Blog Archives

Distribution of a range

Suppose you’re drawing random samples uniformly from some interval. How likely are you to see a new value outside the range of values you’ve already seen? The problem is more interesting when the interval is unknown. You may be trying

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Elementary vs Foundational

Euclid’s proof that there are infinitely many primes is simple and ancient. This proof is given early in any course on number theory, and even then most students would have seen it before taking such a course. There are also

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Rudyard Kipling and applied math

This evening something reminded me of the following line from Rudyard Kipling’s famous poem If: … If all men count with you, but none too much … It would be good career advice for a mathematician to say “Let all

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Commutative diagrams in LaTeX

There are numerous packages for creating commutative diagrams in LaTeX. My favorite, based on my limited experience, is Paul Taylor’s package. Another popular package is tikz-cd. To install Paul Taylor’s package on Windows, I created a directory called localtexmf, set

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Quintic root

Here’s a curious result I ran across the other day. Suppose you have a quintic equation of the form z x5 – x – 1 = 0. (It’s possible to reduce a general quintic equation to this form, known as

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Amazing approximation to e

Here’s an approximation to e by Richard Sabey that uses the digits 1 through 9 and is accurate to over a septillion digits. (A septillion is 1024.) MathWorld says that this approximation is accurate to 18457734525360901453873570 decimal digits. How could

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Independent decision making

Suppose a large number of people each have a slightly better than 50% chance of correctly answering a yes/no question. If they answered independently, the majority would very likely be correct. For example, suppose there are 10,000 people, each with

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Making definitions

“The essential virtue of category theory is as a discipline for making definitions, and making definitions is the programmer’s main task in life.” From Computational Category Theory  

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Intuitionistic logic in Gilbert and Sullivan

In a recent preprint, Philip Wadler introduces intuitionistic logic using the comic opera The Gondoliers. In Gilbert and Sullivan’s The Gondoliers, Casilda is told that as an infant she was married to the heir of the King of Batavia, but

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Log semiring

Here’s a strange way to do arithmetic on the real numbers. First, we’ll need to include +∞ and -∞ with the reals. We define the new addition of two elements x and y to be -log (exp(-x) + exp(-y) ).

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Do the stars go up or down?

The thing that sparked my interest in category theory was a remark from Ted Odell regarding the dual of a linear transformation. As I recall, he said something like “There’s a reason the star goes up instead of down” and

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Imaginary gold, silver, bronze, …

The previous post gave a relationship between the imaginary unit i and the golden ratio. This post highlights a comment to that post explaining that the relationship generalizes to generalizations of the golden ratio. GlennF pointed out that taking the

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Imaginary gold

This morning Andrew Stacey posted a beautiful identity I’d never seen before relating the golden ratio ϕ and the imaginary unit i: Here’s a proof: By De Moivre’s formula, and so Related posts: Golden ratio and special angles Golden strings

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Categories, Birds, and Frogs

Freeman Dyson divided mathematicians into birds and frogs in his essay by that title. Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight

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Beavers and category theorists

“Much as beavers, who as a species hate the sound of running water, plaster a creek with mud and sticks until alas that cursed tinkle stops, so do category theorists derive elaborate and obscure definitions in an attempt to capture

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