Analysis on Omanshu Thapliyal

So does your Car fail? (The Counterexample to HRT Conjecture) - Part 2

Fri, 14 Aug 2026 01:02:49 -0400

Part 1:https://omanshuthapliyal.github.io/blog/hrt-conjecture/

In the previous post, I talked about the HRT conjecture as a rather natural statement about phase space. A signal can be moved in time. It can also be moved in frequency. Moving it in time changes where its envelope lives. Moving it in frequency changes the pitch of its oscillations. If we do both, we obtain a time-frequency shifted version of the original signal.

Using a symmetric convention, one writes this operation as $$(\rho(x,\omega)f)(t)=e^{2\pi i\omega(t-x/2)}f(t-x).$$ The point $$(x,\omega)$$ tells us where we moved the function in time-frequency phase space. The HRT conjecture said that distinct points in phase space should produce distinct, linearly independent copies of a nonzero function. More explicitly, if $$z_1,\ldots,z_N\in\mathbb{R}^2$$ are distinct, and $$f\in L^2(\mathbb{R})$$ is nonzero, then one should never be able to find nontrivial coefficients $$c_1,\ldots,c_N$$ for which $$\sum_{k=1}^{N}c_k \rho(z_k)f=0.$$

On Eigenvalues, Homotopy, and Root Loci

Fri, 05 Sep 2025 11:57:21 -0700

Consider an $$n+1$$ dimensional matrix as follows:

$$ A = \begin{pmatrix} \frac{\pi^2}{6} & 1 & \frac{1}{4} & ... & \frac{1}{n^2} \\ 1 & \frac{\pi^2}{6} & \frac{1}{4} & ... & \frac{1}{n^2} \\ \vdots & & \ddots & & \vdots\\ 1 & \frac{1}{4} & ... & \frac{1}{n^2} &\frac{\pi^2}{6} \\ \end{pmatrix} $$

Is the matrix positive-definite? The problem itself is a dramatization of how my professor would ask us to write arbitrary real matrices and quickly determine its positive-definiteness (most of the times) as a trick, seemingly like in-class wizardry!

MLP Approximation

Thu, 12 Sep 2019 10:27:01 -0400

Almost always we hear about classification or machine learning problems, the go-to methods to solve the problem are neural networks, or multi-layered percetrons (MLP). Now function approximation problems, which is what classification is, are very well defined in terms of consistency, accuracy, and other abilities of the approximator. Why are MLPs then able to approximate functions well? Especially given the fact that most of the problems in coming up with candidate architectures, activation functions, etc. are ‘design problems’.

Real Analysis Notes

Sat, 24 Aug 2019 23:38:20 +0530

When I took the course on Real Analysis, it made me very nervous. I was having fever dreams from the past when I took undergraduate real analysis and the epsilon-delta definitions just did not make a lot of sense. In retrospect, taking MA504 has been one of the best academic decisions, especially as an engineer.

It has helped me a lot in how to concretely formulate definitions, and what ‘approaching’, ’tending’, ’limiting’, and other weasel words mean in a mathematical sense.