Geometry

Stability Theory Motivation We want to know for a dynamic system, in this note majorly autonomous system, when it is stable? The meaning of stability? If it’s stable how to prove so. Majorly we are going to use Lyapnov functions and spectral properties of linearized system to prove.

Apr 9, 2021

Invariance Properties of an Invariance Set Stable and Unstable Manifold Theorem

Apr 9, 2021

Jan 1, 2021

Note on Laplacian-Beltrami (Diffusion) Operator Motivation Laplacian on graph and on discrete geometry (mesh) are very useful tools. One core intuition, just like Laplacian in $\R^n$ space, it’s related to diffusion and heat equation. Recall the diffusion equation is

May 8, 2020

Note on Hyperbolic Geometry Reference Notes 2018 Lec Note 2015 Lecture note Ch5-3 Measurement in Hyperbolic Geometry [Cheatsheet / Note](http://home.iiserb.ac.in/~kashyap/MTH 520/lp.pdf) Motivation Hyperbolic geometry is a great source of inspiration for math art. Besides it is used to model some hierarchical data structure. Here I collected a few models

Apr 10, 2020

Motivation This is a brief analytical note about how physical self movement of eye / camera will induce optic flow in a static environment. And then discuss how a system can separate these two components instantaneously.

Feb 17, 2020

Some Computation on Sphere (Updating) Motivation Recently, in research, we encounter quite a few statistical problems on sphere. For example, Head direction tuning 3d direction of object 3d direction of body parts Some 3d tuning There are many standard statistical operations on Euclidean space, like getting mean, standard deviation and generate uniform distribution, fitting a model etc. We can perform these operation without thinking.

Oct 13, 2019

题记 一直计划着博士期间定期写一点note把自己最近学到的有趣的, 美妙的东西记下来. 如果读得是一个数学物理的博士, 或者理论神经科学的博士, 那这种Note就像是在数学世界中的探险笔记, 可以叫This week’s finding, 大约就是这周看了什么书, 学会了什么数学, 玩儿了什么Model, 发现了什么trick或者math game，做了什么优美的图. (可以参见之前挖出来好些有意思东西的一个站点 This Week’s Finds in Mathematical Physics UCR一个数学物理教授坚持了十多年的每周数学笔记) 不过现实中, 我读的是Neuroscience的博士, 大概只能写learning写不了什么finding了. (而对于Brain一周时间也学不了什么新东西…) 因此, Note的内容就会更庞杂: 一部分是技术性的, 新学会的数学、统计方法、机器学习方法, 也许会有新的实验技术以及相关的物理原理; 另一部分是理念类的，也许有最近听seminar听到的神经或者心理的实验结果，也可能是相关的有趣的哲学讨论。我想我会逐渐发现哪些内容更适合分享, 以及哪些内容写下来对自己以及对读者更有帮助, 经过一段时间的磨合，这个post series应该能形成自己的风格。

Sep 1, 2018