User:Yoni

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משתמש זה דובר עברית כשפת אם.

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Me

I'm Yoni.

Pages I've contributed to

Merged Weyl's criterion into Equidistributed sequence and improved the latter (before, after)
More to come!

My subpages

User:Yoni · talk

Awards

		The E=mc² Barnstar
		message Vitalyb (talk) 20:03, 21 February 2014 (UTC)

Useful math stuff

Analysis

Fubini's theorem and Tonelli's theorem

Let X, Y be measure spaces with measures μ, ν respectively.

Let $f:X\times Y\to \mathbb {R} \cup \left\{+\infty ,-\infty \right\}$ be a measurable function.

Then it is true that

$\int _{X}\left(\int _{Y}f(x,y)\,\mathrm {d} \nu (y)\right)\,\mathrm {d} \mu (x)=\int _{Y}\left(\int _{X}f(x,y)\,\mathrm {d} \mu (x)\right)\,\mathrm {d} \nu (y)=\int _{X\times Y}f(x,y)\,\mathrm {d} \mu \times \nu (x,y)$

provided one of the following criteria:

(Fubini's theorem) The spaces X, Y are complete (all null sets are measurable), and $f\in L^{1}\left(\mu \times \nu \right)$ .
(Tonelli's theorem) The spaces X, Y are σ-finite (a countable union of finite-measure sets)*, and f ≥ 0.

(*) For probability spaces this is automatic.

Convergence of integrals

Let Ω be a measure space with a measure μ.

Let f_n : Ω → ℝ be a sequence of measurable functions that converges pointwise (everywhere, or μ-almost everywhere if μ is a complete measure) to a function f : Ω → ℝ.

Then it is true that $\int _{\Omega }f_{n}\,d\mu \to \int _{\Omega }f\,d\mu$ provided one of the following criteria:

(Monotone convergence theorem)
$0\leq f_{1}\leq f_{2}\leq \ldots$ μ-almost everywhere in Ω.
Note: If additionally $f\in L^{1}(\mu )$ then $f_{n}\to f$ in L¹(μ) by Scheffé’s lemma.
(Dominated convergence theorem)
$\left|f_{n}\right|\leq g$ for some $g\in L^{1}\left(\mu \right)$ (everywhere, or μ-almost everywhere if μ is a complete measure).
Note: This also gives us $f_{n}\to f$ in L¹(μ), and $||f_{n}||_{L^{1}(\mu )}\uparrow ||f||_{L^{1}(\mu )}\leq ||g||_{L^{1}(\mu )}$ .
(Bounded convergence theorem)
$\mu (\Omega )<\infty$ and $\left|f_{n}\right|\leq M$ .
Note: This also gives us $f_{n}\to f$ in L¹(μ), and $||f_{n}||_{L^{1}(\mu )}\uparrow ||f||_{L^{1}(\mu )}\leq M\mu \left(\Omega \right)$ .

Corollary: Differentiation under the integral sign

Let $F(x):=\int _{\Omega }f(x,\omega )\,d\mu (\omega )$ , wherein $x\in \mathbb {R}$ , and if ω is held constant, for all ω (or μ-almost all ω if μ is a complete measure), f is differentiable in x. Suppose F is defined in a neighborhood of 0.

Then it is true that $F'(0)=\int _{\Omega }{\frac {\partial f}{\partial x}}\left(0,\omega \right)\,d\mu (\omega )$ provided one of the following criteria:

${\frac {\partial f}{\partial x}}(0,\omega )\in L^{1}(\mu )$ .
$\mu (\Omega )<\infty$ and $\left|{\frac {\partial f}{\partial x}}(0,\omega )\right|\leq M$ .

Smooth functions

A smooth transition from 0 to nonzero

\varphi (x)={\begin{cases}e^{-{\frac {1}{x}}}&{\mbox{if }}x>0,\\0&{\mbox{if }}x\leq 0\end{cases}}

A bump function - a smooth function with compact support

\psi (x)=\varphi \left(2\left(1+x\right)\right)\varphi \left(2\left(1-x\right)\right)={\begin{cases}e^{-{\frac {1}{1-x^{2}}}}&{\mbox{if }}|x|<1,\\0&{\mbox{if }}|x|\geq 1\end{cases}}

A smooth transition from 0 to 1

This is designed as a partition of unity.

\eta (x)={\frac {\varphi (x)}{\varphi (x)+\varphi (1-x)}}={\begin{cases}0&{\mbox{if }}x\leq 0,\\\left(1+\exp \left({\frac {1-2x}{x(1-x)}}\right)\right)^{-1}&{\mbox{if }}0<x<1,\\1&{\mbox{if }}x>1\end{cases}}

Calculus

Good-to-know changes of variables

List of canonical coordinate transformations

Let σ_d-1 be the uniform probability measure on the d-1-dimensional unit sphere and let κ_d be the volume of the d-dimensional unit ball (so that dκ_d is the surface area of the sphere). Then:

{\frac {1}{d\kappa _{d}}}\int _{x\in \mathbb {R} ^{d},R_{1}<|x|<R_{2}}f(x)\,dx=\int _{R_{1}}^{R_{2}}r^{d-1}\left(\int _{\zeta \in \mathbb {R} ^{d},\left|\zeta \right|=1}f(r\zeta )\,d\sigma _{d-1}\left(\zeta \right)\right)\,dr

Corollary: If f is radial, that is: f(x) = f(|x|), then:

{\frac {1}{d\kappa _{d}}}\int _{x\in \mathbb {R} ^{d},R_{1}<|x|<R_{2}}f(x)\,dx=\int _{R_{1}}^{R_{2}}r^{d-1}f(r)\,dr

Integral convergence

This may be proven using the previously-mentioned change of variables.

Supposing ε > 0, we have

\int _{x\in \mathbb {R} ^{d},|x|<1}{\frac {1}{|x|^{d-\varepsilon }}}\,dx={\frac {d\kappa _{d}}{\varepsilon }}

In particular, $\int _{x\in \mathbb {R} ^{d},|x|<1}{\frac {1}{|x|^{t}}}\,dx<\infty \Leftrightarrow t<d$ .

Probability

Basics

Let (Ω, P) be a probability space.

A real-valued random variable is a Borel-measurable $X:\Omega \to \mathbb {R}$ .
The expected value of X is $\operatorname {E} [X]=\int _{\Omega }X(\omega )\,\mathrm {d} P(\omega )$ .