless , the study of hybrid systems@xcite coupling nanomechanical resonators to majorana qubits is quite limited . this work aims to contribute to this field . in this paper , we propose another majorana qubit - nanomechanical resonator hybrid system in the framework of the spin - boson model,@xcite based on a semiconductor nanowire in proximity to an s - wave superconductor . we show that a strong coupling between a nanomechanical resonator and a majorana qubit can be achieved , allowing an efficient transfer of quantum information between these two quantum systems . further , with braiding operations , it should be possible to realize a universal quantum gate based on a majorana qubit . this paper is organized as follows . first , we describe the majorana qubit and its coupling to a nanomechanical resonator . afterwards , we numerically study the coupling strength and the resonance condition of the hybrid system . then , we solve the qubit - phonon dynamics and achieve a coherent control of the majorana qubit . finally , we summarize our results . as illustrated in fig . [ fig1 ] , we consider a semiconductor nanowire with a rashba soc of strength @xmath2 on the surface of an s - wave superconductor with a superconducting gap @xmath3 . three ferromagnetic gates , fm1 , fm2 and fm3 , are placed on top of and along the nanowire . among these gates , fm1 and fm3 are static while fm2 is free to harmonically oscillate along the nanowire ( with a mass @xmath4 and an oscillation frequency @xmath5 ) . the gates fm1 and fm3 are sufficiently long ( of the order of 1 - 10 @xmath6 m ) while fm2 in between is relatively short ( of the order of 100 nm ) . these ferromagnetic gates induce a local zeeman splitting in the nanowire . for simplicity , we take the zeeman splitting under the three gates to be identical , with a magnitude of @xmath7 . an electric voltage @xmath8 can be applied on the gates to modulate the rashba soc locally , e.g. , from @xmath2 to @xmath9 . in our study , we consider the case with @xmath10 , where @xmath6 is the chemical potential in the nanowire . therefore in the nanowire , the parts subject to the zeeman splitting ( under the three ferromagnetic gates ) are in the topological ( @xmath11 ) region . the remaining parts , without the zeeman splitting , are in the non - topological ( @xmath12 ) region.@xcite as a result , the nanowire has an @xmath12-@xmath11-@xmath12-@xmath11-@xmath12-@xmath11-@xmath12 domain structure where the three @xmath11 domains are under the gates . at the six boundaries between the @xmath12 and @xmath11 domains in the nanowire , mbs arise . as the two outer mbs are far apart , only the four inner ones , schematically labeled as @xmath13-@xmath14 in fig . [ fig1 ] , are coupled due to hybridization arising from their small separation@xcite and are hence relevant to our consideration . m ) and static , while fm2 is relatively short ( of the order of 100 nm ) and free to oscillate as a harmonic oscillator . the ferromagnetic gates induce a local zeeman splitting @xmath7 in the underlying nanowire , and can also be used to modulate the local rashba soc strength by applying an electric voltage @xmath8 . ( b ) wave amplitude @xmath15 of the four coupled mbs at a static state in an insb nanowire with the set - up shown in ( a ) . the red dashed and red solid curves respectively correspond to the wave amplitudes of the lowest two eigenstates ( close to the zero energy ) . these two states constitute the majorana qubit . the dotted curve with the scale on the right - hand side of the frame indicates the profile of the inhomogeneous zeeman splitting along the nanowire . in the calculation @xmath16 nm , @xmath17 nm , @xmath18 mev , and @xmath19 mev . the gate voltage @xmath8 is zero and the rashba soc is homogeneous along the nanowire , with a strength @xmath20 mev nm.,title="fig:",width=321,height=4 ] + m ) and static , while fm2 is relatively short ( of the order of 100 nm ) and free to oscillate as a harmonic oscillator . the ferromagnetic gates induce a local zeeman splitting @xmath7 in the underlying nanowire , and can also be used to modulate the local rashba soc strength by applying an electric voltage @xmath8 . ( b ) wave amplitude @xmath15 of the four coupled mbs at a static state in an insb nanowire with the set - up shown in ( a ) . the red dashed and red solid curves respectively correspond to the wave amplitudes of the lowest two eigenstates ( close to the zero energy ) . these two states constitute the majorana qubit . the dotted curve with the scale on the right - hand side of the frame indicates the profile of the inhomogeneous zeeman splitting along the nanowire . in the calculation @xmath16 nm , @xmath17 nm , @xmath18 mev , and @xmath19 mev . the gate voltage @xmath8 is zero and the rashba soc is homogeneous along the nanowire , with a strength @xmath20 mev nm.,title="fig:",width=321,height=5 ] [ fig1 ] to lowest order , the hybrid system constructed above can be described by the hamiltonian @xmath21 where the mutual coupling hamiltonian of the mbs@xcite @xmath22\gamma_1\gamma_2+ig_{t}(l_{23})\gamma_2\gamma_3+ig_{n}[l_{34}(t)]\gamma_3\gamma_4,\label{hm}\end{aligned}\ ] ] and the nanomechanical oscillator hamiltonian @xmath23 the coupling strengths @xmath24 depend on the domain lengths @xmath25 . due to the oscillation of the gate fm2 , @xmath26 and @xmath27 are time dependent . here , @xmath28 stands for the displacement of the gate fm2 from its balance position , which is much smaller than the static domain lengths @xmath29 . therefore , to first order in @xmath30 , one has @xmath31\gamma_1\gamma_2+ig_t(l_{23})\gamma_2\gamma_3\\&+i[g_n(l_{34}^0)-x_0(t)g^\prime_n(l_{34}^0)]\gamma_3\gamma_4.\end{aligned}\ ] ] the four mbs , satisfying @xmath32 , can be used to construct a majorana qubit as follows.@xcite at first we define two dirac fermion operators@xcite @xmath33 and @xmath34 . the hilbert space of @xmath35 can then be spanned by states @xmath36 , with the fermion occupation numbers @xmath37 and @xmath38 . due to the conservation of fermion parity , the states @xmath39 and @xmath40 form two decoupled ( odd and even ) sectors.@xcite we assume that there is no high - energy excitation ( e.g. , no cooper - pair breaking in the superconducting substrate ) and restrict our study to the odd sector with @xmath41 . for convenience , we define pseudo - spins @xmath42 and @xmath43 , and use them as the two logical states of the majorana qubit.@xcite in this pseudo - spin space , @xmath44 and @xmath45 . the nanomechanical oscillator is quantized in the fock space @xmath46 with @xmath47 is the annihilation operator of phonons . consequently , in the space @xmath48 , the hybrid system can be simply described by the spin - boson hamiltonian,@xcite @xmath49 with the constant omitted . here @xmath50 , @xmath51 , and @xmath52 $ ] , where @xmath53^{1/2}$ ] is the zero - point motion of the oscillator . in this section , we study the mbs and their mutual coupling . in the static state , the inhomogeneous nanowire can be described by a tight - binding model . using the bogoliubov - de gennes basis @xmath54 , where @xmath55 stands for the fermion operator of a spin-@xmath56 ( @xmath57 , @xmath58 ) electron on the @xmath59-th lattice site , the particle - hole hamiltonian reads@xcite @xmath60 , \label{hami}\end{aligned}\ ] ] where @xmath61 in the above hamiltonian , the pauli matrices @xmath62 act on the particle - hole space and @xmath63 act on the real spin space . the spin - diagonal hopping energy is @xmath64 , and the spin - off - diagonal hopping energy is @xmath65 . here @xmath66 and @xmath67 are the on - site zeeman splitting and rashba soc , respectively , @xmath68 is the effective electron mass , and @xmath69 is the lattice spacing in the discretized tight - binding model . in the @xmath11 ( @xmath12 ) domains @xmath70 ( @xmath71 ) and @xmath72 ( @xmath73 ) . when the gate voltage @xmath8 is zero , @xmath74 . here , to lowest order , we follow refs . and to investigate the coupling strength @xmath75 ( @xmath76 ) approximately in an isolated @xmath11-@xmath12-@xmath11 ( @xmath12-@xmath11-@xmath12 ) three - domain structure . in such a simplified model , the inner @xmath12 ( @xmath11 ) domain has a finite length , while the outer two @xmath11 ( @xmath12 ) domains are assumed to be infinitely long . by numerically diagonalizing this three - domain system , one can obtain the energy splitting of the two mbs localized at the two @xmath11/@xmath12 boundaries . this energy splitting is precisely caused by the coupling of the mbs . with @xmath75 and @xmath76 known numerically , the majorana qubit can be well described by @xmath77 and @xmath78 , and the qubit - phonon coupling @xmath79 can be obtained also from @xmath80 [ refer to eq . ( [ he ] ) ] . moreover , by exactly diagonalizing the hamiltonian of the genuine @xmath12-@xmath11-@xmath12-@xmath11-@xmath12-@xmath11-@xmath12 domain structure as shown in fig . [ fig1](a ) , one can obtain the hybrid four mbs under consideration . in this work , we consider an insb quantum wire@xcite with an effective electron mass @xmath81 , a rashba soc @xmath82 mev nm , and a large landau factor @xmath83 . we choose the superconducting gap @xmath19 mev , the local zeeman splitting @xmath18 mev , the chemical potential @xmath84 , and the lattice constant @xmath85 nm . the total lattice site number is chosen as 1000 for the numerical convergence . in fig . [ fig2 ] , we show the dependence of the majorana coupling strength @xmath75 ( @xmath76 ) on the length of the @xmath12 ( @xmath11 ) domain @xmath86 ( @xmath87 ) , as well as the derivative @xmath80 versus @xmath86 . further , as an example , in fig . [ fig1](b ) we present the wave amplitude @xmath15 of the four hybrid mbs , when @xmath16 nm and @xmath17 nm . in fig . [ fig1](b ) , the red dashed and red solid curves stand for the wave amplitudes of the lowest two eigenstates ( close to the zero energy ) in the static inhomogeneous nanowire . the state corresponding to the red solid ( dashed ) curve is mainly contributed by the @xmath88 and @xmath89 ( @xmath13 and @xmath14 ) mbs . here , these two states form the majorana qubit . ( @xmath76 ) versus @xmath86 ( @xmath87 ) , the length of the inner @xmath12 ( @xmath11 ) domain between the two outer @xmath11 ( @xmath12 ) domains . the derivative @xmath80 versus @xmath86 is also shown , with the scale on the right hand side of the frame . the necessary parameters for the calculation are specified in the main text.,title="fig:",width=321 ] we now look into the qubit - phonon coupling and the resonance condition . we assume that the nanomechanical oscillator fm2 has a mass @xmath90 kg and an oscillation frequency @xmath91 mhz . with these parameters , the zero - point motion of the oscillator is calculated to be @xmath92 pm . we consider the symmetric case with @xmath93 nm , and hence we have @xmath94 and @xmath95 mhz in eq . ( [ he ] ) . the longitudinal length @xmath96 of the fm2 gate is chosen as 400 nm , such that the rabi resonance condition @xmath97 can be easily satisfied , e.g. , by further subtly adjusting the gate voltage @xmath8 which controls the local rashba soc strength @xmath98 . in fig . [ fig3 ] , we present the variation of @xmath77 as well as @xmath79 versus @xmath98 . it is shown that when slightly adjusting @xmath8 , and hence @xmath99 , the resonance point @xmath97 can be reached while the qubit - phonon coupling @xmath79 remains almost invariant . this qubit - phonon coupling is relatively strong , in view of the long lifetime of the majorana qubit and the high quality factor of the nanomechanical oscillator . in principle , the qubit - phonon coupling can be stronger when the domain length @xmath100 ( as well as @xmath101 ) becomes smaller ( refer to fig . [ fig2 ] ) . however , if the two edge modes @xmath13 and @xmath88 ( as well as @xmath89 and @xmath14 ) are too close and hence their hybridization becomes quite strong , the model hamiltonian ( [ hm ] ) describing four distinguishable mbs might fail . and the qubit - phonon coupling @xmath79 ( the scale is on the right - hand side of the frame ) versus @xmath98 , the rashba soc strength in the topological ( @xmath11 ) domains modulated by the gate voltage @xmath8.,title="fig:",width=321 ] here we study the dynamics of the qubit - phonon hybrid system . to achieve this , we make use of the python - based qutip software package@xcite to solve the lindblad master equation , @xmath102+\frac{1}{2}\sum_{k}\big\{[l_k,\rho(t)l_k^\dagger]\\&+[l_k\rho(t),l_k^\dagger]\big\}.\label{master}\end{aligned}\ ] ] in this equation , @xmath103 is the density matrix of the qubit - phonon system , and @xmath104 are the lindblad operators accounting for the dissipation of the hybrid system due to its coupling to the environment . the relaxation of the majorana qubit is taken into account by @xmath105 , while the dissipation of the nanomechanical resonator is included by @xmath106 and @xmath107 . here @xmath108^{-1}$ ] is the thermal phonon number in equilibrium with the environmental temperature @xmath109 , @xmath110 is the quality factor of the nanomechanical oscillator , and @xmath111 is the usual relaxation time of the qubit . by solving the master equation , one can obtain the time evolution of the qubit and phonon occupations . in our model , the temperature @xmath109 is set as 10 mk and hence the thermal phonon number @xmath112 is as large as 258 . therefore , an additional cooling of the oscillator@xcite is required , e.g. , as also applied in a proposed nanomechanical resonator nitrogen - vacancy center hybrid system.@xcite we assume that after side - band cooling@xcite the phonons thermally occupy the lowest several quantum states with a small phonon number , e.g. , @xmath113 . the initial state of the majorana qubit is set as @xmath114 , implying that a single electron is splitted into the @xmath13 and @xmath14 majorana fermions . experimentally , this initial state might be realized when only the fm1 and fm3 gates are in proximity to the nanowire before inserting the middle fm2 gate . the relaxation time of the majorana qubit depends on the concrete set - up and environment . following refs . and , we typically set @xmath111 around 100 @xmath6s . in fig . [ fig4 ] , we plot the time evolution of the occupations of the qubit and phonons respectively , with different values of @xmath111 and @xmath110 . as indicated by the figure , quantum information can be effectively transferred back and forth between the majorana qubit and the nanomechanical resonator . during this process , the single electron in the nanowire alternatively occupies ( back and forth ) the pair of mbs : @xmath13 and @xmath14 , or @xmath88 and @xmath89 . inversely , this quantum information transfer can also modulate the motion of the oscillator , e.g. , the oscillation amplitude . in fact , as the nanomechanical resonator is near its quantum ground state , the oscillation amplitude @xmath115 , which might be observable , is almost linearly related to the phonon number . this is because @xmath116 . therefore , the dashed curves in fig . 4 , representing the time evolution of the phonon number , also supply information on the change of the oscillation amplitude of the resonator due to its coupling with the qubit . this phenomenon signifies the presence of a majorana qubit . certainly , for better performance of this hybrid system ( e.g. , with a higher fidelity ) , a higher quality factor of the resonator and a longer relaxation time of the qubit are preferred . s in ( a ) and 150 @xmath6s in ( b ) . the calculations for both ( a ) and ( b ) are performed with two different resonator quality factors : @xmath117 and @xmath118.,title="fig:",width=340 ] here we briefly compare our model to the one proposed by kovalev _ et al._,@xcite where a vibrating cantilever is utilized to rotate a majorana qubit . the effective hamiltonian in their model [ eq . ( 7 ) in ref . ] is in fact equivalent to the one in our manuscript [ eq . this is understandable as both are in the framework of the spin - boson model . note that for both cases there exists a static off - diagonal term [ for our case , that is the @xmath78 term in eq . ( 5 ) ] coupling the two levels of the qubit in the hamiltonian . to neglect this term , in order to simplify the theoretical analysis , some conditions have to be satisfied . specifically , in ref . 5 , a certain equilibrium angle ( @xmath119 there ) of the vibrating cantilever has to be established . in our opinion , exactly solving this angle and then adjusting the experimental setup correspondingly@xcite are challenging . however , in order to neglect the constant off - diagonal term in our case , the experimental setup must be mirror - symmetric about the middle point of the fm2 gate , i.e. , @xmath120 . therefore , we think that our model is more easily accessible by experiments and hence more advantageous . in conclusion , we have proposed a hybrid system composed of a majorana qubit and a mechanical resonator , implemented by a semiconductor nanowire in proximity to an s - wave superconductor . in this proposal , three ferromagnetic gates are placed on top of and along the nanowire ; the two outer gates are static and the inner one is free to oscillate harmonically as a mechanical resonator . these ferromagnetic gates induce a local zeeman splitting and give rise to four majorana bound states , constituting a majorana qubit in the nanowire . the dynamical hybridization of the majorana bound states , arising from the motion of the oscillating gate , results in a coherent coupling between the majorana qubit and the mechanical resonator . this hybrid system can be adjusted to be in resonance , e.g. , with the assistance of a gate voltage on the ferromagnetic gates , which controls the rashba soc locally in the nanowire . our study reveals that under resonance , a strong coupling between the qubit and the resonator can be achieved . consequently , quantum information can be effectively transferred from the majorana qubit to the oscillator and then back to the qubit . this quantum information transfer can manifest itself in modulating the motion of the oscillator , which may conversely signify the presence of the majorana qubit . the authors gratefully acknowledge x. hu , g. giavaras , l. wang and z. li for valuable discussions and comments . p.z . acknowledges the support of a jsps foreign postdoctoral fellowship under grant no . is partially supported by the riken ithes project , muri center for dynamic magneto - 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[ dynamics ] ) . johansson , p.d . nation , and f. nori , _ qutip : a python framework for the dynamics of open quantum systems _ , . commun . * 183 * , 1760 ( 2012 ) . johansson , p.d . nation , and f. nori , _ qutip 2 : a python framework for the dynamics of open quantum systems _ , . commun . * 184 * , 1234 ( 2013 ) .	we propose a hybrid system composed of a majorana qubit and a nanomechanical resonator , implemented by a spin - orbit - coupled superconducting nanowire , using a set of static and oscillating ferromagnetic gates . the ferromagnetic gates induce majorana bound states in the nanowire , which hybridize and constitute a majorana qubit . due to the oscillation of one of these gates , the majorana qubit can be coherently rotated . by tuning the gate voltage to modulate the local spin - orbit coupling , it is possible to reach the resonance of the qubit - oscillator system for relatively strong couplings .
entanglement lies at the heart of quantum mechanics and represents the most characteristic of it @xcite . it is known to be key resource of quantum communication and computation @xcite and has been verified in such protocols as cryptography @xcite and teleportation @xcite . create a large amount of entanglement between distance subsystems is a much desire goal in quantum information tasks . one way to mediate interaction between distant qubits is to use an additional setup , called quantum bus . spin chains are the most common buses where their tunable interaction has motivated researchers to use this permanently potential in the information processes @xcite . due to entanglement fragility under distance in most systems with short range interaction such as spin chains , one has to either delicately engineer the couplings @xcite or switch to super slow perturbative regimes @xcite . in all above studies symmetries of state and hamiltonian seem play the important role versus inherent entanglement for propagating information @xcite . the entanglement inherent in many - body systems has been investigated @xcite and using it for a `` known '' state transferring has been recently proposed in ref . @xcite . sending a `` known '' state makes quantum communication simpler for many communication features such as key distribution @xcite . furthermore , they have shown their proposal is more efficient to state transfer versus the previous scheme which attach a qubit encoding an `` unknown '' quantum state to the system @xcite . the experimental realization of the mott insulator phase for both bosons @xcite and fermions @xcite , with exactly one atom , in optical lattices enables for realizing effective spin hamiltonians @xcite by properly controlling the intensity of laser beam . moreover , single qubit operations and measurements @xcite are available by single site resolution in current experiments @xcite . furthermore , singlet - triplet measurement of simulated spin have been done by using supperlattice@xcite . in this letter , we put forward the approach in ref.@xcite to investigate the amount of entanglement between the ends of a spin chain govern by @xmath0 and @xmath1 hamiltonian . a setup consist of cold atoms trapped in a supperlattice has been introduced as a realization of our model . the structure of this paper is as follows : in section ( [ sec2 ] ) we introduce our setup . in section ( [ sec3 ] ) entanglement generation is investigated with the chain where its dynamics govern by @xmath0 hamiltonian and in section ( [ sec4 ] ) the @xmath1 hamiltonian is considered and variation of obtainable entanglement in all the phase space is discussed . we finally summarize our results in section ( [ sec5 ] ) . we consider a chain of @xmath2 spin @xmath3 particles , where @xmath2 is even , interacting through a dimerized hamiltonian @xmath4\ ] ] where , @xmath5@xmath6 is the strength of coupling in the @xmath7 direction , @xmath8@xmath6 denotes the pauli operators at site @xmath9 and @xmath10 determines the dimerization of the chain . this interaction is called @xmath1 and for @xmath11 is called @xmath12 hamiltonian and reduce to heisenberg hamiltonian with @xmath13 . we assume that the chain is in initial state and alice controls qubits @xmath14 and @xmath15 while bob controls the qubits @xmath16 and @xmath2 . to encode two qubits at the ends of the chain alice and bob apply @xmath17 on first and last qubits ( @xmath18 ) of the chain . after the operation @xmath19 the state of the chain changes to @xmath20 and so the system evolves as @xmath21 . at time @xmath22 the encoding state at each ends of the chain have been swapped while they are entangled via quantum gate at two middle qubits ( @xmath23)@xcite . now alice and bob can localize this information in their single qubits by performing a single - qubit measurement in the computational basis on sites @xmath15 and @xmath16@xcite . after performing the projection amount of entanglement between the ends of the chain can be obtained by calculating the concurrence@xcite @xmath24 where @xmath25 are the eigenvalues of the matrix @xmath26 while @xmath27 is the reduced density matrix of the qubits @xmath14 and @xmath2 . as a physical realization of above hamiltonian we propose the setting of ultracold atoms trapped in an optical supperlattice . an optical lattice mades of an standing wave formed by two different set of laser beams . the resulting potential is @xmath28 where , @xmath29 are the wave lengths , @xmath30 and @xmath31 are the amplitudes . the low energy hamiltonian of atoms trapped by @xmath32 is @xcite @xmath33 where , @xmath34 denotes the nearest neighbor sites , @xmath35 annihilates one atom with spin @xmath36 at site @xmath7 , and @xmath37 . we are interested in the regime where @xmath38 . this choice of hopping terms energetically prohibit the multiple occupancy of any site which corresponds to an insulating phase . the effective hamiltonian is found to be@xcite @xmath39 where@xmath40 and @xmath41 are the pauli s spin operators . the effective couplings @xmath42 and @xmath43 are given by @xmath44 the optical lattice parameters could be engineered such that @xmath45 and @xmath46 . so the effective hamiltonian reduced to the xx spin hamiltonian @xcite . @xmath47 the tunneling @xmath48 s are controlled by the amplitudes @xmath30 and @xmath31@xcite . tuning the intensity of low(high ) frequency trapping laser beam can be controlled independently the even ( odd ) couplings in a superlattice@xcite . so , it is possible to freeze the dynamics(j=0 ) at time @xmath49 with raising the barrier quickly and do the measurement on qubits . a schematic picture of the system is depicted in fig . [ fig1](a ) and ( b ) . atom and the tunnelings @xmath50 and @xmath51 . ( b ) encoding is done through local rotation on qubits @xmath14 and @xmath2 , while decoding is done through measurement on qubits @xmath15 and @xmath16 at time @xmath49.,width=302,height=207 ] in this section we consider a dimerized @xmath0 model defined by eq . ( [ hamiltonian ] ) with @xmath52 so the eq . ( [ hamiltonian ] ) reduced to @xmath53\ ] ] this hamiltonian can then be diagonalized with following the procedure described in@xcite with @xmath54 . the first step is to perform a jordan - wigner transformation@xcite @xmath55 where @xmath56 , @xmath57 and @xmath58 . as a result , the hamiltonian is mapped in the free fermion hamiltonian @xmath59 where @xmath60(@xmath61 ) is the vector of the @xmath2 creation(anihilation ) operators , and @xmath62 is the adjacency matrix . the new fermionic operators are defined as @xmath63 with the eigenvalues @xmath64 and eigenvectors @xmath65 of the matrix @xmath62 . so the hamiltonian ( [ fermion - xy ] ) takes the form@xcite @xmath66 also , dimerized xx hamiltonian ( @xmath67 ) can be diagonalize with the procedure described in @xcite for the odd number of qubits and in @xcite for the chain with even number of spins . the eigenvalues @xmath64 have been introduced in appendix for the nonvanishing amount of @xmath68 . with this dimerized hamiltonian we have a werner state @xmath69 for reduced density matrix of the first or last two qubits@xcite , where @xmath70 , @xmath71 is an @xmath72 identity matrix and @xmath73 if @xmath74 . the projection operators of two qubits have the forms @xmath75 , @xmath76 , @xmath77 and @xmath78 , where @xmath79 means both of two qubits are projected on @xmath80 . alice and bob apply these projections on qubits 2 and @xmath16 . at the first step , the best choice of @xmath81 and @xmath68 should be determined . so the variation of the entanglement at @xmath22 between the ends of the chain with @xmath82 and @xmath83 versus @xmath81 and @xmath68 has been calculated while the projection @xmath79 or @xmath84 has been applied . these results which have been plotted in fig . ( [ theta ] ) show the best amount of @xmath85 and @xmath86 . the concurrence decrease for @xmath87 due to emergence of small couplings(i.e . @xmath51 ) . entanglement at time @xmath49 between ends of the chain with @xmath88 and @xmath86 has been plotted in fig . [ proj ] for different projection operators . so , performing @xmath79(@xmath84 ) is the best choice and we use this projection in the following calculations . versus @xmath81 and @xmath68 for a xx spin chain with n=8 and @xmath83.,width=302,height=207 ] furthermore , we compare the amount of entanglement achieved in our proposal and anti - ferromagnetic chain with attaching a pair of maximally entanglement@xcite in fig [ xx - compare ] . this figure shows that the amount of entanglement in our proposal is higher than attaching scheme and it is in agreement with the compare of the amounts of average fidelity for state transfer in ref @xcite . spin chain with @xmath86 by applying different projection on qubits @xmath15 and @xmath16.,width=302,height=207 ] spin chain namely , entanglement achieved in our scheme(red line ) and entanglement obtained with attaching an extra maximally entangled pair(blue line).,width=302,height=207 ] in this section we consider a dimerized @xmath1 model defined by eq . ( [ hamiltonian ] ) with @xmath89 where @xmath90 is the anisotropy coupling in the @xmath91 direction . the above hamiltonian is a dimerized @xmath1 hamiltonian . the usual @xmath1 hamiltonian has a very rich phase diagram which different phases depend on different range of @xmath92 and @xmath90 . for @xmath90 = 1 and @xmath93 , this interaction is the fm heisenberg chain widely discussed in the context of quantum communication @xcite . more interesting regimes exist for @xmath94 and different values of @xmath90 @xcite . @xmath95 is the fm phase with a simple separable biased ground state with all spins aligned to the same direction . @xmath96 is called xy phase , which is a gapless phase and consists of two different legs , the fm half ( @xmath97 ) and the afm part ( @xmath98 ) . @xmath99 is called nel phase . in the ising limit @xmath100 the ground state is the nel state @xmath101 . we use the same recipe for entanglement generation with this hamiltonian . the concurrence between qubits @xmath14 and @xmath2 for the chain with @xmath88 at time @xmath49 has been plotted in the domain of ( @xmath102 in fig . [ xxz - phase ] with @xmath103 as a best amount of dimerized parameter for this hamiltonian . as we can see in this figure for our proposal works only for the domain @xmath104 . the reduced density matrix of first or last two qubits is the werner state in this domain of phase space . on the other side , there is no local entanglement at ground state for the transition point @xmath105 and the domain @xmath106 so , our mechanism which exploit inherent entanglement between proximally spins in ground state does nt work in this region . also , obtainable entanglement between ends of a chain with different length @xmath2 is enhanced compared to entanglement achieved by attaching a pair of maximally entanglement to the system while @xmath107 as has been shown in fig . [ xxz - compare ] as a function of @xmath90 with dimerized parameter @xmath103.,width=302,height=207 ] spin chain with @xmath107 namely , entanglement achieved in our scheme(black line ) and entanglement obtained with attaching an extra maximally entangled pair(red line).,width=302,height=207 ] in this paper we examined inherent entanglement in ground state of the spin chains for investigation of entanglement generation between the ends of chain . dynamics of the chains govern by dimerized @xmath0 and @xmath1 hamiltonian which can be realized by optical supperlattice . in this scheme local rotation on the ends of the chain encode information in the entangled ground state of the system . we also showed that the obtainable entanglement is higher than entanglement achieved by attaching a pair of maximally entanglement to the system . for @xmath1 hamiltonian this mechanism does nt work for the domain with @xmath108 . authors thank a. bayat for useful discussion and comments at university of ulm . upon the introduction of a nonvanishing @xmath68 the eigenvalues of the adjacency matrix @xmath62 for odd @xmath2 are given by @xcite @xmath109 where @xmath110 the eigenvalues of adjacency matrix @xmath62 for even @xmath2 are given by @xcite @xmath111 where @xmath112 and @xmath113 are the solutions of equation(@xmath114 ) which has different solution for @xmath115 and @xmath116 . d. bouwmeester , j .- w . pan , k. mattle , m. eibl , h. weinfurter , and a. zeilinger , nature ( london ) * 390 * , 575 ( 1997 ) ; d. boschi , s. branca , f. de martini , l. hardy , and s. popescu , phys . lett . * 80 * , 1121 ( 1998 ) . a. bayat and v. karimipour , phys . a * 71 * , 042330 ( 2005 ) . d. burgarth , s. bose , phys . a 73 , 062321 ( 2006 ) ; l. zhou , j. lu , t. shi and c. p. sun , quant - ph/0608135 . d. burgarth and s. bose , phys . a * 71 * , 052315 ( 2005 ) ; m. avellino , a. j. fisher , s. bose , phys . a * 74 * , 012321 ( 2006 ) . l. amico , r. fazio , a. osterloh and v. vedral , rev . phys . * 80 * , 517 ( 2008 ) . s. bose , contemporary physics * 48 * , 13 ( 2007 ) . a. bayat and s. bose , phys . a * 81 * , 012304 ( 2010 ) ; a. bayat , l. banchi , s. bose , p. verrucchi , phys . rev . a * 83 * , 062328 ( 2011 ) . w. s. bakr , et al . , nature 462 , 74 ( 2009 ) ; m. greiner , et al . , nature 415 , 39 ( 2002 ) ; w. s. bakr , et al . , science 329 , 547 ( 2010 ) . r. jordens , et al . , nature 455 , 204 ( 2008 ) ; u. schneider , et al . , science 322 , 1520 ( 2008 ) . p. medley , d. m. weld , h. miyake , d. e. pritchard and w. ketterle , phys . 106 * , 195301 ( 2011 ) . s. r. clark , c. m. alves and d. jaksch , new j. phys . * 7 * , 124 ( 2005 ) . w. k. wootters , phys . rev . lett . * 80 * , 2245 ( 1998 ) . m. christandl , n. datta , a. ekert , and a. j. landahl , phys . lett . * 92 * , 187902 ( 2004 ) ; m. b. plenio and f. l. semiao , new j. phys . * 7 * , 73 ( 2005 ) ; a. wojcik , t. luczak , p. kurzynski , a. grudka , t. gdala , and m. bednarska , phys . rev . a * 72 * , 034303 ( 2005 ) ; a. kay , phys . lett . * 98 * , 010501 ( 2007 ) ; c. di franco , m. paternostro , and m. s. kim , phys . rev . lett . * 101 * , 230502 ( 2008 ) . h. mikeska and a. kolezhuk , lect . notes phys . 645 , 1 ( 2004 ) .	we exploit the inherent entanglement of the ground state of a spin chain with dimerized @xmath0 or @xmath1 hamiltonian to investigate the entanglement generation between the ends of the chain . we follow the strategy has been introduced in ref . @xcite to encode the information in the entangled ground state of the system by local rotation . the amount of achieved entanglement in this scheme is higher than the attaching a pair of maximally entanglement scenarios . also , our proposal can be implemented by using the optical lattices .
electronic charges sometimes show intriguing behavior induced by their mutual interactions in condensed matter . charge ordering is one of the most widely studied phenomena in transition metal oxides@xcite , organic conductors@xcite and rare earth compounds@xcite . normally , charge ordering is seen in materials with low conductivity , and the phenomenon involves a sudden increase in the electrical resistivity . however , ybpd is a good metal in all temperature ranges@xcite with resistivity of @xmath1 cm , and is proposed to have a charge order accompanied by a decrease in resistivity.@xcite ybpd is an interesting compound in many ways . in the yb - pd phase diagram , this cscl - structured intermetallic compound stands at the border of yb@xmath2 and yb@xmath3 , which are stabilized in the yb rich side and the pd rich side , respectively@xcite . the valence of the yb ions in ybpd measured by @xmath4-edge x - ray absorption spectroscopy was actually fractional 2.8 + across all temperature ranges@xcite . in the fractional valence state , it shows magnetic ordering at @xmath51.9 k.@xcite magnetic ordering in the fractional valence state is not very common , and we were therefore interested in studying the origin of this rare phenomenon . mssbauer measurements below 4.2 k show two kinds of yb , one magnetic and the other non - magnetic , coexisting in equal proportions.@xcite there are five phase transitions , at @xmath6=125 k , @xmath7=105 k , @xmath51.9 k , @xmath80.6 k and @xmath90.3 k.@xcite while the low temperature transitions below 2 k are known to be magnetic transitions , the other two transitions have not yet been explained . the high temperature transitions are accompanied by strong anomalies in the material s specific heat , thermal expansion@xcite and elastic constants@xcite . although these observations collectively suggest that either @xmath6 or @xmath7 is the charge ordering temperature , no superstructure has been reported to date . charge ordering can be studied by ordinary x - ray diffraction because of the difference in the ionic radius of yb@xmath2 and yb@xmath3 , which produces a pd displacement in the charge ordering phase . a more relevant probe for the spatial valence arrangement is resonant x - ray diffraction , which is a combination of an x - ray diffraction technique with spectroscopy . using the @xmath4 absorption edge energy , it is possible to study the valence arrangements of @xmath10 electron systems@xcite . we have succeeded in clearly observing the superstructure induced by charge ordering in ybpd by means of these bulk sensitive techniques . single crystal samples were grown by the self - flux method.@xcite the as - grown samples have facets parallel to the \{001}-planes and are cube - shaped . the typical size of these crystals is 1 mm@xmath11 . non - resonant x - ray diffraction measurements were performed with a four - circle x - ray diffractometer attached to a mo @xmath12 x - ray generator . the incident x - ray beam was monochromatized with a bent graphite monochromator , and the scattered beam was then measured with a charge - coupled device ( ccd ) camera or a point detector . to find the superlattice reflections , x - ray photographs were taken with 18 kev intense synchrotron radiation at the bl-8b beamline of the photon factory , kek , japan . the charge order was examined by the resonant x - ray scattering technique at the yb @xmath4 absorption edge . the measurement was performed at the bl-4c beamline of the photon factory with a four - circle diffractometer . the sample temperature was controlled by using a closed cycle refrigerator for all measurements . figure [ fig:006](a ) shows the @xmath13 - 2@xmath13 line profiles of ( 006)@xmath14 bragg reflection at 140 k ( above @xmath6 , high temperature ( ht ) phase ) , 115 k ( between @xmath6 and @xmath7 , medium temperature ( mt ) phase ) and 95 k ( below @xmath7 , low temperature ( lt ) phase ) measured with the mo @xmath12 x - rays . here , the suffix c denotes the index in the high - temperature cubic phase . in addition to the doublet caused by the @xmath15 and @xmath16 x - rays , clear peak splitting with an intensity ratio of 1:2 was found below @xmath6 . this result indicates that the highly symmetrical ybpd deforms into a tetragonal structure below @xmath6 , and the single crystal turns into a multi - domain crystal . panel ( b ) shows the lattice parameters @xmath17 and @xmath18 obtained from the peak positions of the ( 600)@xmath19 and ( 006)@xmath19 reflections , where the suffix t denotes the index in the tetragonal phase . based on the peak profile , we found that @xmath6 is between 130 k and 135 k. above @xmath6 , @xmath17 is equal to @xmath18 because the ht phase is cubic . both of these transitions induce volume expansions with cooling . this behavior agrees well with the slope of the phase boundary in the pressure - temperature phase diagram shown in the inset of panel ( b).@xcite -2@xmath13 line profile of ( 006)@xmath14 bragg reflection at 140 k , 115 k and 95 k measured with mo @xmath12 x - rays . the error bars are shorter than the symbol size . the thick and thin gray curves for the 95 k profile show the results of the peak separation using a double gaussian for the @xmath20 doublet . the dashed lines are intended as visual guides . ( b ) temperature variation of the lattice parameters . inset : phase diagram reported in ref .. ,width=302 ] the isotropic atomic displacement parameters given by @xmath21 , where @xmath22 denotes the atomic displacement from the equilibrium position , for yb and pd at 300 k were estimated from the @xmath23 intensities . the parameter values were 0.7 @xmath24 and 1.4 @xmath24 , respectively . because the lindemann melting criterion predicts a value of 0.8 @xmath24 for @xmath25 at 300 k , the @xmath25 for pd is exceptionally large . the large @xmath25 value for pd may thus originate from the fluctuation of the yb radius caused by the valence fluctuation . the synchrotron diffraction experiment was performed to search for superlattice reflections . figure [ fig : ip ] shows the oscillation photographs around ( 136)@xmath14 taken in ( a ) the ht phase , and ( b ) the lt phase . along with the peak splitting caused by the cubic - tetragonal phase transition , the superlattice reflections characterized by the wavevector ( @xmath26)@xmath14 were observed . taken in ( a ) the ht phase ( 150 k ) and ( b ) the lt phase ( 80 k ) . ( c ) observed intensity at 95 k plotted against the calculated intensity for the best fitting structure . ( d ) schematic view of the lt phase structure.,width=302 ] the positions of the observed superlattice reflections were examined by using the four - circle diffractometer . the superlattice reflections in the lt phase were found to be at @xmath27 away from the fundamental bragg reflections and no reflections were seen at the @xmath28 positions . the peak widths of the superlattice reflections were the same as those of the fundamental bragg reflections , which means that the correlation of the superstructure reaches a long range . a strong ( 0 0 6.5)@xmath19 reflection was observed , whereas ( 6 0 @xmath29)@xmath19 reflection was very weak . this feature indicates that the atomic displacement @xmath30 in the lt phase is parallel to the @xmath31-axis , because the superlattice reflection intensity caused by the small atomic displacement is proportional to @xmath32 , where @xmath33 denotes the scattering vector.@xcite we measured the bragg and superlattice intensities along the ( 00@xmath34)@xmath19 line at 95 k to obtain the structure in the lt phase . figure [ fig : ip](c ) shows the integrated intensities of the ( 00@xmath34)@xmath19 peaks plotted against those fitted by the @xmath25s and the @xmath30s along the @xmath31-axis for both pd and yb . it was found that the pd atoms are displaced by 0.11 along the @xmath31-direction to form a twofold structure in the lt phase , as shown in fig . [ fig : ip ] ( d ) . no atomic displacement was found for the yb ions . this structure implies an alternate arrangement of high- and low - valence yb ions along the @xmath31-direction . to clarify the valence arrangement , the energy spectra of the superlattice reflections were measured . each element has a characteristic absorption edge , and the atomic form factor @xmath35 strongly depends on the x - ray energy @xmath36 around the edge . therefore , @xmath35 is written as @xmath37 , where @xmath38 denotes the thomson scattering term , and @xmath39 and @xmath40 denote the real part and the imaginary part of the anomalous dispersion term , respectively . the difference in the edge energies between the divalent and trivalent yb ions is reported to be 7 ev.@xcite this energy difference enables us to distinguish the valence values of these ions . the structure factor @xmath41 for the superlattice reflections ( @xmath42)@xmath19 is written as @xmath43 where @xmath44 , @xmath45 and @xmath46 denote the atomic form factors for the pd , yb@xmath3 and yb@xmath2 ions , @xmath47 shows the pd displacement in the unit of @xmath18 , @xmath48 is the average hole concentration of the yb @xmath35-orbital with a value close to 0.8,@xcite and @xmath49 denotes the amplitude of the charge ordering . the energy dependence of the scattered intensity is @xmath50 , where @xmath51 denotes the absorption coefficient.@xcite using the theoretically calculated anomalous dispersion term for an isolated yb atom@xcite ( @xmath52 ) and the structure factor from eq.([eq : f ] ) , we analyzed our experimental results . the values of @xmath45 and @xmath46 are obtained from @xmath53 , and @xmath54 was extracted from the fluorescence of the sample ( see fig.[fig : rxs](a ) ) . the theoretically calculated spectra were convoluted with the experimental resolution function , which was a gaussian with a full - width at half maximum of 7 ev . the measured energy spectrum for the ( @xmath55)@xmath19 superlattice reflection is shown in fig . [ fig : rxs](b ) . the upturn in the intensity with increasing energy toward the absorption edge is not expected without charge ordering , as indicated by the thin solid curve , and is therefore a signature of the charge order . and @xmath40 for yb@xmath2 and yb@xmath3 based on hartree - fock calculations . the linear absorption coefficient that was estimated from the fluorescence spectrum is also shown . ( b ) energy spectrum of the ( @xmath55 ) superlattice reflection measured at 80 k. the gray , thick solid , and thin solid curves show the calculated spectra for @xmath49=0.5 , 0.2 and 0 , respectively . , width=302 ] based on a comparison between the experimentally observed spectra and the calculated spectra , we conclude that @xmath56 , i.e. , that the high- and low - valence yb ions are 3 + and 2.6 + , respectively , for the lt phase of ybpd , as shown in fig . [ fig : ip ] ( d ) . in the mt phase , we discovered a splitting of the superlattice reflections . figure [ fig : map](a ) shows the scattering intensity distribution of the ( @xmath57)@xmath19 plane at 115 k. there are four peaks at ( @xmath58 @xmath58 2.5)@xmath19 . panel ( b ) shows the line profiles along ( @xmath59 @xmath59 2.5)@xmath19 ( the dashed line in panel ( a ) ) at 115 k ( mt phase ) , and at 80 k ( lt phase ) . the peak position , i.e. , the incommensurability @xmath60 , changes from 0.07 in the mt phase to 0 with the phase transition at @xmath7 . this result indicates that the charge order in the mt phase is an alternating arrangement along the @xmath31-direction , which is similar to that of the lt phase , but it has an in - plane modulation . similar to the lt phase , the superlattice peaks are as narrow as those of the bragg reflections , and therefore a long - range correlation was established . there are two possible modulation structures , depicted by the stripe and checkerboard structures that are presented in fig.[fig : models ] . the stripe structure is characterized by valence modulation along the [ @xmath61@xmath19 direction with a wavelength of @xmath62 . this modulation structure provides two superlattice peaks at ( @xmath63 @xmath63 2.5)@xmath19 and ( @xmath64 @xmath64 2.5)@xmath19 in the region shown in fig . [ fig : map ] ( a ) . the other two peaks are produced by the twin variant with a modulation vector along [ @xmath65@xmath19 . when we adopt the checkerboard structure , an alternating arrangement of @xmath66-sized tiles is expected to produce the four peaks shown in fig . [ fig : map ] ( a ) . the temperature variations of the incommensurability and the intensity integrated over all the peaks around ( 0 0 2.5)@xmath19 are presented in panel ( c ) . the mt - phase superlattice intensity was found below 140 k , which is slightly above @xmath6 . the inset shows that the peak width of the superlattice intensity above @xmath6 is broader than the instrumental resolution , meaning that the superstructure is short - ranged . the intensity above @xmath6 is therefore given by the fluctuation . the intensity is proportional to the square of the pd displacement , which must be a good indicator of the yb valence . below @xmath7=105 k , the intensity is nearly constant , which means that the charge order amplitude in the lt phase is independent of temperature . the incommensurability varies gradually in the mt phase , and jumps to zero at @xmath7 . the absence of the higher harmonic peaks at ( @xmath67 @xmath67 2.5)@xmath19 ( where @xmath68 is an integer ) and the sudden decrease in the total intensity at @xmath7 with heating indicate that the modulation is sinusoidal . ) @xmath19 plane at 115 k. ( b ) line profiles along the dashed line in panel ( a ) at 115 k and 80 k. ( c ) temperature dependence of the incommensurability @xmath60 and the integrated intensity . , width=302 ] there are two possible interpretations of the intensity distribution in the mt phase : stripe and checkerboard structures . while we have no decisive data to distinguish the two structures , the stripe model is the more plausible of the two . in fig . [ fig : map ] ( a ) , there are some differences in the intensities of the four peaks . the intensity at @xmath69 is equal to that at @xmath70 , and the intensity at @xmath71 is equal to that at @xmath72 . however , the former pair is apparently stronger than the latter pair . this difference is accounted for by the domain ratio in the stripe model , while the checkerboard model can not account for this intensity difference . next , we examine the origins of the incommensurate structure based on the stripe model . the valence states of the ions can be treated as ising pseudospins . it has long been known that an ising spin with a ferro - coupling for the nearest neighbor ( @xmath73 ) and an antiferro - coupling for the second neighbor ( @xmath74 ) , i.e. , the axial next - nearest neighbor ising ( annni ) model , results in an intricate phase diagram that involves wide incommensurate regions and commensurate regions separated by the first order phase transitions.@xcite the annni model has been applied to @xmath10 electron magnets including cesb@xcite , cebi@xcite and upd@xmath75si@xmath75@xcite , to ferroelectrics such as nano@xmath75@xcite and [ n(ch@xmath76)@xmath77@xmath75@xmath78cl@xmath79@xcite , and to charge ordering systems such as nav@xmath75o@xmath80@xcite and euptp@xcite . among these materials , euptp has significant similarities to ybpd , because both are metallic valence fluctuation compounds . let us compare the known properties of the charge ordering in ybpd with those expected from the annni model . first , the characteristic modulation vector varies from ( 0.07 , 0.07 , 0 ) at @xmath6 to ( 0.06 , 0.06 , 0 ) at @xmath7 . according to ref . , the wavelength is stabilized in a narrow temperature range with @xmath81 . when the temperature decreases , the calculation then predicts a first order phase transition to a ferroic arrangement within the @xmath31-plane , which was observed in our experiments at @xmath7 . the ratio @xmath82 is controlled by the application of pressure.@xcite as shown in the inset of fig . [ fig:006](b ) , the lt phase vanishes around 2 gpa , whereas the mt phase remains above that pressure level . when we simply map the annni phase diagram to the ybpd temperature - pressure phase diagram , then the charge order characterized by the wavevector ( @xmath83 ) , which corresponds to @xmath84 in - plane periodicity , is expected for the ground state under pressures of more than 2 gpa . it is also expected that a large number of phase transitions will occur in the mt phase . we can test the applicability of the annni model to ybpd by performing further diffraction measurements in high pressure environments . another common origin of the long wavelength modulation is fermi surface nesting . however , the nesting vector can vary only moderately . the cell volume change at @xmath7 is almost the same as that within the mt phase , which means that the change in the fermi wavevector at @xmath7 should be at a similar level to the change within the mt phase . as shown in the temperature dependence of the incommensurability @xmath60 ( fig . [ fig : map](c ) ) , the jump in @xmath60 at @xmath7 is far greater than its shift in the mt phase . therefore , the long wavelength modulation is unlikely to be caused by nesting of the fermi surface . finally , we discuss the effects of the charge ordering on the magnetic and conductive properties of the material . the application of pressure makes the lt phase unstable , and the ground state charge ordering structure is changed from an @xmath85 structure to an @xmath86 structure ( the mt structure ) above 2 gpa . in accordance with this transition , @xmath87 vanishes.@xcite therefore , the magnetic ordering at @xmath87 requires a charge ordered structure in the lt phase . the charge ordering characterized by @xmath85 makes yb@xmath3 and yb@xmath88 sublattices , which can have different kondo temperatures . the yb@xmath3 sublattice can form a magnetic order because it is not in the mixed valence state . in the @xmath89 structure in which the sinusoidal charge modulation develops , the yb@xmath3 content is small , and therefore the magnetic ordering is suppressed . in general , charge ordering makes the carrier density decrease , and therefore the resistivity increases , as reported in yb@xmath79as@xmath76@xcite and in many charge ordered 3@xmath90 electron systems.@xcite in contrast , the resistivity of ybpd decreases when the charge ordering occurs . this feature originates from the good metallic nature of ybpd . the carrier density is nearly unchanged by the charge ordering because the average valence remains constant , while the randomness of the valence arrangement , which scatters the carriers , is suppressed by the charge ordering . we have performed a series of x - ray diffraction measurements on the valence fluctuating compound ybpd . the material shows two - fold charge ordering characterized by the wavevector @xmath27 below 105 k. between 105 k and 130 k , the charge ordering structure is modulated , and the characteristic wavevector is @xmath91 with @xmath92 . we propose that this long wavelength structure can be described by using the annni model . this work was supported by kakenhi ( grant no . 23684026 ) , the japan securities scholarship foundation and the global coe program ( g10 ) . a. ochiai , t. suzuki , and t. kasuya , j.phys . . jpn . * 59 * 4129 ( 1990 ) . u. staub , m. shi , c. schulze - briese , b.d . patterson , f. fauth , e. dooryhee , l. soderholm , j.o . cross , d. mannix , and a. ochiai phys.rev . b * 71 * , 075115 ( 2005 ) . t. inami , s. michimura , a. mitsuda and h. wada , phys . b * 82 * 195133 ( 2010 ) . r. pott , w. boksch , g. leson , b. politt , h. schmidt , a. freimuth , k. keulerz , j. langen , g. neumann , f. oster , j. rhler , u. walter , p. weidner , and d. wohlleben , phys . lett . * 54 * 481 ( 1985 ) . a. miyake , t. kagayama , k. shimizu , and t. ebihara , j. phys . ser . * 391 * 012045 ( 2012 ) . y. nakanishi , t. kamiyama , k. ito , m. nakamura , m. sugishima , a. mitsuda , h. wada , and m. yoshizawa , chinese j. phys . * 49 * 462 ( 2011 ) canfield and z. fisk , phyl . mag . b * 65 * 1117 ( 1992 ) . the scattering amplitude from a crystal with a structural modulation is @xmath93 , where @xmath94 and @xmath95 denote the average atomic position and atomic displacement of the @xmath68-th atom . the amplitude proportional to @xmath96 corresponds to the superlattice reflection , while the rest of the amplitude corresponds to the bragg reflection . the superlattice intensity is thus proportional to @xmath32 . g. uimin , j. physique lett . * 43 * , 665 ( 1982 ) . t. honma , h. amitsuka , s. yasunami , k. tenya , t. sakakibara , h. mitamura , t. goto , g. kido , s. kawarazaki , y. miyako , k. sugiyama and m. date , j. phys . . jpn . * 67 * 1017 ( 1998 ) . k. ohwada , y. fujii , n. takesue , m. isobe , y. ueda , h. nakao , y. wakabayashi , y. murakami , k. ito , y. amemiya , h. fujihisa , k. aoki , t. shobu , y. noda , and n. ikeda , phys . lett . * 87 * , 086402 ( 2001 ) .	an x - ray diffraction study reveals the charge ordering structure in an intermediate valence magnet ybpd with a cscl - structure . the valence of the yb ions forms an incommensurate structure , characterized by the wavevector ( @xmath00.07 @xmath00.07 1/2 ) below 130 k. at 105 k , the incommensurate structure turns into a commensurate structure , characterized by the wavevector ( 0 0 1/2 ) . based on the resonant x - ray diffraction spectra of the superlattice reflections , the valences of the yb ions below 105 k are found to be 3 + and 2.6 + . the origin of the long wavelength modulation is discussed with the aid of an ising model having the second nearest neighbor interaction .
the solar coronal magnetic field is formed by active regions emerging from the solar interior and becoming distributed throughout the atmosphere by flux transport processes ( e.g. , mackay and yeates 2012 ) . the field can become open to the heliosphere in regions where a single magnetic polarity dominates , including the polar caps and other regions of isolated polarity ( harvey and recely 2002 ) . wherever the coronal fields extend to great heights , the magnetic tension force can be overcome by the thermal pressure of the expanding coronal plasma , and the field dragged out and opened to form a coronal hole ( parker 1958 ) . in extreme ultraviolet ( euv ) and soft x - rays ( sxr ) coronal holes appear as dark regions whereas in helium images they appear bright . harvey and recely ( 2002 ) identified three classes of coronal hole : polar coronal holes confined to high latitudes ( @xmath0 , @xmath1 ) ; isolated coronal holes at active latitudes associated with the remnant fields of decayed active regions ; and transient coronal holes that briefly form after coronal mass ejections . from a study of the polar coronal holes during cycles 22 and 23 , harvey and recely concluded that polar coronal holes are formed from isolated holes associated with decayed active - region field of trailing polarity . earlier , harvey and sheeley ( 1979 ) had shown that during the ascending phase of a solar cycle the leading polarities of active regions may diffuse to form locally unbalanced flux patterns that tend to form coronal holes ahead of active regions . the declining phase of cycle 23 roughly coincided with the beginning of nso s global oscillations network group ( gong ) synoptic magnetogram program and the launch of nasa s solar terrestrial relations observatory ( stereo ) mission ( kaiser et al . 2008 ) in late 2006 . the cycle 23 minimum produced unusually many coronal holes at active latitudes compared to previous cycle minima , mainly owing to the weakness of the polar fields during cycle 23 ( de toma 2011 ) . the decline of solar cycle 23 and the ascent of cycle 24 have provided excellent conditions for studying isolated coronal holes . using mt . wilson observatory and wilcox solar observatory synoptic magnetograms and stereo euv synoptic maps , wang et al . ( 2010 ) studied three instances of small coronal holes appearing at the edges of newly emerged active regions and then expanding and ultimately joining the polar coronal holes . they were able to simulate the formation of three coronal holes using kinematic photospheric flux transport models in combination with extrapolated potential - field source - surface ( pfss ) models for the coronal field . they found that the coronal holes could be explained by a combination of photospheric flux transport processes : turbulent diffusion , differential rotation and the interaction between neighboring active regions . they concluded that the global distribution of photospheric flux determined where the coronal holes formed , in particular the proximity or otherwise of strong flux of the opposite polarity . in the pfss model , coronal loops are assumed to become open if they extend beyond a certain height , hence open field regions form wherever there is insufficient photospheric flux of the opposite polarity to connect to . karachik et al . ( 2010 ) used michelson doppler imager ( mdi ) and gong full - disk magnetograms and extreme - ultraviolet imaging telescope ( eit ) euv images to investigate four cases of isolated polar coronal holes forming on the remnants of decaying active regions . in contrast to wang et al . ( 2010 ) , karachik et al . ( 2010 ) focused on the evolution of individual bipolar active regions . they measured several parameters of the active regions over successive rotations including the imbalance and the compactness of the magnetic flux . they found that less compact field tends to decay faster than more compact field , causing a gradual increase in flux imbalance in the region and the formation of a coronal hole at the location of the more compact flux . karachik et al.s four cases occurred during the ascent of cycle 24 in 2009 and their conclusions are consistent with those of sheeley and harvey ( 1979 ) . using extrapolated pfss models karachik et al . found that in three cases , all of them holes of positive polarity , some field rooted at the coronal hole locations connected back to the photosphere close to the north polar coronal hole . these fields therefore appeared to play a role in the solar cycle process , evolving from toroidal active region field into poloidal field . in this paper we follow karachik et al . ( 2010 ) by focusing on the relationship between the decaying fields of active regions and the formation of coronal holes . we investigate which factors can determine the formation of the coronal holes including active region flux imbalance , field strength , compactness , interactions with other fields , and the structure of the coronal field represented by pfss models . we also try to clarify the role of these fields in the solar activity cycle . most of our examples come from the ascending phase of cycle 24 with a minority of cases representing the declining phase of cycle 23 . the paper is organized as follows . we describe the observational data in section [ sect : data ] . the 14 pairs of active regions and coronal holes will be introduced and their properties summarized in section [ sect : arch ] . pfss models will be used to aid interpretation of coronal hole formation in section [ sect : pfss ] . we will conclude in section [ sect : conclusion ] . full - disk images of the relative doppler shift of the ni i line at 676.8 nm are available from each of the six gong telescopes at a cadence of 1 minute , weather permitting . gong s six stations together provide round - the - clock coverage with approximately an 87% duty cycle . the spatial sampling of the gong images is @xmath2 pixel@xmath3 and the instrumental sensitivity is about 3 g pixel@xmath3 . since the gong instrumentation was upgraded in 2006 , resulting in a great reduction of the zero - point error in the field measurements , synoptic maps for the radial component of the photospheric field have been constructed for every carrington rotation . the stereo sun earth connection coronal and heliospheric investigation ( secchi ) instrument package ( howard et al . 2008 ) has four instruments : an extreme ultraviolet imager , two white - light coronagraphs and a heliospheric imager . the extreme ultraviolet imager , euvi ( wuelser et al . 2004 ) observes in four spectral channels that span the 0.1 to 20 mk temperature range . in this paper we use synoptic maps based on 19.5 nm images , corresponding to about 2.5 mk . coronal holes tend to appear clearest in these maps . we searched for decaying active regions and associated coronal holes by visually inspecting photospheric synoptic magnetograms from gong and the corresponding synoptic euv maps from the stereo a spacecraft s euvi instrument . the gong synoptic map series begins with carrington rotation ( cr ) 2047 ( august 2006 ) whereas the stereo maps begin at cr2051 ( early 2007 ) . from pairs of gong and stereo synoptic maps we identified 14 cases where the decay of an active region could confidently be associated with the formation of a coronal hole . these include four examples from the declining phase of cycle 23 and ten from the ascent of cycle 24 . these are listed in table [ arlist ] and examples are shown in figures [ fig : ex3 ] and [ fig : ex6 ] . we only consider examples where active regions could be identified over at least two consecutive rotations and where the remnant flux of the decaying active region could be associated with a newly formed coronal hole in the corresponding stereo map . gong is a ground - based network of telescopes whereas the stereo a and b spacecraft orbits lead and lag the earth in its orbit around the sun . since their launch in august 2006 , the twin stereo spacecraft have been separating at a rate of @xmath4 per year . the observations analyzed in this paper date from early 2007 until late 2010 , over which time the separation between stereo a and earth grew from a fraction of a degree to about 85@xmath5 , representing a time delay between a gong observation of a given longitude and the corresponding stereo a observation of that same longitude of up to 6.5 days . because of this time delay the gong and stereo maps can not be treated as being nearly simultaneous , but since we are studying the slow decay of active regions and formation of coronal holes over multiple rotations the pairs of gong and stereo synoptic maps are still applicable . our approach differs from that of karachik et al . ( 2010 ) in that we do not estimate coronal hole magnetic fluxes by deriving coronal boundaries from stereo euv data and mapping them onto gong magnetograms . instead we study the magnetic fluxes of the decaying active regions . [ arlist ] + .properties of active regions [ cols="<,^ , < , < , < , < , < , < , < " , ] @xmath6was the leading ( l ) or trailing ( t ) flux more dense / compact ? + @xmath7was the region initially located on the pole - ward ( p ) or equator - ward ( e ) side of the pfss streamer belt , or contained within it ( s ) ? note that none of the cycle 23 examples was initially located pole - ward of the streamer belt , and only one cycle 24 region was initially located equator - ward of the streamer belt . + @xmath8did the coronal hole polarity match the polar field in its own hemisphere ? note that the coronal hole polarity matched the polar field in its own hemisphere in every case except those regions initially located equator - ward of the streamer belt . thus there was no mismatch between a coronal hole magnetic polarity and the polar field polarity on the original active - region s side of the streamer belt . active regions are understood to emerge from the solar interior in the form of closed loops with balanced magnetic flux but with asymmetries between the positive and negative magnetic flux distributions . an active region is usually composed of a magnetic bipole where one polarity clearly trails the other in longitude . it is well known that active regions tend to follow a simple pattern whereby the leading magnetic fluxes of the bipoles in the north and south hemispheres have opposite polarities and that these polarities switch during each activity minimum ( hale and nicholson 1925 ) , known as hale s polarity rule . for example , the leading flux in the north / south hemisphere generally had positive / negative polarity during cycle 23 and negative / positive polarity during cycle 24 . the active regions also tend to be asymmetric in their flux distribution : the leading flux is usually nearer the equator ( hale et al . 1919 , a rule known as joy s law ) , and is also more dense and compact than the trailing flux . the active regions asymmetry has consequences for their decay patterns . the trailing flux tends to decay more quickly than the leading flux and be transported pole - ward , often leaving the leading flux isolated . the leading flux may therefore be expected to open up and form a low - latitude coronal hole under some circumstances ( harvey and sheeley 1979 ) . the asymmetric behavior of the active regions links active and polar fields together in a single magnetic activity cycle ( babcock 1961 ) . during the ascent of an activity cycle the dominant leading polarity of a hemisphere matches the polarity of that hemisphere s polar field , whereas during the declining phase the opposite is true . the fact that the gong and stereo data sets include the declining phase of cycle 23 and the ascent of cycle 24 allows us to investigate whether this difference has consequences for low - latitude coronal hole formation . table [ arlist ] summarizes the quantitative details of the 14 examples of interest in this paper . each example spans 2 - 4 rotations . every active region in the sample followed hale s polarity rule when first observed and all but one followed joy s law for active region tilts . table [ arlist ] also lists the magnetic fluxes and flux densities of the regions during each rotation . we estimate the magnetic flux of a region as follows . we find level curves of field strength 5 g and identify by eye the curves that enclose the active region of interest . in figures [ fig : ex3 ] and [ fig : ex6 ] ( top panels ) the selected level curves for examples 3 and 6 are over - plotted in black on the gong synoptic magnetograms . we then sum by magnetic polarity the flux contained in these curves . also shown are the average location , flux density and compactness of each region . we follow karachik et al . ( 2010 ) in calculating the compactness of the active region magnetic field distribution using their expression for the effective radius , @xmath9 , where @xmath10\|b_r ( x , y)\| / \sum \|b_r ( x , y)\| , \label{eq : reff}\ ] ] where @xmath11 and @xmath12 are the average pixel coordinates of the field distribution . we calculate @xmath9 for the entire flux distribution of each region and also for the positive and negative fluxes separately . table [ arlist ] shows that the total magnetic fluxes of all 14 regions decreased over time , consistent with decay . the average flux densities @xmath13 generally decreased as the regions decayed but sometimes @xmath13 increased when some weak flux decayed to values below the threshold used in the analysis and was no longer included in the calculation . the effective radii @xmath9 tended to increase in time as the flux decayed and became more diffuse , until one of the polarities decayed away . the active regions initially had magnetic fluxes ranging from about @xmath14 mx for a late cycle 23 region to @xmath15 mx for an early cycle 24 region . the late cycle 23 regions were larger than the early cycle 24 regions on average . the first two regions from cycle 24 were small but the regions subsequently increased in size . the last two regions of the 14 were the second and third largest of the study . the larger regions tended to last longer than the smaller regions . a notable exceptions to this rule was example 5 , which was one of the smallest of the 14 but lasted four rotations . figures [ fig : ex3 ] and [ fig : ex6 ] show gong and stereo magnetograms for two typical examples from table [ arlist ] , examples 3 and 6 . the black contour curves over - plotted on the gong maps indicate the decaying active region fields , and the associated coronal holes are indicated by dashed white contour curves on the stereo synoptic maps . the leading fluxes , negative in both cases , were initially denser than the trailing fluxes and the negative fluxes decayed more slowly , creating local negative flux imbalances in both cases , from which coronal holes developed as shown in the plots . this behavior is much as expected from the previous work cited in the first paragraph of this section . karachik et al . ( 2010 ) studied example 3 and reported the same behavior . wang et al . ( 2010 ) analyzed and simulated example 3 and found that the small leading - polarity hole that formed near the equator was balanced by the formation of a trailing - polarity hole near longitude 180 , which subsequently developed into an extension of the south polar hole ( see the arrowed features in figure 2 of wang et al . ) . this positive - polarity hole was located in the trailing flux of the active region just to the east of the active region that contained the leading - polarity hole . in this case the two bipoles were close enough together that their facing polarities connected to each other , leaving their outer polarities to open up . coronal holes form under the constraint that new open flux of one polarity must always be balanced by opening up an equal amount of flux of the opposite polarity , or by closing down an equal amount of flux of the same polarity . this process can be seen in figure [ fig : ex3 ] . of the three types of coronal hole identified by harvey and recely ( 2002 ) , all holes studied here appear to have belonged to the category of holes forming gradually in association with decaying active regions . some of these regions produced minor flaring activity but , browsing through the stereo / euvi sky images , we did not find evidence of a significant eruption associated with sudden formation or development of the holes . the decaying regions appeared to evolve steadily and gradually . in some cases the fully - formed coronal hole would rotate onto the disk where no evidence of coronal hole formation was visible during the previous rotation . in other cases the coronal hole evolved gradually over multiple rotations . table [ chlist ] summarizes the qualitative behavior of all 14 regions . from this table some clear patterns emerge . in all of the active regions except one , the leading polarity was stronger than the trailing polarity when the region first appeared in a synoptic map . the lone exception , example 5 , had marginally more compact leading flux than trailing flux , and it was the leading polarity that was associated with the hole in this case . example 5 was analyzed and simulated by wang et al . ( 2010 ) , who concluded from their models that the formation of the leading - polarity coronal hole was accompanied by some of the negative - polarity flux from the nearby polar hole closing down by connecting to the trailing flux of the mid - latitude active region via interchange reconnection , as is common during the ascending phase . the cause of the fast decay of positive flux between the times that the cr2076 and cr2077 measurements were taken ( see table [ arlist ] ) may be related to the distribution of the fluxes according to the gong synoptic maps . the negative flux was slightly larger but significantly less dense during cr2076 , and was composed of two concentrations to the west and south of the positive flux concentration . there were therefore two neutral lines separating the positive and negative fluxes , which may have allowed much flux cancellation to take place during this time interval . the positive flux may have survived the negative flux simply because there was slightly more of it to begin with . the fact that the net flux of the region stays within the range @xmath16 mx throughout four rotations of decay suggests that flux cancellation was the dominant process in this case . in all but two cases the leading polarity had the more compact distribution . in the exceptional cases , examples 2 and 7 , the leading polarity was the more dense on average but , whereas the leading polarity decayed more slowly and opened in example 2 , it was the trailing polarity that survived and opened in example 7 . example 7 was also analyzed and simulated by wang et al . ( 2010 ) , who explained the formation and rapid decay of this hole in terms of interchange reconnection between the trailing polarity of the active region and the polar hole boundary . this trailing - polarity ( positive ) hole ( which can be seen in the cr2085 euvi / a map in figure 6 of wang et al . 2010 , and which we will model in section [ sect : pfss ] ) is associated with the leading negative flux of the neighboring region just eastward that produced a leading - polarity hole . wang et al . ( 2010 ) showed the roles of diffusion and differential rotation in the formation and evolution of the two holes . we can not generally find observations that catch the active regions at their moment of emergence when their flux is expected to be balanced . as soon as a region emerges it interacts with its surroundings and some reconnection will inevitably take place . according to the estimates in table [ arlist ] the earliest observed fluxes were balanced to within 10% in all regions except one . the exception was the first example which was already a large , mature region at the beginning of the sequence of observations . in most but not all cases the two polarities decayed asymmetrically so that one polarity dominated by the time the coronal hole appeared in the stereo data . in the two cases where the fluxes remained approximately balanced , examples 10 and 12 , the two polarities had become very distant from each other over time , resulting in an isolated unipolar region of field where a coronal hole could form . in 12 out of 14 cases the leading flux opened , whereas the denser polarity opened in only ten out of 14 cases and the sign of the initial flux imbalance matched the final flux imbalance in only eight out of 14 cases . this indicates a strong hemispheric polarity preference in the decay rates , a pattern that is better correlated with the polarities of the coronal holes than the intrinsic asymmetries of the active regions . on the other hand , the polarity of the coronal hole matched the polarity of the polar field in its hemisphere in only ten out of 14 cases , showing that the hemispheric preference is not strict . from the ten cycle 24 active regions the leading flux formed the coronal hole in eight cases . the exceptions are examples 7 and 12 . in example 12 a coronal hole appeared in the stereo cr2091 map near the trailing flux . this may seem surprising because in this example the leading flux was initially stronger and more compact . this trailing flux seemed to open following the emergence of two active regions to the east of the decaying region . the emergence of these regions may therefore have played a key role in the formation of the coronal hole from the trailing flux ( see section [ sect : pfss ] ) . in example 7 the leading negative flux was initially denser and more compact than the trailing flux , and yet the leading flux decayed faster creating a positive flux imbalance and a coronal hole . again the photospheric context of the region will be necessary to explain the formation of this hole ( wang et al . we will return to both examples 7 and 12 in section [ sect : pfss ] . in each of the remaining eight cases from cycle 24 the leading polarity formed the coronal hole , even when the leading flux was not the denser or more compact polarity . the data suggest that the global field structure may have a powerful influence over the polarity distribution of coronal holes . tables [ arlist ] and [ chlist ] show that the coronal holes tend to divide by polarity between the two hemispheres , with negative in the north and positive in the south , matching the polarities of the polar fields . this occurred in most but not all cases : 11 out of 14 . during the ascent of an activity cycle the compact and dense polarity of a typical active region following hale s polarity rule and joy s law would match the polarity of the polar field in its hemisphere , and the resulting low - latitude holes would therefore match the polar hole in polarity . tables [ arlist ] and [ chlist ] hint at a link between the global field structure and the polarity of the field that forms the coronal hole . on the other hand , during the declining phase of cycle 23 this hemispheric bias seems not to have been in strong effect . the leading fluxes of the first four examples were of opposite polarity to the polar fields in their hemispheres . also these four active regions were relatively large and were situated closer to the equator . one might therefore expect the relationship of these declining - phase coronal holes to the global field to be qualitatively different from their ascending - phase counterparts . examples 1 and 3 appear to be analogous to most of the cycle 24 examples in that their leading dense fluxes decayed more slowly , creating an imbalance , and coronal holes form in association with them . examples 2 and 4 appear to have behaved differently . in example 2 the trailing negative flux was initially the more compact and a negative flux imbalance had developed by the next rotation . yet , the coronal hole that formed was associated with the positive flux , the only case among the 14 whose flux imbalance and coronal hole polarities did not agree . example 4 also showed unexpected behavior . here the leading negative flux was initially the denser and more compact and yet it decayed faster , leaving a positive flux imbalance and a coronal hole . these two examples seem difficult to interpret with magnetograms and euv images alone . in the following section we will revisit them using pfss models . while we only have a small number of statistics , it is apparent that there was a larger hemispheric bias in the polarities of the low - latitude holes during the ascent of cycle 24 than during the decline of cycle 23 . the relationship between the active regions and the global coronal field structure is fundamentally different during early phases of the solar cycle compared to late phases . during early / late phases an active region typically has relatively high / low latitude and its lagging flux usually opposes / matches the polarity of the photospheric polar field in its hemisphere . we know that during relatively quiet phases of the solar activity cycle it is the low - order multipoles , in particular the dipole and octupole , that dominate the global coronal field structure , and that these components are well correlated with the photospheric polar fields ( petrie 2013 ) . to search for possible links between the the global coronal field structure and the creation of coronal holes , information for the coronal field is needed . to this end , we will present extrapolated potential field models in the following section . the euv images do not tell us the polarity of a coronal hole . furthermore , as we saw in section [ sect : arch ] , a simple relationship between an euv dark region and the unbalanced magnetic flux can not be assumed to determine the polarity of a coronal hole . to supply this and other important information on the coronal magnetic field we extrapolated pfss models from each gong synoptic magnetogram . for active regions near the beginnings or ends of carrington rotations we used composite synoptic maps formed from half of the rotation containing the region and half of the previous or subsequent rotation . extrapolating coronal pfss field models ( altschuler and newkirk 1969 , schatten et al . 1969 ) from the gong synoptic maps is a simple and effective way to diagnose the response of the coronal magnetic field to the photospheric activity patterns described in section [ sect : arch ] . low in the corona , the magnetic field is sufficiently dominant over the plasma forces that a force - free field approximation is generally applicable . moreover , for large - scale coronal structure the effects of force - free electric currents , which are inversely proportional to length scale , may be neglected . we identify the lower boundary of the model with the solar photosphere at @xmath17 where @xmath18 is the solar radius . we use the photospheric radial field maps from gong to supply the lower boundary data for the radial field component . this use of radial field data , derived from longitudinal measurements of non - force - free photospheric fields , assuming the magnetic vector to be approximately radial in general , has been found to produce models that more successfully reconstruct coronal magnetic structure than the direct application of the longitudinal measurements as boundary data ( wang and sheeley 1992 ) . a synoptic map construction method that derives the radial component from longitudinal measurements without assuming the fields to be approximately radial has been developed and applied to chromospheric data by jin et al . ( 2013 ) with promising results . here we adopt the standard photospheric synoptic maps because measured photospheric fields are found to be approximately radial in general ( svalgaard et al . 1978 , petrie and patrikeeva 2009 , gosain and pevtsov 2012 ) , and the resulting models remain useful and competitive for reasons explained by wang and sheeley ( 1992 ) . above some height in the corona , usually estimated to be between 1.5 and 3.5 solar radii , the magnetic field is dominated by the thermal pressure and inertial force of the expanding solar wind . to mimic the effects of the solar wind expansion on the field , we introduce an upper boundary at @xmath19 , and force the field to be radial on this boundary by setting the potential to zero there , following altschuler and newkirk ( 1969 ) , schatten et al . ( 1969 ) and many subsequent studies . all fields reaching @xmath20 are interpreted as being open to the heliosphere and as representing coronal holes . the usual value for @xmath21 is @xmath22 although different choices of @xmath21 lead to more successful reconstructions of coronal structure during different phases of the solar cycle ( lee et al . 2011 ) . in this paper we adopt the value @xmath23 because this value leads to a reasonable visual resemblance between open fields in the models and coronal hole distributions as seen in the stereo euv synoptic maps . with these two boundary conditions , the potential field model can be fully determined in the domain @xmath24 . we emphasize that each pfss model is therefore completely determined by the corresponding photospheric magnetogram . there is no time - dependence in the pfss model . the model is successful because , to a good approximation , the global coronal field relaxes completely to a near - current - free state independent of the previous magnetic field in the sequence . we computed the pfss models using the national center for atmospheric research s ( ncar ) mudpack package . although pfss models can be calculated analytically , and have been so treated for several decades , we adopt a finite - difference approach in this paper to avoid some problems associated with the usual approach based on truncated spherical harmonic series ( tth et al . mudpack ( adams 1989 ) is a package for efficiently solving linear elliptic partial differential equations ( pdes ) , both separable and non - separable , using multigrid iteration with subgrid refinement procedures . jiang and feng ( 2012 ) have recently presented a high - speed combined spectral / finite - difference pfss solution method using the related ncar fishpack package for separable pdes . in this paper we use the mudpack package to solve laplace s equation in its non - separable form in spherical coordinates , subject to the boundary conditions described above . a successful pfss model tells us not only the polarity of a coronal hole but also the context of the hole within the global coronal field structure , such as its relationship to the polar coronal holes and the streamer belt . figures [ fig : ex3 ] and [ fig : ex6 ] ( third rows of plots ) show the pfss models for examples 3 and 6 . here selected fields and features of the model are over - plotted on the gong synoptic magnetograms . the green and red contours indicate foot - point locations of positive and negative open fields , corresponding to coronal holes . the blue lines delineate the tallest closed field trajectories in the solution , corresponding to the streamer belt , and the black line shows the set of apex locations of these fields , representing the equatorial current sheet . we will refer to this black line as the source - surface neutral line ( ssnl ) in the following . the yellow lines represent the active region fields whose foot - points lie within the black contour curves of figures [ fig : ex3 ] and [ fig : ex6 ] ( top plots ) . the figure also shows plots of the models in spherical coordinates . these plots show magnetic lines selected according to the field intensity at their foot - points . closed lines are plotted in blue and positive / negative open lines in green / red . the coordinate system is rotated so that the decaying region is at central meridian . the plots show the development of open field at the location of the decaying region and enhanced connectivity to the southern polar latitudes . the ssnl is generally confined to low latitudes around solar minimum , but major active regions can cause large excursions of the ssnl from the equator . during the decline of cycle 23 and the ascent of cycle 24 , the axisymmetric dipole and octupole components of the coronal field had the strongest influence over the global coronal field structure , and these components are well correlated with the slowly - evolving photospheric polar fields ( petrie 2013 ) . during this time , the streamer belt separated the corona into two distinct open - field regions : the global corona had negative / positive open field north / south of the ssnl . it is striking that none of the cycle 23 coronal holes formed pole - ward of the ssnl and only one of the cycle 24 holes , example 7 , formed equator - ward , as table [ chlist ] shows . figures [ fig : ex3 ] and [ fig : ex6 ] are illustrative of this pattern , showing the equator - ward example 3 from late in cycle 23 and the pole - ward example 6 from early in cycle 24 . this is perhaps not surprising behavior for the declining / ascending phase of a solar cycle , when active regions emerge at relatively low / high latitudes . the formation of a coronal hole did not result in a significant change in the location of the ssnl in any of the examples . the low - order magnetic multipole fields , corresponding to large spatial scales , were not significantly perturbed by the photospheric active - region decay associated with the formation of the coronal holes . in each case the hole appeared with the same polarity as the pre - existing open field on its side of the streamer belt . this includes the exceptional case , example 7 , where the ssnl looped around the region in the northern hemisphere , placing the region in the southern half of the corona in magnetic terms . in a few late cases , examples 12 - 14 , when the sun was slightly more active and the regions were more numerous and slightly larger , the corona had a more complex structure and the ssnl overlay the active regions . under these conditions the decay rates of the positive and negative polarities seem to have been determined by their relative density and compactness . examples 1 - 4 occurred during the declining phase of cycle 23 and were all located near the equator . two of these holes ( examples 1 and 3 ) appeared equator - ward of the ssnl and two of them ( examples 2 and 4 ) were straddled by it . that none of them occurred pole - ward of the ssnl signifies an important difference between the cycle 23 and cycle 24 coronal holes . table [ chlist ] shows that of the ten out of 14 cases where the polarity of the coronal hole matched the sign of the polar field in its hemisphere , six out of six matched for pole - ward cases and zero out of three for equator - ward cases . these statistics point to a strong influence of the global field structure on the polarity of successfully formed coronal holes . during cycle 23 the leading flux did not match the polarity of the polar field in its own hemisphere , making it much more difficult for it to open and form coronal holes pole - ward of the ssnl than equator - ward . of the equator - ward cases , in the cycle 23 examples 1 and 3 , the leading flux had the same polarity as the polar field on the opposite hemisphere , i.e. , the same polarity as the polar field on its side of the streamer belt . these holes were therefore topologically equivalent to cycle 24 holes pole - ward of the streamer belt . a visual comparison of example 3 in figure [ fig : ex3 ] and example 6 in figure [ fig : ex6 ] illustrates this topological relationship . the major difference between these two examples is that example 3 occurred south of the equator . in terms of magnetic topology they are equivalent . in common with the cycle 24 holes , the equator - ward cycle 23 holes in examples 1 and 3 formed from flux of the same polarity as the pre - existing open field on their side of the streamer belt . in these two cases the leading flux was also more dense and more compact than the trailing flux . the global field structure and the magnetic asymmetry of the active region together seem to have enabled the leading - polarity coronal hole to form . of the five cases occurring near the ssnl , the denser polarity formed the coronal hole in only three of them : zero out of two during cycle 23 and three out of three during cycle 24 . the three cycle 24 cases appear to have evolved straightforwardly according to their asymmetric flux density distributions . the denser leading polarity survived to form the hole in all three cases . as noted in the previous section , the two examples from cycle 23 , examples 2 and 4 , are more difficult to explain . aided by the pfss models , we do so now . example 2 was centered a degree or two north of the equator and had the leading positive polarity of a northern - hemisphere cycle 23 region . the ssnl straddled the active region and the trailing negative flux was initially more compact than the positive flux . a significant negative flux imbalance developed as the positive flux decayed more quickly than the negative flux . according to the pfss models this flux imbalance manifested mostly as increased magnetic connection to the positive south polar latitudes . the coronal hole that did form near the region had positive polarity even though the flux imbalance of the decaying region was clearly negative ( -34% ) . a newly emerged region to the west visible in the cr2053 map ( not shown here ) seems to have been involved in the formation of this coronal hole . its positive trailing polarity combined with the leading polarity of the decaying region to create an area of positive flux where a coronal hole could develop . this is the only example in our sample of 14 where the polarities of the decaying active region flux imbalance and the associated coronal hole did not match . as in example 2 , in example 4 ( figure [ fig : ex4mod ] ) the ssnl straddled the active region but this southern region had leading negative flux that was initially much denser and more compact than its positive flux . and yet the flux balance became overwhelmingly positive over time and a coronal hole of positive polarity formed . in this case the positive flux imbalance and associated asymmetric topology seem to explain the asymmetric development of this region . the active region was initially tucked underneath the streamer belt . the streamer belt separated the closed flux underneath from the negative open flux to the north and the positive open flux to the south . based on its positive flux imbalance , this region might be expected to have had more magnetic connection to the north ( negative ) side half of the corona than the south , consistent with its positive flux imbalance . according to the model , most of these connections occurred within the streamer belt via lines arching from the positive flux of the region over the western half of the region to the distant negative flux of the neighboring region to the west . these connections were discussed in detail by petrie et al . ( 2009 ) and are evident in figure [ fig : ex4mod ] . since the connecting fields extended high in the streamer belt they were particularly susceptible to being reconnected at the ssnl and opened up . according to the plots for the subsequent rotations shown in figure [ fig : ex4mod ] , this seems to be what happened . by the time of cr2070 , some of the positive flux of the region was open and a dark coronal hole corresponding to the positive flux was apparent in the euv map , but the basic global structure of the coronal field was unchanged . figure [ fig : ex5mod ] shows the stereo euv maps and pfss models for example 5 . as noted in the last section , this small region survived several rotations and underwent fast decay between crs 2076 and 2077 , perhaps due to flux cancellation . a sizable coronal hole developed to the west of its main leading - polarity ( negative ) flux concentration between crs 2076 and 2077 . this coronal hole stands out because it developed in association with negative active region flux that was less dense than the corresponding positive flux in the region and yet decayed more slowly and survived to create the coronal hole . we discussed in the last section that this may have been due to the distribution of the negative flux in two concentrations , bordering the positive flux at two neutral lines , visible in figure [ fig : ex5mod ] allowing flux cancellation to dominate the evolution during crs 2076 and 2077 . given that there was slightly more negative than positive flux observed in this region from the start of the sequence of gong observations , flux cancellation alone would ultimately produce a significant negative flux imbalance . the pfss models show that the resulting negative coronal hole also preserved the global coronal structure as a positive hole in this region could not have done . we found no example where the formation of a coronal hole significantly changed the coronal structure . the pfss model plots for example 7 in figure [ fig : ex7mod ] demonstrate the influence active regions can have on the global structure of the corona . the ssnl followed a tight loop - shaped trajectory to straddle the active region whose trailing positive polarity ultimately produced the coronal hole . throughout the three rotations the ssnl maintained this shape and separated this open trailing positive flux from the open leading negative flux of the neighboring region to the east . the ssnl appears to have been locked in this position as the active regions decayed and the euv coronal hole formed from this trailing flux . this is in marked contrast to the effect of this region when it emerged into the corona . as figure [ fig : ex7emer ] shows , there was a major change between cr2082 , when the ssnl lay south of the equator around @xmath25 longitude before the region emerged , and cr2083 , when the ssnl encroached more than @xmath26 northward into the northern hemisphere in response to the emergence of the region and its neighbor to the east . the contrasting responses of the global field structure to active region emergence and decay are due to the effects of these processes on the photospheric field distribution and their timescales . when active regions emerge into the atmosphere at a location where no pre - existing active - region flux is present . the effect on the global field can be significant . the low - order multipole moments of the active region can be large enough to perturb the streamer belt over timescales much less than a rotation as in figure [ fig : ex7emer ] . the effects of the decay of this region was much slower : between crs 2083 and 2085 the region decayed almost completely yet the streamer belt structure hardly changed . the decay processes left a large - scale patch of weak positive flux which , with the large - scale patch of weak negative flux left by the decay of the neighboring region to the east , maintained a low - order magnetic multipole moment in cr 2085 comparable to that of the newly emerged region in cr 2083 . this preserved the kink in the streamer belt even after the active - region - strength fields had almost completely decayed away . in this case the streamer belt did not lose its kink until cr2087 , by which time the weak unipolar photospheric fluxes of the two regions had been transported from the area . in each of our examples the remnant flux of the decayed active region generally maintained the influence of the region on the global coronal field at least a rotation after the region had decayed . in several of the examples , in particular 1 , 2 , 7 and 12 , we see evidence in the models that the active region field was open long before a dark coronal hole was observed in euv images , where the coronal hole did not become visible until the active region decayed . the plots of the models for example 7 , shown in figure [ fig : ex7mod ] , and example 12 indicate that the active regions can contain flux that is open but does not appear as dark regions in the euv images . luhmann et al . ( 2009 ) showed evidence of contributions to the solar wind from low - latitude coronal holes during early 2007 , including the two active regions from examples 1 and 2 . in example 12 , according to the models , the leading flux was partially open during both crs 2090 - 1 but did not appear in the euv observations as a dark region . meanwhile a dark coronal hole appeared in euv in association with the decaying trailing flux as mentioned in the previous section . the overall behavior of the decaying active regions may be summarized as follows . the structure of the global coronal field , including the streamer belt , was preserved during the decay of the active regions and the formation of the coronal holes . thus where the active region was decisively on one side of the streamer belt or the other , the dominant polarity of the open field on that side of the streamer belt invariably matched the polarity of the active - region field that formed the hole . in all cases of isolated active regions within the steamer belt , straddled by the ssnl , the intrinsic magnetic asymmetry of the region became important : the denser , more compact polarity was the one that formed the coronal hole . in cases where the region was within the streamer belt and was not magnetically isolated the coronal hole formed in a complex way particular to the magnetic context of the region , but always without changing the global magnetic structure of the corona . a typical late - phase active region is located closer to the equator and is larger than an ascending phase region , making the structure of the global field somewhat less influential in its topological development . the intrinsic properties of the active region and its complex interactions with neighboring regions therefore appear to be more important factors in determining the behavior of declining - phase regions than of ascending phase regions . nevertheless the global field structure did not change significantly as a result of the decay of even the largest declining - phase regions studied here . only the emergence of new active regions seem to have produced significant global change . active region emergence can significantly change the global structure of the corona , reshaping the streamer belt , and sometimes creating multiple streamer structures . even under the relatively quiet conditions prevailing between 2007 and 2010 when the low - order multipoles dominated the global coronal field , major new active regions produced major changes in the global coronal structure ( see figure [ fig : ex7emer ] ) . yet we have found that active region decay produces no significant change in the the global structure of the corona over multiple rotations , even for large active regions that develop major flux imbalances . the active photospheric fields change significantly from rotation to rotation , as the figures in this paper show . a carrington rotation is plenty of time for the coronal field to adjust to a changed photospheric field . furthermore the pfss model for a given rotation is computed completely independently of the previous rotation . it is therefore striking that the pfss coronal field models show almost global structure change due to coronal hole formation over multiple rotations . two facts seem to explain the continuity of global coronal field structure during coronal hole formation . first , the global coronal field varied gradually over the period of time studied . during the years 2007 - 2010 the low - order multipoles , in particular the dipole and octupole , had the strongest influence over the global coronal field structure , and these components are well correlated with the photospheric polar fields ( petrie 2013 ) . the stability of the large - scale component of the global coronal field is due mostly to their relationship to the slowly - evolving polar fields . the resulting large - scale dipolar and octupolar coronal fields were generally more globally influential than the higher - order coronal fields associated with decaying active regions . photospheric flux transport processes produce unipolar bodies of flux of both polarities in both hemispheres , but the coronal fields of decaying active - regions did not have large enough low - order multipole moments or global influence to cancel the dominant multipoles . they could therefore only open under the condition that they did not significantly change the basic coronal structure dominated by these lower - order fields . unipolar fields of the `` wrong '' polarity generally remained closed ( and unstudied ) . thus the basic global coronal field structure , dominated by low - order , large - scale , slowly - varying fields , was preserved . second , the remnant weak field had a long - lasting effect that remained after each region had decayed away . decaying regions that produced coronal holes tended to leave large - scale patches of unipolar weak field which took several rotations to disperse after the region itself had decayed away . these remnant fields were much weaker than the original active - region fields but they were spread over a much larger area . the weak remnant fields had low - order multipole moments of comparable strength to those of the original active regions , and thereby approximately preserved the global field signature of the active region long after the region itself has decayed away . this is why the global coronal field structure remained almost unchanged during the process of coronal hole formation . we can also comment on the role of these decaying regions in the solar activity cycle . in babcock - leighton models for the solar cycle ( see , e.g. , charbonneau 2010 ) , the global solar field is often idealized as an approximately axisymmetric configuration , with a belt of tilted active regions in each hemisphere and a global coronal ssnl that does not travel far from the equator . in such models the more dense , compact leading active - region fluxes in the two hemispheres interact with each other across the equator while their less dense trailing counterparts decay and are transported pole - ward . both the leading and trailing fluxes would contribute to the activity cycle in converting active - region toroidal flux into poloidal flux : the leading fluxes would form trans - equatorial poloidal loops across the equator and the trailing flux would join the nearly axisymmetric polar flux . it is perhaps not surprising that the fields studied in this paper do not behave in this simple way . the photospheric field rarely featured matching pairs of active regions in the two hemispheres . when the global corona had a simple ssnl and a nearly axisymmetric appearance the active regions were small and distant from the equator , and they did not interact significantly with fields on the opposite side of the streamer belt . their unbalanced flux simply opened up , converting nearly - toroidal active - region flux to nearly - poloidal open flux . in large , magnetically isolated region within the streamer belt the less dense and compact polarity of the region tended to decay faster than the other polarity , leaving it unbalanced and with no partner in the opposite hemisphere to connect with . when a major flux imbalance developed in a region some of its flux necessarily connected elsewhere , usually converting nearly - toroidal active region flux into the nearly - poloidal field of the streamer belt or the open fields . in these ways the decaying active regions participated in the solar activity cycle . we thank the referee and alexei pevtsov for helpful comments . k.j.h . carried out this work through the national solar observatory research experiences for undergraduate ( reu ) site program , which is co - funded by the department of defense in partnership with the national science foundation reu program . this work utilizes data obtained by the nso integrated synoptic program ( nisp ) , managed by the national solar observatory , which is operated by aura , inc . under a cooperative agreement with the national science foundation . the data were acquired by instruments operated by the big bear solar observatory , high altitude observatory , learmonth solar observatory , udaipur solar observatory , instituto de astrofsica de canarias , and cerro tololo interamerican observatory . the stereo / secchi data used here are produced by an international consortium of the naval research laboratory ( usa ) , lockheed martin solar and astrophysics lab ( usa ) , nasa goddard space flight center ( usa ) rutherford appleton laboratory ( uk ) , university of birmingham ( uk ) , max - 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violet ( euv ) synoptic maps and coronal potential - field source - surface ( pfss ) models . we analyze 14 decaying regions and associated coronal holes occurring between early 2007 and late 2010 , four from cycle 23 and 10 from cycle 24 . we investigate the relationship between asymmetries in active regions positive and negative magnetic intensities , asymmetric magnetic decay rates , flux imbalances , global field structure and coronal hole formation . whereas new emerging active regions caused changes in the large - scale coronal field , the coronal fields of the 14 decaying active regions only opened under the condition that the global coronal structure remained almost unchanged . this was because the dominant slowly - varying , low - order multipoles prevented opposing - polarity fields from opening and the remnant active - region flux preserved the regions low - order multipole moments long after the regions had decayed . thus the polarity of each coronal hole necessarily matched the polar field on the side of the streamer belt where the corresponding active region decayed . for magnetically isolated active regions initially located within the streamer belt , the more intense polarity generally survived to form the hole . for non - isolated regions , flux imbalance and topological asymmetry prompted the opposite to occur in some cases . = 1
until recently , it has been a time - consuming , costly and arduous work to collect and analyze data about individual humans at a large scale . with the advent of the digital era , there is a growing amount of data accessible online that enables the analysis and modeling of human behavior . however , our understanding of these digital data sources and the methods that connect the data to real - world outcomes is still limited . several aspects on the possible usage of mobile phone records and social media status updates in the estimation of official data , such as census , demographic or land use records have been discussed in recent papers . a promising approach is the analysis of the diurnal rhythm of humans . due to the 24 hour periodicity of the earth s rotation , we are biologically bound to show daily periodic behavior both at the individual and at the aggregate level . this periodic cycle is governed mainly by internal biochemical processes @xcite , but the impact of external factors and the environment also leaves its imprint on these daily patterns @xcite . as sramaki and moro point out in their paper @xcite , an interesting application is to consider the geospatial aspects of the aggregate level of daily rhythms , as it can provide insight into several different phenomena ranging from the actual land use patterns in a city @xcite and on a campus @xcite , to the tracking of anomalous events @xcite , or the estimation of population size @xcite , mobility patterns @xcite , poverty @xcite or crime rates @xcite in a certain area . because these aggregate patterns always consist of the superposition of the daily rhythms of individuals , it is worth investigating how the main features of the aggregate level form from superposition . if we can cluster individuals into more or less homogeneously behaving groups based on their daily patterns @xcite , then the aggregate pattern can be understood as the combination of the group patterns , and the group that has more individuals dominates the aggregate daily rhythm . the groups of individuals can form along many demographic and/or socioeconomic factors , of which being employed and going to and from work at regular hours is the most determining one with respect to the daily activity patterns . thus , decomposing the groups from the aggregate patterns in different geographical regions may give insight into the estimation of employment statistics in that region . nowcasting or estimating unemployment rates using the digital traces of search engines has already been in the focus of several papers @xcite . it has already been shown , that daily activity patterns of individuals can be linked to the regularity of their working hours @xcite . because the loss of a job has severe psychological consequences @xcite , the effects of a mass layoff can be detected in the unemployment rates and provide a possibility of forecasting macro - economical effects based on observation of several individuals @xcite . in @xcite , there is a strong evidence that aggregated daily activities of certain time intervals of geographical regions can be indicative of unemployment rates . in this paper we obtain 63 million geolocated messages from the publicly available stream of the social network twitter from the area of the united states sent between january and october 2014 . we aggregate monday to friday relative tweeting activity for each hour in each us county to form an average workday activity pattern . we then assume that these activity patterns form a roughly linear subspace of the 24-hour `` timespace '' . by finding this linear subspace , that is , by finding the line on which the county patterns lie , we are able to give a measure that is linked to the ratio of two groups of people tweeting in a county . we then show that this measure correlates significantly with county employment and unemployment rates , and that the average patterns corresponding to the two groups can be linked to lifestyles connected to regular working hours or the lack of them . we thus give a possible framework for decomposing the digital activity patterns of geographical regions and linking the decomposition to employment and unemployment rates . we use the data stream freely provided by twitter through their application program interface , which amounts to approximately 1% of all sent messages . in this study , we focus on the part of the data stream with geolocation information . these geolocated tweets originate from users who chose to allow their mobile phones to post the gps coordinates along with a twitter message . the total geolocated content was found to only comprise of a small percentage of all tweets ; therefore with data collection focusing only on these , a large fraction of all geo - tagged tweets can be gained @xcite . our dataset includes a total of 63 million tweets from the contiguous united states collected between january 2014 and october 2014 . these are all geotagged that is , they have gps coordinates associated with them . we construct a geographically indexed database of these tweets , permitting the efficient analysis of regional features @xcite . using the hierarchical triangular mesh scheme for practical geographic indexing @xcite , we assigned a us county to each tweet . county borders are obtained from the gadm database @xcite . for the population - weighed linear model of the next section , we obtain county - level population statistics from the us 2010 census @xcite . we download the unemployment and labor force data for the time window of the twitter dataset from the local area unemployment statistics page of the bureau of labor statistics @xcite . we take an average of the months ranging from january 2014 to october 2014 for each county . though unemployment levels are defined as the number of unemployed per total labor force in a county , we define the share of employed as the number of employed divided by the whole population of a county . this measure fits the model for the daily rhythm better as discussed in the results section . we define a daily activity pattern with hourly resolution for each county , which are enumerated by @xmath2 . we take all tweets originating from a given county from the period between january 2014 and october 2014 . then we aggregate the number of tweets ( @xmath3 ) in each hour ( the hour range goes from @xmath4 ) on workdays , that is from monday to friday , after correcting for timezone and daylight saving time in each county . because of the differing population and twitter penetration rates ( share of people using twitter ) in each county , we normalize the number of tweets by the total number of tweets counted . thus , each county ( @xmath5 ) is represented by a 24-dimensional vector ( @xmath6 ) , where the elements of @xmath6 are : @xmath7 and obviously , @xmath8 to improve the quality of our dataset , we consider only those counties in which the overall tweet count during the ten month exceeded the threshold of 1800 . thus , we are left with 1884 counties for our analysis . we assume that the tweeting pattern of a county can be represented by the linear combination of only two universal patterns ( @xmath9 and @xmath10 ) that are mixed for each county @xmath5 with a proportion of @xmath11 , and @xmath12 , respectively . thus , we identify the two universal patterns that compose the pattern of a county as corresponding to two differently behaving population groups , whose aggregate tweeting patterns form @xmath9 and @xmath10 . we have no further restriction on these @xmath11 values , they can be any arbitrary real numbers . then the predicted activity @xmath13 of a county @xmath5 in hour @xmath14 would be the following linear combination : @xmath15 let us denote the weight of each county by @xmath16 , which is proportional to its population @xmath17 , such that @xmath18 . we then define the squared error of our model as @xmath19 we would like to minimize this error with subject to the two conditions @xmath20 . it can be shown ( see si ) , that the minimum occurs if @xmath21 is parallel to the eigenvector @xmath22 corresponding to the biggest eigenvalue of the weighed covariance matrix @xmath23 , and that @xmath10 can be chosen as the average of @xmath6s . here , an element of the covariance matrix @xmath23 is @xmath24 where @xmath25 in both cases , we now consider a linear representation of the data with a coordinate system where the mean @xmath26 sets the origin and @xmath22 is the direction of the line . we calculate @xmath11 values for each county by projecting @xmath27 onto this line ( see si ) . a positive @xmath11 means a county , where the majority of people are active on twitter in correspondence with the daily rhythm dictated by @xmath22 , accordingly , negative @xmath11 is in connection with an opposite pattern . because the linear equation system derived from the minimization of the squared error is linearly dependent , the scale on our line is not set ( see si ) , as @xmath28 is only determined up to an arbitrary scaling factor . thus , the @xmath11 values are also determined only up to a scaling factor . let us now choose @xmath9 and @xmath10 to be two standard deviations of @xmath11-s away from the origin @xmath26 in the two directions of our new linear coordinate system : @xmath29 @xmath9 and @xmath10 are both normalized to 1 , where in the 2-dimensional case their components represent the selected two hours , while in the 24 dimensional case they represent the 24 hours of the day . [ sec : methods ] in this section , we present the description and the discussion of the main results of this paper . first , we investigate the correlation between the activities of individual hours and employment and unemployment rates , and choose two dimensions with which employment and unemployment levels have maximum or minimum correlations . we then evaluate to what extent the linear model is a valid description of our data for these most separating dimensions ( 2 ) and then for all possible dimensions ( 24 ) of our dataset . second , we discuss how the linear models in 2 and 24 dimensions separate the two population groups with the two distinct activity patterns , and give a possible interpretation of these patterns . third , we connect the two groups with real - world indicators like share of employed in a county , and discuss the plausibility of the correspondence of the daily patterns of the two separate groups to employment status . we first evaluate population - weighted pearson correlations for each hour @xmath14 between @xmath30 activities for the 1884 counties ( from which we have an adequate number of messages ) and employment and unemployment levels . we calculate the errors of these correlations by bootstrapping our sample for @xmath31 times , the results with errorbars are shown in fig [ fig : hourcorr ] . while unemployment levels are defined in the traditional way of the bureau of labor statistics , we define the share of employed slightly differently , normalizing the number of employed by the entire population of a county . this definition matches the notion of population share of `` active '' people regarding regular working hours better . the hours between 6 am and 8 pm show a significantly positive correlation with employment , and a negative one with unemployment , while during the night , between 9 pm and 5 am , the correlation is reversed . with respect to employment , the correlation peaks at 12 am with @xmath32 and reaches its lowest value at 1 am with @xmath33 . the location of the maximum and minimum of correlation with unemployment are shifted slightly to 0am and 12 am , though exactly with opposite signs ( @xmath34 for 0am and @xmath35 for 12 am ) . the signs of the correlations and the hours of their extreme values indicate that increased daytime activity is associated with higher employment levels , and higher than average nighttime activity corresponds to higher unemployment . to check the linearity of the model described in the methods section , we first choose the coordinate system of the hours having the extreme correlation values with employment levels . fig [ fig : two ] shows the 0am and 1 pm activities of the filtered counties with the dashed line corresponding to the direction of the first eigenvector of the covariance matrix , now calculated only from these two dimensions . if we normalize the eigenvalues by their sum , we see that the first eigenvalue of the covariance matrix carries 0.99 share from all the variance in the data , thus , linearity in this two - dimensional subspace of the whole 24-hour activity space is a good assumption . we continue by assessing the validity of the linear model in all 24 dimensions presented in eq [ eq : linear ] . in fig [ fig : prcomps]a we plot eigenvalues of the covariance matrix @xmath23 again normalized by the sum of all eigenvalues . only the first four eigenvalues correspond to a variance significantly greater than 0 , and the first principal component stands out with a proportion of 0.52 , whereas the other three significant components carry 0.25 , 0.13 and 0.04 share of the variance . thus , our dataset is mostly linear even in the 24-dimensional space , and the representation with eq [ eq : linear ] remains plausible . in the 2-dimensional case , the dashed line of fig [ fig : two ] marks the direction of the first principal vector . the difference between the two vectors @xmath9 ( red ) and @xmath10 ( blue ) representing the two universal patterns ( see methods on p. ) is parallel to this component , let us denote it by @xmath22 . it can be seen in fig [ fig : two ] that the @xmath9 pattern is marked by an increased activity at 1 pm , and a decreased activity at 0am , while pattern @xmath10 is characterized by exactly the inverse relationship . the principal component corresponding to the largest principal value in the 24-dimensional case can be seen in fig [ fig : prcomps ] . as the coordinates represent the hours , it can be seen from fig [ fig : prcomps ] that @xmath22 is positive from 5 am until 8 pm , and negative otherwise . thus , the positive elements of @xmath22 select mainly those hours during which people are awake , and the negative elements correspond to the sleeping hours . we then plot the elements of the 24-dimensional @xmath9 and @xmath10 from eq [ eq : a]-[eq : b ] in fig [ fig : ai ] . by interpreting these patterns as the different average tweeting patterns of two population groups , each @xmath11 is proportional to the share of people in a county in one population group . our hypothesis is that the group more active during the daytime corresponds to people who regularly go to work , school etc . on weekdays , thus their daytime is regulated by the earlier wake - up and bedtime indicated in pattern @xmath9 . on the other hand , pattern @xmath10 could correspond to a group where this regulation factor does not exist due to retirement , unemployment or any other reason , which would allow these people to be more active during nighttime and wake up later . to confirm our hypothesis , we correlate @xmath11 values with labor force and unemployment estimates from the local area unemployment statistics ( see methods on p. ) of the investigated counties . in the 2-dimensional case , these combined values of @xmath11 do not correlate with employment ( @xmath36 ) or unemployment ( @xmath37 ) better than previous activity measures from single dimensions from fig [ fig : hourcorr ] . however , by using all dimensions , we find correlations of @xmath38 and @xmath39 for employment ( see scatterplot in fig [ fig : scatter ] ) and unemployment , respectively . for the employment this is an improvement to that of the single dimensional correlations , while it is not for the unemployment . a possible interpretation is that a stricter daily rhythm is imposed upon those who are employed , as such , the characteristics of their activity curves mean a stronger overall pattern than that of the unemployed . nevertheless , the result shows that high a @xmath11 is significantly bound to higher employment , and lower unemployment rates , and that the overall shape of the activity timeline can give us more information than just using one feature of a whole day . the similarity of the regional distribution of @xmath11 , unemployment and employment rates are visualized on the three maps of fig [ fig : maps ] . our results are in line with previous research carried out for spain in @xcite , where share of twitter activity during a window of the morning hours ( 8 - 11am ) , afternoon hours ( 3 - 5pm ) and of the night hours ( 0 - 3am ) correlated significantly with unemployment rates among 25 to 44-year old inhabitants of spanish administrative areas . high morning and low night activity indicated lower unemployment rates , which is in correspondence with our correlations . although in spain high afternoon activity correlated positively with unemployment levels , we can not observe this phenomenon in the us . due to the bias in the age of twitter users towards younger age groups @xcite , our calculated county activity patterns are not representative of the whole population . we believe that our model could be improved by incorporating labor force data detailed by different age groups . that correlation with unemployment is significantly lower than correlation with labor force share of the population can be related to the fact that the share of employed should overlap more with the population exhibiting the `` working '' pattern @xmath9 , whereas officially registered unemployed people are not distinguishable in this context from those who are on a maternal leave or are retired etc . we also believe that there are other inherent reasons for example the more flexible working hours in the creative industry that limit the power of such a simple model explaining the employment patterns of a geographical area . in this paper we analyzed an extensive collection of geolocated tweets originating from the united states between january 2014 and october 2014 . we assigned a county to each tweet , then aggregated daily tweeting activity patterns for a typical weekday , and investigated to what extent do hourly activities correlate with employment or unemployment levels . we then modelled daily activity patterns as being the superposition of two universal patterns , thus aiming for a simple linear approximation of our dataset . by minimizing the squared error of our estimations , we obtained that the difference of the two patterns should be parallel to the first eigenvector of the covariance matrix of the dataset and that the mean of the data should fit on our line when selecting only 2 dimensions , and when using all 24 dimensions of our data as well . the set of eigenvalues of the covariance matrix in both cases confirmed the validity of our linear model , which captured most ( 0.99,0.52 ) of the variance in the dataset . whereas in the 2-dimensional case the first eigenvector pointed to the direction , where 1 pm activity was increased , and 0am activity decreased , in the 24-dimensional case it had positive elements during the daytime hours ( 6am-8pm ) , and was negative during the most of the night ( 9pm-5am ) . by projecting county activity patterns onto these lines with the mean as the origin , we obtained a measure for each country that indicated the extent to which the tweeting pattern of a county resembles that of the first eigenvector . this measure has been shown to correlate significantly with county labor force shares and unemployment rates , though in the 2-dimension case , these correlations could not enhance the performance of the single hourly correlations . using all 24 dimensions , we obtained a better pearson correlation of @xmath38 and @xmath39 for employment and unemployment , respectively . the signs of the correlations indicate a relationship where counties exhibiting a higher tweeting activity during the daytime ( 6am-8pm ) have higher employment and lower unemployment rates , and counties with increased night activity can be related to lower employment and higher unemployment rates . these correlations show , that even though twitter population is biased towards younger age groups , and employment data was considered for all age groups , the underlying relationship between daily activity patterns and employment data can be captured with plausible outcomes . our results thus showed , that by analyzing a relatively sparse publicly available geolocated dataset , a very simple model can explain to a significant extent such an important socio - economic indicator as employment / unemployment . we believe that our model can be even further improved by incorporating detailed data for different age groups or other datasets from either traditional or digital sources such as mobile traffic data . it would be worth to investigate whether dynamic changes of activity patterns over time can follow employment trends . this kind of analysis would allow policy makers a better insight into the processes connected to employment phenomena , and could form the basis of future datasets , where problems could not only be identified based on officially registered unemployed people , but also on a basis of the digital footprints people leave on different platforms . times . the hours between 6 am and 8 pm correlate significantly positively with employment and negatively with unemployment . this relationship turns out to be exactly the opposite during the night . regarding employment , the most distinguishing hours are 0am ( most negative correlation ) and 1 pm ( most positive correlation ) . ] and @xmath10 vectors , see the linear model part in the methods section . ] to @xmath40 . the vector components are positive from 5 am to 8 pm , and negative otherwise.,title="fig : " ] [ fig : prcomps ] corresponds to the daily activity pattern of a population with regular working hours . the blue line , @xmath10 corresponds to the other group who stay up until later in the evening and wake up later as well . the dashed line marks the average activity of all counties.,title="fig : " ] [ fig : ai ] values with employment and unemployment * ] , employment and unemployment levels . * regional similarities are visualized by plotting * a * @xmath11 measures , employment * b * and unemployment * c * on a us county map . blank counties did not exceed the 1800 tweet threshold described in the methods section . ] and @xmath10 corresponding to the two universal patterns later identified as the active and inactive patterns , the dashed line showing the mean value of activities . the bold vector is the direction of the first eigenvector of the covariance matrix @xmath23 . @xmath41 , @xmath42 and @xmath43 represent three arbitrarily chosen axis corresponding to different hours @xmath14 , @xmath44 and @xmath5 of the day . ] [ fig : vec ] g.v . conceived the experiment , e.b . and z.l . collected the data , e.b . and g.v . analyzed the results , e.b . wrote the manuscript all authors reviewed the manuscript . the dataset supporting the conclusions of this article is available in the following repository : http://www.vo.elte.hu / papers/2016/unemployment/. the authors declare that they have no competing interests . 10 [ 1]#1 urlstyle [ 1]doi # 1 j. aschoff , r. wever , federation proceedings * 35(12 ) , 236 ( 1976 ) a. cagnacci , j.a . elliott , s.s . yen , the journal of clinical endocrinology & metabolism 75(2 ) , 447 ( 1992 ) . pmid : 1639946 r. refinetti , m. menaker , physiology & behavior 51(3 ) , 613 ( 1992 ) . 10.1016/0031 - 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ssdbm 14 pp . 14 ( 2014 ) . 10.1145/2618243.2618245 global administrative areas . accessed 21 november 2016 united states census 2010 . http://www2.census.gov / census_2010/. accessed 21 november 2016 local area unemployment statistics . / lau/. accessed 21 november 2016 m. duggan , n.b . ellison , c. lampe , a. lenhart , m. madden , l. rainie , a. smith , pew social media report 2015 . january , pew research center ( 2015 ) * + eszter boknyi , [email protected]^^ , + zoltn lbszki , [email protected] , + gbor vattay , [email protected] + department of physics of complex systems + pzmny pter stny 1/a , etvs lornd university , budapest h-1117 , hungary corresponding author we define a daily activity pattern with hourly resolution for each county that are enumerated by @xmath2 . thus , each county ( @xmath5 ) is represented by a 24-dimensional vector ( @xmath6 ) , where the elements of @xmath6 are aggregated normalized hourly tweeting activities . we assume that the tweeting pattern of a county can be represented by the linear combination of only two universal patterns ( @xmath9 and @xmath10 ) that are mixed for each county @xmath5 with a proportion of @xmath11 and @xmath12 , respectively . we have no further restriction on these @xmath11 values , they can be any arbitrary real numbers . @xmath9 and @xmath10 are both 24-dimensional vectors normalized to 1 , the 24 dimensions representing the 24 hours of the day . summing eq [ eq : sys1 ] and eq [ eq : sys2 ] for @xmath44 yield 0 for the lagrange multipliers @xmath46 and @xmath47 . thus , the problem reduces to minimizing @xmath50 , which actually measures the sum of squared distances from the line parametrized by @xmath21 , @xmath10 and @xmath11 for a county @xmath5 . since @xmath51\cdot ( a_j - b_j ) = \sum_m ( \ref{eq : sys3}),\ ] ] the equation system is not linearly independent . thus , we can not obtain all exact values for @xmath52 , @xmath53 and @xmath11 , they will be dependent on each other . the line from which the summed distance of the datapoints is minimal is the line whose direction is parallel to the eigenvector ( @xmath22 ) corresponding to the largest eigenvalue of the covariance matrix @xmath23 , where @xmath55 if @xmath56 denotes the weighted mean ( @xmath57 , @xmath58 ) @xmath25	by modeling macro - economical indicators using digital traces of human activities on mobile or social networks , we can provide important insights to processes previously assessed via paper - based surveys or polls only . we collected aggregated workday activity timelines of us counties from the normalized number of messages sent in each hour on the online social network twitter . in this paper , we show how county employment and unemployment statistics are encoded in the daily rhythm of people by decomposing the activity timelines into a linear combination of two dominant patterns . the mixing ratio of these patterns defines a measure for each county , that correlates significantly with employment ( @xmath0 ) and unemployment rates ( @xmath1 ) . thus , the two dominant activity patterns can be linked to rhythms signaling presence or lack of regular working hours of individuals . the analysis could provide policy makers a better insight into the processes governing employment , where problems could not only be identified based on the number of officially registered unemployed , but also on the basis of the digital footprints people leave on different platforms .
the fact that gauge coupling unification is a `` near - miss '' within the standard model ( sm ) provides an indication in favor of the idea of unification @xcite . likewise , the existence of neutrino masses , required to account for neutrino oscillation data @xcite , also provides another motivation towards unified or gut ( grand unified theory)-like extensions of the sm . however , the most characteristic feature of gut - type unification , namely matter instability , has so far defied experimental confirmation @xcite . on the other hand , neither the generation of neutrino masses nor the tilting in the evolution of the gauge couplings require unification in the conventional sense . for instance , it is well known that the gauge couplings merge in the minimal supersymmetric extension the sm , provided that supersymmetric states lie around the tev scale @xcite . so far , though , there has been no trace of such states in the lhc data @xcite . here we consider an alternative approach in which new physics at the tev scale realizes an extended electroweak gauge structure with perturbatively conserved baryon - number . for definiteness we consider the @xcite ( 3 - 3 - 1 ) framework , which implies that the number of generations equals the number of colors , in order to cancel anomalies . this scheme has attracted attention recently also in connection with b physics @xcite , or flavor symmetries @xcite . in this letter we present a model in which the gauge couplings can naturally unify at some accessible energy , and where small calculable neutrino masses are induced by new gauge bosons exchange , in the absence of supersymmetry . neutrino masses arise at the tev scale @xcite instead of the conventional high - scale seesaw mechanism @xcite . we first recall that , by adding three gauge singlet fermions @xmath0 , the light neutrinos acquire mass only at one - loop order @xcite . unfortunately , however , unification does not occur , as can be seen in fig . [ fig : unificationplots0 ] . this is mostly due to fact that the new gauge bosons make @xmath1 weaker at high energies , while the new colored particles strengthen @xmath2 . hence we contemplate the possibility of unifying the gauge couplings in such a scheme by promoting the three fermion singlets to three octets of the enlarged electroweak symmetry . the new variant not only opens the possibility of reconciling neutrino mass generation with gauge coupling unification but also provides a novel radiative seesaw mechanism . the gauge group is broken down to the standard model at some scale @xmath3 characterizing the new gauge boson masses . this scale is found to lie in the @xmath4 range , with a plethora of new states expected to be directly accessible to lhc searches . + scale is set to @xmath5 tev . ] we consider a simple variant of the model introduced in @xcite , where the fermion singlet is now promoted to an octet representation of @xmath6 . the model is based on the same gauge symmetry , extended with a global @xmath7 , which is necessary in order to consistently define lepton number , and an auxiliary parity symmetry whose purpose will be made clear below . .[tab : field_content]field content of the model . [ cols="^,^,^,^,^,^,^,^,^,^,^,^ " , ] the model contains three generations of lepton @xmath6 anti - triplets , two generations of quark triplets and one of anti - triplets ( quarks and charged leptons are accompanied with their right - handed @xmath6 singlet partners ) , three generations of fermion octets , and finally three scalar boson anti - triplets . we summarize the particle content of the model in table [ tab : field_content ] . the allowed lepton interactions compatible with the quantum number assignments given in table [ tab : field_content ] are the following : @xmath8 the components of @xmath9 and the @xmath10 are written as : @xmath11 the scalars take vacuum expectation values ( _ vevs _ ) in the directions @xmath12 and @xmath13 . as for the octets , one can write @xmath14 such that @xmath15 is transformed into @xmath16 under an @xmath17 gauge transformation , where @xmath18 is the transformation matrix of the triplet representation . + under , each @xmath15 breaks into the representations @xmath19 , @xmath20 , @xmath21 , @xmath22 , so there are four new charged leptons ( @xmath23 , @xmath24 , @xmath25 , @xmath26 ) and four new neutral fermion states ( @xmath27 , @xmath28 , @xmath29 , @xmath25 ) in each generation . + note that the @xmath7 charge assignment of @xmath30 and @xmath31 differs from the one of the singlet of ref . @xcite so as to allow for a ( vector - like ) octet mass term , @xmath32 , in eq . [ eq : lag ] . on the other hand , the @xmath33 symmetry forbids a @xmath34 coupling , which leads to the existence of one massless neutrino state in each generation , at the tree level . tev mass . ] note also that the electric charge and lepton number assignments of the particles of the model follow from is a gauge generator , there is no physical goldstone boson associated with spontaneous lepton number violation @xcite . ] @xmath35 where @xmath36 and @xmath37 are the diagonal generators of @xmath6 . the one - loop renormalization group equation of the @xmath38 is given by @xcite : @xmath39 where @xmath40 is the logarithm of the energy scale , and the @xmath41 coefficients are functions of the casimir of the gauge group , @xmath42 , and of the dynkin index of the scalar and ( weyl ) fermion representations , @xmath43 and @xmath44 , respectively : @xmath45 values of the gauge coupling constants near unification , assuming that the 3 - 3 - 1 scale is 1 tev . the thickness of the @xmath46 line reflects the @xmath47 uncertainty in the measurement of the strong coupling constant at the @xmath48 scale @xcite , while the octet mass affects the running of @xmath49 . one sees how unification prefers @xmath50 to lie roughly between 1 and 6 . ] for the sm , the @xmath41 are @xmath51 , while in the phase they read @xmath52 , for @xmath53 active fermion octets @xmath30 . it should be noted that while we do not speculate here about the possible embedding of into some bigger group ( for example @xmath54 ) , it can be shown on very general grounds that the @xmath55 charge normalization should be @xmath56 . given the relation between @xmath57 and the sm hypercharge indicated by eq . [ eq : ql ] , it follows that @xmath58 at the 3 - 3 - 1 breaking scale . figure [ fig : unificationplots ] illustrates the running of the gauge coupling constants in our model , with the 3 - 3 - 1 scale fixed at 1 tev and the three octets @xmath30 integrated out at 3 tev . the exotic scalar states are also integrated out at the 1 tev scale , although the running of the gauge couplings is not very sensitive to this value . given that the @xmath59 coefficients are not very different from @xmath60 with the three octets , unification is sensitive mostly to the ratio @xmath50 , and not to the 3 - 3 - 1 scale _ per se_. the effect is shown in fig . [ fig : unificationplot_zoomed ] ; allowing for threshold and 2-loop effects , one can see that unification constrains @xmath61 to lie between 1 and 6 . lepton mass matrices arise from the lagrangian in eq . [ eq : lag ] after breaks to @xmath62 , @xmath63 in the basis where @xmath64 and @xmath65 , the charged leptons mass matrix reads @xmath66 where the entries are given by @xmath67 note that the _ vev _ @xmath68 ( together with @xmath69 ) sets the breaking scale . in contrast , the _ vevs _ @xmath70 and @xmath71 must lie at the electroweak scale since they belong to @xmath72 doublets after the breaking of @xmath6 . + from these expressions it follows that amongst the charged leptons , there is only a pair which is light in each generation , namely : @xmath73 where @xmath74 and @xmath75 . here the parameter @xmath76 is constrained by unification to lie in the range @xmath77 . these two 2-component states form the standard dirac charged lepton which now has a squared mass given by @xmath78 , an expression that differs from the sm one . notice that the presence of states which do not come from the @xmath79 electroweak representation is @xmath80-suppressed . the two remaining pairs of charged leptons are heavy , with octet - scale masses . + turning now to the neutral fermions , their mass matrix reads : + + in the eigenbasis @xmath81 . in the one - family approximation , @xmath82 has a null eigenvector @xmath83 with @xmath84 given as @xmath85 , where @xmath86 is some normalization factor.note that the observed neutrinos are mainly a mixture of @xmath87 and @xmath29 which are both in the @xmath79 representation of the group . the admixture of the remaining neutrino states are suppressed by at least a factor @xmath80 , which can be vanishingly small . we have verified that in the multi - generation case @xmath82 has a null eigenvector associated to each of the three generations of leptons . as a result neutrinos are massless in the tree level approximation . this property forms the basis of the radiative mechanism discussed below . + diagrams contributing to light neutrino mass . ] at the one - loop level , the exchange of gauge bosons will give rise to a dimension - nine operator which , after symmetry breaking , yields a small neutrino mass through diagrams such as those displayed in fig . [ fig : massinserionexample ] . in order to understand the result of a detailed exact calculation here we simply focus on a typical contribution illustrated in fig . [ fig : massinserionexample ] , which is found to be of the form @xmath88 where @xmath89 is some loop function with dimensions of @xmath90 , depending on the internal masses . this approximation captures the key features of the exact result such as ( @xmath91 ) the gauge nature of the underlying radiative dimension - nine seesaw mechanism , requiring two bosonic and three fermionic mass insertions in the internal lines ; as well as ( @xmath92 ) the symmetry structure , both gauge as well as lepton number , as can be seen explicitly . one also sees that no mass is generated in the limit where @xmath93 is set to zero . ] . with a _ alignment which minimizes this factor , neutrinos get a sub - ev mass even for large @xmath94 values . should @xmath95 be significantly smaller than unity , one must suppress the mixing between @xmath96 and @xmath97 , since in this case one of the neutrino mass eigenstates is mainly @xmath96 , with approximate mass @xmath98 . this is readily achieved by setting @xmath99 . in this letter we have proposed an electroweak gauge extension of the sm in which neutrinos are massless at tree - level . even though the neutral fermion mass matrix has a seesaw structure , the messengers only provide mass at the loop level , thanks to the symmetry protection . gauge mediated radiative corrections generate small calculable neutrino masses . the physics responsible for providing small neutrino masses is also responsible for gauge coupling unification , which can be achieved at a characteristic scale of order tev in the absence of supersymmetry and of gut - like interactions . a plethora of new states such as new gauge bosons and fermions makes the model directly testable at the lhc , with a non - trivial interplay between the quark sector and the lepton sector . the presence of such new features is presently under investigation . + we thank martin hirsch for valuable discussions . this work was supported by spanish grants fpa2011 - 22975 and multidark csd2009 - 00064 ( _ mineco _ ) , and prometeoii/2014/084 ( _ generalitat valenciana _ ) . fgc acknowledges support from _ conacyt _ under grant 208055 . rf was supported by _ fundao para a cincia e a tecnologia _ through grants cern / fp/123580/2011 and expl / fis - nuc/0460/2013 . 21ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop ( ) , http://stacks.iop.org/1674-1137/38/i=9/a=090001 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevd.22.738 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1007/jhep08(2014)039 [ * * , ( ) ] , link:\doibase 10.1007/jhep02(2014)112 [ * * , ( ) ] , link:\doibase 10.1007/jhep02(2013)023 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.81.053004 [ * * , ( ) ] link:\doibase 10.1007/jhep04(2014)133 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.90.073001 [ * * , ( ) ] , link:\doibase 10.1155/2014/831598 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.22.2227 [ * * , ( ) ] link:\doibase 10.1103/physrevd.90.013005 [ * * , ( ) ] link:\doibase 10.1142/s0217751x07036142 [ * * , ( ) ] , \doibase http://dx.doi.org/10.1016/0370-2693(81)90011-3 [ * * , ( ) ] link:\doibase 10.1103/physrevd.25.774 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.30.1343 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.30.1346 [ * * , ( ) ]	the physics responsible for gauge coupling unification may also induce small neutrino masses . we propose a novel gauge mediated radiative seesaw mechanism for calculable neutrino masses . these arise from quantum corrections mediated by new ( 3 - 3 - 1 ) gauge bosons and the physics driving gauge coupling unification . gauge couplings unify for a 3 - 3 - 1 scale in the tev range , making the model directly testable at the lhc .
two dimensional quantum field theories with supergroup symmetries have played an increasingly important role in our attempts to understand phase transitions in 2d disordered systems - some recent works in this direction are @xcite . these theories however prove quite difficult to tackle . attempts at non perturbative approaches using conformal invariance @xcite or exact s matrices @xcite have been popular recently , but so far , very few complete results are available . this paper is the second of a series ( started with @xcite ) on models with orthosymplectic symmetry . our goal is to relate and identify the different pieces of the theoretical puzzle available - sigma models , wess zumino witten ( wzw ) models and gross neveu ( gn ) models , integrable lattice models , and exactly factorized s matrices - and to find out which physical systems they describe , and which peculiarities arise from the existence of supergroup symmetries . in our first paper @xcite , we studied among other things the @xmath1 gross neveu model and the @xmath5 supersphere sigma model . a physical realization for the latter was identified in @xcite in terms of a lattice loop model with self intersections , based on an earlier work of @xcite . other such realizations for different models or supergroups have yet to be made . in the case of ordinary algebras , integrable lattice models do provide such realizations , and are closely related with wzw and gn models based on the corresponding groups @xcite . this relation is also important , for technical reasons , in the solution of the principal chiral models ( pcm ) @xcite . the main result of this paper is an analysis of integrable lattice models based on the @xmath6 superalgebra , and the associated field theories . while the general pattern is not unlike the case of ordinary groups , important differences are also encountered . in section 2 , we show that the continuum limit of the model based on the fundamental representation is not the gn ( or wzw model ) but the supersphere sigma model , generalizing the observation of @xcite . in section 3 and 4 we show that that , for integer spin , the continuum limit is the @xmath3 wzw model at integer level - in particular , the spin @xmath7 quantum spin chain flows to the @xmath8 level one model . this provides , to our knowledge , the first physical realization of a super wzw model . we also find that for odd spin @xmath9 , the continuum limit , like for @xmath10 , is not a wzw model . attempts are made in section 6 to identify the corresponding field theories , based on the expectation that in these cases , the orthosymplectic symmetry is realized non linearly . the @xmath3 pcm model is discussed in section 5 , and the @xmath11 models and associated parafermions in section 7 . our conventions for the @xmath6 algebra @xcite are summarized in the appendix . we start with the integrable model based on the fundamental representation @xmath12 . the highest weight vector is denoted by @xmath13 , and we shall treat it as fermionic , so the super dimension of this representation is equal to @xmath14 . ' model , and does not change any of the physical results . the grading we chose is simply more convenient , as it is well adapted to the structure of the symmetry algebra . ] the product of two spin @xmath15 representations decomposes into a spin 0 , a spin 1/2 and a spin 1 representation . their highest weights are respectively bosonic , fermionic , and bosonic . the graded permutation operator reads @xmath16 and the casimir @xmath17 the hamiltonian of the integrable model is defined on the space @xmath18 as @xcite @xmath19 ( where @xmath20 denotes the projector onto spin @xmath21 in the tensor product of the representations at site @xmath22 , @xmath23 is a normalization constant related with the sound velocity ) is integrable , and corresponds to the anisotropic limit of the integrable @xmath6 vertex model one can deduce from the scattering matrix of @xcite . the bethe ansatz equations for this model read schematically @xmath24 ( where the @xmath25 s are the roots ) and the energy @xmath26 the sign @xmath27 depends on the boundary conditions for the hamiltonian , and has not , in our opinion , always been correctly interpreted in the literature @xcite . the point is that a hamiltonian with @xmath6 symmetry will be obtained by having the last term in the sum involve the projectors @xmath28 , and identifying the states in the @xmath29 space with the ones in the first space . in the case of superalgebras , this is not exactly the same as having the projectors @xmath30 : the difference involves ` passing generators ' through the @xmath31 first states in the tensor product , and this can of course generate signs . the hamiltonian with @xmath6 symmetry corresponds to the bethe equations with @xmath32 in ( [ bethe ] ) . this agrees with the original results in @xcite . antiperiodic boundary conditions for the fermions would correspond to @xmath33 instead . according to martins @xcite , when @xmath32 , the ground state of the @xmath34 coincides with the one of the @xmath35 sector , leading to a degeneracy of 4 for the state @xmath36 . the central charge read in that sector is @xmath37 . the total partition function ( that is , the trace of @xmath38 , @xmath39 the momentum , and for @xmath32 again ) reads from @xcite @xmath40 this is in agreement with the interpretation of the low energy limit of this lattice model with a symplectic fermion theory , as was proposed in @xcite . in the latter paper , this identification was made by using the fact that the hamiltonian is the anisotropic limit of a vertex model which can be reinterpreted as a loop model , and thus as a model of classical @xmath1 spins in two dimensions , similar to the one used in the analysis of the usual @xmath41 model . it was then argued that the integrable hamiltonian lies in the broken symmetry goldstone phase , and that the low energy limit is the weak coupling limit of the supersphere sigma model , whose target space is @xmath42 $ ] ( the equivalent of @xmath43 for @xmath44 ) . recall one can easily parametrize this target space using @xmath45 such that @xmath46 . the sigma model action ( boltzmann weight @xmath47 ) is @xmath48\label{basicsigma}\ ] ] with the beta function @xmath49 . at small coupling , the action reduces to the symplectic fermions theory , and the partition function ( [ parfun ] ) coincides with the determinant of the laplacian with periodic boundary conditions in the space direction and antiperiodic boundary conditions in the `` time '' direction ( along which the trace is taken ) . for @xmath50 negative , the model flows to weak coupling in the uv , and is massive in the ir , where symmetry is restored . the action reads then , in terms of the fermion variables , and after trivial rescalings , @xmath51\ ] ] notice that the relative normalization of the two terms can be changed at will by changing the normalization of the fermions . the relative sign can also be changed by switching the fermion labels @xmath52 . however , the sign of the four fermion term can not be changed , and determines whether the model is massive or massless in the ir . for @xmath50 positive , the model flows ( perturbatively ) to weak coupling in the ir . this is the case of the lattice model introduced in @xcite . it is possible to generalize the integrable model by introducing heterogeneities in a way well understood for ordinary algebras @xcite . in doing so , the source term in the equations ( [ bethe ] ) is replaced by @xmath53 where @xmath54 is a parameter measuring heterogeneities , and the energy becomes @xmath55 we will not discuss complete calculations here , but simply derive some essential features of the associated thermodynamics bethe ansatz ( tba ) . the ground state is made of real particles , and excitations are holes in the ground state . after introducing the frourier transforms @xmath56 the physical equations read @xmath57 and the energy , up to a constant @xmath58 the interesting way to proceed then is to take the limit @xmath59 , @xmath60 ( @xmath61 the lattice spacing ) , such that @xmath62 finite . we then take the limit @xmath63 with @xmath64 finite . in that limit , excitations at finite rapidity acquire a relativistic dispersion relation , with rapidity @xmath65 . the scattering of these excitations with themselves corresponds to the @xmath66 matrix element : @xmath67\label{sigma0elt}\ ] ] and the latter coincides with @xmath68 , the scattering matrix element of particle 1 with itself in the sigma model ( [ basicsigma ] ) , as discussed in @xcite ( this matrix element is called @xmath69 there ) instead . ] . in fact , one can check that the thermodynamics of the spin chain , in this limit , coincides with the thermodynamics of the field theory for the supersphere sigma model discussed in @xcite : the introduction of heterogeneities provides thus a regularization of this field theory . as always - and this can be related @xcite to the nielsen - ninomiya theorem @xcite - the massive degrees of freedom near vanishing bare rapidity in the model with heterogenities are completed by massless degrees of freedom at large bare rapidities ( edges of the brillouin zone ) . these are the same massless modes that would be present in the homogeneous chain obtained by letting @xmath70 . the dynamics of these massless modes decouples entirely from the dynamics of the massive ones , and one can identify the associated cft with the weak coupling limit of the supersphere sigma model , that is , the symplectic fermion theory . it is tempting to carry out the same procedure for the case of higher spin . unfortunately , not much is known about the higher spin integrable @xmath71 spin chains in explicit form . it is fair to expect , based on analogies with other cases - in particular the @xmath72 case - that such chains do exist , and are described by changing the source terms and energy terms as @xmath73 where @xmath9 is the higher spin . the thermodynamics of the massive field theory limit is described by the equations @xmath74 where @xmath75 and @xmath76 . the boundary condition @xmath77 must be imposed . the free energy reads then @xmath78 the thermodynamics of the lattice model is described by similar equations , but different source terms . it allows one in particular to determine the entropy per site of the chain in the large @xmath79 limit . one finds that this entropy corresponds , for @xmath9 half integer , to a mix of representations @xmath80 , and for @xmath9 integer , a mix of representations @xmath81 . the integrable models must therefore involve these mix of representations on every site , and presumably must be considered as having @xmath71 super - yangian symmetry , in analogy with the @xmath72 case @xcite . in particular , the extension of the adjoint by a scalar representation to form an irreducible representation of the yangian is typical . calculations with a twist angle giving antiperiodic boundary conditions to the kinks shows that the representations with half - integer spin have superdimension @xmath14 , while those with integer spin have superdimension @xmath82 . some of these results have been obtained independently and using a different approach in @xcite . it is easy to check that the central charge of these models is @xmath83 as in the usual @xmath84 case , one can deform the models by considering @xmath85 matrices with @xmath86 symmetry , and one can truncate them in the case @xmath87 a root of unity . the resulting tba s have the form shown in figure 1 ( with a total number of nodes equal to @xmath31 ) , and central charge @xmath88 most of the following is devoted to understanding the field theories associated with ( [ firstceff ] ) and ( [ secondceff ] ) . the basic field theory we have introduced so far is the @xmath89 non linear sigma model ( [ basicsigma ] ) . another type of sigma model plays a major role in the analysis : the @xmath90 wess zumino witten model . details about @xmath91 and @xmath8 are furnished in the appendix : the bosonic part of @xmath3 is @xmath92 , and the group is compact . the level @xmath93 is quantized ( for the normalization of @xmath93 , we use the level of the sub @xmath92 , like for instance in the works @xcite . the same model would be called the @xmath94 model following the conventions used in the literature on disordered systems ( see eg @xcite , as well as in our previous paper ) . the model is not expected to be a unitary conformal field theory : this is clear at the level of the action , where for instance the purely fermionic part is closely related to the @xmath95 system , a non unitary theory . this is also expected on general grounds , since , for instance , there is no way to define a metric without negative norm ( square ) states in some representations . it turns out however that the @xmath90 wzw theories are relatively simple , at least at first sight . the best way to understand them is to use a remarkable embedding discovered by fan and yu @xcite . these authors made the crucial observation that @xmath97 where the branching functions of the latter part define a virasoro minimal model , with @xmath98 only for @xmath93 an integer does the action of the wess zumino model make sense , and we will restrict ourselves to this case in the following . the virasoro models which appear there have @xmath99 ; they are non unitary , and their effective central charge is @xmath100 . these models can thus be considered as @xmath101 coset models ! the perturbation of these models by the operator @xmath102 ( here , the labels refer to the description as a virasoro minimal model ) with dimension @xmath103 is well known to be integrable ( the @xmath7 comes from the @xmath91 , the @xmath104 from the @xmath92 ) . the tba has the form shown in figure 2 @xcite . as observed in @xcite , it can be obtained after a q - deformation and a truncation of the basic supersphere sigma model tba . the corresponding s matrices can thus easily be deduced , and follow rsos restrictions of the q - deformed @xmath105 s matrices , or , equivalently , q - deformed @xmath106 s matrices . the simplest and most interesting case corresponds to the model of virasoro minimal series @xmath107 . its central charge is @xmath108 while @xmath109 . the tba for a perturbation by the operator @xmath102 of weight @xmath110 is described by the diagram in the figure in the particular case where the number of nodes is two . the s matrix has been worked out in details in @xcite . an amusing consequence of this observation is that the supersphere sigma model appears as the limit @xmath111 of a series of coset models . this is quite similar to the way the ordinary sphere sigma model appears as the limit of a series of parafermion theories @xcite , this time of type @xmath112 . an important difference between the two cases is that , since the three point function of @xmath102 vanishes , the perturbation of the coset models is independant of the sign of the coupling , and thus always massive . the situation was different in the case of parafermionic theories @xmath113 , where one sign was massive ( and corresponded , in the limit @xmath111 , to the case @xmath114 ) , but the other was massless @xcite ( and corresponded in the limit @xmath111 , to the case @xmath115 ) . for the supersphere , there is no theta term , so it is natural that we get only one flow . for @xmath116 , @xmath117 for @xmath118 . ] an interesting consequence of the embedding is that we can deduce the effective central charge of the @xmath3 wzw model at level @xmath93 . using that for the virasoro model , @xmath119 , one finds @xmath120 this result will be compatible with all the subsequent analysis , but it is in slight disagreement with @xcite . in the latter papers , conjectures are made that the spectrum closes on primary fields of spin @xmath121 with dimension @xmath122 . if this turned out to be true , the models we identify would not exactly be the wzw models , but maybe some `` extensions '' of these - at the present time , this issue is not settled , but it seems simpler to assume the value ( [ ceff ] ) is indeed the effective central charge of the wzw model . we consider now tba s with a total number of nodes @xmath124 . if the massive node is the @xmath125 one , the uv central charge is @xmath126 suggesting that the model can be understood as a coset model @xmath127 . assuming the tba corresponds to a theory perturbed by an operator whose odd point functions vanish , we find the dimension of the perturbing operator to be @xmath128 . this is compatible with taking the spin @xmath15 field in the denominator of the coset . if the massive node is the @xmath129 one meanwhile , the central charge is @xmath130 suggesting similarly that the model can be understood as a coset @xmath131 perturbed by the operator of dimension @xmath128 . of course the two cases are actually equivalent by taking mirror images , but it is convenient to keep them separate to study the large @xmath132 limit later . we now consider instead tba s with a total number of nodes @xmath133 . if the massive node is the @xmath125 one , the uv central charge is found to be @xmath134 suggesting that the models can be interpreted as coset @xmath135 . assuming the tba corresponds to a theory perturbed by an operator whose odd point functions do not vanish , we find the dimension of the perturbing operator to be @xmath136 . this is compatible with taking the spin @xmath15 field in the denominator of the coset . note that , since we have assumed the three point function of the perturbing operator does not vanish , switching the sign of the perturbation should lead to a different result . it is natural to expect that one has then a massless flow , whose tba and s matrices are readily built by analogy with the @xmath92 case @xcite : we leave this to the reader as an exercise . finally , we notice that the @xmath3 coset model with @xmath137 was first identified in the paper @xcite . the last possible case we can obtain out of this construction corresponds to a tba s with an odd number of nodes ( say , @xmath138 ) , and the mass on an odd node , too . the effective central charge is @xmath139 . the models can be considered as virasoro models with @xmath140 , and the tba corresponds to perturbation by the @xmath141 field now , of dimension @xmath142 . we have not found any convincing way to interpret this in terms of @xmath1 cosets ; maybe it is not possible . notice that the @xmath143 is a weight for @xmath144 , which , since it appears with a minus sign in @xmath145 , should be in the denominator of the sought after coset . notice also that , by using the remark at the end of the previous paragraph , we expect flows between the models we have interpreted in terms of @xmath1 and @xmath92 cosets and these unidentified models . this could be a useful hint . taking @xmath146 for the class of models where the massive node is an even one , we obtain theories with central charge @xmath147 . this value coincides with the result obtained in the first section for @xmath148 . we therefore suggest that the continuum limit of the lattice models with _ integer spin _ @xmath9 are the @xmath149 models . introducing heterogeneities then gives rise to the current - current perturbation of these models . the s matrix is the tensor product of the rsos s matrix for the virasoro model @xmath150 perturbed by @xmath102 ( which we saw can be reinterpreted as an @xmath8 rsos matrix ) and the supersphere sigma model s matrix . these results apply to the ns sector of the model , where the fermionic currents have integer modes , and are periodic . the ramond sector can be obtained by spectral flow ; one has in particular @xcite @xmath151 while the true central charge seems inaccessible from the tba , one can follow the spectral flow by giving a fugacity to the solitons , as was discussed in our first paper , ie calculating @xmath152 $ ] , where @xmath87 is the topological charge of the solitons , normalized as @xmath153 . antiperiodic boundary conditions correspond to @xmath154 , and are found to give , using the system of equations ( 38,39 ) of our previous paper @xmath155 in agreement with ( [ spectralflow ] ) . finally , it is easy to check from the tba that the dimension of the perturbing operator has to be @xmath156 . this gives strong support to our conjecture . we stress that , as far as we know , none of the perturbed @xmath90 wzw models can be interpreted as a gross - neveu model . the @xmath91 gn models correspond to models with , formally , level @xmath157 , and have a different physics , and different scattering matrices , as discussed in @xcite . we will get back to this issue in the conclusion . if we take the limit @xmath146 for models which have the mass on an odd node , the central charge as well as the interpretation of the coset models are consistent with a theory of the form @xmath158 , of which the supersphere sigma model was just the simplest ( @xmath159 ) version . it would be most interesting to find out the action describing these models , but we have not done so for now - we will comment about the problem below . in the @xmath92 case for instance , the limit @xmath111 of the wzw model with a current current perturbation coincides with the scattering theory for the pcm ( principal chiral model ) model @xcite . it is natural to expect that the same thing will hold for the @xmath3 case . the tba looks as in figure 8 , and the scattering matrix has obviously the form @xmath160 , where @xmath66 is the s matrix for the supersphere sigma model , up to cdd factors we will discuss below let us study this pcm model more explicitely . it is convenient to write an element of @xmath3 as @xmath161 with the constraint @xmath162 . in a similar way , the conjugate of the matrix , @xmath163 , reads @xmath164 the action of the pcm model reads , after a rescaling of the fermions @xmath165 @xmath166 we note that the @xmath3 group manifold can be identified with the supersphere @xmath167@xcite , that is , the space @xmath168 . the pcm model , however , can not be expected to coincide with the sigma model on @xmath167 : the symmetry groups are different , and so are the invariant actions . for instance , in the pcm model , the group @xmath3 acts by conjugation , leaving the identity invariant . in the vicinity of the identity , under the @xmath169 , the fermionic coordinates transform as a doublet , and the bosonic coordinates transform as a triplet . in the sigma model , the coordinates near the origin transform as the fundamental of @xmath170 . under the @xmath171 of the @xmath170 , the bosonic coordinates transform as a triplet _ but _ the fermionic coordinates now transform as a singlet ( they form a doublet under a different @xmath172 , which leaves the sphere @xmath173 invariant ) . the groups acting differently , the invariant actions can be expected to be different . this is confirmed by explicit calculation . the supersphere @xmath167 can be parametrized in terms of coordinates @xmath174 , @xmath175 and @xmath176 . the constraint @xmath177 gives rise to @xmath178 the sigma model action @xmath179 becomes then @xmath180 the two equations ( [ eqi],[eqii ] ) are similar , but exhibit a major difference in the sign of the four fermion term . the physics of the two models is considerably different . for the supersphere sigma model , the @xmath181 function is exactly zero to all orders , and the theory is exactly conformal invariant for any value of the coupling constant ( like in the @xmath182 case ) . for the pcm , the @xmath181 function follows from wegner s calculations in the case @xmath183 @xcite @xmath184 to be compared eg with the @xmath92 case @xmath185 the conventions here are that the boltzmann weight is @xmath186 , and @xmath187={1\over 2\lambda } \int tr \left[g^{-1}\partial_\mu g\right]^2\ ] ] in the @xmath92 case , the massive theory corresponds to @xmath188 . by contrast , for the @xmath1 case , the massive direction corresponds to @xmath189 . however , since one takes then a supertrace instead of a trace , the @xmath92 part of the pcm action has the _ same _ sign as in the @xmath92 pure case , with boltzmann weight @xmath190 $ ] , and the functional integral is well defined . note that the symplectic fermion part of the boltzmann weight is of the form @xmath191 $ ] , and also exhibits the same sign as the action of the supersphere sigma model in the massive phase ( where the symmetry is restored ) . the exact s matrix can be deduced from the tba by noticing that , for the matrix @xmath192 , the presence of the self coupling for the first node in the sigma model tba would lead to a _ double _ self coupling . this has to be removed , and the usual calculation gives @xmath193 where the cdd factor @xmath194 , @xmath195 cancels the double poles and double zeroes in @xmath196 ( [ sigma0elt ] ) . let us recall for completeness the sigma model s matrix . @xmath197 where we have set @xmath198 while @xmath39 is the graded permutation operator @xmath199 the indices @xmath200 take values in the fundamental representation of the @xmath6 algebra , @xmath201 . we set @xmath202 , @xmath203 . the factors @xmath204 in ( [ maini ] ) read @xmath205 for the value @xmath206 characteristic of the @xmath1 case . in section 4 , we have found two families of models whose @xmath66 matrix has @xmath3 symmetry . the models based on the lattice tba for @xmath9 integer correspond to @xmath149 wzw models peturbed by a current current interaction . the uv theory is a current algebra , in which the symmetry is locally realized by two sets of currents , @xmath207 and @xmath208 . what happens in the other family of models is less clear . an exception to this is the case @xmath209 , ie the @xmath210 sigma model . in this case , the symemtry is realized non linearly , and it is worthwhile seeing more explicitely how this works . consider thus the supersphere sigma model . this model for positive coupling describes the goldstone phase for @xmath1 symmetry broken down spontaneously to @xmath172 ( possible since the group is not unitary compact ) . for negative coupling , it is massive , and the @xmath1 symmetry is restored at large distance . in either case , the action is proportional to ( we have slightly changed the normalizations compared with the previous paper ) @xmath211 with @xmath212 . we can find the noether currents with the usual procedure . an infinitesimal @xmath1 transformation reads x&=&-_1_1+_2_2 + _ 1&=&-_2x+a_1 + c_2 + _ 2&=&-_1x+b_1-a_2 where @xmath213 are ` small ' fermionic deformation parameters , @xmath214 small bosonic parameters . by definition , this change leaves @xmath215 invariant . in terms of the fermion variables , the symmetry is realized non linearly : _ 1=-_2(1-_1_2)+ a_1+c_2 + _ 2=-_1(1-_1_2)+ a_1-a_2 performing the change in the action , and identifying the coefficients of linear derivatives @xmath216 with the currents gives five conserved currents . three of them generate the sub @xmath84 : j^+&=&-12_1_1 + j^-&=&12_2_2 + j^3&=&14(_1_2-_1_2 ) the two fermionic currents meanwhile are j^+=_x _ 1-x _ _ 1= _ _ 1(1 - 2_1_2 ) + j^-=_x _ 2-x _ _ 2=__2(1 - 2_1_2 ) these five currents should be present in the uv limit of the sigma model , which coincides with symplectic fermions . the latter theory has been studied a great deal . of particular interest is the operator content , which is conveniently encoded in the generating function ( [ parfun ] ) . recall that the `` ground state '' ( that is , fields of weight @xmath217 ) is degenerate four times , while there are eight fields of weight @xmath218 ( and eight fields of weight @xmath219 ) . it has sixteen fields of weight @xmath156 . we can understand these multiplicities easily by using the sigma model interpretation . from the @xmath1 symmetry , we expect to have , by taking the weak coupling limit of the foregoing currents , five fields @xmath218 and five fields @xmath219 ( these fields are not chiral currents , because of some logarithmic festures : more about this below ) . meanwhile , the broken @xmath1 symmetry implies the existence of three non trivial fields with weight @xmath217 , whose derivatives are also necessarily ` currents ' . we therefore expect _ eight _ fields ( @xmath220 fundamental @xmath221 adjoint ) @xmath218 and @xmath219 , in agreement with the known result . note that fields with weights @xmath218 and @xmath219 can have some common components due to the presence of fields with vanishing weights . it follows that many of their products do actually vanish , leading to a multiplicity of sixteen for fields @xmath156 , and not @xmath222 , as one could have naively assumed . an interesting question is now what remains of the @xmath1 symmetry _ right at the weak coupling fixed point _ , that is , in the symplectic fermions theory itself . there , it turns out that only the sub @xmath172 can be observed , as the bosonic currents @xmath223 are still conserved in the symplectic fermion theory . this conservation boils down to the equations of motion @xmath224 . if one naively tries to check the conservation of the fermionic currents , it seems one needs @xmath225 , which is manifestly wrong ! so these currents , which are conserved in the sigma model at any non zero coupling , are not strictly speaking conserved right at the weak coupling fixed point . the explanation of this apparent paradox lies in the role of the coupling constant and how exactly one can obtain the conformal limit . the best is to take the boltzmann weight as @xmath47 with @xmath66 as above , @xmath226 and put the coupling constant in the radius of the supersphere @xmath227 , which now leads to @xmath228 . the equations of motion are _ ^ x&=&x + _ ^ _ 1&=&_1 + _ ^ _ 2&=&_2 where @xmath229\ ] ] leading , as usual , to the conservation of @xmath230 . the conformal symplectic fermion theory is then obtained in the ( singular ) limit @xmath231 , where the field @xmath232 formally becomes a constant , and @xmath233 a triviality . within this limit , the @xmath1 symmetry is lost , but one gets as its remnant the two fermionic `` currents '' , @xmath234 and @xmath235 . it is interesting finally to discuss the algebra satisfied by the @xmath172 currents right at the conformal point ( a related calculation has been presented in @xcite , but we do not think its interpretation - based on rescaling the currents- is appropriate ) . the ope s are rather complicated : j^+(z)j^-(w)&=&141 + 2\|z - w\|+_2_1(z - w)^2 -18 ( _ 1_2 ) + \|z-\|wz - w\|(_1_2)z - w -12\|z - w\|_1_2 + & + & 32j^3 + 1 2z - w\|z-\|z\|j^3z - w + j^3(z)j^(w)&= & + j^3(z)j^3(w)&=&181 + 2\|z - w\|+_2_1(z - w)^2 -116 ( _ 1_2 ) + \|z-\|wz - w\|(_1_2)z - w -14\|z - w\|_1_2and we see that the notation @xmath236 is abusive : the field has weights @xmath218 but the opes involve @xmath237 terms . the commutators of charges are only affected by the @xmath238 term , and the @xmath84 relations are recovered not through a rescaling but because of the presence of other non trivial opes between the ` left ' and ` right ' components . for instance , writing only the relevant term , one has j^+(z)\|j^-(w)&=&12(z - w)\|j^3 + 1 2(\|z-\|w)j^3 + where @xmath239 . amusingly , the @xmath240 part of the opes corresponds to the normalization @xmath241 , so the uv limit of the sigma model does contain a `` logarithmic @xmath242 '' @xmath84 current algebra . the evidence from the tba is that the pcm model can give rise to two kinds of models ( more on this in the conclusion ) : either the @xmath2 wzw models like in the usual case , but also the @xmath243 model , which presumably involves some sort of term changing the @xmath92 part of the action into the wzw one with a current current perturbation , but leaving the symplectic fermionic part essentially unaffected . we do not know how to concretely realize this though . another interesting aspect stems from the fact that the central charge obtained by giving antiperiodic boundary conditions to the kinks reads , after elementary algebra , @xmath250 this is precisely the central charge of the models @xmath251 , of which the first two have @xmath252 and @xmath253 . we are thus led to speculate that the @xmath251 models - or rather , their proper ` non minimal ' versions ( studied in @xcite , although we do not necessarily agree with the conclusions there ) , as the minimal models are entirely empty in this case , are models with spontaneously broken @xmath3 symmetry . it would be very interesting to look further for signs of an @xmath8 structure in these models , and to study their ` logarithmic ' @xmath92 algebra . note that these models are obtained by hamiltonian reduction of the @xmath244 model . in this reduction @xcite , an auxiliary @xmath95 system is introduced to play the role of fadeev - popov ghosts , so these models are indeed naturally related to the product of @xmath244 and @xmath254 as we observed earlier . instead of factoring out the @xmath92 , one can of course also factor out the @xmath254 and get an @xmath11 sigma model . this is especially interesting since the standard argument to derive the continuum limit of the @xmath6 spin chains would lead to a sigma model on the manifold parametrizing the coherent states , and this is precisely @xmath11 @xcite . note however that the manifold @xmath11 is not a symmetric ( super ) space ( this can easily be seen since the ( anti ) commutator of two fermionic generators does not always belong to the lie algebra of @xmath254 ) . as a consequence , sigma models on this manifold will have more than one coupling constant . to proceed , a possible strategy is to follow @xcite and consider for a while models @xmath255 , that is graded parafermionic theories . graded parafermions @xcite theories are constructed in a way similar to the original construction of fateev and zamolodchikov , with the additional ingredient of a @xmath256 grading . they obey the ope rules @xmath257 their dimensions are @xmath258 , where @xmath32 , @xmath132 half an odd integer , @xmath259 otherwise . of particular interest is the ope @xmath260\nonumber\\ \psi_{1}(z)\psi_{-1}(w)&=&(z - w)^{{2\over k}-2 } \left[1+(z - w)^2 o^{(1)}+\ldots\right]\nonumber\\\end{aligned}\ ] ] here , the operators @xmath261 have dimension 2 , and must obey @xmath262 , @xmath79 the stress energy tensor . the simplest parafermionic theory for @xmath263 has @xmath264 , and seems to coincide with the model @xmath265 . can be represented in terms of a free boson , the cosets @xmath266 and @xmath267 are equivalent there . ] for @xmath93 an integer , @xmath132 runs over the set @xmath268 , @xmath269 . parafermions with integer @xmath132 are bosonic , the others are fermionic . for @xmath263 , @xmath270 , and there is only a pair of parafermionic fields , of weight @xmath271 . it can be shown that the parafermionic theories just defined coincide with @xmath272 coset theories . like in the @xmath273 case , the @xmath90 model with a current - current pertubation can be written in terms of the graded parafermions and a free boson @xmath274 . it is then easy to find an integrable anisotropic deformation @xmath275 ( in the case @xmath263 , the perturbation reads @xmath276 . ) the non local conserved currents @xcite are @xmath277 and @xmath278 ( where @xmath279 denotes the right component of @xmath280 ) . the tba and s matrices are rather obvious : we take the same left part of the diagram as for the @xmath281 case , but replace the infinite right tail by the ubiquitous , finite and anisotropic part discussed in our first paper . in the isotropic limit @xcite @xmath282 , the rg generates the other terms necessary to make ( [ parafpert ] ) into a whole current current perturbation . taking the limit @xmath283 would then lead to the tba for the parafermionic theory . this would require an understanding of the scattering in the attractive regime where bound states exist , but we have not performed the related analysis . it is possible however to make a simple conjecture based on numerology , and analogies with the @xmath92 case . consider indeed the tba in figure [ fig8 ] where the box represents the set of couplings discussed in our first paper @xcite . in the uv , the diagram is identical to the one arising in the study of the @xmath105 toda theory . the central charge is @xmath284 as discussed in @xcite . in the ir , the diagram is identical to the ones arising in the @xmath101 coset models , and @xmath285 . the final central charge is thus @xmath286 , and concides with the effective central charge for @xmath11 parafermions of level @xmath93 . we conjecture this tba describes the perturbation of these parafermionic theories by the combination of graded parafermions @xmath287 the effective dimension of the perturbation deduced from the tba is @xmath288 , and this coincides with the combination @xmath289 . note that we have not studied what kind of scattering theory would give rise to the tba in figure [ fig8 ] , and whether it is actually meaningful . still , taking the limit @xmath111 , we should obtain the tba for something that looks like an @xmath11 sigma model . notice that the bosonic part of this model is identical with the @xmath113 sigma model , and thus there is the possibility of a topological term . it is not clear what the low energy limit of the model with topological angle @xmath115 would be . the results presented here presumably have rather simple generalization to the @xmath290 case , even though details might not be absolutely straightforward to work out - for instance , we do not know of embeddings generalizing the one discussed in the first sections . the supersphere sigma model for @xmath50 positive in the conventions of section 2 , flows in the ir to weak coupling , at least perturbatively . it is expected that the phase diagram will exhibit a critical point at some value @xmath291 and that for larger coupling , the theory will be massive . the critical point presumably coincides with the dilute @xmath292 theory first solved by nienhuis @xcite . this theory is described by a free boson with a charge at infinity , and is closely related with the minimal model @xmath265 . in fact the partition function of the dilute @xmath292 model provided one restricts to _ even _ numbers of non contractible loops can be written in the coulomb gas language of di francesco et al . @xcite as @xmath293\ ] ] and coincides with the partition function of the minimal model . earlier in this paper , we have identified this model with the @xmath294 parafermionic theory . the full @xmath292 theory , however defined , has a considerably more complex operator content @xcite . note that antiperiodic boundary conditions for the fermions , which give an effective central charge equal to @xmath295 in the supersphere sigma model give , in the critical theory , a highly irrational value @xmath296 . there are no indications that an integrable flow from the critical theory to the low temperature generic theory exists . an integrable flow is known to exist in the special case where the symmetry is enhanced to @xmath297 . in that case , the ir theory is the so called dense @xmath292 model , which has @xmath253 , and is closely related with the minimal model @xmath298 . note that this model is the second model of the unidentified series in section 4 , and bears some formal resemblance to the model @xmath299 . what this means remains one of the many open questions in this still baffling area . * acknowledgments : * we thank g. landi , n. read and m. zirnbauer for useful remarks and suggestions . we especially thank p. dorey for pointing out the discussion of @xmath91 coset models in @xcite . the work of hs was supported in part by the doe . we collect in this appendix some formulas about @xmath6 , the associated current algebra and groups . the supergroup @xmath1 is the group of ` real ' matrices @xmath50 obeying ( basic references are @xcite ) @xmath300 where a bosonic matrix , @xmath301 , recall that @xmath302 . ] @xmath303 elements of the group preserve the quadratic form , if @xmath304 , @xmath305 . they can be parametrized by @xmath306 with @xmath307 here no complex conjugation is ever needed : @xmath308 are real numbers , and @xmath176 are ` real ' grassman numbers . the group @xmath3 in contrast is made of complex supertransformations satisfying @xmath309 to define the adjoint @xmath310 , we first need to introduce a complex conjugation denoted by @xmath311 . it is , technically , a graded involution , which coincides with complex conjugation for pure complex numbers , @xmath312 , @xmath313 , and obeys in general with the usual conjugation . ] @xmath314 one then sets @xmath315 operation obeys the usual properties , @xmath316 . it can be considered as the combination of the @xmath317 operation in the lie algebra ( see the appendix ) , and the @xmath311 operation on ` scalars ' . ] , so @xmath50 in @xmath3 preserves in addition the form @xmath318 . one has now @xmath306 with @xmath319 with @xmath61 real , @xmath320 . the fermionic content of the supergroup is essentially unchanged , with @xmath321 , @xmath322 . but the bosonic content is different : the non compact bosonic subgroup @xmath172 has been replaced by the compact one @xmath92 . the algebra @xmath6 is generated by operators which we denote @xmath323 ( bosonic ) and @xmath230 ( fermionic ) . their commutation relations can be obtained from the current algebra given below by restricting to the zero modes . the casimir reads @xmath324 the representations of the super lie algebra are labelled by an integer or half integer @xmath21 , and are of dimension @xmath325 . the fundamental representation is three dimensional , and has spin @xmath326 . it does contain a sub @xmath327 fundamental representation , following the pattern of @xmath328 . the generators @xmath329 are bosonic . the fermionic generators are given by @xmath330 the only metric compatible with @xmath6 requires the definition of a generalized adjoint satisfying ( here @xmath331 denotes the parity ) @xcite @xmath332 and thus @xmath333 it follows that @xmath334 , @xmath335 , while there remains some freedom for the fermionic generators , @xmath336 , @xmath337 . it is in the nature of the algebra that negative norm square states will appear whatever the choice . indeed , let us choose for instance @xmath338 it then follows that the norm square of the state @xmath339 is @xmath340 here , @xmath341 if the highest weight state @xmath342 is bosonic , @xmath343 if it is fermionic . even if we start with the fundamental representation @xmath326 with @xmath13 bosonic , in the tensor product of this representation with itself , representations where the highest weight is fermionic will necessary appear . these do contain negative norm square states . in this paper , we will always choose the gradation for which @xmath13 is fermionic , and thus the fundamental representation has superdimension equal to @xmath14 . the current algebra is defined by @xmath344=\pm j^\pm_{n+m } & \left[j_n^3,j_m^0\right]={k\over 2 } n\delta_{n+m}\nonumber\\ \left[j_n^+,j_m^-\right]&=kn\delta_{n+m}+2j^3_{n+m}\nonumber\\ \left[j_n^3,j_m^\pm\right]=\pm { 1\over 2}j^\pm_{m+n } & \left[j_n^\pm , j_m^\pm\right]=0\nonumber\\ \left[j_n^\pm , j_m^\mp\right]=-j_{n+m}^\pm & \left\{j_n^\pm , j_m^\pm\right\}=\pm 2j_{n+m}^\pm\nonumber\\ \left\{j_n^+,j_m^-\right\}&=2kn\delta_{n+m}+2j_{n+m}^3\end{aligned}\ ] ] normalizations are such that the algebra contains a sub @xmath327 current algebra at level @xmath93 . as commented in the text , the supersphere @xmath167 is the supermanifold of the supergroup @xmath3 . it is also the total space of a principal fibration with structure group @xmath254 and the quotient of this action is just the supersphere @xmath345 . the explicit realization is as follows @xcite . setting @xmath346 ( these obey @xmath347 , and @xmath348 ) we obtain points in @xmath349 , since @xmath350 . conversely , for a given point @xmath351 of @xmath349 one gets @xmath352\nonumber\\ bb^\diamond&=&{1\over 2}\left[1-x_0(1+{1\over 2}\eta_1\eta_2)\right]\nonumber\\ ab^\diamond&=&{1\over 2}(x_1-ix_2)(1+{1\over 2}\eta_1\eta_2)\nonumber\\ \eta a^\diamond&=&-(x_1+ix_2)\eta_1+(1+x_0)\eta_2\nonumber\\ \eta b^\diamond&=&(x_1-ix_2)\eta_2-(1-x_0)\eta_1\end{aligned}\ ] ] define finally @xmath353 . since the parametrization of ( [ sphparam ] ) is invariant under @xmath354 , this proves the statement . i.p.ennes , a.v . ramallo and j. m. sanchez de santos , `` osp(1/2 ) conformal field theory '' , hep - th/9708094 , in `` trends in theoretical physics ( la plata , 1997 ) '' , conf . 419 , amer . woodbury , ny , ( 1998 ) ; nucl . b491 ( 1997 ) 574 ; nucl . b502 ( 1997 ) 671 ; phys . b389 ( 1996 ) 485 .	as a step to understand general patterns of integrability in @xmath0 quantum field theories with supergroup symmetry , we study in details the case of @xmath1 . our results include the solutions of natural generalizations of models with ordinary group symmetry : the @xmath2 wzw model with a current current perturbation , the @xmath3 principal chiral model , and the @xmath4 coset models perturbed by the adjoint . graded parafermions are also discussed . a pattern peculiar to supergroups is the emergence of another class of models , whose simplest representative is the @xmath5 sigma model , where the ( non unitary ) orthosymplectic symmetry is realized non linearly ( and can be spontaneously broken ) . for most models , we provide an integrable lattice realization . we show in particular that integrable @xmath6 spin chains with integer spin flow to @xmath3 wzw models in the continuum limit , hence providing what is to our knowledge the first physical realization of a super wzw model . # 1#2#3#4 # 1#2#3#4
thermal radiation from the surfaces of neutron stars ( ns ) provides important information about these elusive compact objects . in particular , neutron stars thermal histories are already being probed by x - ray observations ( gelmann 1995 ) ; these data place useful constraints on the interior physics and the equation of state of matter at super - nuclear densities ( tsuruta 1995 ) . significant limits on heating processes such as precipitation of magnetospheric particles and rotational energy dissipation in the crust can also be obtained . to interpret the observed x - ray fluxes , it is often assumed that the ns spectrum is blackbody . however in general we expect the emergent flux to be reprocessed by the ns atmosphere , and romani ( 1987 ) showed that the departures from blackbody emissivities in the observed bands can be quite substantial . in romani ( 1987 ) opacities from the los alamos opacity library ( laol ) ( huebner et al . 1977 ) were used to generate model atmospheres and emergent spectra for non - magnetic neutron stars with a variety of surface compositions . more recent work ( shibanov et al . 1992 ; potekhin & pavlov , 1993 ) has treated the equation of state and opacity of pure h atmospheres in strong ( @xmath1 g ) magnetic fields . this allowed pavlov et al . ( 1995 ) to produce atmosphere models for high field ns ; substantial departures from blackbody spectra were found , albeit not as large as in the non - magnetic case . since the atomic composition of a ns surface is quite uncertain it is also important to treat heavy element atmospheres . miller & neuhauser ( 1991 ) have calculated wave functions and energies of some magnetized heavy elements , and miller ( 1992 ) produced polarization averaged bound - free cross sections and computed approximate model atmospheres . however , separate transport of the polarization modes dramatically affects the emergent spectrum ( pavlov et al . 1995 ) , while at zero field bound - bound opacity can exceed bound free by large factors ( iglesias , rogers & wilson 1992 ) , so these heavy element atmospheres may not adequately model the emergent spectrum . thus since detailed treatments for elements heavier than h in strong fields are not yet available , comparison with low - field results remains useful . to delimit the range over which our non - magnetic atmospheres can be applied , we must estimate the field required to perturb the opacity significantly at frequencies of interest . in the presence of strong magnetic fields , the free - free opacities for one of two polarization modes perpendicular to the field and for radiation parallel to the field are strongly suppressed . for thompson scattering , the cross section decrease occurs below the electron cyclotron energy , i.e. @xmath2kev , where @xmath3 g ( canuto , lodenquai & ruderman , 1971 ) . for inverse bremsstrahlung suppression , both @xmath4 and @xmath5 should be less than @xmath6 ( pavlov & panov , 1976 ) ; at typical ns temperatures of @xmath7k , these effects would affect the rosat pspc band ( 0.1 - 2.4kev ) when @xmath8 . magnetic effects on the bound - free and bound - bound opacities depend on the full quantum states of magnetized atoms , but some generalizations are possible . in the case of hydrogen , detailed calculations are available ( e.g. rsner et al . 1984 ) ; for the field @xmath9 g , at which @xmath10 , these authors find the ground state binding energy @xmath11 is increased by @xmath12 . thus this bound - free feature too is altered when @xmath13 . the effect vanishes rapidly at lower fields , while for large fields we find their results give @xmath14 . to estimate the minimum perturbing field in general , consider the magnetic contribution to the hamiltonian of a hydrogenic atom ( rsner et al . 1984 ) : @xmath15 where @xmath16 , @xmath17 is the quantum number for angular momentum in the field direction , @xmath18 is for the spin , and @xmath19 ( in units of bohr radii ) is the atomic radius perpendicular to @xmath20 . here @xmath21 is the zeeman effect , while @xmath22 is sometimes referred to as the quadratic zeeman effect . for an energy level @xmath11 with principal quantum number @xmath23 and a nucleus with charge @xmath24 , we define the critical field @xmath25 to be that which gives @xmath26 . using @xmath27 and @xmath28 we find that @xmath29 ( for @xmath30 of order unity ) , and @xmath31 . when @xmath32 , @xmath33 and so the critical field scales as @xmath34 . this scaling is borne out by the results of rsner et al . ( 1984 ) for excited states of hydrogen . however , in ns atmosphere conditions only modest excitation of a species is obtained before it is further ionized , so for larger @xmath35 the perturbation @xmath21 dominates in almost all cases . its dependence implies that spectral features will be perturbed when @xmath13 , in agreement with miller & neuhauser ( 1991 ) , who find for hydrogenic atoms of atomic number z that the lowest energy levels follow @xmath36 , where @xmath37 is the corresponding hydrogen energy level at field @xmath38 . they also find the helium ground state to follow this scaling , with a screened @xmath39 . line positions , which represent a difference between two energy states , will deviate substantially from their zero field values and ordering even before @xmath6 reaches either energy , but a shell feature such as the iron l - edge at 0.7 kev in our spectra should persist to @xmath40 . though the atmospheric structure will be affected for @xmath41 as the peak flux is altered , gross opacity features above the thermal peak will persevere to higher fields . our non - magnetic atmospheres are therefore entirely suitable for low - field ( @xmath42 g ) ns such as millisecond pulsars , and indicative of some spectral features at higher fields . two developments suggest that a re - examination of the heavy element , non - magnetic ns atmospheres is timely . first , improved treatments of astrophysical opacity using the opal code ( iglesias et al . 1992 ; and references therein ) and by the op collaboration ( seaton et al . 1994 ) have shown that the laol results substantially underestimate the opacity , especially for iron - group elements in the @xmath43k temperature range . use of the new opacities already seems to resolve several puzzles in stellar astrophysics , including discrepancies between observations of cepheid pulsations and models based on laol data ( see iglesias et al . the improvements to the opacity and equation of state ( eos ) are particularly important to the ns spectrum problem , as the photosphere in these atmospheres forms precisely at the densities and temperatures where the changes to the laol results are largest . secondly , recent observations have resulted in a number of strong ( rosat ) and potential ( euve ) detections of apparently thermal emission from nearby neutron stars , providing limited spectral information in the euv / soft x - ray band . in particular , some low field ( millisecond ) pulsars have now been detected . it is clear that the next generation of space facilities in this band will provide significant constraints on ns thermal compositions and emissivities . thus to provide an improved baseline of models that illustrate spectral changes with surface composition and are directly applicable to low field neutron stars , we have constructed neutron star atmospheres and emergent spectra using the new opacities and consistent eos , for pure iron , pure hydrogen , and the solar abundances of grevesse & noels ( 1993 ) . improvements in the radiative transfer are also introduced and the resulting spectra should superceed those of romani ( 1987 ) . initial results of these computations have been reported in romani , rajagopal , rogers & iglesias ( 1995 ) . in section ii we describe the calculation of the model atmosphere structures ; in section iii we discuss convectional stability of the atmospheres , and present emergent spectra . spectra for various compositions are compared to black bodies and the high field hydrogen atmospheres of pavlov et al . ( 1995 ) . in section iv we provide an initial application , fitting the spectra to rosat observations of the millisecond pulsar j04374715 . section v describes implications of these results and prospects for future work . because the eos treatment allows extension to high density , opal eos and opacity tables ( kindly supplied by f. rogers and c. iglesias ) are our primary data source for these models . opal opacity computations are based on the method of detailed configuration accounting using ls coupling ( iglesias , rogers , & wilson , 1987 ) , and now use full intermediate coupling for iron to incorporate spin - orbit interactions ( iglesias et al . abundances of all possible ions are obtained from an activity expansion of the grand canonical ensemble ( rogers 1986 ) , and tables of the equation of state ( eos ) completely consistent with that used for the opacity calculations are available ( rogers , swenson , & iglesias , 1995 ) . for the present models , the eos tables provide pressure and other thermodynamic variables as a function of temperature and density , interpolated using routines supplied with the data . as an example , for pure iron the opacity grid was obtained for each decade of @xmath44 , where @xmath45 , from -5 to + 1 ; with @xmath46 from 4.5 to 7.5 every 0.25 , and with @xmath47 photon energies linearly spaced over @xmath48 . as needed , we interpolate @xmath49 linearly against @xmath50 and @xmath51 and linearly against @xmath52 ( at constant @xmath53 and @xmath4 ) . to allow extension to high photon energies , we use additional tables for the same @xmath54 and @xmath55 spaced linearly in @xmath4 from 1ev to 10kev at 1ev intervals . these tables , used only when @xmath56 , allow us to solve the transfer equation for @xmath4 up to 10 kev , through the lower temperatures present in the outer atmospheres . the data for the pure h and solar ( grevesse & noels 1993 ) abundances are similar , densely covering the thermal peak with extension up to 1 kev . we define _ ab initio _ a vector of 120 mean optical depths @xmath57 , logarithmically spaced between @xmath58 and @xmath59 . these correspond to physical depths through @xmath60 , where @xmath61 is the total mean opacity in @xmath62 including electron conductivity ( more below ) . the temperature and density at these grid points are adjusted until both hydrostatic equilibrium and energy steady state ( total astrophysical flux @xmath63 ) are achieved at all depths . for each of the three compositions , we generate six model atmospheres with @xmath64 from @xmath65 to @xmath66 , four per decade . for the light element mixtures , the eos tables stop before @xmath67 in the two coolest atmospheres , and for iron in the one coolest and two hottest . in these cases , the depth grid extends to @xmath68 ; this affects only the highest @xmath4 slightly . at each frequency @xmath69 , the outward flux @xmath70 at the frequency - specific optical depth @xmath71 is obtained from the milne integral @xmath72 where @xmath73 is the radiation source function , and @xmath74 is the second exponential integral , which has maximum of one at zero , with decay length of order unity ( mihalas 1978 ) . similarly , the intensity @xmath75 is given by @xmath76 where @xmath77 is the first exponential integral . we assume local thermodynamic equilibrium , taking the source of radiation at each depth to be a planck function at the local temperature @xmath78 . the atmosphere models begin with the grey opacity solution @xmath79 , using tabulated values for @xmath80 . in general , in a non - grey purely radiative atmosphere , at depths so large that radiative flux is given at all frequencies by the diffusion approximation @xmath81 the grey solution is obtained on the optical depth scale defined by the rosseland mean opacity . in the dense atmosphere of a neutron star , however , energy transport by electron conduction can be significant . we thus define an equivalent conduction opacity @xmath82 by analogy with equation ( [ diffusion ] ) so that @xmath83 , where @xmath84 is the thermal conductivity . here we use @xmath85 where @xmath86 is the number of free electrons per amu , @xmath87 is the average atomic mass , @xmath88 the average ionic charge , @xmath89 is in @xmath90 k , and @xmath53 is in @xmath91 . adding harmonically with the radiative opacity provides a total mean opacity @xmath92 on whose depth scale the initial grey model provides the correct atmosphere solution at large @xmath93 . we start with the grey atmosphere temperature structure , and impose hydrostatic equilibrium starting at the surface ( @xmath94 ) . for each @xmath95 layer , using an initial density @xmath96 from the layer above , we calculate the pressure at the base of that layer from @xmath97 . we then obtain a corrected @xmath98 from the eos tables , and iterate to convergence . at each iteration , we extract the opacity on an 800 bin logarithmic frequency grid extending to 10kev , saving harmonic means of 50 opal opacities sampled within the bin . this grid of @xmath99 values is used to produce the rosseland mean opacity @xmath100 . once convergence is reached at each depth , the final @xmath99 and @xmath100 are saved , and similar arithmetic bin averages used to produce the planck mean @xmath101 . once hydrostatic equilibrium is achieved for the whole atmosphere , we solve the transfer equation ( [ transfer ] ) for each of the same 800 frequencies , first computing @xmath71 corresponding to each of the 120 @xmath57 by integrating @xmath102 down from the surface . at very large depth , where @xmath103 varies slowly , we are in the diffusion limit and @xmath104 is obtained from equation ( [ diffusion ] ) ; the intensity @xmath75 always comes directly from equation ( [ j_trans ] ) . total flux @xmath105 and intensity @xmath106 are obtained by summing over the frequency grid , while the flux and absorption mean opacities @xmath107 and @xmath108 are obtained by appropriately weighted sums over the @xmath99 . the temperature structure is then corrected using the lucy - unsld procedure to drive towards the desired steady state flux : @xmath109 , \ ] ] ( mihalas , 1978 ) where @xmath110 as defined above , and @xmath111 . the corrected temperature run is then smoothed and any temperature inversions in the outer atmosphere are removed . the procedure is iterated to convergence @xmath112 at all depths ) . we show the converged temperature vs. density runs for iron in figure 1 . open circles indicate regions of convective instability , by definition where @xmath113 . the value of @xmath114 for iron from the eos is indicated by the size of the background circles ; instability in the radiative / conductive solutions occurs chiefly where it is low , in ionization zones . to gauge the importance of this instability we compute the energy gain ratio of a rising , but radiating , bubble @xmath115 ( _ e.g. _ bhm - vitense , 1992 ) where @xmath116 is the mean bubble speed and we take the bubble mean free path @xmath117 to be the atmospheric scale height @xmath118 . for small @xmath119 , departures from the radiative temperature gradient scale as @xmath120 . if the bubbles rise freely and quasi - adiabatically , then equating the potential energy gain to the kinetic energy gives @xmath121^{1/2 } l_{\rm mfp}$ ] as an upper limit to the bubble velocity . in this worst case picture , @xmath119 is as large as @xmath122 at large depth in the @xmath123 atmosphere . accordingly the emergent flux at several kev , where @xmath71 is modest at large depth , will be slightly affected . however , even the lowest known magnetic fields for neutron stars are @xmath124 g . if @xmath125 , which is true for @xmath126 g throughout all the atmospheres we have computed , then the magnetic field will suppress the convection . mass motions can then only occur on the diffusion time scale , which for a partly ionized plasma allows @xmath127 . under these conditions @xmath119 is no larger than 0.04 except in the coldest iron atmosphere , in which convection might cause departure from the radiative temperature gradient at the 10% level . accordingly , even when the magnetic field is too small to affect the eos and opacities it should strongly suppress convection in most cases . we calculate emergent spectra by solving the transfer equation at the edge of the atmosphere . to resolve lines from intermediate depths , we compute emergent spectra on a @xmath128 point energy grid , logarithmically spaced between 1ev and 10kev . the spectra , binned down to @xmath129 energies , are shown in figure 2 . much coarser binning still is appropriate to the resolution of current ns observations , and as argued above , broad - band features will be stable even to moderate ns fields . the iron spectra all show features at the k , l and m edges , which are also present in the solar abundance spectra . narrow absorption lines are formed at small physical depth , while strong pressure broadening affects lines from the deep layers . all three compositions are compared in figure 3 ( left ) ; the hydrogen spectra show the expected hardening due to the @xmath130 dependence of the opacity , which allows flux to shift above the wien peak . the two major spectral effects of magnetic field on hydrogen are visible in figure 3 ( right ) . first , the high energy hardening is not nearly as marked , since the diminished magnetic opacities which dominate transfer fall much less steeply than @xmath130 ( shibanov et al . 1992 , fig . 1 ) . secondly , the higher photoionization threshold increases absorption between @xmath131 0.1 and 0.3 kev for the fields shown ; the sharp onset is smoothed out by pressure effects on the atomic initial state ( pavlov et al . 1995 ) . at the lower field , this effect causes the prominent dip in the @xmath132 spectrum ; at @xmath133 the dip is minimal , due to the lesser abundance of atomic hydrogen . at the higher field , only the onset is seen in the @xmath132 spectrum , and it is clear this edge could dominate a fit . to apply these spectral results to soft x - ray data , we convert our spectra for each composition into a tabulated model for the xspec package . our model supplies unredshifted flux at the ns surface . these spectra are subjected to a specified gravitational redshift , and overall normalization is fit as a free parameter . model surface flux @xmath134 is related to model observed flux @xmath135 by : @xmath136}{(1+z ) } \left ( \frac{a_{\rm em}}{\omega d^2 } \right ) \left ( \frac{1}{1+z } \right),\ ] ] where area @xmath137 of the neutron star at distance @xmath138 emits into solid angle @xmath139 ; @xmath35 is the gravitational redshift , and the last factor due to time dilation . for full surface emission @xmath140 ; if the star has a small emission region such as a heated polar cap we take @xmath141 , noting that limb - darkening would tend to decrease this value while gravitational light bending near the ns surface would cause it to increase . as an example , we fit our models and magnetic hydrogen atmosphere emergent spectra for @xmath142 g and @xmath143 g , ( pavlov _ et al . _ , 1995 ) , to a hypothesized thermal component in the x - ray emission of the nearby millisecond ( p = 5.7 ms ) pulsar j04374715 measured by rosat . due to its low inferred surface dipole field of @xmath144 g this is an ideal candidate , though its large spindown age @xmath145yr ( johnston et al . 1993 ) suggests that any thermal emission should come from reheating . we have re - analyzed the september 20 - 21 1992 pspc data studied by becker & trumper ( 1993 ) . this data set is in three segments , each of about 2000 seconds duration , the last two about one hour apart and separated from the first by about one day . we use 1300 photons from a circle 2 arcmin in radius centered at the x - ray position . arrival times at the spacecraft clock are barycentred using standard iraf pros routines , and corrected for the pulsar s orbital position using an updated radio ephemeris ( johnston et al . 1993 ; bell , 1995 ) . the nominal phase of the x - ray pulse in the last two segments aligns with the radio peak to within the rosat clock accuracy . an apparent rosat clock error after the first data segment required us to phase it separately , and make the necessary shift to align it with the other two . based on the resulting light curve , we divide the data set into on- and off - pulse phases of equal duration ; the on - pulse interval contains @xmath146 of the counts . while the clock error decreases confidence in absolute phasing somewhat , the measured alignment of the x - ray pulse with the radio peak ( inferred from polarization data to be coincident with the neutron star magnetic axis ; manchester & johnston 1995 ) supports the picture that this pulse represents thermal emission from a reheated polar cap . hypothesizing that cap emission is superimposed on an unpulsed background magnetospheric or plerionic in origin , we fit a power law to the off - pulse data , finding a power law index of 2.5 , with absorption column density of @xmath147 , in agreement with the results of becker & trmper ( 1993 ) . we then added each thermal candidate model to the frozen power law with absorption held fixed , finding the best - fit parameters to the on - pulse data for blackbody spectra , atmosphere spectra for our three compositions , and spectra from the magnetic hydrogen atmospheres . all models are processed through the current pspc response matrix for comparison to the pi - channel data , and given @xmath148 redshift for a standard @xmath149 neutron star 10 km in radius . fit parameters are determined via the maximum likelihood method using the c - statistic on unbinned data ; the @xmath150 for these fits are then determined from data binned to at least 20 counts per bin , with bin - width at least 1/3 the half - max width of the rosat energy response ( table 1 ) . there are 12 dof for the magnetic models , 11 for the others . figure 4 shows the best fit models for black body and for each composition , smoothed through the detector response . given the fitted normalization of the thermal component , we use equation ( [ fluxnorm ] ) , the radio dispersion pulsar distance of 140pc , @xmath148 as above , and @xmath151 to infer the emission areas ( table 1 ) . table 1 : fits to psr j04374715 pspc data [ cols="^,^,^,^,^ " , ] the fit for both atmospheres containing iron is significantly poorer because the strong l - edge ( see fig . 2 ) provides excess flux near 0.7 kev where the data show a deficit . further , the model counts fall sharply above the edge where the data show an excess . the problem is exacerbated by redshifting the model edge to 0.6 or 0.5 kev . the bb and h fits are acceptable . following our hypothesis that the thermal emission comes from re - heated polar caps , we can check the fitted effective areas against the cap area expected in the aligned dipole rotator model : the polar cap radius is @xmath152 , where r is the pulsar radius and @xmath153 the light cylinder radius . in the case of j04374715 , for @xmath154 we have @xmath155 , giving @xmath156 . only the h atmosphere fits give areas close to this standard polar cap ; bb and fe fits imply areas @xmath157 times too small . while the magnetic h fits and areas are acceptable , the @xmath158 g light cylinder field inferred from spindown torques would not allow surface @xmath159 g unless the magnetic structure were of octupole order or higher . the resulting cap structure and thermal pulse would differ greatly from the dipole values inferred for psr j04374715 . the behavior of ns spectra at energies high in the rosat band is crucial for temperature determination , as interstellar absorption often leaves only the wien - like tail detectable . in that case , the inferred temperature depends strongly on the composition model adopted : in most cases the atmospheric temperatures are substantially lower than those inferred for blackbodies , although high temperature fe atmospheres may require a @xmath64 slightly higher than black body . even with present limited spectral information and moderate absorption this composition dependence is significant . as an illustration , we have drawn sample rosat pspc data sets as seen with absorption of @xmath160 from our redshifted ( @xmath148 ) models , and fit them with black body spectra . figure 5 shows the ratios of fitted to actual temperature . for the @xmath161 hydrogen atmosphere at it s lowest two temperatures , the black body temperature is driven down by the photoabsorption edge ( above , and figure 3 ) . standard cooling curves for neutron stars ( e.g. modified urca processes ) run about four times hotter than curves for more exotic processes ( e.g. direct urca , pion condensates ) before photon cooling takes over ( gelman , 1995 ; tsuruta , 1995 ) at @xmath162y . black body fits to thermal spectra from gelman s four initial cooling candidates place them quite near the standard curve , but figure 5 demonstrates that even magnetic light element atmospheres could move their correct positions close enough to the cooler exotic process cooling to confuse matters considerably . our emergent spectra are available in electronic form for the reader interested in determining flux implications for a particular band or detector , either as text files or as xspec models . for the pulsar j04374715 , enough spectral information exists to break the composition/@xmath64 degeneracy , favoring a pure hydrogen atmosphere quite strongly . the implied polar cap area is in much better agreement with theoretical expectation than that inferred from a blackbody fit . future missions ( e.g. axaf & xmm ) should be able to make similar measurements of other millisecond pulsars . interestingly , non - magnetic fits to the ` thermal ' emission of young , high - field pulsars seem in many cases to favor h atmosphere models to blackbody spectra ( gelman 1995 and references therein ) . while this agrees with the the compositional inference from psr j04374715 , comparison with heavy element magnetic atmospheres is needed to substantiate this result . a correct treatment of the detailed spectrum of such high @xmath20 neutron star atmospheres will require extensive improvements to present atomic structure and opacity computations , although approximate models following broad - band features may now be feasible ( work in progress ) . for younger pulsars , such models together with data from the next generation of x - ray satellites should enable a serious study of ns surface conditions , including composition , magnetic field and even redshift , with important implications for ns evolution and the eos of matter at very high densities . we are indebted to forrest rogers and carlos iglesias for computation of the opal data for the fe models and assistance with their interpretation . we also thank g. g. pavlov for providing high - field hydrogen spectra , i .- a . yadigaroglu for assistance with the j0437 data , and the referee for a careful reading . rwr was supported in part by an alfred p. sloan fellowship and nasa grant nagw-2963 , and mr in part by a fellowship from the national science and engineering research council of canada . this research has made use of data obtained through the high energy astrophysics science archive research center online service , provided by the nasa - goddard space flight center .	we compute model atmospheres and emergent spectra for low field ( @xmath0 g ) neutron stars , using new opacity and equation of state data from the opal project . these computations , incorporating improved treatments of flux transport and convective stability , provide spectra for hydrogen , solar abundance and iron atmospheres . we compare our results to high field magnetic atmospheres , available only for hydrogen . an application to apparently thermal flux from the low field millisecond pulsar psr j04374715 shows that h atmospheres fit substantially better than fe models . we comment on extension to high fields and the implication of these results for neutron star luminosities and radii .
quantum chromodynamics predicts a new form of matter , a deconfined state of quarks and gluons called quark - gluon plasma ( qgp ) . it can be created by heavy ion collisions at relativistic energies experimentally @xcite . in a collision , particles are produced by two main processes : soft hadrons are generated from the hot dense medium created by energy sedimentation , while jets are produced from initial hard scatterings . being relevant to the two processes respectively , the discovery of the large elliptic anisotropy ( @xmath2 ) @xcite and the jet quenching @xcite have been taken as important evidences of qgp at the relativistic heavy ion collider ( rhic ) @xcite . particles produced by different processes carry different information of the system . for example , the soft hadron azimuthal anisotropy at low transverse momentum , called elliptic flow , is formed due to the hydrodynamic pressure buildup in the initial almond overlap region of the colliding nuclei @xcite . at large transverse momentum , energetic partons are predicted to lose energy by induced gluon radiation @xcite . this energy loss depends strongly on the traversed path length of the propagating parton , thus also leads to azimuthal anisotropy @xmath2 the @xmath2 measured from final state particles is the combination of these two effects . however , the elliptic flow @xmath2 and azimuthal anisotropy @xmath2 of jets individually might not be the same . trying to separate the two , previous work @xcite in this regard has focused on the covariance of @xmath2 from forward and backward regions , which has been found later that the method is complicated by @xmath2 fluctuations @xcite , and the jet contribution still can not be directly accessed from final state particles . in this paper , we approach the problem differently by studying the multiplicity correlation . the procedure is similar to the previous study @xcite . to insure that jets only contribute to one region while collective particles contribute to both in a single event , two regions with the same size in phase space are chosen . since jets are produced locally , following ref . @xcite , we make forward - backward rapidity bins with a gap between them . the proposed new observable @xmath3 is defined as the standard correlation function : @xmath4 where @xmath5 refers to the particle yields in the forward ( backward ) region , and @xmath6 means taking the average over events . [ fig : fig1 ] ( color online ) the dependence of forward - backward multiplicity correlation function @xmath3 on jet fraction @xmath7.,title="fig:",width=3 ] to count the contribution of jets to the total multiplicity , we define a @xmath7-parameter : @xmath8 here @xmath9 refers to the multiplicity of flow ( jets ) . since the experimental efficiency does not bias flow or jet particles , forward or backward regions , it is not expected to affect the measurement . similar to @xcite , in each event , the yields in forward ( backward ) region can be written as @xmath10 with @xmath11 in the above , @xmath12 is the total multiplicity in an event , and @xmath13 is the event - by - event multiplicity fluctuation of flow particles in each region . the jet multiplicity may also fluctuates event - by - event , thus no average is taken for @xmath14 . after taking the average over all events , we will get @xmath15 . for the final state particles , the correlation between flow particles and jets is believed to be small thus ignored , and this is justified by that they are produced at different moments during the collision , i.e. , @xmath16 . @xmath17 describes the jet contribution to the forward and backward region . since jets only fall in one side in an event , the assumed value is 1 or 0 . the location is randomly decided , so the probability is the same for each side , i.e.,@xmath18 while @xmath19=0 . combine eq . ( [ gfun ] ) and ( [ nfun ] ) and insert into eq . ( [ cfb ] ) , we find the relation for @xmath7-parameter and @xmath3 : @xmath20 @xmath7 is the contribution of jets and @xmath21 is a measure of correlated forward - backward fluctuations . both of them affects the behavior of @xmath3 . figure [ fig : fig1 ] shows the relation between @xmath3 and the @xmath7-parameter . in the ideal case , if flow particles from the two regions fluctuate independently , i.e. , @xmath22 , as assumed in ref @xcite , eq . ( [ c2 ] ) can be simply written as @xmath23 then , @xmath7 can be extracted from the value of @xmath3 . if the system is hydro dominated , @xmath24 and @xmath25 . while if the system is jets dominated , @xmath26 and @xmath27 . on the other hand in a realistic case , if flow particles from the two regions do not fluctuate independently ( @xmath28 ) , when @xmath7 is small , @xmath3 will be larger than 0 , and when @xmath7 is large , the effect of @xmath21 can be ignored . ( color online ) @xmath3 as a function of @xmath29 for flow - only simulation . the black dash curve is from eq . ( [ c4]).,width=3 ] the eq . ( [ c2 ] ) works in the low @xmath29 with relatively large number statistic . however , if there is only one flow particle produced a particular @xmath29 bin in a event , it can only fall into one side and behaves like a jet . in that case , the multiplicity correlation function will automatically decrease when the particle yield is much smaller than 1 . this effect has not been discussed in previous studies @xcite , and can be easily confused with the contribution from jet . to explore this effect , a simulation was performed as followed . the flow particles were generated with a exponentially decreases distribution , and randomly fell into forward / backward region . adopting the consistent terminology as in @xcite , we call them `` flow - only '' particles throughout this paper . the red curve in fig . [ fig : fig2 ] shows the result of flow - only simulation . it can be seen that although there is no jet , @xmath0 is still approaching to @xmath30 at large @xmath29 . this phenomena is solely due to , for a given @xmath29 bin , the exclusion of events for which both @xmath31 and @xmath32 for that @xmath29 bin are zero , from the @xmath33 calculation . this happens even when the two samples are completely independent of each other . with the simplest assumption that when particle yield @xmath34 is smaller than 1 , maximum one particle can be found in either forward or backward region , the probability of finding an empty @xmath29 bin is @xmath35 . if one calculates @xmath36 without the exclusion of events with empty bins as mentioned above , for two independent samples it should be at unity . considering the exclusion , one needs to scale each averaging term in @xmath36 by a factor of @xmath37 , then @xmath38 can be written as : @xmath39 thus even for independent samples , with the exclusion of empty bins , @xmath38 is decreasing with increasing @xmath34 . the black dash line in fig . [ fig : fig2 ] is a plot of eq . ( [ c4 ] ) . we can see that when @xmath29 is larger than 2.2 gev/@xmath40 , the equation can well describe the simulation result . while when @xmath29 is smaller than 2.2 gev/@xmath40 , the simulation result is lower than the black dash curve . this is because when particle yield is not so small , there might be more than one particle generated in the forward / backward region for the flow - only simulation . therefore , from the eq . ( [ c3 ] ) and fig . [ fig : fig2 ] we can conclude that both of the jets and the scarcity of particles can cause the decreasing trend of @xmath38 . we may also expect that it will decline even faster when jets are involved in the particle - rare case . the forward - backward multiplicity correlation function @xmath0 is studied in a multiphase transport model ( ampt ) @xcite . there are four main components in this transport model : the initial conditions , the parton - parton interactions , the conversion from the partonic to the hadronic matter and the late hadronic interactions . the initial conditions are based on the hijing model @xcite in which the eikonized parton model is employed . it includes the spatial and momentum distributions of minijet partons from hard processes and strings from soft processes . the parton - parton interactions and the time evolution of partons is then treated by the zhang s parton cascade ( zpc ) @xcite model . the hadronization process is described by a combined coalescence and string fragmentation model . a relativistic transport ( art ) model @xcite which includes baryon - baryon , baryon - meson and meson - meson elastic and inelastic scattering is employed to describe the late hadronic process . in our study , we analyzed the events from string melting ampt model for au + au collisions at @xmath1 gev with parton cross sections equal to 10 mb . pseudorapidity regions @xmath41 $ ] and @xmath42 $ ] are chosen as forward and backward region respectively . ( color online ) @xmath3 as a function of @xmath29 for au+au collision at @xmath43=200 gev in ampt string melting model . the red dash curve is the fitting of data with the form of eq . ( [ fit1 ] ) , and the green dash - dot curve is a plot of eq . ( [ fit2 ] ) . the blue curve is the flow - only simulation . , width=3 ] ( color online)(a ) ( @xmath44 ) ( solid points ) and ( @xmath45 ) ( red dash curve ) from ampt are divided by ( @xmath44 ) from flow - only simulation . ( c ) @xmath7-parameter as a function of @xmath29.,width=3 ] the @xmath29 dependence of multiplicity correlation function is shown in fig . [ fig : fig3 ] . we can see that @xmath3 almost stays as a constant when @xmath46 gev/@xmath40 . since particles from high energy jets give only a small contribution at those @xmath29 , the value of @xmath3 reflects the flow multiplicity fluctuation , @xmath47 . although the particle yield decreases exponentially , @xmath47 shows little @xmath29 dependence for this region . to obtain the jet contribution , @xmath3 is fitted by the function : @xmath48 and @xmath49 is parameterized using a tanh function to describe threshold behavior between the two regimes , as suggested in @xcite : @xmath50\}/2 . \label{fit}\ ] ] here @xmath49 is not simply @xmath51 anymore . instead , it is a convolution of both @xmath51 ( dominated at low @xmath29 ) and the automatic decrease of @xmath33 ( dominated at large @xmath29 ) . after fitting , we could obtain the first part of eq . ( [ fit1 ] ) : @xmath52 where the effect of flow multiplicity correlation between the forward and backward regions has been canceled . in fig . [ fig : fig3 ] , we present the fitting of ampt data by the red dash curve . we can see that this form can well describe the trend of the data , and we obtain that @xmath53 . for comparison , we fit the particle distribution with levy function , which can well describe the data , and generates flow particles with the same distribution . as discussed before , the flow particles will randomly fall into each side in this simulation . the result of this flow - only simulation is shown as the blue curve in fig . [ fig : fig3 ] . from the plot , we can see that the simulation stays zero at low @xmath29 as expected since there is neither jet nor correlation . while at intermediate @xmath29 range , both of the ampt data and the flow - only simulation decreases as @xmath29 increasing . we found that the ampt result including jet contribution is only slightly lower than the flow - only simulation . therefore , we argue that the decreasing trend of the forward - backward correlation function as shown in @xcite may not only due to the jet contribution , but mostly due to the artificial decrease for the particle - rare case . to see the details clearly , the results from ampt model are divided by that from the flow - only simulation . here we define @xmath54 the deviation from unity of this ratio reflects the dilution of @xmath55 due to pairs containing jets . from the plot , we can see that the ratio begins to deviate from 1 around 2 gev/@xmath40 , which indicates that the jet contribution begins to join in this region . the difference between the points and the curve comes from flow multiplicity correlation . for qualitatively study , we assume that the square root of the ratio is proportional to the particle yield which is expected to work for @xmath56 gev/@xmath40 as shown in fig . [ fig : fig2 ] , and finally @xmath7 can be extracted from @xmath57 . [ fig : fig4](b ) shows that the @xmath7-parameter is about 10% at 2 gev/@xmath40 and reaches 30% at 3 gev/@xmath40 . our method works best for central collisions in which away - side production of a jet is suppressed thus those particles can be considered as being `` melted '' into flow particles . while for the non - central collision , the @xmath7 quantity will be reduced due to the not quenched away - side jets . in that case , the correlation function measures the @xmath58 jets in one side . the forward - backward elliptic anisotropy correlation has been discussed in ref @xcite . this correlation function also subjects to the small number statistic discussed before . here we redefine the elliptic correlation function : @xmath60 where @xmath61 refers to the sum of the @xmath2 of particles in forward ( backward ) region in each event . the @xmath62 we measured in experiments is constituted of two parts : flow @xmath63 and jet @xmath64 : @xmath65 @xmath66 describe the collective behavior of hydro , and @xmath64 is related to the parton energy loss in different directions . the relation between @xmath59 and pure jet ( flow ) @xmath2 can then be written as : @xmath67 the differences between this and ref @xcite is coming from the @xmath30 in the definition . one should pay attention that @xmath21 here is not the same as in the multiplicity correlation . for multiplicity correlation , @xmath13 refers to the total multiplicity fluctuation in each side . while for @xmath2 correlation , @xmath13 have azimuthal dependence , i.e. , the in - plane and out - of - plane multiplicity fluctuation may cause the statistic fluctuation of @xmath2 . to be clearer , the eq . ( [ v2 ] ) can be written as @xmath68 therefore , for the hydro dominate case , i.e. , @xmath24 , @xmath69 . the @xmath59 describes the @xmath2 fluctuation at low @xmath29 , and the large value of the @xmath59 observed in ref @xcite could be understood according to the magnitude of the flow fluctuation studied in ref @xcite . for intermediate @xmath29 , however , the decreasing trend of @xmath59 observed @xcite are mostly due to the rare flow particles as discussed before . therefore , it is interesting to check the behavior of @xmath59 in the lhc energy where particle yield is much higher . in summary , we have proposed a forward - backward multiplicity correlation function @xmath0 to study the jet contribution . we find that the @xmath0 is sensitive to the jet contribution , and its value will be enlarged by the forward - backward multiplicity correlation . in the intermediate @xmath29 , we find that the @xmath0 is sensitive to small number statistic and will automatically decrease even there are no jets involved . this effect should also be present in previous study of @xmath70 , and might be misunderstood as the contribution of jets . we study the @xmath0 in au+au collisions at @xmath1 gev with ampt string melting model . when @xmath29 is lower than 1.4 gev/@xmath40 , @xmath0 is independent of @xmath29 , and @xmath21 is about 0.01 . at intermediate @xmath29 range , @xmath0 decreases with @xmath29 , and the result from ampt model is lower than the flow - only simulation of the same yield . it indicates that both of the the jet contribution and the procedure of excluding events with empty bins will cause the decrease of @xmath0 . the estimated jet fraction is much smaller than previous study . finally , we discussed the connection of our study to the forward - backward elliptic anisotropy correlation function . the large value of @xmath70 is found due to @xmath2 fluctuation , and the decrease of @xmath70 observed previously is caused by both of the jet and the tail of flow particle distribution , with the latter dominates . the authors thank l. x. han , j. liao and g. wang for the discussion . the work was supported in part by huazhong university of science and technology foundation under grant no . 2011qn195 , the national natural science foundation of china under grant no . 11147196 , 11105060 and 10835005 , and by the office of nuclear physics , us department of energy under grants de - ac02 - 98ch10886 and de - fg02 - 89er40531 . 99 i. arsene @xmath71 ( brahms collaboration ) , nucl . phys . a * 757 * , 1 ( 1990 ) ; b. b. back @xmath71 ( phobos collaboration ) , nucl . a * 757 * , 28 ( 2005 ) ; j. adams @xmath71 ( star collaboration ) , nucl . phys . a * 757 * , 102 ( 2005 ) ; s. s. adcox @xmath71 ( phenix collaboration ) , nucl . phys . a * 757 * , 184 ( 2005 ) . c. adler _ ( star collaboration ) , phys . lett . * 87 * , 182301 ( 2001 ) ; k. h. ackermann _ ( star collaboration ) , phys . 86 * , 402 ( 2001 ) ; b. i. abelev _ ( star collaboration ) , phys . 99 * , 112301 ( 2007 ) ; m. gyulassy and m. plmer , phys . b * 243 * , 432 ( 1990 ) ; x. n. wang and m. gyulassy , phys . d * 44 * , 3501 ( 1991 ) ; x. n. wang and m. gyulassy , phys . 68 * , 1480 ( 1992 ) ; r. baier , d. schiff and b. g. zakharov , ann . nucl . part . sci . * 50 * , 37 ( 2000 ) .	we propose a forward - backward multiplicity correlation function @xmath0 , which is experimentally accessible , to measure the noncollectivity contribution . it is found that the function is sensitive to both of the jet contribution and the small number statistics . we point out that the effect of the latter one is also involved in the previous studies of the forward - backward elliptic correlation function but was confused as the contribution from jets . we study the @xmath0 in au+au collision at @xmath1 gev with a multiphase transport model ( ampt ) . the result shows that the estimated jet contribution is much smaller than previous study due to the finite number statistics which has not been noticed before . the connection between this study and the forward - backward elliptic correlation function is also discussed .
it is well known that the jaynes - cummings model @xcite plays an important role in investigation of interaction between atoms and quantum radiation field ( e.g. , see @xcite ) . the point is that the model describes fairly well the physical processes in the system and , at the same time , allows an exact solution . in the usual formulation of the jaynes - cummings model @xcite , the atom is considered as though it consists of two or very few non - degenerated levels . in fact , the radiative transitions in real atoms occur between the states with given angular momentum @xmath0 and its projection @xmath1 ( e.g. , see @xcite ) . this means that even in the case of only two levels , the degeneration with respect to the quantum number @xmath2 taking @xmath3 different values should be taken into account . the simplest example is provided by a dipole transition between the states @xmath4 and @xmath5 when the excited atomic state is a triply degenerate one ( see figure 1 ) . let us stress one more important difference . the radiation field in conventional jaynes - cummings model is represented by the plane waves of photons with given linear momentum and polarization . at the same time , the multipole transitions in real atoms emit the multipole photons represented by the quantized spherical waves with given angular momentum and parity @xcite . although there is no principle difference between the plane and spherical waves within the classical domain since both represent the complete orthogonal sets of solutions of the homogeneous wave equation @xcite and can be re - expanded with respect to each other , the quantum counterparts of these two representations are non - equivalent because they describe the physical quantities ( the linear and angular momenta respectively ) which can not be measured at once . moreover , the multipole field is characterized by more quantum degrees of freedom than the plane waves of photons . in fact , the monochromatic pure @xmath6-pole multipole radiation of a given type ( either electric or magnetic ) is specified by the @xmath3 different values of the quantum number @xmath2 . since @xmath0 , the total number of degrees of freedom here is not less than three . at the same time , the monochromatic plane waves of photons are described by only two different polarizations . in particular , the increase of the number of degrees of freedom can lead to an increase of the zero - point oscillations @xcite . the multipole generalization of the jaynes - cummings model has been discussed in @xcite . let us stress that similar models have been considered in different problems of interaction of quantum light with matter ( e.g. , see @xcite and references therein ) . the main objective of this paper is to examine the quantum properties of light emitted by a dipole atom at any distance from the source , depending on the boundary conditions . the paper is arranged as follows . in section 2 we review the properties of quantum multipole field in comparison with these of plane waves of photons . in section 3 we briefly discuss the multipole generalization of the jaynes - cummings model . then , in section 4 we consider the polarization of multipole radiation and introduce a novel general polarization matrix . this object permits us to take into account the spatial anisotrophy of both the electric and magnetic fields at once . in section 5 we examine the spatial properties of multipole photons emitted by an atom in an ideal cavity as well as in empty space . in particular , we show that the polarization properties of quantum multipole radiation changes with distance from the atom . in section 6 we briefly discuss the obtained results . following @xcite , we list below some important formulas describing the quantum multipole field . it is usually considered in the so - called helicity basis @xcite @xmath7 it is clear that @xmath8 formally coincides with the three eigenstates of spin @xmath9 of a photon . since the polarization is defined to be the spin state of photons @xcite , one can choose to interpret @xmath10 as the unit vector of circular polarization with either positive or negative helicity , while @xmath11 gives the linear polarization in the @xmath12-direction . to within the sign at @xmath10 the helicity basis ( 1 ) coincides with the so - called polarization basis usually used in optics @xcite . in the basis ( 1 ) , the positive - frequency part of the operator vector potential of multipole field can be expanded as follows @xcite @xmath13 where @xmath14 is the photon annihilation operator which obeys the following commutation relations @xmath15= \delta_{\lambda \lambda ' } \delta_{kk ' } \delta_{jj ' } \delta_{mm'}. \nonumber\end{aligned}\ ] ] here @xmath16 is the wave vector , @xmath17 denotes the type of radiation ( parity ) , index @xmath0 gives the angular momentum , index @xmath18 , and @xmath19 is the annihilation operator of the corresponding photon . the mode functions @xmath20 in ( 2 ) are represented in the following way @xmath21 , \nonumber \\ v_{mkjm \mu } & = & \gamma_{mkj } f_j(kr ) \langle 1,j , \mu , m- \mu \|jm \rangle y_{j , m- \mu } ( \theta , \phi ) \label{3}\end{aligned}\ ] ] for the electric and magnetic radiation respectively . here @xmath22 are the normalization constants , @xmath23 is the volume of quantization , @xmath24 denotes the clebsch - gordon coefficient of vector addition of the spin and orbital parts of the angular momentum of a multipole photon , and @xmath25 is the spherical harmonics . the radial contribution into the mode function ( 3 ) depends on the boundary conditions . in the standard case of quantization in terms of standing spherical waves in a spherical cavity @xcite , we have @xmath26 where @xmath27 is the spherical bessel function . the positive frequency parts of the operator field strengths obey the following relations @xmath28 it can be easily seen from ( 2 ) and ( 5 ) that the electric multipole field always has longitudinal component of the electric field strength in addition to the two transversal components , while it is completely transversal with respect to magnetic induction . at the same time , the magnetic multipole field has all three components of magnetic induction and only two transversal components of the electric field strength . the position dependence of the mode functions ( 3 ) is not an unusual fact . in reality , the mode functions of the plane waves also depend on position : @xmath29 here we choose the basis ( 1 ) with @xmath30 and @xmath31 denotes the photon annihilation operator , corresponding to the states with given linear momentum ( direction of propagation ) and transversal polarization with either helicity . the third projection of the photon spin is forbidden in this case @xcite . to complete the discussion of the quantum multipole field , consider now the zero - point or vacuum oscillations . since the free - field hamiltonian has the form @xcite @xmath32 for the energy of the vacuum state in the whole volume of quantization we get @xmath33 for comparison , we show here well known expressions valid in the case of plane waves of photons ( 6 ) : @xmath34 due to the definition of @xmath16 , both expressions ( 7 ) and ( 8) give an infinite energy and , at first sight , can not be compared with each other . in fact , this infinity is inessential because of the following reason . the contribution of zero - point oscillations can be observed only via measurement which implies an averaging of physical quantities over a finite `` volume of detection '' and exposition time of detector @xcite . such an averaging plays a part of filtration leading to a selection of a certain finite transmission frequency band . it is therefore seen from ( 7 ) and ( 8) that , even if the filtration process leads to selection of the electric dipole photons only , the ratio of contributions of the zero - point oscillations of the multipole field and plane waves is equal to @xmath35 . from the physical point of view , this result is caused by the more number of quantum degrees of freedom in the case of multipole photons @xcite . much more interesting and important result can be obtained from consideration of the spatial properties of the zero - point oscillations . the energy density of the field is represented as follows @xmath36= \frac{1}{16 \pi } \left\ { [ { \vec e}^+({\vec r})+{\vec e}({\vec r})]^2+[{\vec b}^+({\vec r})+{\vec b}({\vec r})]^2 \right\ } . \nonumber\end{aligned}\ ] ] then , the zero - point contribution is @xmath37 consider first the case of plane waves . in view of ( 6 ) and symmetry relations @xmath38 , we get @xmath39 thus , the zero - point oscillations of the energy density of the plane waves of photons are homogeneous in the space of quantization in spite of the position dependence of the mode functions in ( 6 ) . in turn , employing the relations ( 5 ) , we get @xmath40 in view of ( 3 ) , we obtain @xmath41 . \label{12}\end{aligned}\ ] ] taking into account that @xmath42 it is straightforward to show that the zero - point energy density of the multipole field is independent of the angular variables @xmath43 and @xmath44 , while depends on the distance from the origin ( the singular point corresponding to the source location ) : @xmath45 thus , the density of the zero - point oscillations of multipole field has the spherical symmetry with respect to the singular point ( atom ) , while manifests the radial dependence . taking into account the radial dependence ( 4 ) and making use of the following formula @xcite @xmath46 it is easy to prove that the multipole zero - point oscillations ( 12 ) vanish at far distance . thus , the vacuum fluctuations of the multipole field are concentrated near the atom . further employing the properties of spherical bessel functions shows that the principal contribution into ( 12 ) at @xmath47 comes from the term with @xmath48 , corresponding to the electric dipole radiation . the radial dependence of ( 12 ) at fixed @xmath16 and @xmath49 is shown in figure 2 . it is seen that the zero - point oscillations of the multipole field are concentrated in some vicinity of the singular point where they strongly exceed the level ( 10 ) predicted within the framework of the representation of plane waves of photons . from the figure 2 , the radius of the region of concentration of the zero - point oscillations can be estimated as follows : @xmath50 where @xmath51 is the wavelength . thus , the atom condenses the zero - point oscillations in the near and intermediate zones . let us stress that in the number of modern experiments on engineered entanglement in the systems of trapped rydberg atoms , the interatomic distances are of the same order @xcite . the above result ( 13 ) should be considered as an estimation from below because it corresponds to the first term in ( 12 ) . successive taking into account of further terms leads to a certain shift of @xmath52 into the intermediate zone . the main aim of this section is to emphasize that the interaction between the photons and electrons on atomic sub - levels with the same @xmath6 and different @xmath2 is specified by a certain coupling constant independent of @xmath2 . consider a two - level atom with the electric dipole transition @xmath53 . the coupling constant of the atom - field interaction can be found by calculating the matrix element @xcite @xmath54 obtained from the expression @xmath55 describing the interaction between the atomic electron with linear momentum @xmath56 , charge @xmath57 , and mass @xmath58 and radiation field specified by the vector potential @xmath59 . here @xmath60 is the dipole moment of the atomic transition with the resonance frequency @xmath61 . assuming the central symmetry of atomic field and taking into account the fact that the spin state of an atom does not change under the electric dipole transition , we can represent the atomic states in ( 14 ) as follows @xmath62 where @xmath63 is the radial part of the atomic wave function of either excited ( @xmath57 ) or ground ( @xmath64 ) state . expanding the dipole moment @xmath65 over the helicity basis ( 1 ) , substituting ( 2 ) , and carrying out the calculations of integrals in ( 14 ) over a small volume occupied by the atom , for the coupling constant ( 14 ) we get @xmath66 where @xmath67 is the effective dipole factor which , by construction , is independent of the quantum number @xmath2 . taking into account the properties of the clebsch - gordon coefficients and spherical harmonics , for the position - dependent mode function in ( 2 ) we get @xmath68 this means that the electric dipole transition @xmath69 at any given @xmath2 creates a photon with helicity ( polarization ) @xmath70 . finally , the jaynes - cummings hamiltonian of the electric dipole transition in the rotating - wave approximation @xcite takes the form @xcite @xmath71 to simplify the notation , hereafter we omit insignificant indices . here the atomic operators are defined as usually @xcite in terms of the projections on the atomic states : @xmath72 the hamiltonian ( 16 ) describes the creation and absorption of the single cavity - mode photons _ at the atom location_. everywhere in the surrounding space , we have to take into account the spatial dependence of the radiation field described by the vector potential ( 2 ) . similar model can be constructed in the case of magnetic dipole radiation as well as in the case of other high - order atomic multipoles . it is known that the polarization defines the direction of oscillations of the field strengths . within the classical picture based on the consideration of plane waves , the polarization is defined to be the measure of _ transversal _ anisotrophy of the electric field strength @xcite . in turn , the quantum mechanics interprets the polarization as given spin state of photons @xcite . in the usual approach , the quantitative description of polarization is based either on the hermitian polarization matrix or on the equivalent set of real stokes parameters . in the standard case of plane waves , we get the @xmath73 polarization matrix and four stokes parameters @xcite , while the description of multipole radiation requires for the @xmath74 polarization matrix and nine stokes parameters @xcite . moreover , the electric- and magnetic - type radiation fields are usually described in terms of different polarization matrices , taking into account the spatial anisotrophy @xcite . here we construct a more general novel object , describing in a unique way the polarization properties of multipole radiation of either type both classical and quantum as well as these of the plane waves and other forms of electromagnetic radiation ( e.g. , of the cylindrical waves ) . in general , the field can be described in terms of the field - strength tensor which can be chosen as follows @xcite @xmath75 it seems to be tempting to introduce the general quantitative description of polarization using ( 17 ) . since the polarization is specified by the intensities of different spatial components of the radiation field and by the phase differences between these components @xcite , it should be described in terms of bilinear forms in the field strengths . the simplest bilinear form in the field - strength tensor is @xmath76 which differs from the energy - momentum tensor by a scalar . in some sense it is similar to the ricci tensor considered in the general relativity @xcite . it is easily seen that ( 18 ) has the following structure @xmath77 where @xmath78 is a scalar , @xmath79 , apart from an unimportant factor , coincides with the poynting vector , and @xmath80 is the hermitian @xmath81 matrix of the form @xmath82 here @xmath83 and @xmath84 we note here that ( 20 ) has been proposed in @xcite in order to describe the spatial anisotrophy of the electric dipole radiation , while ( 21 ) is similar to the object has been discussed in @xcite . we choose to interpret ( 19 ) as the general polarization matrix , while the terms ( 20 ) and ( 21 ) give the electric and magnetic field contributions respectively . to justify this statement , consider first the case of plane waves propagating in the @xmath12-direction when @xmath85 and @xmath86 . then , the matrix(20 ) takes the form @xmath87 it is seen that the non - zero submatrix in ( 22 ) coincides with conventional @xmath73 polarization matrix of plane waves @xcite . in turn , ( 21 ) takes the form @xmath88 where corresponding @xmath73 submatrix in the top left corner coincides with ( 22 ) . thus , the general polarization matrix ( 19 ) describes the polarization of plane waves adequately . consider now the multipole radiation . in the case of electric - type radiation when @xmath89 everywhere , the matrix @xmath90 has the general form ( 20 ) , while the magnetic polarization matrix ( 21 ) is reduced to @xmath91 the polarization of magnetic - type radiation is described by ( 20 ) with @xmath92 which coincides with ( 22 ) and by the general form ( 21 ) with @xmath93 . in this case , ( 21 ) coincides , within the transposition of lines and columns , with the polarization matrix considered in @xcite . it is natural that the general polarization matrix ( 19 ) reflects the three - dimensional structure of the radiation field . the diagonal terms in ( 20 ) and ( 21 ) give the radiation intensities . their angular and radial dependence corresponds to the radiation patterns of the multipole field . the off - diagonal terms give the phase information as in the case of plane waves @xcite . in contrast to the standard case of plane waves , there are the two independent phase differences @xmath94 instead of only one phase difference because @xmath95 since @xmath96 at any point , the magnetic part ( 21 ) of the general polarization matrix ( 19 ) contains the same phase differences as ( 20 ) . similar expressions for the polarization matrix ( 19 ) can also be obtained in the helicity basis ( 1 ) . for example , the matrix ( 20 ) takes the form @xmath97 the quantum counterpart of ( 19 ) can be obtained by formal substitution of the operators instead of the classical field strengths ( compare with @xcite ) . averaging of the corresponding operator matrix over a given state of the radiation field gives the polarization matrix . by construction , the operator matrices ( 19)-(24 ) correspond to the normal ordering in the creation and annihilation operators : @xmath98 in addition , one can define the anti - normal polarization matrix @xmath99 by a simple change of order of product of the field strengths in all elements of the matrices ( 19)-(24 ) . it is then clear that the matrix @xmath100)= \langle 0\|p^{(an)}\|0 \rangle \equiv p_{vac } \label{26}\end{aligned}\ ] ] determines the zero - point ( vacuum ) contribution into the polarization . following the ideas and results of section 2 , it is a straightforward matter to show that the vacuum polarization of plane waves of photons is uniform in the space , while the multipole vacuum polarization concentrates near the atom and exceeds the level predicted by the representation of plane waves . it was shown in section 3 that an atomic electric dipole transition with given @xmath2 emits the photon with given polarization @xmath101 . we now show that the polarization changes with the distance from the source . in other words , the polarization is not a global property of the field , while changes from point to point , at least in the case of multipole radiation . assume , for example , that the atom emits the electric dipole photon with @xmath102 , i.e. circularly polarized with positive helicity . consider the polar direction @xmath103 , corresponding to the maximum of the radiation pattern in this case @xcite . then , the matrix ( 25 ) averaged over the photon state @xmath104 takes the form @xmath105 ^ 2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array } \right ) \nonumber\end{aligned}\ ] ] thus , there is only one polarization @xmath106 in the polar direction . in the less probable case of the equatorial direction @xmath107 , from ( 25 ) we get @xmath108 ^ 2 & 0 & \frac{-3}{4 } j_{2}(kr ) [ \frac{1}{4 } j_{2}(kr)+j_{0}(kr)]e^{2i \phi } \\ 0 & 0 & 0 \\ \frac{-3}{4 } j_{2}(kr ) [ \frac{1}{4 } j_{2}(kr)+j_{0}(kr)]e^{-2i \phi } & 0 & \frac{9}{16 } [ j_{2}(kr)]^2 \end{array } \right ) \label{27}\end{aligned}\ ] ] so that there are the two circularly polarized components with opposite helicities . comparison of intensities of the two components shows that the positive helicity dominates at short distances ( @xmath109 ) , while both components contribute equally at far distances ( @xmath110 ) ( see figure 3 ) . it is clear that any deviation from the polar direction leads to creation of polarizations additional to @xmath106 . thus , the polarization o radiation under consideration strongly depends on the direction and distance from the source . similar picture can be obtained for polarization of photons with @xmath111 and @xmath112 . the above results were obtained in the case of standing waves in an ideal spherical cavity when the radial dependence of the mode functions ( 3 ) is specified by equation ( 4 ) . in this case , the radiation field is subjected to the rabi oscillations which can be described through the use of the steady - state time dependent wave function for the system with hamiltonian ( 16 ) : @xmath113 where we choose the initial state as the vacuum state of the cavity field and excited state of the atom is specified by given @xmath2 . then , the elements of the polarization matrix ( 19 ) obtained by averaging or corresponding operator matrix over the state ( 28 ) should be multiplied by an additional factor of @xmath114 , describing the steady - state time dependence of polarization . consider now the radiation by a dipole atom in empty space . then , the hamiltonian ( 16 ) should be generalized as follows @xmath115 to take into account the @xmath16-dependence of the radiation field . let us again choose the initial state as the vacuum state of photons and excited atomic state with given @xmath2 @xmath116 . \label{30}\end{aligned}\ ] ] it is then clear that the radiation field is represented by the outgoing spherical waves of photons which stipulates the choice of the radial dependence in ( 3 ) in terms of the spherical hankel function of the first kind @xmath117 instead of ( 4 ) @xcite . this choice assumes that the atom occupies a small but finite spherical volume of radius @xmath118 at the origin to avoid the divergence at @xmath119 . the elements of the polarization matrix ( 25 ) of the electric dipole radiation in the equatorial direction take the form @xmath120 ^ 2 + [ \gamma_- ( kr ) ] ^2 \right\ } , \nonumber \\ e_-^+e_- & = & \frac { \hbar \omega}{3v } \frac{9}{16 } \left\ { [ j_{2}(kr)]^2+[j_{-3}(kr)]^2 \right\ } , \nonumber \\ e^+_+e_- & = & \frac{-\hbar \omega}{4v } \ { [ j_2(kr)\gamma_+ ( kr)+ j_{-3}(kr)\gamma_- ( kr)]^2 \nonumber \\ & + & [ j_{2}(kr)\gamma_-(kr)-j_{-3}(kr)\gamma_+(kr)]^2\}^{1/2 } e^{i \varphi } , \label{32}\end{aligned}\ ] ] where @xmath121 and @xmath122 all elements containing @xmath123 are equal to zero . unlike the case of radiation in an ideal cavity ( 27 ) , the phase difference @xmath124 between the components with opposite helicity depends here on the distance from the source . other cases can be examined in the same way . for the polarization matrix ( 25 ) in the polar direction , we again get only one non - zero element @xmath125 ^ 2 + [ \xi_- ( kr)]^2 \right\ } , \nonumber\end{aligned}\ ] ] where @xmath126 the time evolution can be described with the aid of approach proposed in @xcite . then , the matrix elements of the polarization matrix should be multiplied by the following factor @xmath127 + e^{- \eta t } , \nonumber\end{aligned}\ ] ] where @xmath128 and @xmath129 denotes the principle value of corresponding integral . again , any deviation from the polar direction changes the picture of polarization at intermediate and far distances with respect to that at the atom location . let us briefly discuss the obtained results . in this paper we have concentrated on the description of the spatial properties of multipole radiation by a single atom . the consideration is based on sequential use of the representation of multipole photons corresponding to the radiation of real atoms . it is shown that the zero - point oscillations of the multipole field are concentrated near the atom , while exceed the level predicted by the model of plane waves of photons everywhere . in a certain neighborhood of the atom , the effect is strong enough . although the effect can be observed at the relatively short distances , it seems to be important for the near- and intermediate - field quantum optics as well as for the experiments with trapped rydberg atoms when the typical interatomic distances are of the same order @xcite . in particular , it can be important for engineered entanglement in the system of two atoms in a cavity proposed in @xcite , for experiments with single - atom laser as well as for the estimation of casimir effect in atomic systems . let us note in this connection that possible influence of an atom on the electromagnetic vacuum state in the absence of radiation has been discussed in quantum electrodynamics for a long time ( e.g. , see @xcite ) . the new element here is the spatial inhomogeneity of the vacuum noise . unlike the effects discussed in @xcite , the specific distance dependence of the zero - point oscillations in the presence of atom has the geometrical nature and is independent of the atom - field interaction . to describe the polarization of multipole field , we proposed in section 4 a new definition of the polarization matrix based on the bilinear form in the field - strength tensor . the generalized polarization matrix ( 19 ) is additive with respect to the contributions coming from the electric field and magnetic induction . it reflects the three - dimensional nature of polarization connected with the three possible states of spin of photon . in special case of plane waves , when the third spin state is forbidden , it reduces to the conventional @xmath73 hermitian polarization matrix . in the case of multipole radiation , it combines together the objects considered earlier in @xcite . by construction , the generalized operator polarization matrix is a local object in spite of the global nature of the photon operators of creation and annihilation . the spatial properties of polarization are caused by the mode functions and changes with distance and direction from the source . for example , the electric dipole radiation from the excited atomic level with @xmath102 , corresponding to the creation of a circularly polarized photon with positive helicity , is transformed , at far distances , into the radiation with both helicities . in the case of radiation in empty space , the phase difference between the modes with opposite helicities is also a function of distance and direction from the source . both the distance and direction dependence of the polarization seems to be very important . in fact , any real measurement of intensity assumes the finite aperture of a detecting device @xcite . this means that , in the case of radiation by the atomic transition @xmath130 considered in section 5 , the precision of measurement of the polarization @xmath106 in the polar direction would be influenced by the zero - point oscillations of all three polarizations . let us stress that locality of polarization discussed in this paper can also be interpreted in terms of the photon localization . it is well known that , while the photon operators of creation and annihilation are defined in the whole space at once , the notion of photon localization is not deprived of physical meaning @xcite . the photodetection process provides a quite certain example of the photon localization @xcite . there are also known attempts to interpret the photon localization as the specific fall - off of the photon energy density @xcite . it is also clear that emission and absorption of radiation by atoms can also be interpreted as the photon localization @xcite . our results based on consideration of the representation of multipole photons shows the rapid fall - off of the density of zero - point oscillations and of the vacuum noise of polarization with the distance from atom as well as the spatial dependence of a certain characteristics of the radiation field ( e.g. , polarization ) . jaynes , e.t . and cummings , f.w . , 1963 , _ proc . ieee _ * 51 * , 89 . yoo , h.i . and eberly , j.h . , 1985 , _ phys . _ * 118 * , 239 kien , f.l . and shumovsky , a.s . , 1991 , _ int . _ b * 5 * , 2287 shore , b.w . and knight , p.l . , 1993 , _ j. mod . optics _ * 40 * , 1195 aliskenderov , e.i . , dung , h.t . , and shumovsky , a.s . , 1993 , _ phys . * 24 * 177 . condon , e.u . and shortley , g.h . , 1987 , _ the theory of atomic spectra _ ( new york : cambridge university press ) . heitler , w. , 1984 , _ the quantum theory of radiation _ ( new york : dover ) . cohen - 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birula , i. , 1998 , _ phys . _ , * 80 * , 5247 . shumovsky , a.s . and klyachko , a.a . , 2000 , _ laser physics _ * 10 * , 1 .	we discuss the spatial properties of quantum radiation emitted by a multipole transition in a single atom . the qualitative difference between the representations of plane and spherical waves of photons is examined . in particular , the spatial inhomogeneity of the zero - point oscillations of multipole field is shown . we show that the vacuum noise of polarization is concentrated in a certain vicinity of atoms where it strongly exceeds the level predicted by the representation of the plane waves . a new general polarization matrix is proposed . it is shown that the polarization and its vacuum noise strongly depend on the distance from the source .
in any cdm or @xmath15 cosmology , the first generation of stars is expected to form at redshifts 1030 in small ( @xmath16 ) bound objects and is assumed to imprint the intergalactic medium by both enriching and ionizing it . this enrichment may now have been detected in keck spectra of the lyman forest ( cowie et al . 1995 ; tytler et al . 1995 ; songaila & cowie 1996 [ sc96 ] ) , and various attempts have been made to use the first crude measurements of the forest metallicity ( sc96 ; haehnelt , steinmetz , & rauch 1996 ; rauch , haehnelt & steinmetz 1997 ) to place upper limits on the amount of early star formation in small galaxies ( sc96 ; miralda - escud & rees 1997 ; gnedin & ostriker 1997 ; haiman & loeb 1997 ) and to infer the rate of the high redshift type ii supernovae that may be our best hope of detecting objects in these early stages of the galaxy formation era ( miralda - escud & rees 1997 ) . the critical quantity for normalizing the amount of early star formation is the density of metals at early epochs . this has generally been obtained ( e.g. miralda - escud & rees 1997 ; haiman & loeb 1997 ) by combining the rough estimates of metallicity in forest clouds with the total baryon density ( @xmath17 ) inferred from big bang nucleosynthesis ( sbbn ) . apart from the uncertainty in the value of @xmath17 from sbbn , this methodology assumes a uniform enrichment of metals in the igm and also entails considerable uncertainty in the determination of individual cloud metallicities caused by poorly known ionization corrections and abundance patterns in the heavy elements . it is possible to avoid many of these problems by directly integrating the observed ion column densities in the forest clouds to obtain @xmath18 and , within plausible limits , @xmath19 . carbon and silicon are both well suited to this approach : as discussed below , their dominant ionization stages in the igm are and , strong doublets that are easy to find and measure with 10 m class telescopes ( [ obs ] ) ; they both have strong accessible lines in other ionization stages with which to assess ionization corrections ( [ ion ] ) ; and the ratio of @xmath12-process silicon to fe - coproduction carbon is useful for determining the abundance pattern at early epochs and to give some idea of the nature of the initial mass function ( [ discuss ] ) . letter _ reports the results of such a direct integration . the data used comprises high resolution observations of eleven quasars with @xmath20 made with the hires spectrograph on the keck i telescope , for a variety of programs , between 1994 november and 1997 april . all spectra have a resolution @xmath21 and have variable wavelength coverage in the range @xmath22 . full details of the extraction and processing of these spectra are given in a companion paper ( songaila 1997 ) . seven of the spectra have complete wavelength coverage between the quasar s @xmath2 and emission lines , whereas the remainder have only partial coverage redward of @xmath2 emission or have remaining inter - order gaps from hires s incomplete coverage above @xmath23 . the sample used to construct the column density distribution consists of all doublets detected redward of @xmath2 emission in those spectra with complete coverage . the doublets were found by inspection of the spectra , and confirmed by consistency of the column density and velocity structure in the two members of the doublet . systems within @xmath24 of the quasar s emission redshift were excluded from the sample to avoid including proximate systems which have ion ratios dominated by photoionization from the quasars themselves . also excluded were 10 specifically targeted partial lyman limit systems ( plls ) in 9 quasars that were chosen to be observed in a program to look for suitable systems in which to measure the primordial ratio of deuterium to hydrogen . the final sample consists of 81 doublets , with @xmath25 and @xmath26 , free from proximity or observational selection bias . the column density distribution was similarly determined from all doublets detected in the same subsample of quasar spectra as used for the distribution . only systems blueward of the quasars rest - frame but redward of the lyman forest were included , again excluding proximate systems and targeted plls . this resulted in 35 systems in seven lines of sight between @xmath27 and @xmath28 , with @xmath29 . in all cases , ion column densities were determined by fitting up to ten voigt profiles to each redshift system , defined in this context to be all absorption near a given fiducial redshift with gaps in velocity space of no more than @xmath30 . the column density at a given redshift is then the total column density of all such components . in all but a few systems , individual lines were unsaturated , so fitted column densities are insensitive to @xmath31-value . with these samples , the column density distribution function @xmath32 was determined for each ion . @xmath32 is defined as the number of absorbing systems per unit redshift path per unit column density , where at a given redshift @xmath33 the redshift path , @xmath34 , is defined as @xmath35 $ ] for @xmath36 or by @xmath37 $ ] for @xmath38 ( tytler 1982 ) . the and distribution functions are shown in figure 1 , over the redshift ranges , @xmath39and @xmath40 for , and @xmath41 and @xmath42 for . the data were calculated with @xmath36 and are plotted with @xmath9 error bars calculated from the poisson errors based on the number of systems in each bin . because of the somewhat heterogeneous nature of the parent observations , the signal - to - noise of the final sample is quite variable , in some cases because the low quasar emission redshift entailed observing below @xmath43 where the ccd efficiency dropoff is quite severe , in others because of variable coverage at the red end of the spectrum to fill in inter - order gaps . exposure times were also variable for the usual observational reasons . the amount of incompleteness at low column density was assessed by recalculating the column density distributions using only systems drawn from the highest signal - to - noise spectrum , toward q1422 + 231 , which has an exposure time of 580 minutes and reaches a @xmath9 limiting column density of @xmath44 for and @xmath45 for for @xmath46 . comparing this with the distributions obtained from the full sample , it is found that the turnover at low column density ( @xmath47 ) in the full sample is almost entirely a result of incompleteness . to take this into account , power laws were fitted only above @xmath48 for and @xmath49 for , with best - fit indices of @xmath50 for both low and high redshift , and @xmath51 for low redshift and @xmath52 for high redshift . these values are quite similar to those measured in in the forest at these redshifts ( petitjean et al . 1993 ; hu et al . 1995 ; kim et al . 1997 ) and in higher column density samples ( petitjean & bergeron 1994 ) . @xmath53 and @xmath54 were calculated from the column density distributions of figure 1 according to @xmath55 where @xmath56 is the cosmological closure density , @xmath57 is the ion s mass , and @xmath58 ( e.g. , lanzetta et al . values of @xmath53 and @xmath54 were calulated for @xmath59 in individual lines of sight to the five quasars in the sample with @xmath60 and complete spectral coverage between the quasar s @xmath2 and emission lines . the results are tabulated in table [ tbl:1 ] for @xmath61 and @xmath62 ( @xmath63 scales as @xmath64^{1/2}$ ] ) and give some idea of the uncertainty in calculating @xmath63 . formal mean values , weighted by @xmath65 , are @xmath66^{1/2}$ ] and @xmath67^{1/2}$ ] , where the errors are @xmath9 . the mean redshift of the sample is @xmath68 . a similar procedure applied to the @xmath69 systems gives @xmath70^{1/2}$ ] and @xmath71^{1/2}$ ] at @xmath72 . because of the rarity of high column density systems ( @xmath73 ) considerably longer path lengths are required to determine the number density of such systems , and the present data measure @xmath18 only for absorption with column density less than this value . ( this corresponds roughly to @xmath74 , from sc96 . ) the metal densities of the stronger systems are addressed in a companion paper ( songaila 1997 ) . the contribution of systems to @xmath75 and @xmath54 converges at the low column density end , and systems weaker than those observed will not contribute significantly to the density unless there is a very rapid upturn below the observed range . a less restricted sample was used to assess the ionization balance correction to be applied to the and sample . this was drawn from the full set of eleven quasar spectra , excluding proximate systems . in cases where there was no apparent absorption at the redshift in the other ionic species , an upper limit to the absorption in other ions was found by formally fitting the component model to the local continuum . this approach results in conservative upper limits , especially to lower ionization species , since absorption is likely to be more widespread in velocity space than lower ionization absorption . as is shown in figure 2 , is a trace ion relative to with nearly all systems having @xmath76 0.1 . an additional direct search in the spectra for systems with no strong yielded no additional systems . for photoionization models , this in turn yields a high photoionization parameter ( @xmath77 ) which , for a wide range of photoionizing spectra , implies that @xmath78 and @xmath78 ( e.g. , steidel 1990 ; giroux & shull 1997 ) . for this can be directly verified from observations ( middle panel of figure 2 ) which show that , even with @xmath2 forest contamination , the upper limits on the column density are generally compatible with measurements . ionization by a starburst galaxy spectrum with no high energy photons could result in a much higher ratio of to while maintaining the low / and / ratios , but it would also produce much in excess of , which is not observed : at @xmath79 / has an average value of about 0.2 ( sc96 ; songaila 1997 ) . finally , neither nor is strong in the -selected systems . the right panel of figure 2 shows that @xmath80 0.05 in the small number of systems measured . the high observed ratio of to combined with the low observed values of / and / suggests that forest clouds at @xmath59 are ionized by a broken power low spectrum with relatively few high energy photons and that it is unlikely that there is much material above the and levels ( sc96 ; giroux & shull 1997 ) . the best guess , therefore , would give @xmath81 and @xmath82 in these clouds . converting @xmath83 and @xmath84 to the @xmath19 of all metals requires an assumption of an abundance pattern in the @xmath59 forest . as in the old metal - poor halo stars , silicon is overabundant with respect to carbon relative to solar . however , assuming that the carbon and silicon abundances trace the universal abundances of iron coproduction and @xmath12-process elements , respectively ( e.g. , timmes , lauroesch , & truran 1995 ) , then for the fe - coproduction elements ( c , n , fe ) , @xmath85 , and for the @xmath12-process elements ( o , ne , si , mg , s ) , @xmath86 , and setting @xmath87 and @xmath88 gives @xmath89^{1/2}$ ] ( relative abundances from anders & grevesse 1989 ) . assuming a value of @xmath17 from sbbn of @xmath90 ( songaila , wampler , & cowie 1997 ) gives @xmath91^{1/2}$ ] . with @xmath7 , @xmath8 , and a solar metallicity of 0.019 , this implies a minimum universal metallicity relative to solar , at @xmath92 , in the range [ @xmath93 to [ @xmath94 . ( the value of @xmath95 given in tytler , fan & burles ( 1996 ) has now been revised upwards to @xmath96 ( tytler 1997 ) which would imply @xmath97 and a metallicity of [ @xmath93 , while the value of @xmath98 obtained by webb et al.(1997 ) would give a metallicity of [ @xmath99 [ @xmath100 . ) this is a minimum range of metallicity since there may be additional metals in higher column density clouds or galaxies as well as in ionization states that were not sampled by these observations . furthermore , adoption of @xmath38 would roughly double the metallicity . however , much of @xmath17 inferred from sbbn is believed to reside in the forest clouds at this time ( rauch et al . 1997 ; kim et al . 1997 ) , suggesting that this calculated metal density should be a good estimate of the total metals . interestingly , since a metallicity of @xmath101 gives rise to one ionizing photon per baryon ( e.g. , miralda - escud & rees 1997 ) , the metallicity measured here is just sufficient to preionize the igm . however , this metallicity range is substantially lower than the value of @xmath102 assumed by miralda - escud & rees ( 1997 ) and haiman & loeb ( 1997 ) and implies that their predicted rate of very high-@xmath33 type - ii supernovae could be lowered by as much as an order of magnitude , to a value of around 1 sn per 10 arcmin@xmath103 per year . i am grateful to the many people at the keck telescopes who made these observations possible , and to len cowie and esther hu for obtaining some of the observations on which this work is based . the research was supported by the national science foundation under grant ast 96 - 17216 . haiman , z. , & loeb , a. 1997 , in proceedings of the 18th texas symposium on relativistic astrophysics , ed . a. olinto , j. frieman , & d. schramm ( singapore : world scientific ) , in press [ preprint astro - ph/9701239 ] ccccccc 0014 + 813 & 3.384 & 1.29 & 0.62 & 0.42 & 1.2 & 1.8 0302@xmath104003 & 3.286 & 0.88 & 0.51 & 0.17 & 0.9 & 0.74 0956 + 122 & 3.301 & 0.94 & 2.4 & 3.7 & 4.5 & 16.0 1159 + 123 & 3.493 & 1.76 & 0.62 & 0.63 & 1.2 & 2.7 1422 + 231 & 3.620 & 2.32 & 1.3 & 2.7 & 2.4 & 12.0 mean & & & @xmath105 & @xmath106 & @xmath107 & @xmath108	column density distribution functions of with @xmath0 and with @xmath1 have been obtained using 81 absorbers and 35 absorbers redward of the @xmath2 forest in the lines of sight to seven quasars with @xmath3 . these distribution functions have been directly integrated to yield ion densities at @xmath4 to 3.5 of @xmath5 and @xmath6 with @xmath7 and @xmath8 ( @xmath9 errors ) . a larger sample of 11 quasar lines of sight was used to measure / , / , and / ratios , which suggest that and are the dominant ionization stages and that corrections to @xmath10 and @xmath11 are no more than a factor of two . normalizing the @xmath12-process elements to silicon and the fe - coproduction elements to carbon gives a density of heavy elements in these forest clouds of @xmath13(@xmath14 . the implications for the amount of star formation and for the ionization of the igm prior to @xmath4 are discussed . 2 cm^-2 # 110^#1 accepted for publication in _ astrophysical journal letters _
environment - induced decoherence is a vexed issue in confronting the challenges to quantum information processes . simple model calculations thus assume significance , to grasp the underlying parameter - regimes that can be manipulated , in order to minimize the effect of the environment . in quantum optics and solid state physics , the device that has gained popularity in recent times is a qubit ( which can be approximately represented by a pair of two quantum dots)@xmath0 . this system is equivalent to a superconducting josephson junction@xmath1 or a spin-1/2 nmr nucleus@xmath2 . a qubit can be described by a two - level hamiltonian , the quantum mechanics of which is rather straightforward . environmental influences can also be easily incorporated in this hamiltonian . there have been two distinctive attempts in modeling the environment , either in terms of ( @xmath3 ) a classical stochastic process , such as a gaussian or a telegraph one , or ( @xmath4 ) an explicit quantum collection of bosonic oscillators . one of the main objectives in this paper is to seek a unification of these two apparently disparate approaches . we consider an electron hopping between two single level quantum dots , coupled via a tunneling coefficient @xmath5 and an asymmetric bias @xmath6 . denoting by @xmath7 and @xmath8 the left and right dot states , the qubit hamiltonian , indicated by the subscript @xmath9 , can be written as , @xmath10 if we introduce the so - called bonding and antibonding states as the spin up and spin down states of the pauli matrix @xmath11 , denoted by @xmath12 and @xmath13 respectively , as @xmath14 can be cast into more familiar notation : @xmath15 where @xmath16 is the other pauli matrix that is off - diagonal in the representation in which @xmath11 is diagonal . in order to incorporate the effect of the environment on the qubit our strategy is to expand the hilbert space to re - express the system hamiltonian as @xmath17 where @xmath18 is yet another pauli matrix , which could represent a spin-1/2 impurity , for instance . the idea is to couple the @xmath18-system to a bath of bosonic oscillators , thereby causing fluctuations in @xmath18 . the resulting full hamiltonian can be written as @xmath19 here @xmath20(@xmath21 ) are boson creation ( annihilation ) operators , @xmath22 are coupling constants and @xmath23 is the harmonic oscillator frequency of the @xmath24 mode . it is interesting to physically assess the effect of the coupling term ( proportional to @xmath25 ) on the system hamiltonian @xmath26 . because @xmath27 is purely off - diagonal in the @xmath18- representation , it would cause spin - flips in @xmath18 between two allowed values + 1 and -1 , as in the celebrated glauber model of ising kinetics@xmath28 . these flips would occur with an amplitude field that would be proportional to @xmath25 and time - varying bosonic operators @xmath29 , in the interaction representation of the last term in eq . the net effect on @xmath26 is a quantum noise , encapsulated by @xmath30 . in a suitable limit , when the latter , i.e. @xmath30 , could be replaced by a classical noise @xmath31 that jumps at random between @xmath32 , the system hamiltonian would be stochastic , given by , @xmath33 where @xmath31 is a two - state jump process or a telegraph process . while we will present below a fully quantum mechanical treatment of eq . ( 5 ) , our aim will also be to set conditions under which a classical stochastic description via a telegraph process of @xmath31 , as expounded in detail by tokura and itakura@xmath34 , would ensue from the quantum formulation . the form of the hamiltonian in eq . ( 5 ) falls under the general scheme of a system - plus - bath approach to nonequilibrium statistical mechanics , in which the full hamiltonian is written as , @xmath35 where @xmath36 is the interaction hamiltonian and @xmath37 is the bath hamiltonian . in the present instance , @xmath38 as we will be employing a liouvillean approach to the dynamics governed by @xmath39 , it is useful to explain the notation . the liouville operator @xmath40 , associated with @xmath39 , is defined by @xmath41,\ ] ] where @xmath42 is an arbitrary but ordinary operator . thus a liouville operator yields an ordinary operator when it operates on an operator ( as in the right hand side of eq . ( 9 ) ) , just as an operator , operating on a wavefunction , yields another wavefunction . we will designate the states of a liouville operator by round brackets : @xmath43 , in analogy with the dirac ket vectors : @xmath44 for the wavefunctions . the decomposition in eq . ( 7 ) further allows @xmath40 to be split as @xmath45 the liouville dynamics effected by @xmath40 on the density operator @xmath46 , leading to a non - markovian master equation , in the context of the decoherence of a qubit ( as well as a collection of qubits ) have been extensively studied by kurizki and collaborators@xmath47 . they have also considered the additional influence of external time - varying fields for controlling decoherence , dynamically . while we will be employing a very similar approach , our treatment will differ from gordon etal@xmath47 . the latter utilize the born approximation , valid for weak coupling constants @xmath25 , whereas we will consider strong coupling in which @xmath25 will be treated to all orders . such strong coupling considerations are the hallmark of functional integral treatments of quantum dissipative systems eg . that of a spin - boson hamiltonian@xmath48 . furthermore our focus in the classical limit of the quantum noise will be a telegraph process as opposed to the much - studied gaussian stochastic processes . similar spin - boson hamiltonians , akin to eq . ( 5 ) , have been considered in the past@xmath49 , though the comparison has not been dealt with , to the best of our knowledge . with these preliminaries the paper is section wise organised as follows . in sec . ii we outline the mathematical steps for a resolvent expansion of the averaged time - development operator in the laplace transform space , the average being carried out over the hilbert spaces of @xmath18 and @xmath37 . the latter is expressed in terms of a self - energy whose form is provided . in sec . iii explicit results are presented for the much explored ohmic - dissipation model and comparisons are drawn with the classical telegraph process , results of which are given in the appendix . for facilitating comparison we discuss fluctuation in bias and hopping separately . in a subsection of sec . iii , we indicate results for non - ohmic situations as well . finally in sec . iv , we discuss the issue of partial decoherence , recently studied by us in the context of the telegraph process@xmath50 , to put it under the perspective of quantum noise . the section v contains a few concluding remarks . in order to incorporate strong - coupling effects it is convenient to transform the hamiltonian @xmath39 in eq . ( 5 ) with the aid of a unitary transformation defined by the operator : @xmath51\exp\left[-i\frac{\pi}{2}\tau_y\right].\ ] ] the transformed hamiltonian is given by @xmath52 what the second term in @xmath53 does is to cause a rotation in the @xmath54-space by an angle of @xmath55 about the y - axis , in the anti - clockwise direction such that @xmath56 and @xmath57 . the first term in @xmath53 then eliminates the interaction term @xmath36 but puts the onus of coupling on @xmath26 itself ( besides generating an innocuous counter term that is constant , and can be dropped ) . the net result is @xmath58 where @xmath59.\end{aligned}\ ] ] the transformed hamiltonian @xmath39 is now endowed with a new interaction term that may be written as @xmath60 the important point however is that any perturbation treatment of @xmath61 is tantamount to treating the coupling to all orders as @xmath62 now occurs in the exponent as is evident from the structure of @xmath63 ( cf . the equation ( 13 ) is again of the generic form of eq . ( 7 ) except that the system is now the qubit itself as @xmath26 is replaced by @xmath64 ! with reference to the liovillean in eq . ( 10 ) the equation of motion for the density operator @xmath46 reads @xmath65 unlike other approaches@xmath66 we find it convenient to work with the laplace transforms , defined by @xmath67 from eq . ( 16 ) then @xmath68 we employ the usual factorization approximation that at @xmath69 the qubit and the bath are decoupled@xmath70 , so that @xmath71 where @xmath72 is the qubit density operator at time @xmath69 , @xmath73 , is the canonical density operator for the bath and @xmath74 is the corresponding partition function . underlying the prescription in eq . ( 19 ) is the assumption that the bath always remains in quilibrium at a fixed temperature t while the qubit evolves from an arbitrary state . for most of our derived expressions we shall assume that the electron is on the left dot at @xmath69 i.e. @xmath75 translated to the bonding and anti - bonding basis , this implies @xmath76 ( we shall return in sec . iv to a more general initial condition . ) the laplace transform of the so - called reduced density operator can be written as @xmath77}\rho_q(0),\ ] ] where @xmath78 denotes trace over the eigenstates of @xmath18 whereas @xmath79 specifies trace over the bath ( i.e. eigenstates of @xmath37 ) . because the trace is invariant under the unitary transformations ( eg . in eq . ( 11 ) ) , eq . ( 22 ) can be equivalently expressed as @xmath80}\rho_q(0),\ ] ] where the tilde on @xmath81 implies that the corresponding time - evolution is now governed by @xmath82 of eq . ( 13 ) . denoting the liouville operator associated with @xmath83 as @xmath84 , we have upto second order in @xmath85 , @xmath86,\ ] ] where the self - energy @xmath87 is @xmath88}.\ ] ] the z - dependence ( or frequency dependence ) of @xmath87 implies that the underlying dynamics is non - markovian , in general . we emphasize once more that though the self - energy is computed to second order in @xmath84 , the theory is valid to arbitrary orders in the original coupling constants @xmath25 . hence , the resultant master equation in the time domain that emanates from eq . ( 24 ) is of more general validity than the one obtained in the born - approximation@xmath47 . our next step is to write the matrix elements of the self - energy @xmath87 . it is clear that @xmath87 is a liouvillean in the @xmath89-subspace alone because the @xmath54 and the bath variables are averaged over , in eq . hence , @xmath87 has a @xmath90 matrix representation in the eigen basis @xmath91 ( note the round brackets used for states of the liouvillean ) , wherein @xmath92 , @xmath93 , ... are the eigenkets of @xmath11 . clearly then , the rows and columns of the @xmath90 matrix would be labelled by @xmath94 , @xmath95 , @xmath96 and @xmath97 , respectively . additionally , while performing the traces in eq . ( 25 ) , we would need to sum over the eigenstates of @xmath18 and @xmath37 , as in the following : @xmath98 etc , with these explanatory remarks about the notation , the matrix elements of @xmath87 turn out to be generalized forms of those given in ref . [ 13 ] : @xmath99\langle n_b\|\rho_b\|n_b\rangle.\end{aligned}\ ] ] from the structure of @xmath83 in eq . ( 15 ) it is evident that the bath variables enter only through the operators @xmath63 . therefore , when we sum over the bath indices @xmath100 , @xmath101 , etc . , with the boltzmann weight @xmath102 , we end up with correlation fucntions of @xmath63 , as in the following : @xmath103 as it happens , the only non - zero components of the correlation function are @xmath104\right\}.\end{aligned}\ ] ] an essential attribute of quantum dissipative system is that the system @xmath37 is taken to the limit of an infinitely large number of bosonic modes . this implies that the discrete sum over @xmath9 has to be replaced by an integral over continously varying frequencies @xmath105 with an appropriate weight called the spectral density @xmath106 , thus @xmath107\right\},\ ] ] where @xmath108 in the literature on dissipative quantum systems one model for the spectral density @xmath106 that has received the maximum amount of attention is what gives rise to ohmic dissipation@xmath109 . in this , @xmath106 is assumed to have a linear frequency dependence with an exponential cut - off @xmath110 @xmath111 where @xmath112 is a phenomenological damping parameter that measures the strength of the coupling with the heat bath . this form allows for an analytic expression for the laplace transform as defined in eq . ( 17 ) , for @xmath113 as @xmath114 where @xmath115 denotes gamma functions . the significance of the condition @xmath113 can be ascertained from the fact that the quantity @xmath116 ( note : @xmath117 has been set to unity , hitherto ) sets the time scale over which quantum coherence is maintained . thus , for all frequencies greater than the cut - off @xmath110 , quantum fluctuations remain strictly coherent . before we take up the comparison with the telegraph noise it is pertinent to point out that it is meaningful to think of a quantum bath as classical only in the incoherent regime i.e. over time scales larger than @xmath118 . in other words , the classical limit of the bath obtains for frequencies @xmath119 , i.e. the temperature is significantly large , but not so large as to violate the condition @xmath120 , if we want to restrict our discussion within the domain of validity of the analytical result of eq . if the restriction @xmath121 does apply , the frequency or z - dependence in the arguments of the gamma functions in eq . ( 33 ) can be dropped yielding @xmath122 the parameter @xmath123 will turn out later to be related to the jump rate of the telegraph process . with this background we are ready to provide explicit results for the different elements of the density matrix in order to examine decoherence as well as to draw comparison with the results for the telegraph noise . because the latter have been extensively studied recently by itakura and tokura@xmath34 , we will follow their lead in ignoring the bias term in the original qubit hamiltonian @xmath64 , i.e. set @xmath124 though it should be evidently clear that the formalism can easily embrace asymmetric cases ( @xmath125 ) as well . with this , the relevant hamiltonian for further considerations can then be written as ( cf . ( 3 ) and ( 13 ) ) @xmath126 for ease of discussion and for remaining close to the treatment of itakura and tokura we take up the cases of fluctuation in bias ( @xmath127 ) and fluctuation in hopping ( @xmath128 ) separately below . the appropriate interaction hamiltonian that has to be substituted in th expression for the self energy @xmath129 in eq . ( 25 ) is now given by @xmath130 carrying out the summations implied in eq . ( 25 ) we arrive at @xmath131,\ ] ] where @xmath132 , a_2(z)=2\zeta_{\epsilon}^{2}[\tilde{\phi}(z_{-})+\tilde{\phi}'(z_{+ } ) ] \nonumber \\ a_3(z)&=&2\zeta_{\epsilon}^{2}[\tilde{\phi}(z)+\tilde{\phi}'(z ) ] , \quad z_\pm = z\pm2i\delta.\end{aligned}\ ] ] here @xmath133 and @xmath134 are the laplace transforms of the correlation functions @xmath135 and @xmath136 , respectively . the matrix element of @xmath137 , required in eq . ( 24 ) , is obtained from a combination of eq . ( 27 ) and the definition of the matrix elements of the liouville operator @xmath138 a la eq . the resultant matrix is of a block - diagonal form and can be easily inverted to yield @xmath139}.\end{aligned}\ ] ] $ ] is plotted for @xmath140 , @xmath141 and @xmath142 is plotted for @xmath140 , @xmath141 and @xmath142 is plotted for @xmath140 , @xmath141 and @xmath142 it is of course possible to go back to the dot - basis with the aid of eq . results are shown in figs . 1 , 2 and 3 in which @xmath143 $ ] , @xmath144 and @xmath145 are plotted versus time t for a certain choice of parameters . while @xmath143 $ ] oscillates and decays to zero as @xmath146 , the diagonal elements , proportional to the populations of the two dot states , oscillate and settle to the common value of 0.5 , as we expect for full decoherence . if we compare figs 1 - 3 with fig a.1 for the telegraph process below , it is evident that the quantum case exhibits persistence of oscillations over longer time scales . the appropriate interaction hamiltonian that has to be substituted in the expression for the self energy @xmath87 in eq . ( 25 ) is now given by @xmath147 following the formalism developed in sec . ii and using eq . ( 25 ) , the self energy matrix @xmath87 is given by @xmath148,\ ] ] where @xmath149 , \nonumber \\ b_4(z)&=&4\zeta_{\delta}^{2}[\phi'(z_-)+\phi(z_- ) ] , \quad z_\pm = z\pm2i\delta , \end{aligned}\ ] ] where @xmath150 and @xmath151 have been defined in the earlier section . using the form @xmath152 , it is evident from the diagonal form of the matrix that the population of the bonding and anti bonding state remains constant for all @xmath153 . using @xmath154 the matrix elements are given by , @xmath155.\end{aligned}\ ] ] however in the dot basis we expect the usual saturation to the common value of 0.5 for both left and right dot populations . ( 4 ) shows the variation of left and right dot populations with time and also the loss of coherence over time from the decay of @xmath156 . ( solid ) , @xmath157 ( dashed ) and @xmath158 ( dotted ) are plotted for @xmath140 , @xmath141 and @xmath142 we are now in position to make a connection between the results of quantum and classical regimes considered separately in sec.iii and appendix a. there are two critical distinctions between the two regimes . at the outset , the quantum behavior is non - markovian , reflected in the z ( or frequency)-dependence of the self - energy @xmath87 in eq . ( 25 ) . on the other hand , in the classical stochastic case , the markovian assumption is invoked , to begin with , as seen in eq . ( a.2 ) below . the second crucial ingredient in the quantum situation , even after the markovian approximation is incorporated , is in the temperature - dependence of the relaxation rate @xmath123 . while the latter , in the classical domain , is usually endowed with an exponential dependence on temperature , with a barrier - activation in mind , it has a richer structure in the quantum regime , as discussed below . we mention below in appendix . a.2 that in the long time region the off - diagonal element of the system density operator follows an exponential decay with time constant @xmath159 . from eq . ( 43 ) , as it can be clearly seen from the laplace transform of @xmath160 , the decay is exponential only if we assume @xmath161 to be independent of @xmath162 , wherein the time constant is just the inverse of @xmath163 $ ] . thus the comparison with the classical telegraph process is meaningful only if one takes the markovian limit of heat bath induced relaxation , at the outset . at sufficiently high temperatures given by @xmath164 we can see from eq . ( 33 ) that @xmath133 can be approximated as @xmath165 which leads to a rather simple expression for @xmath166 ( for @xmath167 ) in eq . ( 42 ) as @xmath168 , that is real . hence @xmath123 , which appears as the jump rate in the case of telegraph noise , discussed in the appendix in the sequel , can be compared with the relaxation rate @xmath163 $ ] in the case of quantum noise , is now a function of @xmath112 and @xmath169 . also , as it is obvious from the expression of the relaxation rate that the dependence on temperature is not an exponential , rather it shows a power law behavior . however , it is pertinent to emphasize that these conclusions hinge on our assumption that quantum dissipation is ohmic . in the more general case , discussed below in sec . iii d , we will see that the temperature - dependence of the relaxation rate can in fact be exponential , under certain special situations . at low temperatures and weak damping i.e. small @xmath112 , @xmath133 is given by @xmath170 where @xmath171 is a renormalized form of tunneling frequency given by @xmath172 . using this form of @xmath133 , and eq . ( 43 ) we can deduce , @xmath173.\ ] ] from eq . ( 44 ) , it is evident that the relaxation rate becomes @xmath174 , which is far from an exponential in temperature . thus it may be stressed that even in the markovian limit the temperature dependence of both the relaxation rate as well as the rabi frequency is much more complex than an exponential , as is usually assumed for classical activated processes . these conclusions are similar to those obtained earlier in the context of spin relaxation of a muon , tunneling in a double well@xmath175 . we had in the text presented results based on the assumption of ohmic dissipation i.e. choosing @xmath176 . the ohmic model is a rather limited one , not applicable to phonons but is relevant to a bath consisting of electron - hole excitations near the fermi surface , as in metals.@xmath177 in this subsection we will allow for a general spectral density function , and will be analysing the relaxation rate in the case of quantum noise . we will try to make a connection with the classical telegraph process and show that in the high - temperature limit , the relaxation rate indeed follows an exponential dependence on temperature with however quantum corrections , which was not the case in the ohmic limit . our starting point is eq.(30 ) which upon clubbing the oscillating terms together , can be rewritten as , @xmath178\right].\ ] ] by making a change of variable from @xmath179 to @xmath180 in eq . ( 45 ) , @xmath150 can be written as , @xmath181\right],\ ] ] in the high temperature regime@xmath182 , the integrand in the second term in eq . ( 46 ) , can be replaced by its short time limit as , @xmath183\right].\ ] ] also in the high temperature regime i.e. low @xmath169 , @xmath184 and @xmath185 can be replaced by the argument i.e. @xmath186 , and hence the integrations in eq . ( 47 ) can be carried out easily to yield , @xmath187 where @xmath188 . also @xmath151 is obtained by replacing @xmath3 by @xmath189 in eq . thus the relaxation rate which is given by re(@xmath166 ) , for the quantum case in sec . iii.c , is obtained by imposing the markovian limit i.e. putting @xmath167 , leading to @xmath190 finally the lower limit in eq . ( 49 ) can be set to zero and thus the integral is just a function of temperature which follows a power law . thus the dominant behavior of the relaxation rate is an exponential dependence on temperature , which is what we expect at high temperatures . this analysis therefore provides a regime where we can more meaningfully compare the relaxation rate @xmath123 for the case of classical telegraph process with the laplace transform of the bath correlation function in the quantum case . note that the present treatment does not depend on any specific assumption for the spectral density @xmath106 but is more generally couched , irrespective of whether the phonons are acoustic or optic . in the context of fluctuation in hopping , it is evident that the system hamiltonian commutes with the interaction hamiltonian , which means there is no energy exchange between the system and the bath . such cases are important for studying partial decoherence@xmath70 . thus we generalize our discussion to a more general initial state of a qubit ( than of eq . ( 20 ) ) given by , @xmath191 in such a scenario it can be established that the density matrix does not evolve to a completely mixed state , rather it approches a limiting value , which does contain the off - diagonal components . in this situtation coherence is not completely lost and the off - diagonal terms give information about the initial state of the system . we note that the initial qubit information can be retrieved , either from experiments related to persistent current or by measuring the population of states . the former path is not applicable in our case as we have not allowed for the existence of an aharonov - bohm flux@xmath70 . the density matrix ( in @xmath7 and @xmath8 basis ) at time @xmath69 is given from eq . ( 50 ) by , @xmath192,\ ] ] which can be translated to the bonding - antibonding basis , via the transformation given by eq . ( 2 ) . the off - diagonal term @xmath193 , which measures coherence , does not go to zero unlike in the cases discussed in sec.(iii.c and a.2 ) . these particular symmetric cases of @xmath124 and @xmath194 show partial coherence which is important for decoherence - free quantum computation protocols . $ ] for quantum ( solid ) and classical ( dashed ) cases are plotted versus time t. ] $ ] for quantum ( solid ) and classical ( dashed ) cases are plotted versus time t. ] the above formalism is general and is valid for any kind of environment , as long as the commutator of the system hamiltonian and the interaction hamiltonian is zero . as shown in the cases of classical and quantum noises ( figs . 5 and 6 ) , the imaginary and real parts of @xmath195 as @xmath196 , whereas the populations of bonding and anti - bonding states stay constant , as is evident from eq . ( 43 ) . thus , in general , the off - diagonal elements of the density matrix ( in @xmath7 and @xmath8 basis ) approach limiting values , emphasizing that the asymptotic state is not a fully mixed state : @xmath197.\ ] ] at this stage it is also worthwhile to underscore the importance of time scales involved in attaining partial decoherence . for this we define a function @xmath198 as : @xmath199 this function is 1 at time @xmath69 and 0 at time @xmath200 . when integrated over all times , we get a relaxation time for substenance of decoherence as : @xmath201 we carry out an explicit calculation for @xmath54 in both classical and quantum regimes . using eq . ( a.14 ) and ( 43 ) and the initial density matrix given by eq . ( 45 ) , we can calculate @xmath202 to obtain : @xmath203 where , for the quantum case , @xmath204 on the other hand , for the classical telegraph process , @xmath205 where @xmath166 is given by the @xmath167 limit of eq . it is evident from fig . ( 6 ) that the partial decoherence is attained more rapidly in the case of classical noise . our main emphasis in this paper has been a relative assessment of quantum and classical noise sources attached to a pair of quantum dots or a qubit . the analysis has been made with the aid of a unified formalism , comprising a resolvant expansion of an averaged time - development operator . the quantum formalism enabled us to provide a microscopic meaning and detailed temperature - dependence to the phenomenologically introduced parameter of the relaxation rate , that appears in the classical case of a telegraphic noise . our treatment of the quantum noise has included the much studied ohmic model of dissipation that characterizes electron - hole excitations off the fermi surface , in a metal , as well as non - ohmic dissipation which covers both acoustic and optic phonons@xmath109 . it is eventually in the phonon model of dissipation that the usually assumed exponential temperature dependence of the relaxation rate ( of a telegraph process ) is realized , making the comparison between the classical and quantum cases more direct . in the last subsection of the paper ( see iv ) we focussed on the important issue of partial decoherence that can be utilized for quantum computation , which has received recent attention@xmath70 . this case was analyzed when fluctuation is ascribed only to the hopping term , yielding a situation in which the system hamiltonian commutes with the coupling to the heat bath . here , the comparison between classical and quantum noises is quite striking coherence persists over longer time scales for the quantum case . this attribute can be effectively exploited in the context of quantum computation , in which it is essential to be able to retrieve information about the initial quantum state , notwithstanding heat bath - induced effects . ek thanks the inspire support of the department of science and technology for an ms thesis project that contains the present contribution . sd will like to record his gratitude to amnon aharony , ora entin - wohlman and shmuel gurvitz for many helpful discussions . where the stochastic states @xmath209 , @xmath210 ... are associated with the two possible values @xmath32 of @xmath31 ( over which the summations in eq . ( a.2 ) are performed ) , @xmath211 is the a - priori probability of the occurence of the state @xmath210 , @xmath138 is the liouville operator accompanying @xmath212 , and @xmath213 is the liouville operator associated with an ordinary operator @xmath214 ( j=+1 or -1 ) such that it remains then to define the stochastic matrices @xmath216 and @xmath217 . the matrix @xmath216 is a projection governed by @xmath218 whereas w is a jump matrix whose element @xmath219 , for instance , is the rate of jump of the process @xmath31 from the state @xmath210 to the state @xmath209 . the underlying markovian assumption is manifest in the frequency independence of @xmath217 . the properties of the telegraph process allow us to calculate the average over the stochastic indices a , b , ... in eq . ( a.2 ) in closed form , leading to @xmath220\rho_q(0),\ ] ] where @xmath221 @xmath123 being the jump rate . in calculating the matrix elements of @xmath222 we ned to use the properties of liouville operators given by@xmath207 @xmath223 using ( a.7 ) , it is straightforward to obtain @xmath224,\ ] ] where @xmath225 the matrix of @xmath226 is therefore of a simplified block - diagonal form from which matrix under the square brackets in eq . ( a.5 ) can be evaluated easily . combined with the initial condition in eq . ( 21 ) we find , after some straight forward algebra , @xmath227\left(\rho_\mathcal{r}\right)_{++}(0 ) \nonumber \\ \left(\rho_\mathcal{r}\right)_{--}(z)&=&\left[\frac{2(a+b)-\lambda ( a^{2}-b^{2})}{d_{1}}\right]\left(\rho_\mathcal{r}\right)_{--}(0).\end{aligned}\ ] ] and @xmath228\left(\rho_\mathcal{r}\right)_{+-}(0)\ ] ] where @xmath229 } , \nonumber \\ b&=&\frac{4\zeta_\epsilon^2}{\bar{z}[\bar{z}^2 + 4(\zeta_\epsilon^2+\delta^2 ) ] } , \nonumber \\ c&=&\frac{2[\zeta_\epsilon^2+\bar{z}(\bar{z}-2i\delta)]}{\bar{z}[\bar{z}^2 + 4(\zeta_\epsilon^2+\delta^2 ) ] } , \nonumber \\ d_1&=&(2-\lambda a)^2-(\lambda b)^2 , \nonumber \\ d_2&=&(2-\lambda c)(2-\lambda c^*)-(\lambda b)^2.\end{aligned}\ ] ] with the chosen initial condition in eq . ( 21 ) , eq . ( a.10 ) simplifies to @xmath234 it is of course a matter of ease to go from eqs . ( a.10 ) and ( a.11 ) to the dot basis , with the aid of the transformation in eq . ( 2 ) to rederive the elements of the density operator , results of which are presented in fig . the latter are in complete conformity with those of ref . [ 7 ] and display full decoherence at infinite times i.e. @xmath235 and @xmath236 as expected . [ [ fluctuation - in - hopping - zeta_epsilon0-under - telegraph - process ] ] fluctuation in hopping ( @xmath128 ) under telegraph process ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ the relevant stochastic hamiltonian is now given by ( cf . , eq . ( 6 ) ) @xmath237 in this case the fluctuating part of the hamiltonian commutes with the system part of the hamiltonian , a case which has received attention recently@xmath50 . in this case now there is no energy transfer between the system and the bath , allowing for partial decoherence ( see sec . iv ) . in this case the calculations are simpler than in a.i as the matrix of @xmath226 is diagonal in the @xmath11 representation . therefore , the diagonal elememts of @xmath238 do not evolve at all from their initial values of one - half whereas the off - diagonal element becomes @xmath239^{-1}\ ] ] the time dependence of the off - diagonal term @xmath240 is obtained from the inverse laplace transform of @xmath241 to yield , for @xmath242 , this expression coincides with the one derived by itakura and tokura@xmath34 following an elaborate dyson series - type time - domain treatment . the offdiagonal term initially follows a gaussian decay , @xmath244 , discussed in [ 7 ] , and it becomes exponential in the long time regime . the time constant appearing in the exponential decay is referred to as @xmath245 following the nmr literature @xmath246 , turning out to be @xmath159 for @xmath247 . transforming to the dot basis the behaviour of the elements of the density operator is exhibited in fig . 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the hartree fock bogoliubov ( hfb ) approximation was outlined more than twenty years ago @xcite for infinite systems and almost immediately was introduced in nuclear physics @xcite . in the case of infinite systems the hfb procedure is well studied and the character of the wave functions ( w.f . ) is well understood . however , in the case of finite systems ( nuclei ) , things are not so clear . the hfb equations in the case of nuclei are meaningful provided the boundary conditions for the single quasiparticle ( s.qp . ) are correctly formulated in order to describe a genuine finite state . the goal of the present paper is to provide the correct formulation of the hfb approximation in the case of finite systems . it is well known that pairing correlations always appear whenever a pole ( i.e. a bound state ) is present in the two - body green function of the many - body system @xcite . in this case , corrections to the single particle ( s.p . ) green function of the type = 3.5 cm give rise to diagrams of the type = 3.5 cm when only the contribution of the pole is taken into account @xcite . here @xmath1 stands for the energy of the two particle bound state and @xmath2 and @xmath3 are the particle and hole energies respectively . the process represented by diagram ( 1 ) leads to a mixing between particle and hole states and as a result to a smearing of the fermi surface @xcite . this mixing has the special feature that a hole ( particle ) can transform into a particle ( hole ) , due to the presence of the pair condensate , provided their energies are related by the energy conservation law @xmath4 whenever the energy of the hole state is less than @xmath1 the corresponding particle state to which the hole state is coupled lies in continuum . this situation formally resembles the case of an electron in a very strong field @xcite or the case of a bound state embedded in continuum @xcite . consequently , a sufficiently deep hole state becomes unstable with respect to decay into a particle state by an interaction with the pair condensate . the same thing happens in the case of a deep hole decaying into a less deeply bound hole , with the excitation of a phonon , which can further decay by particle emission . this process is described by the diagram = 3.5 cm where the wavy line represents a phonon . the physical situation is thus not new . new is the fact that due to the presence of the pair condensate , some hole states acquire a width and therefore the s.qp.w.fs . can no longer be treated as corresponding to bound states , as has hitherto been done , but as continuum states . how to introduce this property of the s.qp.w.fs . into the hfb approximation for finite systems and therefore how to define the boundary conditions correctly is the main goal of the present discussion . in this section the well known hfb equations will be derived for the sake of completeness . the emphasis will be on the correct definition of s.qp.w.fs . so as to describe genuine finite systems , i.e. systems with finite matter distribution . the forces between particles will not be specified , except for some general properties , like the finite range . by analogy with the usual hf approximation , the hfb ground state w.f . @xmath5 is defined as the vacuum for the fermi quasiparticles @xcite @xmath6 where @xmath7 , \\ \alpha_i^\dagger & = & \int dx [ v_i^ * ( x ) \psi ( x)+ u_i ( x)\psi^\dagger ( x ) ] , \nonumber \end{aligned}\ ] ] and @xmath8 and @xmath9 stand for field operators for annihilation and creation of a particle with space spin coordinates @xmath10 , which satisfy the usual anticommutation relations @xmath11 by requiring that @xmath12 and @xmath13 represent fermion operators , i.e. @xmath14 one easily obtains the relations @xmath15 dx = \delta_{ij } , \\ & & \int [ u_i^ * ( x ) v_j ( x ) + v_i ( x ) u_j^ * ( x ) ] dx = 0 , \nonumber \end{aligned}\ ] ] @xmath16 the constraints ( 56 ) ensure the unitary character of the transformations ( 3 ) . the total energy of the many body system and the mean number of particles are @xmath17 and @xmath18 where @xmath19 and @xmath20 stand for the hamiltonian and the number operator in the second quantization representation . the mean values for the energy and for the particle number can be expressed through the densities @xmath21 @xmath22 in the following way @xmath23 @xmath24 if only two body interactions are present . @xmath25 and @xmath26 stand for the kinetic energy and the two body interaction respectively . a shorthand notation was used for the traces in rels . ( 9 ) and ( 10 ) . the hfb equations for the two component s.qp.w.fs . @xmath27 are derived from the stationarity condition of the total energy ( 9 ) under the subsidiary conditions ( 10 ) and ( 5 ) . these equations are @xmath28 @xmath29 where @xmath30 is the chemical potential , and @xmath31 and @xmath32 are the s.p . hamiltonian and the pairing field respectively and @xmath33 stands for the s.qp . energies . when performing the summations in rels . ( 7 ) and ( 8) one must include only those solutions of the nonlinear system ( 1112 ) with @xmath34 , which define the operators @xmath35 . the solutions with @xmath36 correspond to operators @xmath37 . let us analyse in more detail these equations in the case of a finite system . the problem which arises is the meaning of the normalization condition ( 5 ) ; namely , if the right hand side of rel . ( 5 ) should be a @xmath38function or a kronecker symbol , as is usually the case @xcite , in complete analogy with the hf approximation . a finite system is characterized by a finite matter distribution and therefore by a finite range of the s.p . field ( except in the case of the coulomb interaction ) . naturally , being determined by the matter distribution , the pairing field must also have a finite range ( an infinite range of the pairing field can only occur if the system under consideration is unstable with respect to two particle decay . ) the problem is determining the mechanism which leads to finite matter distribution when s.qp . states no longer have the character of discrete states , as was discussed in the introduction . when pairing is turned off ( i.e. @xmath39 ) the spectrum of eqs . ( 1112 ) looks like the one show in fig . the left hand side corresponds to the spectrum of eq . ( 11 ) , while the right hand side corresponds to eq . ( 12 ) . = 7.5 cm upon turning the pairing on , the two spectra mix and the discrete states with energies outside the interval @xmath40 will lie in the continuum . only the states with energy within the interval ( 13 ) will preserve the bound state character . the continuum part of the spectrum for @xmath41 and @xmath42 can disappear only if the s.p . potential is finite and positive at infinity as in the case of a harmonic oscillator . can also be preserved if @xmath43=0 $ ] , a condition which is not fulfilled in fact . this condition is satisfied in the case of constant pairing approximation @xmath44 , an approximation which leads to an unphysical density distribution . ] there is no physical reason for this to be true in the case of finite systems . a glance at fig.1 . reveals why pairing correlations lead to a significant increase of the level density in the vicinity of the fermi surface . pairing appears when the chemical potential has a value within a shell , as illustrated in fig . 1 . in such a case , the total number of hole ( the left hand side of fig . 1 ) and particle ( the right hand side of fig . 1 ) states , which is practically equal to the number of s.qp . states with pairing included , is almost double the number of s.p . states in hf approximation . in order to determine what happens in the case of a bound state with an energy @xmath45 , when pairing is turned on , we shall use perturbation theory . for the sake of simplicity , the nucleus will be assumed to be spherical and the pairing field real @xmath46 . also , the spin and angular variables are assumed to be already separated from eqs . ( 11 ) and ( 12 ) and the corresponding geometrical factors included in the definition of the single particle fields . from eq . ( 11 ) one easily obtains in the vicinity of a bound state the relation @xmath47 where @xmath48 the other component of the single quasi particle wave function becomes then @xmath49 where @xmath50 and @xmath51 is a normalization constant . from rels . ( 14 ) and ( 15 ) one obtains @xmath52 in order to determine the normalization constant @xmath51 , we shall use the representation of the green function through regular and irregular solutions of eq . ( 16 ) @xmath53 where @xmath54 stands for the wronskian @xmath55 the asymptotic behavior for the regular @xmath56 and irregular @xmath57 solutions is given by @xmath58 where @xmath59 . the asymptotic behavior of the @xmath60component is @xmath61,\ ] ] and therefore @xmath62 ^ 2 } { [ e - e_0 + < \phi_0 \mid \delta \frac{1}{-e + \lambda -h } \delta \mid \phi_0 > ] ^2 + \pi^2 \mid < \phi_0 \mid \delta \mid u_{0e } > \mid^4}.\ ] ] consequently , the normalized solutions are @xmath63 @xmath64 where @xmath65 and @xmath66 the matrix elements in the above relations can be calculated without any significant loss of accuracy for @xmath67 . as one can observe , due to the coupling with the continuum , the bound state spreads over the entire spectrum ( see rel . ( 21 ) ) . now the quantity @xmath68 has thus to be interpreted as the occupation number probability density over a unit energy interval . the solution is formally equivalent to the solution of a coupled channel problem with a bound state embedded in the continuum@xcite . it displays a well defined resonant character with a width @xmath69 and a shift @xmath70 . the case of a resonant state can be treated in a similar way . far from the resonance energy @xmath71 , the amplitude of the @xmath60component is very small , while the @xmath60component is practically equal to the unperturbed solution @xmath56 . in the vicinity of the resonance @xmath72 , the amplitude of the @xmath60component increases significantly inside the potential well ( see rels . ( 18 ) , ( 20 ) ) , while the phase of the @xmath60component changes by @xmath73 . the phase of the @xmath60component is @xmath74 the solution just described is characterized by the fact that the @xmath75component is square integrable , even though the single quasi particle wave function solution @xmath76 represents a continuum state . the question is : does this feature hold true for the selfconsistent solution of the hfb equations ? the density distribution @xmath77 satisfies rel . ( 10 ) ( i.e. the diagonal part of @xmath77 is integrable ) , while the density of the pair condensate satisfies the condition @xmath78 < n,\ ] ] which can be easily derived by means of rels . ( 56 ) @xcite . therefore , one can expect that @xmath77 and @xmath79 both fall down quickly enough outside the system . the density distribution @xmath77 determines the asymptotic behavior of the single particle selfconsistent potential @xmath80 , while the density @xmath79 defines the pairing potential @xmath81 . the nonlocality of these potentials is governed by the range of the two body interaction @xmath26 , assumed to be finite . we shall now show that the following asymptotic behaviors take place @xmath82\right ) \ ] ] and @xmath83 \right ) , \ ] ] when @xmath84 tends to infinity and @xmath85 remains finite ( practically of the order of the range of @xmath26 ) . ( in the above rels . ( 2324 ) the symbol @xmath86 means that the quantities on the left hand side behave like the corresponding arguments of the @xmath86function . ) as one can observe , the pairing field @xmath87 has a longer tail than the s.p . selfconsistent field @xmath80 . using rels . ( 23 ) and ( 24 ) one can show , using hfb eqs . ( 1112 ) , that the @xmath75 and @xmath60components behave asymptotically as @xmath88 @xmath89 if @xmath90 and @xmath91 @xmath92 if @xmath93 outside the potential well , for energies in the interval ( 27 ) , the two eqs . ( 1112 ) decouple and the corresponding asymptotic behavior of the @xmath75 and @xmath60components of the single quasi particle wave function is determined by the `` energies '' @xmath94 and @xmath95 , respectively . for energies in the interval ( 30 , ) the asymptotic behavior of the @xmath75component is governed by the inhomogeneous part of the eq . ( 11 ) ( i.e. by the term @xmath96 ) , which can not be neglected in this case , as was possible for energies in the interval ( 27 ) . on the other hand , the term @xmath97 falls down exponentially and does not influence the asymptotic behavior of the @xmath60component . the asymptotic behavior of the @xmath60component is fully determined by the `` energy '' @xmath98 , which is positive in the interval ( 30 ) . now , if one takes into account the definitions of densities @xmath77 and @xmath79 ( rels . ( 7 ) and ( 8) respectively ) , one can easily notice that the asymptotic behavior of the @xmath75 and @xmath60components ( see rels . ( 2530 ) ) completely agrees with the asymptotic behaviors ( 23 ) and ( 24 ) . this means that the corresponding asymptotic behaviors are selfconsistent . it is physically natural to expect that the asymptotic behavior of the density @xmath77 is controlled by the chemical potential @xmath99 , i.e. by the energy of the least bound particle . the density of the pair condensate @xmath79 can be interpreted as the wave function of a bound state of two interacting particles in an external field with energy @xmath100 . using hfb eqs . ( 11 ) and ( 12 ) and the definition ( 8) , one can show that the density @xmath79 satisfies the equation @xcite @xmath101 _ other terms negligible outside the system _ . from this equation it follows that at large distances @xmath79 behaves as @xmath102 where @xmath103 which also agrees with rel . ( 24 ) . strictly speaking , this asymptote is correct only outside the range of @xmath26 . if , however , one takes into account the fact that a system of two identical nucleons does not have bound states , the asymptote is valid everywhere outside the s.p . potential well . summing up , in the normalization conditions ( 5 ) for the single quasi - particle wave functions , the right hand side has to be interpreted as a kronecker symbol if @xmath104 and as a dirac @xmath38function if @xmath105 ( as it is well known @xcite the system ( 1112 ) has the property that if @xmath106 is a solution , then @xmath107 is a solution as well ) . furthermore , for @xmath108 the @xmath75component of the single quasi particle wave function is always square integrable and its norm has to be interpreted as the occupation number probability . on the other hand , the relation @xmath109 is valid only for @xmath110 . if the energy @xmath45 is outside this interval , the integral @xmath111 should be interpreted as an occupation number probability density per unit energy interval . this section is devoted to some simple examples of hfb equations , which although somewhat unrealistic , lead to a better understanding of various issues arising while solving eqs . ( 1112 ) . this approximation is also known as the bcs approximation @xcite . the s.qp.w.f . in this case is @xmath113 where @xmath114 and @xmath115 usually , the pairing field @xmath112 const is taken to be different from zero in a limited energy interval around the fermi surface . however , there is no recipe to determine this energy band in a unique way . it is obvious that the density @xmath77 can not be integrable if solutions belonging to the continuum part of the spectrum of the eq . ( 32 ) are included ( i.e. when the pairing field is nonvanishing for such states ) . furthermore , if one considers that @xmath87 is acting for all energies , then the density @xmath79 is @xmath116 - \frac{\delta}{2 } g ( x , y,\lambda ) , \nonumber\end{aligned}\ ] ] where @xmath117 stands for the single particle green function of the eq . as is well known , the green function diverges like @xmath118 when @xmath119 . therefore , a finite density @xmath79 can not be defined in this case because of this divergence . one notices that the divergence is not logarithmic , as is usually stated in textbooks @xcite . a local pairing field corresponds to a zero range two body interaction . then , as one can easily show by using eq . ( 31 ) , the density @xmath79 will always be singular for coinciding arguments . for the entire hfb procedure to be meaningful , the two body interaction , which is responsible for pairing , must have a finite range . we assume that @xmath120 and look for solutions of eqs . ( 1112 ) with zero orbital momentum @xmath121 . for @xmath122 the solution reads @xmath123 where @xmath124 and @xmath125 stand for some constants , which have to be determined from matching the interior with the exterior solutions and normalization . for @xmath126 @xmath127 or @xmath128 if @xmath129 where @xmath130 and @xmath131 have to be determined from matching and normalization . if @xmath104 , the spectrum is discrete and the energies have to be determined from the matching condition @xmath132 if @xmath41 , then one deals with a state lying in continuum . in contrast to the case of constant pairing , one now has @xmath133 \neq 0 $ ] and sufficiently deeply bound hole states acquire width . inside the potential well , the two components of the s.qp.w.f . form a superposition of two s.p.w.f . with energies equal approximately to @xmath134 and @xmath135 , respectively ( as a rule @xmath136 is very small and can be neglected in the determination of @xmath137 ) . in the vicinity of a hole state ( the left hand side of fig . 1 ) @xmath138 has a zero and one can show that @xmath139 similar to the bcs approximation . this relation holds only inside the potential well and it is not valid outside it . far from the resonance this ratio becomes @xmath140 unlike the bcs approximation , the radial behaviors of the @xmath75 and @xmath60components are no longer identical . in this case , the density @xmath77 has the correct asymptotic behavior , but due to the local character of the pairing potential , the density @xmath79 has the same divergence as in the case of bcs approximation . . this is another approximation which has been used in nuclear physics . the solution of eqs . ( 1112 ) is now @xmath142 where @xmath143 @xmath144 is a normalization constant and @xmath145 is given by the equation @xmath146 ( @xmath145 has to be included only for @xmath147 ) the density @xmath77 has a correct asymptotic behavior , but the same problems arise with the density @xmath79 as above . the density of the pair condensate @xmath79 is singular for @xmath148 ( see eq . ( 31 ) ) . even though the examples discussed here have little in common with the real selfconsistent solution of eqs . ( 1112 ) , in our opinion however , they lead to a deeper understanding of the structure of the hfb equations in the case of finite systems . especially instructive in this sense is the role played by the nonlocality of the pairing field and consequently by the range of the two body forces . we have examined the hfb approximation in the case of finite systems , when the two body interaction between particles has a finite range . special attention was paid to the asymptotic behavior of the s.qp.w.fs . it was shown that the s.qp . states located in spectrum sufficiently far from the fermi surface have the character of continuum states . e.g. the deep hole states acquire a width corresponding to the decay into a particle state and the pair condensate . this width has to be interpreted as a contribution to the imaginary part of the s.p . optical potential . these features of the general solutions of the hfb equations have to be included in any hfb calculations . to what extent the available hfb results @xcite correspond to the real solution , is still an issue which needs further studies .	_ the original preprint was never published , due to various circumstances beyond the author s control . the initial preprint has been known to a number of people , the results have been used in published literature and the preprint is still been referred to . the present text has been edited somewhat and a number of typos have been corrected ; however , no significant changes have been incorporated into the text . no formulas have been added or deleted and the figures have been redrawn . a scanned copy of the original preprint in jpeg format can be found at + @xmath0 _ the author thanks f.m . edwards for helping to edit the text and yongle yu for bringing to his attention a number of typos . some general features of the spectrum of the hartree fock bogoliubov equations are examined . special attention is paid to the asymptotic behavior of the single quasiparticle wave functions ( s.qp.w.fs . ) , matter density distribution and density of the pair condensate . it is shown that due to the coupling between hole and particle states , the deeply bound hole states acquire a width and have to be treated as continuum states . the proper normalization of the s.qp.w.fs . is discussed .
controlling chaotic transport is a key challenge in many branches of physics like for instance , in particle accelerators , free electron lasers or in magnetically confined fusion plasmas . one way to control transport would be that of reducing or suppressing chaos . there exist numerous attempts to control chaos ( see refs . @xcite for a rather extended list of references ) . most of the methods for controlling chaotic systems is done by tilting targeted trajectories . however , for many body experiments like the magnetic confinement of a plasma or the control of turbulent flows , such methods are hopeless due to the high number of trajectories to deal with simultaneously . for these systems , it is desirable to control transport properties without significantly altering the original system under investigation nor its overall chaotic structure . here we focus on another strategy which is based on building barriers by adding a small apt perturbation which is localised in phase space , hence confining all the trajectories . the main motivations for a _ localised control _ are the following ones : very often the control of a physical system can only be performed in some specific regions of phase space . this is in particular the case in thermonuclear fusion devices where the electric potential can only be modified near the border of the plasma . for some purposes it is sometimes desirable to stabilize only a given region of phase space without modifying the major part of phase space in order to preserve some specific features of the system . this method can be used to bound the motion of particles without changing the perturbation inside ( and outside ) the barrier . also , using a localised control means that one needs to inject much fewer energy than a global control in order to create isolated barriers of transport . in this article , we consider the class of hamiltonian systems that can be written in the form @xmath0 i.e. an integrable hamiltonian @xmath1 ( with action - angle variables ) plus a small perturbation @xmath2 . the idea is to _ slightly _ and _ locally _ modify the perturbation and create regular structures ( like invariant tori ) : the aim is to devise a control term @xmath3 such that the dynamics of the controlled hamiltonian @xmath4 has more regular trajectories or less diffusion than the uncontrolled one . for practical purposes , the control term should be small with respect to the perturbation @xmath2 , and localised in phase space ( i.e. the subset of phase space where @xmath3 is non - zero is finite and small ) . in refs . @xcite , an explicit method of control was provided in order to construct a control term @xmath3 of order @xmath5 such that the controlled hamiltonian @xmath6 is integrable . the main drawback of this approach is that the control term has to be applied on all the phase space . here we provide a method to construct control terms @xmath3 of order @xmath5 with a finite support in phase space , such that the controlled hamiltonian @xmath7 has isolated invariant tori . for hamiltonian systems with two degrees of freedom , these invariant tori act as barriers in phase space . for higher dimensional systems kam tori act as effective barriers of diffusion . the main result of the paper is stated as follows : for a hamiltonian system written in action - angle variables with @xmath8 degrees of freedom , the perturbed hamiltonian is @xmath9 where @xmath10 and @xmath11 is a non - resonant vector of @xmath12 . we consider a region near @xmath13 and the perturbation @xmath14 has constant and linear parts in actions of order @xmath15 , i.e. @xmath16 where @xmath17 is of order @xmath18 . we notice that for @xmath19 , the hamiltonian @xmath20 has an invariant torus with frequency vector @xmath21 at @xmath13 for any @xmath17 not necessarily small . the controlled hamiltonian we construct is @xmath22 where @xmath23 is a smooth characteristic function of a region around a targeted invariant torus ( the size of its support is of order @xmath15 ) . it is sufficient to have @xmath24 for @xmath25 . for instance , @xmath26 would be a possible and simpler candidate , however representing a long - range control . we notice that the control term @xmath3 we construct only depends on the angle variables and is given by @xmath27 where @xmath28 is a linear operator defined below as a pseudo - inverse of @xmath29 . note that @xmath3 is of order @xmath5 . for a sufficiently small perturbation , hamiltonian ( [ eqn : gene ] ) has an invariant torus with frequency vector close to @xmath21 . after proving this result , we check numerically that the controlled hamiltonian is more regular than the uncontrolled one , i.e. the invariant tori of the controlled hamiltonian persist to higher values of the amplitude of the perturbation than in the uncontrolled case . in sec . [ sec:1 ] , we explain the theory of the localised control of hamiltonian systems and in particular we prove eqs . ( [ eqn : gene])-([eqn : exf ] ) . in sec . [ sec:2 ] , we give some applications of the localised control on the following models : a forced pendulum hamiltonian , the delta - kicked rotor ( standard map ) and a non - twist hamiltonian model . for these systems , we show numerically that the localised control is able to create isolated invariant tori beyond the values of the parameters for which there are no invariant tori in the absence of control . we first recall the global control theory as explained in refs . @xcite in order to define the main operators that will be used for the localised control . + let us fix a hamiltonian @xmath1 . we define the linear operator @xmath30 by @xmath31 where @xmath32 is the poisson bracket . the operator @xmath33 is not invertible , e.g. , @xmath34 . we consider a pseudo - inverse of @xmath35 , denoted by @xmath28 , satisfying @xmath36 if the operator @xmath28 exists , it is not unique in general . we define the _ resonant _ operator @xmath37 as @xmath38 we notice that eq . ( [ gamma ] ) becomes @xmath39 . a consequence is that any element @xmath40 is constant under the flow of @xmath1 . _ notation : _ in what follows , we will use the notation @xmath41 for an operation between @xmath42 and @xmath43 which can be vectors or covectors . for instance , if @xmath42 is a covector and @xmath43 a vector , @xmath41 is the usual scalar product @xmath44 . if @xmath42 is a vector and @xmath43 a covector , @xmath41 is the matrix whose elements are @xmath45_i^j= a_i b^j$ ] . for a vector @xmath42 and a matrix @xmath46 , @xmath47 is a vector . in the same way , if @xmath42 is a covector , @xmath48 is a covector . also we denote @xmath49 the matrix @xmath50 with elements @xmath51_i^j=\partial a^j/\partial \theta^i$ ] . for clarity we also denote @xmath52 the operator @xmath53 . + let us now assume that @xmath1 is integrable with action - angle variables @xmath54 where @xmath55 is the @xmath8-dimensional torus . here , @xmath56 is a @xmath8-dimensional vector and @xmath57 is a @xmath8-dimensional covector . the poisson bracket between two functions @xmath20 and @xmath14 is given in the usual form @xmath58 we assume that @xmath1 is linear in the actions variables , so that @xmath59 , where the frequency vector @xmath60 is any co - vector of @xmath12 . in this paper , we assume that @xmath60 is non - resonant , i.e. there is no vector @xmath61 such that @xmath62 . the operator @xmath30 acts on @xmath14 given by @xmath63 as @xmath64 a possible choice of @xmath28 is @xmath65 we notice that this choice of @xmath28 commutes with @xmath30 . + the operator @xmath66 is the projector on the resonant part of the perturbation : @xmath67 since @xmath60 is non - resonant . we also define the projector on the non - resonant part of the perturbation @xmath68 the global control follows directly from the definition of these operators @xmath28 , @xmath66 and @xmath69 : we construct a global control term for the perturbed hamiltonian @xmath70 , i.e. we construct @xmath3 such that the controlled hamiltonian @xmath71 is canonically conjugate to @xmath72 . this conjugation is given by the following equation @xmath73 where @xmath74 we notice that if @xmath14 is of order @xmath15 , the control term @xmath3 is of order @xmath5 . in general , the control term depends on all the variables @xmath56 and @xmath75 , and acts _ globally _ on all phase space . since @xmath60 is non - resonant , @xmath76 only depends on the actions and thus @xmath77 is integrable . the derivation of eqs . ( [ prop1])-([eqn : ctf ] ) is given in refs . @xcite . starting from this global control , we derive a localised control such that the control term only acts in a given region of phase space around a selected invariant torus . we consider a nearly integrable hamiltonian system : @xmath78 we assume that @xmath1 has the invariant torus with a non - resonant frequency vector @xmath60 at @xmath79 . for @xmath14 sufficiently small , the kam theorem ensures that this invariant torus is preserved under suitable hypothesis . we expand hamiltonian ( [ eqn : h0v ] ) around @xmath79 and we translate the actions such that the invariant torus with frequency @xmath21 is located at @xmath13 for @xmath1 , and around @xmath13 for the perturbed hamiltonian . hamiltonian ( [ eqn : h0v ] ) becomes ( up to a constant ) @xmath80 where @xmath17 is of order @xmath18 , i.e. , @xmath81 and @xmath82 . without any restriction , we assume that hamiltonian ( [ eqn : hcf ] ) is such that @xmath83 and @xmath84 : the mean value of @xmath85 is absorbed into the total energy and the mean value of @xmath86 into the frequency vector @xmath60 . for @xmath56 sufficiently small , the perturbation @xmath87 is small . we apply eq . ( [ eqn : fctf ] ) in order to get the control term @xmath3 . however , for larger @xmath56 , the control term is no longer small . therefore we localise it in a region close to @xmath13 , i.e. we consider the following controlled hamiltonian : @xmath88 where @xmath23 is a smooth characteristic function such that @xmath89 if @xmath90 , and @xmath91 if @xmath92 . the main drawback of this approach is that the control term is a priori of order @xmath15 even if it is small since it is localised in a region near @xmath13 . in the next section we develop another approach where the control term @xmath3 is of order @xmath5 and does no longer depend on @xmath56 . as in the previous section , we consider the family of hamiltonians ( [ eqn : hcf ] ) . for @xmath93 , hamiltonian ( [ eqn : hcf ] ) has an invariant torus with frequency vector @xmath60 located at @xmath13 . the problem of control we address is to slightly modify hamiltonian ( [ eqn : hcf ] ) near @xmath13 in the following way : @xmath94 such that the invariant torus with frequency @xmath60 exists for the controlled hamiltonian @xmath95 for higher values of the parameter @xmath15 than in the uncontrolled case . here @xmath23 denotes a smooth step function , meaning that the control only applies in a small part of phase space ( of size @xmath15 ) : for instance , @xmath23 is a sufficiently smooth function such that @xmath91 if @xmath96 and @xmath89 if @xmath90 . moreover , we notice that the control term @xmath3 we apply is only a function of the angles in the region around the invariant torus . the main proposition of the _ localised _ control of hamiltonian systems is the following one : _ proposition 1 : _ if @xmath85 and @xmath86 are sufficiently small and if @xmath85 , @xmath86 and @xmath17 are smooth , there exists a control term @xmath3 such that the controlled hamiltonian @xmath97 is canonically conjugate to @xmath98 where @xmath99 with a constant covector @xmath42 and @xmath100 is of order @xmath18 , i.e. @xmath101 and @xmath102 . the control term @xmath3 is given by @xmath103,\ ] ] which can also be written as @xmath104.\ ] ] the important feature of the control term @xmath3 is that it does only depend on the angle variables . since the hamiltonian @xmath105 has an invariant torus with frequency @xmath106 at @xmath13 , the controlled hamiltonian @xmath95 has also this invariant torus in the region where @xmath56 is close to @xmath107 . _ proof : _ we consider the following transformations @xmath108 acting on functions @xmath109 like @xmath110,\ ] ] where the covector @xmath111 and the vector @xmath43 are functions of @xmath57 from @xmath55 into @xmath12 . the operator @xmath112 is the linear operator from @xmath12 to @xmath113 acting on a vector @xmath114 as @xmath115 . here @xmath116 is the linear operator from @xmath12 to @xmath12 which is the product of the two linear operators acting on a vector @xmath114 as @xmath117 in [ sec : appa ] , we check that the transformations @xmath108 are canonical if @xmath43 derives from a scalar function . we perform a transformation @xmath108 on the controlled hamiltonian @xmath95 given by eq . ( [ eqn : hc ] ) and we determine the functions @xmath111 and @xmath43 in the following ways : @xmath118 the function @xmath43 is determined such that the order @xmath15 of the constant term in actions vanishes . + @xmath119 the function @xmath111 is determined such that the linear term in actions ( which is of order @xmath15 ) vanishes . + @xmath120 the control term @xmath3 is determined such that the constant term in actions [ which is now of order @xmath5 after @xmath118 ] vanishes . we perform a transformation @xmath122 which is @xmath15-close to the identity : the expression of @xmath123 is @xmath124,\end{aligned}\ ] ] where the covector @xmath125 is defined by @xmath126 the function @xmath125 is @xmath15-close to @xmath111 : @xmath127 and satisfies @xmath128 so that @xmath129 where @xmath130 is a matrix , function of @xmath131 , which results from the action of the operator @xmath132 on the constant function 1 . first we notice that the scalar function @xmath133 is of order @xmath134 . this can be seen from the equations @xmath135 where @xmath136 is the transposed covector of @xmath43 . the function @xmath137 will be chosen such that @xmath138 for all @xmath57 . therefore we have @xmath139 and @xmath140 , according to the hypothesis on the smooth step function @xmath23 . + next , we expand the function @xmath141 around @xmath13 : @xmath142 where @xmath143 is of order @xmath144 , i.e. @xmath145 and @xmath146 . we notice that @xmath147 is of order @xmath5 and the covector @xmath148 is of order @xmath15 since @xmath17 is of order @xmath134 . the hamiltonian @xmath105 becomes @xmath149.\end{aligned}\ ] ] the canonical transformation is determined by two equations : @xmath150={\bf 0}. \label{eqn : detm}\end{aligned}\ ] ] the control term is chosen such that @xmath151.\ ] ] equations ( [ eqn : detb ] ) is solved in fourier space . we expand the function @xmath43 : @xmath152 and the coefficients @xmath153 are given according to eq . ( [ eqn : detb ] ) : @xmath154 when @xmath155 , and @xmath156 when @xmath157 . thus the vectorial function @xmath43 is chosen to be @xmath158 we recall that we require that @xmath159 which is ensured if @xmath85 is sufficiently small and smooth and if @xmath21 satisfies a diophantine condition ( see the usual kam proofs like for instance in ref . in particular , we notice that this choice of @xmath43 satisfies @xmath160 . equation ( [ eqn : detm ] ) is solved by choosing @xmath161 , where @xmath162 satisfies @xmath163 it is straightforward to check that eq . ( [ eqn : detm ] ) is satisfied if @xmath164 is a solution of eq . ( [ eqn : muti ] ) . if the operator @xmath165 , which is @xmath15-close to the identity and then invertible , eq . ( [ eqn : detm ] ) has a solution @xmath166^{-1 } ( { \bf w}+\varepsilon^{-1}\partial_{\bf a } q(-\varepsilon \gamma \partial v,{\bm \theta})).\ ] ] the hypothesis on invertibility is fulfilled if @xmath86 as well as @xmath85 are small enough and smooth and if @xmath21 satisfies a diophantine condition ( see again ref . the covector @xmath125 which is @xmath15-close to @xmath111 has the following expansion @xmath167 we notice that @xmath168 since @xmath28 and @xmath66 commute . the resulting hamiltonian is then given by eq . ( [ eqn : htildc ] ) where @xmath169 and @xmath170 which is of order @xmath18 . note that we have dropped some additive constants of order @xmath5 , @xmath171 and @xmath172 since we recall that @xmath66 is the mean value with respect to the angles . the equations of motion for @xmath173 are @xmath174 since @xmath101 and @xmath102 , we see that @xmath13 is invariant , and that the evolution of the angles is linear in time with frequency vector @xmath106 . therefore , the hamiltonian @xmath173 has an invariant torus located at @xmath13 with frequency vector @xmath106 . more precisely , the flow of the controlled hamiltonian @xmath95 on @xmath13 is @xmath175 where @xmath176 the equation of the torus @xmath13 is thus @xmath177 @xmath178 + _ remark 1 : _ we notice that if @xmath179 , the control term @xmath3 given by eq . ( [ eqn : cth ] ) is zero . in this case , the original hamiltonian already has the invariant torus at @xmath13 . + _ remark 2 : addition property of the control term _ in the case where more than one invariant torus needs to be created , we can add the control terms localised in non - overlapping regions of phase space . this is a straightforward extension to the previous case . the controlled hamiltonian becomes @xmath180 where @xmath181 is defined for each region of phase space by eq . ( [ eqn : cth ] ) . we notice that in each region of phase space , the operators @xmath28 , @xmath66 and @xmath69 are different since they are defined from the frequency vector of a given invariant torus . we consider the following forced pendulum model : @xmath182.\ ] ] figure [ fig1 ] depicts a poincar section of hamiltonian ( [ eqn : fp ] ) for @xmath183 . we notice that for @xmath184 there are no longer any invariant rotational ( kam ) torus @xcite . ( 7.5,6)(0,0 ) ( 0,0 ) ( 3,3.9 ) first , this hamiltonian with 1.5 degrees of freedom is mapped into an autonomous hamiltonian with two degrees of freedom by considering that @xmath185 is an additional angle variable . we denote @xmath186 its conjugate action . the autonomous hamiltonian is @xmath187.\ ] ] the aim of the localised control is to modify locally hamiltonian ( [ eqn : h2dof ] ) in order to reconstruct an invariant torus with frequency @xmath188 . we assume that @xmath188 is sufficiently irrational in order to fulfill the hypotheses of the kam theorem . first , the momentum @xmath189 is shifted by @xmath188 in order to define a localised control in the region @xmath190 since the invariant torus is located near @xmath191 for hamiltonian ( [ eqn : h2dof ] ) for @xmath15 sufficiently small . the operators @xmath28 and @xmath66 are defined from the integrable part of the hamiltonian which is linear in the actions @xmath192 : @xmath193 and hamiltonian ( [ eqn : h2dof ] ) is @xmath194 where @xmath195+\frac{p^2}{2}.\ ] ] the action of @xmath28 , @xmath66 and @xmath69 on a function @xmath196 of @xmath189 , @xmath197 and @xmath198 given by @xmath199 are given by @xmath200 the actions of @xmath28 , @xmath66 and @xmath69 on @xmath14 given by eq . ( [ eqn : vfp ] ) are @xmath201,\\ & & { \mathcal r } v=\frac{p^2}{2},\\ & & { \mathcal n } v=\varepsilon \left[\cos x+\cos(x - t)\right].\end{aligned}\ ] ] since @xmath202 depends only on @xmath197 and @xmath198 , and since @xmath14 and @xmath203 are quadratic in @xmath189 , it is straightforward to check that only the first two terms of the series ( [ eqn : ctf ] ) are non - zero . the global control term reduces to @xmath204 its explicit expression is given by @xmath205 we notice that the control term is of order @xmath15 , i.e. of the same order as the perturbation . however , the control term @xmath3 acts only in a region where @xmath206 since it is multiplied by a function @xmath207 such that @xmath208 when @xmath209 , and @xmath210 when @xmath211 . consequently , the controlled hamiltonian @xmath212 is locally integrable ( since it is locally conjugate to @xmath213 ) provided that the canonical transformation is well defined ( which is obtained when @xmath15 is sufficiently small ) . a phase portrait of hamiltonian ( [ eqn : fp ] ) with the control term ( [ eqn : ffp ] ) shows a very regular behaviour which persist for high values of @xmath15 . however we notice that for @xmath15 greater than one , the control term is no longer small compared with the perturbation . in order to apply the localised control as in sec . [ sec:2b ] , we notice that hamiltonian ( [ eqn : h2dof ] ) is of the form ( [ eqn : hcf ] ) with @xmath214 , @xmath215 and @xmath216 . in this case the control term given by eq . ( [ eqn : cth ] ) is equal to @xmath217 therefore the control term is equal to @xmath218 this control term has four fourier modes with frequency vectors @xmath219 , @xmath220 , @xmath221 and @xmath222 . we consider the region in between the two primary resonances located around @xmath223 and @xmath224 . the control term given by eq . ( [ eqn : fpa ] ) can be simplified by considering the region of phase space around @xmath225 . by keeping the main fourier mode of this control term , i.e. the one with frequency vector @xmath221 which has the largest amplitude for @xmath188 close to @xmath226 , the control term becomes @xcite @xmath227 for the numerical computations we have chosen @xmath228 ( golden - mean invariant torus ) which is the last invariant torus to break - up for hamiltonian ( [ eqn : fp ] ) . a poincar section of hamiltonian ( [ eqn : fp ] ) with the approximate control term ( [ eqn : f2fpa ] ) for @xmath229 shows that a lot of invariant tori are created with the addition of the control term precisely in the lower region of phase space where the localisation has been done ( see fig . [ figcg ] ) . using the renormalization - group transformation @xcite , we have checked the existence of the golden - mean invariant torus for the hamiltonian @xmath230 is given by eq . ( [ eqn : fpa ] ) with @xmath231 . by using the approximate and simpler control term @xmath232 given by eq . ( [ eqn : f2fpa ] ) the existence of the invariant torus is obtained for @xmath233 . however , we have checked using laskar s frequency map analysis @xcite that invariant tori and effective barriers to diffusion ( broken tori ) persist up to higher values of the parameter ( @xmath234 ) . + the next step is to localize @xmath3 given by eq . ( [ eqn : fpa ] ) around a chosen invariant torus created by @xmath3 : we assume that the controlled hamiltonian @xmath230 has an invariant torus with the frequency @xmath188 . we locate this invariant torus using frequency map analysis . then we construct an approximation of the invariant torus of the hamiltonian @xmath230 of the form @xmath235 . we consider the following localised control term : @xmath236 where @xmath23 is a smooth function with finite support around zero . more precisely , we have chosen @xmath91 for @xmath237 , @xmath89 for @xmath238 and a third order polynomial for @xmath239\alpha,\beta[$ ] for which @xmath23 is a @xmath240-function , i.e. @xmath241 . the function @xmath242 and the parameters @xmath243 , @xmath244 are determined numerically ( @xmath245 and @xmath246 ) . the support in momentum @xmath189 of the localised control is of order @xmath247 compared with the support of the global control which is of order 1 . ( 15,6.3)(0,0 ) ( 0,0 ) ( 3,3.9 ) ( 8,0 ) figure [ fig2 ] shows that the phase space of the controlled hamiltonian is very similar to the one of the uncontrolled hamiltonian . we notice that there is in addition an isolated invariant torus . using frequency map analysis @xcite , we check that this invariant torus corresponds to the one where the control term has been localised , i.e. its frequency is equal to @xmath248 . we notice that the perturbation has a norm ( defined as the maximum of its amplitude ) of @xmath249 whereas the control term has a norm of @xmath250 for @xmath183 . the control term is small ( about 4% ) compared to the perturbation . we notice that there is also the possibility of reducing the amplitude of the control ( by a factor larger than 2 ) and still get an invariant torus of the desired frequency for a perturbation parameter @xmath15 significantly greater than the critical value in the absence of control . we consider the standard map : @xmath251 this map is obtained by a poincar section of the following hamiltonian @xmath252 figure [ figsmc0 ] depicts a phase portrait of the standard map for @xmath253 . we notice that there are no kam tori ( dividing phase space ) at this value of @xmath254 ( the critical value of the parameter for which all the kam tori are broken is @xmath255 ) . similarly to the forced pendulum , we consider the invariant torus with frequency @xmath188 . by translating the momentum , we map hamiltonian ( [ eqn : hsm ] ) into @xmath256 where @xmath257 . using the same computations as for the forced pendulum , the controlled hamiltonian obtained by the procedure described in sec . [ sec:2b ] is @xmath258 which is integrable and canonically conjugate to @xmath259 . we notice that although the control term is of the same order as the perturbation , it is more regular than the perturbation since its fourier coefficients decrease like @xmath260 . hamiltonian ( [ eqn : hsml ] ) is of the form ( [ eqn : hcf ] ) with @xmath261 , @xmath215 and @xmath216 . the control term given by eq . ( [ eqn : cth ] ) becomes @xmath262 again we notice that this control term is more regular than the perturbation since its fourier coefficients decrease like @xmath260 . in particular , it is bounded in space and time , piecewise continuous in time . for instance , for @xmath263 , the control term ( [ eqn : fsma ] ) is equal to @xmath264 the phase portrait of hamiltonian ( [ eqn : hsml ] ) with the control term ( [ eqn : smt ] ) for @xmath265 is depicted on fig . [ figsmct ] . we notice that in this case , the controlled kicked rotor is now a kicked pendulum : instead of the rotor @xmath259 , the integrable part becomes a pendulum @xmath266 and the perturbation is a periodic @xmath267-kick . we notice that the controlled hamiltonian has invariant tori in the region near @xmath225 ( where the control has been localised ) . these invariant tori persist up to high values of the parameter @xmath268 larger than 10 which has to be compared with @xmath269 in the absence of control . note that the control term we use is bounded ( conversely to the perturbation ) and its amplitude is small compared with the amplitude of the fourier coefficients of the perturbation ( with a factor smaller than 10% depending on @xmath254 ) . ( 17,7 ) ( 0,0 ) ( 8.5,0 ) in order to recover a map , we need to locate the control at each @xmath270 for @xmath271 . for a given @xmath272 , the approximate control term is @xmath273 since each of these control term act on different regions of phase space , we sum all these control term to obtain a more global stabilization and recover a map . the control term becomes @xmath274 next , we perform an inverse fourier transform . the hamiltonian becomes @xmath275.\ ] ] the controlled standard map is thus obtained by performing additional kicks : each @xmath276 , in addition to the kicks of strength @xmath277 , one has to perform kicks of strength @xmath278 , and each @xmath279 , one has to perform kicks of strength @xmath280 . this leads to the following form for the controlled standard map @xmath281 it has a more compact form by using @xmath282 and @xmath283 : @xmath284,\\ & & x_{n+1}=x_n+p_{n+1}+ \frac{k^2}{32}\sin ( 2x_n+p_n+k\sin x_n+(k^2\sin 2x_n)/16 ) . \label{eqn : smct1b}\end{aligned}\ ] ] if we neglect the kicks every @xmath279 , we obtain the following map @xmath285 conversely , if we neglect the kicks every @xmath286 , we get @xmath287 a phase portrait of these maps are depicted on figs . [ figsmc1 ] , [ figsmc2 ] and [ figsmc3 ] . the most efficient control is obtained for the map ( [ eqn : smct1])-([eqn : smct1b ] ) by considering the two additional kicks of order @xmath288 . the control term which only add the negative kicks does not lead to an efficient control although it appears slightly more regular than the uncontrolled case . using frequency map analysis @xcite , we have computed the critical thresholds for the break - up of the last kam tori : there are invariant tori for the map ( [ eqn : smct1])-([eqn : smct1b ] ) up to @xmath289 which is more than twice the uncontrolled case ( @xmath269 ) . the map ( [ eqn : smct2])-([eqn : smct2b ] ) which is simpler than the map ( [ eqn : smct1])-([eqn : smct1b ] ) has invariant tori up to @xmath290 . in this section , we have used the control for hamiltonian flows in order to derive control terms for area - preserving maps . we note that a control method has been developed directly for area - preserving maps in refs . @xcite . we consider the following hamiltonian @xmath291 where @xmath292 . a poincar section of this hamiltonian with @xmath293 is depicted on fig . [ fig : cub ] . the invariant torus with frequency @xmath188 is located at @xmath223 for @xmath19 . we notice that this invariant torus is shearless since the second derivative of @xmath20 with respect to @xmath189 is zero on this torus . hamiltonian ( [ eqn : hcub ] ) is of the form given by eq . ( [ eqn : hcf ] ) with @xmath294 , @xmath215 and @xmath295 . the control term is given by eq . ( [ eqn : cth ] ) : @xmath296 the noticeable feature is that the modification of the hamiltonian is of order @xmath297 compared with the forced pendulum or the standard map where the control term is of order @xmath5 . a poincar section of the controlled hamiltonian ( [ eqn : hcub ] ) with the control term ( [ eqn : hcubc ] ) is depicted on fig . [ fig : cubc ] . we notice that there are invariant tori in the region near @xmath223 that have been created with the addition of the small control term of order @xmath297 . for instance , for @xmath293 and @xmath292 , the control term has a norm which is about 10% of the perturbation . in order to obtain a localised control , one has to apply the control term only in the region where the kam invariant tori have been recreated , i.e. , in the region @xmath190 . we acknowledge useful discussions with ph ghendrih , m pettini and v zagrebnov . this work is supported by euratom / cea ( contract @xmath298 ) . first we notice that @xmath108 can be written as @xmath299 where @xmath300 is a dilatation and @xmath301 a translation of the actions acting on a function @xmath302 as @xmath303 a translation of the actions by a function @xmath43 of @xmath75 ( i.e. independent of @xmath56 ) is obviously a canonical transformation if @xmath43 derives from a scalar function , i.e. @xmath304 . thus it is sufficient to prove that @xmath305 is also a canonical transformation . we notice that @xmath305 is an automorphism since it is the product of two automorphisms : an exponential of a derivation and a dilatation . in what follows we prove that @xmath306 which is equivalent to say that @xmath305 is a lie transform generated by the scalar function @xmath307 . in other terms , eq . ( [ eqn : eqap ] ) can also be written as [ cf . ( [ eqn : wf ] ) ] @xmath308 since these operators are automorphisms , it is sufficient to check that eq . ( [ eqn : eqap ] ) is satisfied on the basis @xmath309 . first , we expand the operator @xmath310 : @xmath311 and we use the trotter - kato formula @xcite to express the exponential of the sum of two operators : @xmath312 any function of @xmath75 is invariant under the action of an operator @xmath313 . therefore it is straightforward to check that @xmath314 for all @xmath315 . it follows that eq.([eqn : eqap ] ) is satisfied on @xmath75 . for @xmath56 , we use the identity @xmath316 where @xmath46 and @xmath317 are functions of @xmath318 . this identity follows from @xmath319 . in particular , we have @xmath320 using eq . ( [ eqn : ema ] ) we prove that @xmath321 and it follows recursively that @xmath322 where the product is taken from right to left , i.e. @xmath323 . concerning the operator @xmath324 , the trotter - kato formula leads to the following expansion @xmath325 since @xmath326 and @xmath327 [ see eq . ( [ eqn : dmu ] ) ] . using the same type of computations as for @xmath305 , we have @xmath328 if we multiply the above expression by @xmath329 and change the index of the product ( @xmath330 ) , it leads to @xmath331 and hence eq . ( [ eqn : eqap ] ) is proved . + _ remark : _ we notice that eq . ( [ eqn : eqap ] ) implies that @xmath332 which leads to the expression of the inverse of @xmath333 : @xmath334 99 chen g and dong x 1998 _ from chaos to order _ ( singapore : world scientific ) gauthier d j 2003 resource letter : cc-1 : controlling chaos _ am . j. phys . _ * 71 * 750 ciraolo g , chandre c , lima r , vittot m , pettini m , figarella c and ghendrih ph 2004 controlling chaotic transport in a hamiltonian model of interest to magnetized plasmas _ j. phys . a : math . gen . _ * 37 * 3589 ciraolo g , briolle f , chandre c , floriani e , lima r , vittot m , pettini m , figarella c and ghendrih ph 2004 control of hamiltonian chaos as a possible tool to control anomalous transport in fusion plasmas _ phys . e _ * 69 * 056213 ( archived in ` arxiv.org/nlin.cd/0312037 ` ) vittot m 2004 perturbation theory and control in classical or quantum mechanics by an inversion formula _ j. phys . a : math . ( in press , archived in ` arxiv.org/math-ph/0303051 ` ) benettin g , galgani l , giorgilli a and strelcyn j m 1984 a proof of kolmogorov s theorem on invariant tori using canonical transformations defined by the lie method _ nuovo cimento _ * 79b * 201 chandre c and jauslin h r 2002 renormalization - group analysis for the transition to chaos in hamiltonian systems _ phys . rep . _ * 365 * 1 ciraolo g , chandre c , lima r , vittot m and pettini m 2004 control of chaos in hamiltonian systems _ celest . mech ( in press , archived in ` arxiv.org/nlin.cd/0311009 ` ) ciraolo g , chandre c , lima r , vittot m , pettini m , figarella c and ghendrih ph 2004 tailoring phase space : a way to control hamiltonian chaos ( preprint , archived in ` arxiv.org/nlin.cd/0405010 ` ) laskar j 1999 introduction to frequency map analysis _ hamiltonian systems with three or more degrees of freedom _ ed sim c ( dordrecht : nato asi series , kluwer academic publishers ) chandre c , vittot m , elskens y , ciraolo g and pettini m 2004 controlling chaos in area - preserving maps ( preprint , archived in ` arxiv.org/nlin.cd/0405008 ` ) trotter h f 1959 on the product of semi - groups of operators _ proc . * 10 * 545 ; kato t 1974 on the trotter - lie product formula _ proc . acad . _ * 50 * 694 ; zagrebnov v 2003 _ topics in the theory of gibbs semi - groups _ ( leuven : leuven univ .	we present a method of _ localised _ control of chaos in hamiltonian systems . the aim is to modify the perturbation _ locally _ by a small control term which makes the controlled hamiltonian more regular . we provide an explicit expression for the control term which is able to recreate invariant ( kam ) tori without modifying other parts of phase space . we apply this method of localised control to a forced pendulum model , to the delta - kicked rotor ( standard map ) and to a non - twist hamiltonian .

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