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Why polynomial coefficients?
2+ hour, 26+ min ago (207+ words) Second order linear differential equations with polynomial coefficients form their own area of study. This seems like a narrow class of equations, but it’s very important in applications. This class of equations seems like a mathematically natural topic, but why…...
Runge-Kutta: over versus stages | Butcher barrier
6+ hour, 47+ min ago (292+ words) The textbook version of the Runge-Kutta method for solving differential equations has 4 stages and has 4th order error. For lower order versions of RK the number of stages s also matches the order of the error p. But in order to achieve…...
Cryptographic Keys and Decks of Cards
4+ day, 10+ hour ago (314+ words) The previous post looked at the idea of storing a cryptographic key in the order of a deck of cards. A deck of 52 cards can store 225 bits of data because Here ⌊x⌋ is x rounded down to the nearest integer....
Hiding data in permutations
4+ day, 23+ hour ago (373+ words) The latest issue of Paged Out! has an article by Stephen Hewitt “An off-line backup of your cryptographic key using playing cards.” The idea is to use a deck of 52 to store a 128-bit cryptographic key. To erase the key, shuffle…...
Counting permutations with roots
5+ day, 6+ hour ago (212+ words) My post from yesterday on permutation roots ends with a Mathematica code for finding the probability that a permutation of n elements has a kth root. This is done by finding the coefficient of xn in the generating function I…...
Printing floating point numbers in binary
5+ day, 7+ hour ago (215+ words) It’s well known that you can convert the base 16 (hex) representation of an integer to the base 2 (binary) representation by simply converting each digit from hex to binary. For example, CAFEhex = 1100 1010 1111 1110two I imagine it’s less well known that you can…...
Forensic statistics in Python
1+ week, 4+ day ago (264+ words) I recently had a project in which I had to reverse engineer a data analysis. There was some ambiguity regarding which of several possibilities someone chose for several of the variables, something analogous to the following example. Suppose you have…...
Generating Lean 4 and SWI Prolog code with Grok 4.5
1+ week, 5+ day ago (237+ words) I’ve written several posts about using Claude or ChatGPT to generate Prolog or Lean code to solve a chess puzzle. I didn’t think Grok would be up to the task, though I didn’t try it. I’ve heard good things about…...
Fitting a regular expression to a list of words
1+ week, 6+ day ago (420+ words) Suppose you want to search for a list of words. If you’re using grep, you can add the -f flag provide a file of regular expressions, and you can add the -F to tell it that the regular expressions are…...
Visualizing medical code hierarchy with treemaps
2+ week, 1+ day ago (128+ words) Quick follow up to the previous two posts on ICD-10 codes and HCPCS codes. This post uses Python’s squarify library to create treemaps visualizing how many codes begin with each letter. Here’s the treemap for HCPCS codes. And here’s the treemap…...