<s>
In	O
mathematics	O
,	O
an	O
ultrametric	B-Algorithm
space	I-Algorithm
is	O
a	O
metric	O
space	O
in	O
which	O
the	O
triangle	O
inequality	O
is	O
strengthened	O
to	O
.	O
</s>
<s>
Sometimes	O
the	O
associated	O
metric	O
is	O
also	O
called	O
a	O
non-Archimedean	B-Algorithm
metric	I-Algorithm
or	O
super-metric	O
.	O
</s>
<s>
Although	O
some	O
of	O
the	O
theorems	O
for	O
ultrametric	B-Algorithm
spaces	I-Algorithm
may	O
seem	O
strange	O
at	O
a	O
first	O
glance	O
,	O
they	O
appear	O
naturally	O
in	O
many	O
applications	O
.	O
</s>
<s>
}	O
(	O
strong	O
triangle	O
inequality	O
or	O
ultrametric	B-Algorithm
inequality	O
)	O
.	O
</s>
<s>
An	O
ultrametric	B-Algorithm
space	I-Algorithm
is	O
a	O
pair	O
consisting	O
of	O
a	O
set	O
together	O
with	O
an	O
ultrametric	B-Algorithm
on	O
,	O
which	O
is	O
called	O
the	O
space	O
's	O
associated	O
distance	O
function	O
(	O
also	O
called	O
a	O
metric	O
)	O
.	O
</s>
<s>
If	O
satisfies	O
all	O
of	O
the	O
conditions	O
except	O
possibly	O
condition	O
4	O
then	O
is	O
called	O
an	O
ultrapseudometric	B-Algorithm
on	O
.	O
</s>
<s>
An	O
ultrapseudometric	B-Algorithm
space	I-Algorithm
is	O
a	O
pair	O
consisting	O
of	O
a	O
set	O
and	O
an	O
ultrapseudometric	B-Algorithm
on	O
.	O
</s>
<s>
In	O
the	O
case	O
when	O
is	O
an	O
Abelian	O
group	O
(	O
written	O
additively	O
)	O
and	O
is	O
generated	O
by	O
a	O
length	O
function	O
(	O
so	O
that	O
)	O
,	O
the	O
last	O
property	O
can	O
be	O
made	O
stronger	O
using	O
the	O
Krull	B-Algorithm
sharpening	I-Algorithm
to	O
:	O
</s>
<s>
From	O
the	O
above	O
definition	O
,	O
one	O
can	O
conclude	O
several	O
typical	O
properties	O
of	O
ultrametrics	B-Algorithm
.	O
</s>
<s>
All	O
balls	O
of	O
strictly	O
positive	O
radius	O
are	O
both	O
open	O
and	O
closed	O
sets	O
in	O
the	O
induced	O
topology	B-Architecture
.	O
</s>
<s>
All	O
directly	O
derive	O
from	O
the	O
ultrametric	B-Algorithm
triangle	O
inequality	O
.	O
</s>
<s>
The	O
intuition	O
behind	O
such	O
seemingly	O
strange	O
effects	O
is	O
that	O
,	O
due	O
to	O
the	O
strong	O
triangle	O
inequality	O
,	O
distances	O
in	O
ultrametrics	B-Algorithm
do	O
not	O
add	O
up	O
.	O
</s>
<s>
The	O
discrete	O
metric	O
is	O
an	O
ultrametric	B-Algorithm
.	O
</s>
<s>
The	O
p-adic	O
numbers	O
form	O
a	O
complete	O
ultrametric	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
resulting	O
metric	O
is	O
an	O
ultrametric	B-Algorithm
.	O
</s>
<s>
The	O
set	O
of	O
words	O
with	O
glued	O
ends	O
of	O
the	O
length	O
n	O
over	O
some	O
alphabet	O
Σ	O
is	O
an	O
ultrametric	B-Algorithm
space	I-Algorithm
with	O
respect	O
to	O
the	O
p-close	O
distance	O
.	O
</s>
<s>
If	O
r	O
=	O
(	O
rn	O
)	O
is	O
a	O
sequence	O
of	O
real	O
numbers	O
decreasing	O
to	O
zero	O
,	O
then	O
|x|r	O
:=	O
lim	O
supn	O
→	O
∞	O
|xn|rn	O
induces	O
an	O
ultrametric	B-Algorithm
on	O
the	O
space	O
of	O
all	O
complex	O
sequences	O
for	O
which	O
it	O
is	O
finite	O
.	O
</s>
<s>
If	O
G	O
is	O
an	O
edge-weighted	O
undirected	O
graph	O
,	O
all	O
edge	O
weights	O
are	O
positive	O
,	O
and	O
d(u,v )	O
is	O
the	O
weight	O
of	O
the	O
minimax	O
path	O
between	O
u	O
and	O
v	O
(	O
that	O
is	O
,	O
the	O
largest	O
weight	O
of	O
an	O
edge	O
,	O
on	O
a	O
path	O
chosen	O
to	O
minimize	O
this	O
largest	O
weight	O
)	O
,	O
then	O
the	O
vertices	O
of	O
the	O
graph	O
,	O
with	O
distance	O
measured	O
by	O
d	O
,	O
form	O
an	O
ultrametric	B-Algorithm
space	I-Algorithm
,	O
and	O
all	O
finite	O
ultrametric	B-Algorithm
spaces	I-Algorithm
may	O
be	O
represented	O
in	O
this	O
way	O
.	O
</s>
<s>
p-adic	O
analysis	O
makes	O
heavy	O
use	O
of	O
the	O
ultrametric	B-Algorithm
nature	O
of	O
the	O
p-adic	O
metric	O
.	O
</s>
<s>
In	O
condensed	O
matter	O
physics	O
,	O
the	O
self-averaging	O
overlap	O
between	O
spins	O
in	O
the	O
SK	O
Model	O
of	O
spin	O
glasses	O
exhibits	O
an	O
ultrametric	B-Algorithm
structure	O
,	O
with	O
the	O
solution	O
given	O
by	O
the	O
full	O
replica	O
symmetry	O
breaking	O
procedure	O
first	O
outlined	O
by	O
Giorgio	O
Parisi	O
and	O
coworkers	O
.	O
</s>
<s>
Ultrametricity	B-Algorithm
also	O
appears	O
in	O
the	O
theory	O
of	O
aperiodic	O
solids	O
.	O
</s>
<s>
In	O
taxonomy	O
and	O
phylogenetic	O
tree	O
construction	O
,	O
ultrametric	B-Algorithm
distances	O
are	O
also	O
utilized	O
by	O
the	O
UPGMA	B-Algorithm
and	O
WPGMA	B-Algorithm
methods	O
.	O
</s>
<s>
When	O
DNA	O
,	O
RNA	O
and	O
protein	O
data	O
are	O
analyzed	O
,	O
the	O
ultrametricity	B-Algorithm
assumption	O
is	O
called	O
the	O
molecular	O
clock	O
.	O
</s>
<s>
Models	O
of	O
intermittency	O
in	O
three	O
dimensional	O
turbulence	O
of	O
fluids	O
make	O
use	O
of	O
so-called	O
cascades	O
,	O
and	O
in	O
discrete	O
models	O
of	O
dyadic	O
cascades	O
,	O
which	O
have	O
an	O
ultrametric	B-Algorithm
structure	O
.	O
</s>
<s>
In	O
geography	O
and	O
landscape	O
ecology	O
,	O
ultrametric	B-Algorithm
distances	O
have	O
been	O
applied	O
to	O
measure	O
landscape	O
complexity	O
and	O
to	O
assess	O
the	O
extent	O
to	O
which	O
one	O
landscape	O
function	O
is	O
more	O
important	O
than	O
another	O
.	O
</s>
