Mathematics

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 $\nabla, \nabla \cdot, \nabla^2$....

Notice that for this course (and all UCLA courses) the slides are intellectual properties of the instructor; therefore the post is just my own curated notes and projects based on the course (which are therefore my own intellectual properties). The notes here should consist of two parts: My learning journey, taking this challenging course as someone who has no fluid mechanics background (although plenty of backgrounds in applied mathematics and machine learning)....

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: $$\int_U u_{x_i} v dx + \int_U v_{x_i} u dx = \int_{\partial U} uv \hat{n}^i dS, \quad (i = 1,\cdots, n)$$...

Week 1 This week, we introduced Fluid Mechanics from a “third grade perspective”. Professor Roper introduced to us fluid mechanics in a lively manner; in the first lecture (referred to as “3rd grade definition”, intended for kids in 3rd grade), he defined fluid as that which takes shape of its container. This definition sounds like one of the natural philosophers from ancient Greece. This class is about “what makes fluid flow” and “what stops them from flowing”, and we listed the following:...

Problem Formulation