Category Archives: Blog

The Shapes and Structures of Molecules Also, spin angular momentum coupling between \(^{13}\text C\) nuclei and \(^1\text H\) nuclei are typically suppressed via broadband proton decoupling. Also, due to low isotopic abundance, spin angular momentum coupling among \(^{13}\text C\) nuclei … Continue reading →

If one has a collection \(Q=\int\rho d^3x>0\) of positive charge \(\rho>0\) in some region of space, then the electric dipole moment \(\textbf p\) of \(Q\) may be viewed as \(\textbf p=Q\textbf X\), where \(\textbf X=\frac{1}{Q}\int\textbf x\rho d^3x\) is the center … Continue reading →

Problem: Describe the wrong way to define a (Bravais) lattice. Solution: A wrong way to define a Bravais lattice proceeds by \(3\) steps. First, one constructs the concept of a Delone set as any subset of a metric space with … 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 →