I consider myself an expert in regular expressions, yet finding out that a very simple regex can determine whether a number is prime just exploded my brain. Note that while the regex given in the article matches strings of 1s, that's totally incidental. You can easily modify the regex to use strings of any single repeated character. What's really being tested is whether the length of a string is prime.
He is best known for working closely with Isaac Newton by proofreading the second edition of his famous book, the "Principia", before publication. He also invented the quadrature formulas known as Newton–Cotes formulas (a group of formulas for numerical integration based on evaluating the integrand at equally spaced points), and made a geometric argument that can be interpreted as a logarithmic version of Euler's formula. via @wikipedia
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. He made important contributions in the theory of differential equations. He also made contributions to applied mathematics, including the theories of telegraphy and elasticity. via @wikipedia
It seems to me that #clarity and #communication is so important in #mathematics, but this need constantly breaks against the language and terminology specific to and embedded in each subfield…
I want to graphically demonstrate the effect of some algebraic operators in a high-dimensional complex-valued vector space. I will be picking out a small number of example vectors, applying the operators to yield new vectors, and looking at what has happened in terms of the angles between the vectors.
The source vector space may be very high dimensional (say, 1000). Each element of the source vector is constrained to have magnitude 1. That is, each complex value has only one degree of freedom - the phase angle.
I am interested in the angles between vectors in the source space. In these high-dimensional spaces two vectors chosen at random are almost always very close to orthogonal.
I am interested in the angular relations (measured in the high-dimensional source space) between a a small number of vectors and 2 or 3 mutually orthogonal reference vectors.
I want to project the high-dimensional source space onto a real-3-sphere so that the angles of interest are maintained sufficiently well to be visually interpretible. I don't really care what happens to the angle between the other vectors.
I would greatly appreciate any pointers to how I might define and implement such a projection.
(Bonus if you can suggest an R package or code to do this.)
#TIL about the #HyperLogLog algorithm and I think it's a damn brilliant way to estimate the number of unique elements of a potentially gargantuan set of items and only running in O(n) time and O(1) space. The fact that variants of the algorithm can be done in parallel makes it even more awesome!
I'm so old I can remember when they wouldn't even let you present slides in a #mathematics talk unless you could find a Mac to borrow, since they were the only things that worked easily. Today I saw my first catastrophic failure in a presentation on a Mac, while my PC set itself up in three seconds and had no issues.
He worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electricity and magnetism, thermodynamics, elasticity, and fluid mechanics. Moreover, he predicted the Poisson spot in his attempt to disprove the wave theory of Augustin-Jean Fresnel, which was later confirmed. via @Wikipedia
Congratulations to our colleagues David Bate & Marie-Therese Wolfram for being awarded London Mathematical Society (LMS) Whitehead Prizes!
▶️ David Bate is awarded a Whitehead Prize for his fundamental contributions to the development of Geometric Measure Theory in the metric setting, including the characterisations of rectifiability in terms of projections and in terms of tangent planes.
▶️ Marie-Therese Wolfram is awarded a Whitehead Prize for her groundbreaking contributions to applied partial differential equations, mathematical modelling in socio-economic applications and the life sciences, and numerical analysis of partial differential equations.
Peer review of Enflo’s earlier proof, for Banach spaces in general, took several years. However, that paper ran to more than 100 pages, so a review of the 13 pages of the new paper should be much speedier.
A #RosettaCode contribution for #ATS -- the old insideness of a convex hull algorithm. I decided to do this because I am likely to stick the algorithm within my next Bézier intersection algorithm (which will be coded in Ada using homogeneous geometric algebra, not in ATS using euclidean, but whatever) --
#mathematics is probably about structure (relations) and invariants (identities). Theoretical #physics knows the notion of an „unphysical solution“ - not all implications of a theory make physically sense. To decide, one puts the theory into the context of experiment (practice).
What if a #LLM is a collection of structure and invariants? And what if the „inaccurate“ outputs (bias, „hallucinations“) are like „unphysical solutions“? They need practical constraints.