URL : | http://www.johndcook.com/blog/ | |
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Filed Under: | Academics | |

Posts on Regator: | 1442 | |

Posts / Week: | 4.7 | |

Archived Since: | April 26, 2011 |

This morning I was reading Terry Tao’s overview of the work of Yves Meyer and ran across this line: The powers ?, ?2, ?3, … of the golden ratio lie unexpectedly close to integers: for instance, ?11 = 199.005… is unusually close to 199. I’d never heard that before, so I wrote a little code […]

In a recent interview, Tyler Cowen discusses complacency, (neruo-)diversity, etc. Let me give you a time machine and send you back to Vincent van Gogh, and you have some antidepressants to make him better. What actually would you do, should you do, could you do? We really don’t know. Maybe he would have had a […]

Radim ?eh??ek of Rare Technologies just posted an interview he and I did a few days ago. We mostly talk about consulting: getting started, finding work, good and bad leads, etc.

The canonical examples of natural and unnatural transformations come from linear algebra, namely the relation between a vector space and its first and second duals. We will look briefly at the finite dimensional case, then concentrate on the infinite dimensional case. Two finite-dimensional vector spaces over the same field are isomorphic if and only if […]

The ladder of abstractions in category theory starts with categories, then functors, then natural transformations. Unfortunately, natural transformations don’t seem very natural when you first see the definition. This is ironic since...Show More Summary

Eight blog posts based on the work of the intriguing mathematician Srinivasa Ramanujan: Ramanujan series for computing ? Ramanujan ? approximation Ramanujan’s nested radical Ramanujan approximation for circumference of an ellipse Ramanujan’s most beautiful identity Ramanujan’s factorial approximation Sums of fourth powers Algorithm used for world record pi calculations

Larry Wall, creator of the Perl programming language, created a custom degree plan in college, an interdisciplinary course of study in natural and artificial languages, i.e. linguistics and programming languages. Many of the features of Perl were designed as an attempt to apply natural language principles to the design of an artificial language. I’ve been […]

A blog post by Evelyn Lamb yesterday introduced Thomas Baruchel’s web site by of images from complex analysis. I wondered what a collage of these images would look like, so I used the ImageQuilts software by Edward Tufte and Adam Schwartz to create the image below. Related: Applied complex analysis

In my previous post, I discussed how number theory and topology relate to other areas of math. Part of that was to show a couple diagrams from Jean Dieudonné’s book Panorama of Pure Mathematics, as seen by N. Bourbaki. That book has only small star-shaped diagrams considering one area of math at a time. I’ve created a […]

Areas of math all draw on and contribute to each other. But there’s a sort of trade imbalance between areas. Some, like analytic number theory, are net importers. Others, like topology, are net exporters. Analytic number theory uses the tools of analysis, especially complex analysis, to prove theorems about integers. The first time you see […]

Yesterday I ran across Askar Safin’s blog post The Collapse of the UNIX Philosophy. Two quotes from the post stood out. One was from Rob Pike about the Unix ideal of little tools that each do one job: Those days are dead and gone and the eulogy was delivered by Perl. The other was a […]

The trapezoid rule is the most obvious numerical integration technique. It comes directly from the definition of a definite integral, just a Riemann sum. It’s a very crude technique in general; you can get much more accuracy with the same number of function evaluations by using a more sophisticated method. But for smooth periodic functions, […]

I’d never given any thought to the order of polar coordinates until yesterday. They’re written (r, ?). That’s just how it is. But a friend pointed out two reasons why this bothers him. First, r is typically a function of ?, just as y is typically a function of x. But in rectangular coordinates, the […]

Someone mailed me this afternoon asking if I knew what function was graphed on the cover of Abramowitz and Stegun’s famous Handbook of Mathematical Functions. Here’s a close-up of the graph from a photo of my copy of A&S. (I actually have two copies. Before I started working from home, I kept a copy at […]

Fourier series express functions as a sum of sines and cosines of different frequencies. Bessel series are analogous, expressing functions as a sum of Bessel functions of different orders. Fourier series arise naturally when working in rectangular coordinates. Bessel series arise naturally when working in polar coordinates. The Fourier series for a constant is trivial. […]

The web is all abuzz about how SHA1 is “broken”, “a failure,” “obsolete”, etc. It is supposed to be computationally impractical to create two documents that have the same secure hash code, and yet Google has demonstrated that they have done just that for the SHA1 algorithm. I’d like to make two simple observations to […]

In one section of his book The Great Good Thing, novelist Andrew Klavan describes how he bluffed his way through high school and college, not reading anything he was assigned. He doesn’t say what he majored in, but apparently he got an English degree without reading a book. He only tells of one occasion where […]

I posted a couple things on Twitter today about micro-resumés. First, here’s how I’d summarize my work in a tweet. What I’ve done: Math prof, programmer, statistician What I do now: Consulting #microresume — John D. Cook (@JohnDCook) February 21, 2017 (The formatting is a little off above. It’s leaving out a couple line breaks […]

You can associate a matrix with a graph and find the eigenvalues of that matrix. This gives you a spectrum associated with the graph. This spectrum tells you something about the graph analogous to the way the spectrum of a star’s light tells you something about the chemical composition of the start. So maybe it […]

In 1838, Rev. Henry Moseley discovered that a large number of mollusk shells and other shells can be described using three parameters: k, T, and D. First imagine a thin wire running through the coil of the shell. In cylindrical coordinates, this wire follows the parameterization r = ek? z = Tt If T = 0 this is a […]

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