Misc Notes


Fourier

differentials $$ (\mathcal{F} f^{(n)})(s) = (2 \pi i s)^n \mathcal{F} f (s) $$ $$ P(\frac{d}{dx}) = a_n(\frac{d}{dx})^n + ... + a_1 \frac{d}{dx} + a_0 $$ $$ (\mathcal{F} (P(\frac{d}{dx}) f))(s) = P(2 \pi i s) \mathcal{F} f (s) = (\sum_{i=0}^{n} a_i (2 \pi i s)^i) \mathcal{F} f (s) $$
duality & transformations $$ \mathcal{F} f = (\mathcal{F}^{-1} f)^{-} $$ $$ \mathcal{F} f^{-} = \mathcal{F}^{-1} f $$ shift $$ f(t \pm b) \rightleftharpoons e^{\pm 2 \pi i s b} F(s) $$ stretch $$ f(a t) \rightleftharpoons \frac{1}{|a|} F(\frac{s}{a}) $$
coefficients $$ c_n = \frac{1}{T} \int_{-T/2}^{T/2} e^{-2 \pi i n t / T} f(t) dt $$ Often put $ \frac{1}{\sqrt T} $ into harmonic $ \frac{1}{\sqrt T} e^{-2 \pi i n t / T} $

Machine Learning

common distributions (less familiar ones)
Multinomial $$ Mu(\mathbf{x} \mid n, \mathbf{\theta}) \triangleq {{n}\choose{x_1...x_K}} \prod_{j=1}^{K} \theta_{j}^{x_j} $$ where $$ {{n}\choose{x_1...x_K}} \triangleq \frac{n!}{x_1!...x_K!} $$ Rolling $K$ sided die once ("Multinoulli"): $$ Cat(x \mid \mathbf{\theta}) \triangleq Mu(x \mid 1, \mathbf{\theta}) $$ Gamma (for $T, a, b$ in $ \mathbb{R}^+ $) $$ Ga(T \mid shape = a, rate = b) \triangleq \frac{b^a}{\Gamma(a)} T^{a-1} e^{-Tb} $$ where $$ \Gamma(x) \triangleq \int_{0}^{\infty} u^{x-1} e^{-u} du $$ $$ Exp(x \mid \lambda ) = Ga(x \mid 1, \lambda)$$ $$ \chi^2(x \mid \nu) = Ga(x \mid \frac{\nu}{2}, \frac{1}{2})$$ Beta $$ Beta(x \mid a, b) = \frac{1}{B(a,b)} x^{a-1} (1-x)^{b-1} $$ where $$ B(a,b) \triangleq \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)} $$
transformations $$ p_y(y) = p_x(x) \left\lvert \frac{dx}{dy} \right\rvert $$

Information Theory

Physics

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