Monthly Archives: May 2024

Problem: What is a lattice \(\Lambda\)? What does it mean to say that a lattice is Bravais? Give an example of a Bravais lattice and a non-Bravais lattice. Solution: A lattice \(\Lambda\) is any periodic set of lattice points in … Continue reading →

Problem: What does it mean for a collection of feature vectors \(\mathbf x_1,…,\mathbf x_{T}\) to represent a form of sequence data. Give some examples of sequence data. Solution: It means that the feature vectors are not i.i.d.; indeed, they are … Continue reading →

Problem: Where and why should one create an __init__.py file? Solution: Inside a folder/directory that’s meant to be a Python package containing a bunch of Python modules with useful functions, etc. that other Python scripts would be importing from (not … Continue reading →

The purpose of this post is to compile solutions to select exercises from Steven Strogatz’s textbook Chaos and Nonlinear Dynamics. Chapter #\(1\): Chapter #\(2\): Chapter #\(3\): Chapter #\(4\): Chapter #\(5\): Chapter #\(6\): Chapter #\(7\): Chapter #\(8\):

The purpose of this post is to review how the fields of an electrostatic dipole \(\boldsymbol{\pi}\) and magnetostatic dipole \(\boldsymbol{\mu}\) arise. For the electrostatic dipole, “fields” means the electrostatic potential \(\phi\) and by extension the electrostatic field \(\textbf E=-\partial\phi/\partial\textbf x\) … Continue reading →

Given a flow field \(\textbf v(\textbf x,t)\), the vorticity \(\boldsymbol{\omega}\) of \(\textbf v\) is defined by taking its curl \(\boldsymbol{\omega}:=\frac{\partial}{\partial\textbf x}\times\textbf v\). For a flow field rotating rigidly with angular velocity vector \(\boldsymbol{\omega}_0\) so that \(\textbf v=\boldsymbol{\omega}_0\times\textbf x\). The vorticity … Continue reading →

Often in biochemistry, if a single substrate \(\text S\) needs to be become a product \(\text P\) via a chemical reaction of the form \(\text S\to \text P\). Assuming this is a first-order elementary chemical reaction, it would merely have … Continue reading →

The free, dispersionless wave equation on \((ct,\textbf x)\in\textbf R\times \textbf R^d\) is: \[\frac{\partial^2\psi}{\partial (ct)^2}-\frac{\partial^2\psi}{\partial|\textbf x|^2}=0\] or more simply as \(☐^2\psi=0\), where the d’Alembert operator is defined by \(☐^2:=\frac{\partial^2}{\partial (ct)^2}-\frac{\partial^2}{\partial|\textbf x|^2}=\partial^{\mu}\partial_{\nu}=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\) with the usual metric \(\eta=\text{diag}(1,-1,-1,-1)\) defining the hyperbolic geometry of … Continue reading →

Problem #\(1\): Find the mistake in this derivation of the Green’s function for the inhomogeneous Helmholtz equation subject to the boundary condition of an outgoing wave: Solution #\(1\): First of all, the correct answer should be \(G(r)=\frac{e^{ikr}}{4\pi r}\), not this … Continue reading →