I find it to be widely enlightening to learn about probability distributions beyond the scope of a standard undergraduate curriculum. It turns out, as Taleb would agree. Only knowing distributions from undergraduate statistics can hurt your intellectual ability Also, even for lots of graduate students in quantitative disciplines, it is rare to go beyond normal distributions, which gives all kinds of nice formulas and analytical forms. This post summarizes some more exotic distributions that are quite useful....

Introduction In this post I go over the basics of index notation for calculus. This is the notation that was invented by Einstein and also known in machine learning community as einsum. It serves as a convenient way to supress summations in formulas, by viewing repeated indices as being summed over. In the field of tensor calculus and in particular fluid dynamics, this notation can come in handy when deriving complex formulas involving $\nabla, \nabla \cdot, \nabla^2$....

Green’s Three Identities are the fundamental results for vector calculus, which is widely used in basic PDE theories and fluid mechanics. Here I present the results. Assume that $U \subset \mathbb{R}^n$ be open and has $C^1$ boundary, then: Integration by Part Result 1 (Integration by Part): for $u,v \in C^1(\bar{U})$, we have: $$\int_U u_{x_i} v dx + \int_U v_{x_i} u dx = \int_{\partial U} uv \hat{n}^i dS, \quad (i = 1,\cdots, n)$$...

Problem Formulation