John D. Cook

The previous post looked at an inequality discovered by Dmitri Mendeleev and generalized by Andrey Markov: Theorem (Markov): If P(x) is a real polynomial of degree n, and |P(x)| ≤ 1 on [−1, 1] then |P′(x)| ≤ n² on [−1, 1]. If P(x) is a trigonometric polynomial then Bernstein proved that the bound decreases from n² to n. Theorem […] The post From Mendeleev to Fourier first appeared on John D. Cook.

Dmitri Mendeleev is best known for creating the first periodic table of chemical elements. He also discovered an interesting mathematical theorem. Empirical research led him to a question about interpolation, which in turn led him to a theorem about polynomials and their derivatives. I ran across Mendeleev’s theorem via a paper by Boas [1]. The […] The post Mendeleev’s inequality first appeared on John D. Cook.

A few years ago I wrote about comm, a utility that lets you do set theory at the command line. It’s a really useful little program, but it has two drawbacks: the syntax is hard to remember, and the input files must be sorted. If A and B are two sorted lists, comm A B […] The post Set intersection and difference at the command line first appeared on John D. Cook.

The hardest part of using regular expressions is not crafting regular expressions per se. In my opinion, the two hardest parts are minor syntax variations between implementations, and all the environmental stuff outside of regular expressions per se. Embedded regular expression modifiers address one of the environmental complications by putting the modifier in the regular expression […] The post Embedded regex flags first appeared on John D. Cook.

The gamma function Γ(z) extends the factorial function from integers to complex numbers. (Technically, Γ(z + 1) extends factorial.) There are other ways to extend the factorial function, so what makes the gamma function the right choice? The most common answer is the Bohr-Mollerup theorem. This theorem says that if f: (0, ∞) → (0, […] The post A lesser-known characterization of the gamma function first appeared on John D. Cook.

The alternating series test is part of the standard calculus curriculum. It says that if you truncate an alternating series, the remainder is bounded by the first term that was left out. This fact goes by in a blur for most students, but it becomes useful later if you need to do numerical computing. To […] The post Tighter bounds on alternating series remainder first appeared on John D. Cook.

If a number has a finite but nonzero fractional part, so do all its powers. I recently ran across a proof in [1] that is shorter than I expected. Theorem: Suppose r is a real number that is not an integer, and the decimal part of r terminates. Then rk is not an integer for any positive integer […] The post Powers don’t clear fractions first appeared on John D. Cook.

The previous post introduced the idea of a twelve-tone row, a permutation of the twelve pitch classes of a chromatic scale. The earlier post also introduced a group of operations on a tone row with elements P, R, I, and RI. Here P, which stands for “prime”, is the identity operator: it leaves the tone […] The post Tone row operations first appeared on John D. Cook.