<s>
Criteria	O
for	O
unimodality	O
can	O
also	O
be	O
defined	O
through	O
the	O
characteristic	O
function	O
of	O
the	O
distribution	O
or	O
through	O
its	O
Laplace	B-Algorithm
–	I-Algorithm
Stieltjes	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
where	O
the	O
median	O
is	O
ν	O
,	O
the	O
mean	O
is	O
μ	O
and	O
ω	O
is	O
the	O
root	B-General_Concept
mean	I-General_Concept
square	I-General_Concept
deviation	I-General_Concept
from	O
the	O
mode	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
for	O
a	O
unimodal	O
distribution	O
that	O
the	O
median	O
ν	O
and	O
the	O
mean	O
μ	O
lie	O
within	O
(	O
3/5	O
)	O
1/2	O
≈	O
0.7746	O
standard	B-General_Concept
deviations	I-General_Concept
of	O
each	O
other	O
.	O
</s>
<s>
A	O
similar	O
relation	O
holds	O
between	O
the	O
median	O
and	O
the	O
mode	O
θ	O
:	O
they	O
lie	O
within	O
31/2	O
≈	O
1.732	O
standard	B-General_Concept
deviations	I-General_Concept
of	O
each	O
other	O
:	O
</s>
<s>
Rohatgi	O
and	O
Szekely	O
claimed	O
that	O
the	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
of	O
a	O
unimodal	O
distribution	O
are	O
related	O
by	O
the	O
inequality	O
:	O
</s>
<s>
where	O
κ	O
is	O
the	O
kurtosis	B-Error_Name
and	O
γ	O
is	O
the	O
skewness	B-General_Concept
.	O
</s>
<s>
A	O
general	O
method	O
based	O
on	O
derivatives	B-Algorithm
exists	O
,	O
but	O
it	O
does	O
not	O
succeed	O
for	O
every	O
function	O
despite	O
its	O
simplicity	O
.	O
</s>
<s>
Examples	O
of	O
unimodal	O
functions	O
include	O
quadratic	O
polynomial	O
functions	O
with	O
a	O
negative	O
quadratic	O
coefficient	O
,	O
tent	B-Algorithm
map	I-Algorithm
functions	O
,	O
and	O
more	O
.	O
</s>
<s>
An	O
example	O
of	O
a	O
weakly	O
unimodal	O
function	O
which	O
is	O
not	O
strongly	B-Algorithm
unimodal	I-Algorithm
is	O
every	O
other	O
row	O
in	O
Pascal	O
's	O
triangle	O
.	O
</s>
<s>
For	O
example	O
,	O
local	B-Algorithm
unimodal	I-Algorithm
sampling	I-Algorithm
,	O
a	O
method	O
for	O
doing	O
numerical	O
optimization	O
,	O
is	O
often	O
demonstrated	O
with	O
such	O
a	O
function	O
.	O
</s>
<s>
One	O
important	O
property	O
of	O
unimodal	O
functions	O
is	O
that	O
the	O
extremum	O
can	O
be	O
found	O
using	O
search	B-Application
algorithms	I-Application
such	O
as	O
golden	B-Algorithm
section	I-Algorithm
search	I-Algorithm
,	O
ternary	B-Algorithm
search	I-Algorithm
or	O
successive	B-Algorithm
parabolic	I-Algorithm
interpolation	I-Algorithm
.	O
</s>
<s>
A	O
function	O
f(x )	O
is	O
"	O
S-unimodal	O
"	O
(	O
often	O
referred	O
to	O
as	O
"	O
S-unimodal	O
map	O
"	O
)	O
if	O
its	O
Schwarzian	O
derivative	B-Algorithm
is	O
negative	O
for	O
all	O
,	O
where	O
is	O
the	O
critical	O
point	O
.	O
</s>
