This article will probably only be of interest to a small number of readers. Those unfamiliar with category theory may find it bewildering, and those well versed in category theory may find it trivial. My hope is that someone in between, someone just starting to get a handle on category theory, will find it helpful. […]
Dimension 2 The equation for the perimeter of an ellipse is where a is the semimajor axis, e is eccentricity, and E is a special function. The equation is simple, in the sense that it has few terms, but it is not elementary, because it depends on an advanced function, the complete elliptic integral of the […]
When is the discrete Fourier transform of a vector proportional to the original vector? And when that happens, what is the proportionality constant? In more formal language, what can we say about the eigenvectors and eigenvalues of the DFT matrix? Setup I mentioned in the previous post that Mathematica’s default convention for defining the DFT […]
I was flipping through Gravitation [1] this weekend and was curious about an illustration on page 309. This post reproduces that graph. The graph is centered at Cairo, Egypt and includes triangles whose side lengths are the distances between cities. The triangles are calculated using only distances, not by measuring angles per se. The geometry […]
Many numerical integration formulas over a finite interval have the form That is, the integral on the left can be approximated by evaluating the integrand f at particular nodes and taking the weighted sum, and the error is some multiple of a derivative of f evaluated at a point in the interval [a, b]. This […]
A few days ago I wrote about using the CMU Pronouncing Dictionary to search for words that decode to certain numbers in the Major mnemonic system. You can find a brief description of the Major system in that post. As large as the CMU dictionary is, it did not contain words mapping to some three-digit […]
Jaccard index is a way of measuring the similarity of sets. The Jaccard index, or Jaccard similarity coefficient, of two sets A and B is the number of elements in their intersection, A ∩ B, divided by the number of elements in their union, A ∪ B. Jaccard similarity is a robust way to compare […]
I have a little script that will print the frequency of the most common characters in a file and the number of lines. All numbers are displayed along with their factorizations. It also prints the number of non-ASCII characters. CSV files These simple statistics are surprisingly useful. For example, when I ran it on an […]
Here’s a simple but surprising theorem from digital signal processing: linear, time-invariant (LTI) operators commute. The order in which you apply LTI operators does not matter. Linear in DSP means just you’d expect from seeing linear defined anywhere else: An operator L is linear if given any two signals x1 and x2, and any two […]
Suppose you have a database of personally identifiable information (PII) and you want to allow someone else to query the data while protecting the privacy of the individuals represented by the data. There are two approaches: Deidentify, then query Query, then deidentify The first approach is to do whatever is necessary to deidentify the data—remove […]
