I'm just yet another postdoc researcher lost somewhere between mathematics and signal processing.
My infamous but useful mathematical formula list
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Some equations useful in my research. Most of them can be collected from wikipedia.
∑
k
=
n
+
1
∞
1
k
2
≤
1
n
{\displaystyle \sum _{k=n+1}^{\infty }{1 \over k^{2}}\leq {1 \over n}}
ln
(
1
−
x
)
≤
−
x
−
x
2
/
2
,
∀
x
≥
0
{\displaystyle \ln(1-x)\leq -x-x^{2}/2,\quad \forall x\geq 0}
ln
(
1
+
x
)
≤
x
−
x
2
/
2
+
x
3
/
3
,
∀
x
≥
0
{\displaystyle \ln(1+x)\leq x-x^{2}/2+x^{3}/3,\quad \forall x\geq 0}
x
1
−
x
+
ln
(
1
−
x
)
≥
x
2
/
2
,
∀
x
∈
[
0
,
1
)
{\displaystyle {x \over 1-x}+\ln(1-x)\geq x^{2}/2,\quad \forall x\in [0,1)}
Math 710: Measure Concentration, A. Barvinok )
x
+
ln
(
1
−
x
)
≤
−
x
2
/
2
,
∀
x
∈
[
0
,
1
)
{\displaystyle x+\ln(1-x)\leq -x^{2}/2,\quad \forall x\in [0,1)}
Math 710: Measure Concentration, A. Barvinok )Combinatorics
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(
n
k
)
≤
n
k
k
!
{\displaystyle {n \choose k}\leq {\frac {n^{k}}{k!}}}
(
n
k
)
≤
(
n
⋅
e
k
)
k
{\displaystyle {n \choose k}\leq \left({\frac {n\cdot e}{k}}\right)^{k}}
(
n
k
)
≤
e
n
H
(
k
/
n
)
,
w
h
e
r
e
H
(
q
)
=
−
q
log
q
−
(
1
−
q
)
log
(
1
−
q
)
{\displaystyle {n \choose k}\leq e^{\,n\,H(k/n)},\ {\rm {where}}\ H(q)=-q\log q-(1-q)\log(1-q)}
Binary entropy (for 0<q<1).
(
n
k
)
≥
(
n
k
)
k
{\displaystyle {n \choose k}\geq \left({\frac {n}{k}}\right)^{k}}
more )From Sondow (2005) [1] and Sondow and Zudilin (2006) [2], and here on Nuit-Blanche :
(
n
k
)
≤
(
n
k
)
k
(
n
n
−
k
)
n
−
k
{\displaystyle {n \choose k}\leq \left({\frac {n}{k}}\right)^{k}\left({\frac {n}{n-k}}\right)^{n-k}}
(
n
k
)
≥
1
4
(
n
−
k
)
(
n
k
)
k
(
n
n
−
k
)
n
−
k
{\displaystyle {n \choose k}\geq {\frac {1}{4(n-k)}}\,\left({\frac {n}{k}}\right)^{k}\left({\frac {n}{n-k}}\right)^{n-k}}
[1] Sondow, J. "Problem 11132." Amer. Math. Monthly 112, 180, 2005.
[2] Sondow, J. and Zudilin, W. "Euler's Constant, q-Logarithms, and Formulas of Ramanujan and Gosper." Ramanujan J. 12, 225-244, 2006.
Statistics
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If
X
∼
N
(
0
,
1
)
{\displaystyle X\thicksim N(0,1)}
E
[
e
t
X
2
]
=
1
/
1
−
2
t
{\displaystyle E[e^{tX^{2}}]=1/{\sqrt {1-2t}}}
t
∈
(
−
∞
,
1
2
)
{\displaystyle t\in (-\infty ,{1 \over 2})}
(see "Database friendly random projection" by Achlioptas, or "An elementary proof of JL Lemma" , by S. Dasgupta and A. Gupta)
Markov Inequality
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In the language of measure theory , Markov's inequality states that if (X ,Σ,μ) is a measure space , f is a measurable extended real -valued function, and t > 0, then
μ
(
{
x
∈
X
|
|
f
(
x
)
|
≥
t
}
)
≤
1
t
∫
X
|
f
|
d
μ
.
{\displaystyle \mu (\{x\in X|\,|f(x)|\geq t\})\leq {1 \over t}\int _{X}|f|\,d\mu .}
For the special case where the space has measure 1 (i.e., it is a probability space ), it can be restated as follows: if X is any random variable and a > 0, then
Pr
(
|
X
|
≥
a
)
≤
E
(
|
X
|
)
a
.
{\displaystyle {\textrm {Pr}}(|X|\geq a)\leq {\frac {{\textrm {E}}(|X|)}{a}}.}
Laplace transform
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In the language of measure theory (see also Math 710: Measure Concentration, A. Barvinok ), if (X ,Σ,μ) is a measure space , f is a measurable extended real -valued function, and t > 0, then
μ
(
{
x
∈
X
|
f
(
x
)
≥
a
}
)
≤
e
−
t
a
∫
X
e
t
f
d
μ
.
{\displaystyle \mu (\{x\in X|\,f(x)\geq a\})\leq e^{-ta}\int _{X}e^{tf}\,d\mu .}
μ
(
{
x
∈
X
|
f
(
x
)
≤
a
}
)
≤
e
t
a
∫
X
e
−
t
f
d
μ
.
{\displaystyle \mu (\{x\in X|\,f(x)\leq a\})\leq e^{ta}\int _{X}e^{-tf}\,d\mu .}
![{\displaystyle E[e^{tX^{2}}]=1/{\sqrt {1-2t}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/694ab9aa7530e668ed91ee2040feb463046d5154)
