Monthly Archives: August 2025

Problem: At a high level, what is the goal of classical hydrodynamics? Solution: The program of classical hydrodynamics seeks to bridge the physics of a many-body system at different length scales. The idea is to start from microscopics (i.e. Newton’s … Continue reading →

Problem: Prove the Sokhotski-Plemelj theorem on the real line \(\textbf R\): \[\frac{1}{\omega-\omega’\pm i0^+}=\mathcal P\frac{1}{\omega-\omega’}\mp i\pi\delta(\omega-\omega’)\] Solution: (in this solution, the function \(\chi(\omega)\) is assumed to be analytic on the integration interval in \(\textbf R\) to be able to apply the … Continue reading →

Problem: Distinguish between a supervised data set and unsupervised data set. Solution: Supervised data is of the form \(\{(\textbf x_1,y_1),…,(\textbf x_N,y_N)\}\). Unsupervised data consists of stripping away the target labels \(y_1,…,y_N\), leaving behind just the feature vectors \(\{\textbf x_1,…,\textbf x_N\}\). … Continue reading →