Monthly Archives: February 2026

Problem: Deduce the Hamilton-Jacobi equation of classical mechanics. Solution: Instead of viewing the action \(S=S[\mathbf x(t)]\) as a functional of the particle’s trajectory \(\mathbf x(t)\), it can be viewed more simply as a scalar field \(S(\mathbf x,t)\) in which the … Continue reading →

Problem: Define the signature of a matrix. Hence, state and prove Sylvester’s law of inertia. Solution: The signature of an \(n\times n\) matrix \(A\) is a \(3\)-tuple \((n_+,n_-,n_0)\) where \(n_+\) is the number of positive eigenvalues of \(A\) (including multiplicity), … Continue reading →

Problem: How does the paradigm of reinforcement learning (RL) fit within the broader context of machine learning? Solution: It is instructive to compare/contrast reinforcement learning with supervised learning. In this way, it will be seen that RL can in fact … Continue reading →