Of all the formal constructions of the set of the real numbers, Eudoxus reals are to me the most elegant, even if completely impractical. The idea, due to Schnauel, is that any line in the plane, be its slope rational or irrational, can be described using only points with integer coordinates: taking the floor (or ceil or round or worse, doesn’t matter) of the y-values associated with integer x-values does characterize uniquely, asymptotically, any line....
Is fire hot or cold? I mean, relatively to the avergage temperature of the observable Universe. Well, according to this the average temperature by volume is determined by cosmic microwave background, around 2.7 K. the average temperature by observable mass is determined by intergalactic material, that is about 10'000 to 100'000 K. if there’s dark matter out there, that goes up to about 1'000'000 K. A professor I asked told me something about the temperature of the cosmological constant, and that in different periods of the universe different contribution were predominant, but I didn’t understand it much....
Some ideas. https://www.thingiverse.com/thing:5209797 custodie per cose strane gomboc scacchi divertenti appendini giacche ricardo fermalibri case hdd freezer hdd zip felpa lacci mascherina
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)?