I'm Yoni.

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Awards

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


Useful math stuff

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Analysis

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Fubini's theorem and Tonelli's theorem

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

Let   be a measurable function.

Then it is true that

 

provided one of the following criteria:

  1. (Fubini's theorem) The spaces X, Y are complete (all null sets are measurable), and  .
  2. (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

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Let Ω be a measure space with a measure μ.

Let fn : 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   provided one of the following criteria:

  1. (Monotone convergence theorem)

      μ-almost everywhere in Ω.

    Note: If additionally   then   in L1(μ) by Scheffé’s lemma.

  2. (Dominated convergence theorem)

      for some   (everywhere, or μ-almost everywhere if μ is a complete measure).

    Note: This also gives us   in L1(μ), and  .

  3. (Bounded convergence theorem)

      and  .

    Note: This also gives us   in L1(μ), and  .

Corollary: Differentiation under the integral sign
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Let  , wherein  , 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   provided one of the following criteria:

  1.  .
  2.   and  .

Smooth functions

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A smooth transition from 0 to nonzero
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The function φ
 
A bump function - a smooth function with compact support
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The function ψ
 
A smooth transition from 0 to 1
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This is designed as a partition of unity.

 
The function η
 

Calculus

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Good-to-know changes of variables
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List of canonical coordinate transformations

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

 

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

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

Supposing ε > 0, we have  

In particular,  .

Probability

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Basics

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Let (Ω, P) be a probability space.

  • A real-valued random variable is a Borel-measurable  .
  • The expected value of X is  .

Geometry

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Euclidean balls

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Denote by κd the volume of the d-dimensional unit ball. Then

 

Denote by sd-1 the surface area of the d-1-dimensional unit sphere (the boundary of the d-dimensional unit ball). Then

 

Proof.

Let Bd(r) be the d-dimensional Euclidean ball centered at the origin with radius r. Then the following inclusion is true:

 

(TODO: The more general result with Hölder's inequality, inclusions of Lp spaces, etc.)