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ZX calculus is a graphical calculus (looks like colored birdtracks) for qubit manipulation. Here is a nice introduction. The idea is that you can write out every gate in terms of simple operations in the two standard bases, i.e. eigenvectors of the Pauli matrices z and x.

Uncle Arnold’s talk about mathematical trinities is a very fun read. Here a summary: Real-complex dualities free will vs predestination pi0 vs pi1 Z2 vs Z modes and quasimodes vs Berry phase and integer quantum Hall effect S1 (projective variety) vs S2 S1 (group) vs S3 real vs complex trigonometric polynomial stratification Real-complex-quaternion trinities: R, C, H E6, E7, E8 tetrahedron, octahedron, icosahedron A3, B3, H3 D4, F4, H4 first three Hopf bundles polynomials, Laurent polynomials and modular polynomials, aka with poles at 0, 1, infinity numbers, trigonometric numbers, elliptic numbers quadratic, hermitian, hyperhermitian forms flat connection monodromy, vector bundle curvature, ?...

I was looking for some examples of Monte-Carlo methods, as the only one I had concretely in mind was the sausage method for pi. Applications? Not many. I run into a way to solve the (say, 2d) Laplace equation with Dirichlet boundary conditions. Writing out the null Laplacian condition on a square lattice can be seen as stating random walks are isotropic in every direction. I think the method was derived by contemplating these two facts:...

Dirac’s derivation of his famous equation seems to involve finding some king of “splitting algebra”. Are there other equations that need such treatment?

It is often said that epicycles, a fundamental mathematical tool of the Ptolemaic planetary model, which I was always taught were as primitive, obsolete and embarassing as the conception of our solar system, are nothing more and nothing less than Fourier series expansions. I realize, with much consternation, that this is true, but the question arises: Has anybody done the calculations and got that Fourier series for Kepler orbits do agree with the epicycle description?...

I heard a rumor from a friend of mine who talked to one of our professors. He said the professor told him about how his highschool background on basic electronics circuits let him easily “imagine” the integral representation of Bessel Functions. How?

The Dirac measure looks like the Lebesgue measure composed with the Dirac delta distribution. Does this generalize to other measures (on spaces that admit a Lebesgue measure)?

This one’s diehard. Every two norms on a finite dimensional vector space are equivalent: that is to say, each one can be rescaled by a constant to bound the other from both above and below. A consequence of this is that all norms induce the same topology. In infinite dimension, this is not true anymore. But so far I have only seen examples of pair of norms where one beats the other on the whole space....

Some years ago I studied Japanese for some months, out of sheer desire to consume anime in its raw form. The Japanese writing system is one of those systems that are so inefficient one eventually begins to love them. They’re a mess, but dually offer a sadistic pleasure for those who master them. Other than two “alphabets”, hiragana and katakana, which really aren’t a problem, Japanese uses kanji, repurposed Chinese characters....