Monthly Archives: May 2026

Problem: State the Schwarzschild metric solution to the Einstein field equations and the assumptions underlying it. Solution: The Schwarzschild metric is given by: \[ds^2=\left(1-\frac{r_s}{r}\right)d(ct)^2-\frac{dr^2}{1-\frac{r_s}{r}}-r^2d\Omega^2\] (where \(r_s:=2GM/c^2\) is the Schwarzschild radius) and is (by Birkhoff’s theorem) the unique isotropic solution of … Continue reading →

Problem: Explain why for a random walk in \(\mathbf R^d\), the probability distribution of the random vector sum \(\mathbf x:=\mathbf x_1+…+\mathbf x_N\) of \(N\) i.i.d. random vectors \(\mathbf x_i\) each with identical mean \(\boldsymbol{\mu}:=\langle\mathbf x_i\rangle\) and identical covariance matrix \(\sigma^2:=\langle\mathbf … Continue reading →

Problem: Define the \(2\) words in the phrase “ideal paramagnet“. Show that a classical ideal paramagnet of \(N\) spins each with the same fixed magnetic dipole moment \(\mu:=|\boldsymbol{\mu}|\) placed in a uniform external magnetic field \(B:=|\mathbf B|\) will develop a … Continue reading →

Problem: Train a physics-informed neural network (PINN) on both the van der Pol oscillator and the drift-free Fokker-Planck diffusion equation. Solution: report Spectral Bias of Physics-Informed Neural Networks¶ 1. Introduction and Background¶ The physics-informed neural network (PINN) paradigm is a … Continue reading →