18 posts in this category
Motivation How to compute determinant or inversion of matrix with a low rank modificaiton? This is a very interesting and important math technique in statistical methods, since people frequently model covariance matrix or connectivity matrix as such: a matrix plus a low rank modification.
Apr 16, 2022
Motivation Here we summarize a few common probabilistic neural population models. Adapted from reading notes and class presentations from Neuro QC316 taught by Jan Drugowitsch. LNP, GLM These are the simplist models of neurons.
Motivation As mentioned in our high-dimensional PCA note, understanding the spectrum of Toeplitz matrix is important. The subject itself is a bit technical, but the analytical techniques involved in it are splendid and general. So here I took note from this paper and present way to calculate the spectrum on paper (or by mathematica).
Feb 4, 2022
Motivation When you think about random walks, what shape do you think about? Is it like this? Or this? These are good examples of random walks in two or three dimensions. But what about random walks in higher dimensions?
Jan 22, 2022
Stability Theory When a system has control, then comes the questions of whether we could make it stable under the control. Problem Setup A control affine system is this $$ \dot x =f(x)+\sum_i g_i(x)\bar u_i=f(x)+g(x)\bar u $$Interpretation:
May 10, 2021
Basic Notions Def Topological Equivalence: 2 dynamic systems are topological equivalence when there is a homeomorphism between their solutions. Def Conjugate: Two maps are connected by $$g=h^{-1}\circ f \circ h$$.
May 9, 2021
Bifurcation Normal Form Invariance and Stable Manifold Lyapnov Stability Theory Feedback Stablization
Apr 9, 2021
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.
Invariance Properties of an Invariance Set Stable and Unstable Manifold Theorem
Problem Statement Given a bunch of noisy data, you want a smooth curve going through the cloud. As the points are noisy, there is no need to going through each point.
Oct 13, 2020