Frans #Faase’s blog, discussing Modbus, the new #Lean #math proof of Fermat’s Last Theorem, laser-cutting puzzle pieces, etc.
on 02026-09-08the Leiden Declaration about how #AI could be bad in #math.
on 02026-06-04“Dispatches from the possibly last days of human relevance” on Erdos’s Unit Distance Problem and how “a system called AlphaProof Nexus to settle nine more (!) Erdös problems, many of them in additive combinatorics, along with miscellaneous other open #math problems. Notably, in this case the AI also fully formalized its proofs in #Lean.” #AI #Scott-Aaronson
on 02026-06-01longer explanation of #Kullback-Leibler divergence. #math #statistics #toread
on 02026-04-08pretty explanation of #Kullback-Leibler divergence. #math #statistics #toread
on 02026-04-08page from #Leibniz’s first published paper on the differential calculus, “#Nova-methodus pro maximis & minimis, itemque tangentibus, quae nec fractas, nec irrationales quantitates moratur, & singulare pro illis calculi genus”. #math #history
on 02026-04-08Terry Tao on #Mastodon explaining that #AI solved Erdos problem 728. “Interestingly, the proof contained some minor errors in it, but the AI tool Aristotle was able to automatically repair these gaps and produce a Lean-verified proof.” #math #LEAN
on 02026-01-16in the “power-limited regime” of the Shannon–Hartley channel capacity C = B lg (1 + S/N) bits per second, ln(1 + S/N) ≈ S/N because S/N is very small. Consequently C ≈ B S/N/ln 2 = B 1.44 S/N, 1.44 bits per second per hertz times the signal-to-noise ratio, so if you have additive white noise (whose power is proportional to bandwidth B) your channel capacity is independent of the signal bandwidth for fixed signal power. You start to lose power efficiency once you cram your transmit power into such a narrow bandwidth that ln(1 + S/N) stops being a good approximation to S/N. When S/N = ¼ (-6dB) you’re still getting 89% power efficiency, which rises to 95% at -10dB. #communications #math
on 02026-01-16#paper about a “single-scale nomogram” by Richard K. Guy from 01953. The Mathematical Gazette, Vol. 37, No. 319 (Feb. 1953), p. 39, DOI: 10.2307/3609499. Using almost an #elliptic-curve and a straightedge to multiply, divide, and extract square roots. #Analog computation. The explanation of why it works uses elementary algebra: “Since the equation x³ + ax + b = 0 has zero for the sum of its roots, the x-coordinates of the three intersections of the line y = mx + c and the curve y = x³ + px + q add to zero.” #nomography #math
on 02026-01-10#Numberphile #video of #Cliff-Stoll with a hole in a hole in a hole. #math #topology
on 02026-01-07a very poor #ebook scan, missing pages, of Edwin Bidwell Wilson’s “Advanced Calculus” from 01912 #math #toread
on 02026-01-06“Feynman’s Trick: a.k.a. Differentiation under the Integral Sign & Leibniz Integral Rule” #math #toread
on 02026-01-06new #Segerman #video extending his expanding-racks approach to three-dimensional expanding objects. Lots of #math about three-dimensional symmetry groups and crystal structures.
on 02026-01-06“Word Processing in Groups is a monograph in mathematics on the theory of automatic groups, a type of abstract algebra whose operations are defined by the behavior of finite automata.” #math #toread
on 02025-12-11#paper on the “tropical approach” in #math, starting with the #tropical semiring, then presenting phylogenetic tree derivation. #toread
on 02025-12-11very nice #math #video about deriving the derivative of ln x, for example using implicit differentiation in calculus.
on 02025-11-26#Segerman #video on the Cannon–Thurston map and the complementary space of the figure-eight knot. #math #toread
on 02025-10-15#Segerman #video on “wild” knots such as the wild slipknot which cannot be topologically deformed into polygons. Introduction to knot theory. #toread #math
on 02025-10-15#Segerman #video showing Sydler’s π/4 polyhedron from 01965, constructed by Matthias Goerner, in which all dihedral angles are 90° except for one, and talking-heading about the Dehn invariant. #math #geometry
on 02025-10-15#3Blue1Brown #video on the Laplace transform. #math #toread
on 02025-10-14Adaptive quadrature. #math
on 02025-09-09Richardson extrapolation is a convergence-accelerating technique. #math
on 02025-09-09Quadrature on evenly spaced points uses the Newton–Cotes formulas, a generalization of the trapezoid rule and Simpson’s rule. #math
on 02025-09-09#Posits for computer #math
on 02025-09-07#PDF of #ebook of introductory #coding-theory #textbook "Essential Coding Theory". 566 pp. #math
on 02025-08-29Eleanor Robson #paper #PDF on #Plimpton-322 #math #history #toread
on 02025-08-27some kind of weird transformation of the complex plane using projections onto a sphere? Representable as a projective matrix multiplication? #math #toread
on 02025-08-27#PDF in French of #Bourbaki on “Noetherian rings”. Possibly useful for the #history of #well-founded things in #math.
on 02025-08-02Was #well-founded induction discovered by Noether? “#Bourbaki’s first edition of Algèbre commutative dates from 1968. But Noetherian topological spaces are already defined in Grothendieck-Dieudonné’s Éléments de géométrie algébrique, I (1960), and the « principe de récurrence noethérienne » is stated (and proved) in Chapter 0, (2.2.2). (...) Meanwhile, the Wikipedia entry on the axiom of foundation asserts that “the concept of well-foundedness and rank of a set were both introduced by Dmitry Mirimanoff (1917).” #math #history
on 02025-08-02I didn’t realize there was a ProofWiki. #math
on 02025-08-02#math #video #toread about #optimization in #C of Fibonacci number computation
on 02025-03-18a new #3Blue1Brown #math #video about the colliding blocks computing π, making a connection to Grover’s algorithm in #quantum-computation #toread
on 02025-03-15#convolution #algorithms using polynomial multiplication for Transformer #neural-networks, going back and forth between the coefficient representation of a polynomial and value representations of it, especially evaluating it at n+1 complex roots of unity. #math #toread
on 02024-12-12#PDF #paper on #history of #math showing a 45th-degree Chebyshev polynomial in the 16th-century works of Adrianus Romanus, who calculated π to 16 decimals in 01593
on 02024-11-27#graphics with spherical harmonics for Lambertian lighting with light probes #math
on 02024-06-18#video about #mechanical #gears sliding; #math says that if you take a parametric curve γ(s) and add a parameter t for its motion over time γ(s,t) the Jacobian of that function will have determinant zero at its envelope (where movement through time is the same as a movement along the surface), and from that this video derives a closed-form parametric representation of that envelope curve, which is the mating gear.
on 02024-04-28Johan de Jong’s #Stacks-Project is a single-maintainer Wiki (?) of 7300 pages about #algebraic-geometry (including #3D), of proofs, theorems, lemmas, etc., looking like “a traditional textbook”, winning de Jong the AMS’s Steele Prize #math
on 02024-04-15From Newton’s method to Newton’s fractal #video #math #toread
on 02024-04-03John Baez comments on Dan Christensen’s plot of the roots of all quintics with integer coefficients in [-4, 4] #math #pretty
on 02024-03-06an extreme version of #math #constructivism in the #foundations-of-mathematics
on 02024-02-06overview of Brouwer's #intuitionism, part of #constructivism in the #foundations-of-mathematics #math
on 02024-02-06Phil Wilson explains the #foundations-of-mathematics debate betweeen #formalism, #constructivism (including #intuitionism), #platonism, and #logicism. #math
on 02024-02-06handbook of domain theory, explaining Dana Scott’s logical foundation for computation and computability, and specifically defining the notion of “continuity” which is equivalent to computability. A fairly dense 167-page #PDF, but at least the parts I’ve read seem to be pretty penetrable. #math
on 02024-01-22"Arb" is “a C library for #arbitrary-precision ball arithmetic”. #interval-arithmetic on complex numbers. #math
on 02024-01-14different approaches to handling infinity in #math CASes for symbolic computation of limits in the author’s systems Calcium and Fungrim.
on 02024-01-14How to solve quartic equations. #math
on 02023-09-19The quartic formula. #math
on 02023-09-19Hammack's #ebook "Book of Proof" has been approved by the American Institute of Mathematics' Open Textbook Initiative. #logic #math
on 02023-06-17#CC-BY-NC licensed blog of #math cartoons
on 02022-02-09V.I. Arnold says #math is a branch of physics; he is very anti-Bourbaki
on 02022-02-09another #elliptic-curve nomogram, with some historical notes. #nomography #math
on 02022-02-09#Elliptic-curve nomogram calculator, similar to Guy’s 01953 “single-scale nomogram” but actually different #nomography #math
on 02022-02-09new home page for Jim Hefferon’s well-regarded CC-BY-SA #linear-algebra #ebook #math
on 02021-12-05#linear-algebra #ebook Git repo #math
on 02021-12-05Jim Hefferon’s well-regarded CC-BY-SA #linear-algebra #ebook, accompanied by a lab manual using Sage. #math
on 02021-12-05#numerical-methods #ebook by Giray Ökten, CC-BY-NC-SA, using Julia. Root-finding (bisection, Newton’s method, method of secants, Muller’s method (the parabolic version of the method of secants), fixed-point iteration, high-order fixed-point iteration), interpolation (polynomial, Hermite, spline), numerical quadrature and differentiation (Gaussian quadrature), and least-squares approximations. Notably omits matrix algorithms and everything about differential equations: no ODEs, no PDEs. 224 pp. #math
on 02021-12-05Leon Q. Brin’s "Tea Time Numerical Analysis" #numerical-methods #ebook home page (CC-BY-SA). Very good coverage, including historical documents. Starts a bit more basic than the other alternatives. Covers root finding (bisection, fixed-point iteration, Steffensen’s method for accelerating it, Newton’s method, synthetic division, Müller’s method), interpolation (Lagrange polynomials, Neville’s method, Newton polynomials, divided differences, Bèzier curves, splines), quadrature, ordinary differential equations (Taylor methods, #Runge-Kutta, adaptive Runge-Kutta). Includes proofs and error bounds, with more of a mathematical orientation than the alternatives. #math 375 pp.
on 02021-12-05review of #Tea-Time-Numerical-Analysis #numerical-methods textbook #math
on 02021-12-05another #numerical-methods #ebook, this one using Python, including floating-point error, bisection, regula falsi, Newton’s method, the method of secants, numerical optimization including Simpson’s rule, LU factorization, QR factorization, least-squares curve fitting, eigenvalue computation, ODE integration (the midpoint method, #Runge-Kutta, backwards Euler), PDEs (the heat equation and the wave equation), Vandermonde interpolation, Lagrange interpolation, and Chebyshev points #math
on 02021-12-05#numerical-methods #ebook and online courseware; covers least squares, numerical integration of ODE with #Runge-Kutta and #leapfrog-integration; ODEs with boundary conditions; PDEs including elliptic, parabolic, and hyperbolic; successive over-relaxation (SOR); fluid dynamics; Navier-Stokes; Boltzmann’s equation; neutron transport; Monte Carlo techniques, including for radiation transport; etc. #math
on 02021-12-05The Russian #math #pedagogy magazine Quantum was translated into English for several years, 01990 to 02001, and is available as #PDF #ebook files.
on 02021-01-21the Bernoullis emigrated from Antwerp to Frankfurt in 1570, later Basel in 1620, to escape the Spanish persecution of the Protestants. #history #math #Switzerland
on 02021-01-20More on the "Minsky cirle algorithm". #math
on 02018-11-23A summary of Minskys and Trinskys. #math #minsky-circle-algorithm #graphics #fractals
on 02018-11-23How to typeset #math in #TeX (specifically #LaTeX).
on 02018-11-23Jacob Rus’s #explorable-explanations of some spline #math.
on 02018-11-03Wildberger’s rational trigonometry makes a surprising appearance in Babylonian clay tablet of Pythagorean triples, or perhaps their squares. #math
on 02017-11-25Brownian motion is recurrent in 1-D or 2-D but transient in 3-D or more. #math
on 02017-07-18#PDF of a Henry Baker #paper about complex #math #algorithms and CORDIC and whatnot. #geometry #graphics
on 02017-05-27#math #statistics proof by Thomas Royen of the Gaussian correlation inequality; nearly ignored because he wrote it in Word and published it in “the Far East Journal of Theoretical Statistics, a periodical based in Allahabad, India...which...listed Royen as an editor”. Not a joke!
on 02017-05-02D3 #animation of Bézier curves. #math #explorable-explanations
on 02017-04-26#Statistics #math #paper #toread on a very generic explanation for Zipf’s law
on 02017-03-11#math #paper #PDF on “unbounded spigot #algorithms” for digits of π
on 02017-03-11987-page #pdf of “Mathematics for Computer Science”, a cc-by-sa #math #ebook covering things like graphs, satisfiability, and linear recurrences
on 02017-03-07#PDF #paper on lattices of N-ary relations — natural-join is meet, an "inner" generalization of union is join, because that’s the lattice-dual of natural join. #math
on 02017-03-07"constant-recursive sequences" or “C-finite sequences” are I think sequences of linear homogeneous recurrences. #math
on 02017-03-02how to use #IPython/#Jupyter with #SymPy to do symbolic #math in a convenient notebook format.
on 02016-10-11Gowers on #math #education
on 02016-10-07Nadia, Joshua Fried, and some French guys demonstrate a discrete logarithm against a 1024-bit trapdoored prime field using the special number field sieve. If people are using such a field for DH or DSA, they are vulnerable to such an attack. #crypto #math
on 02016-10-06#interval-arithmetic #math #algorithms in ℤ/Nℤ #toread by #ryg
on 02016-09-29a simple #introduction to the Fourier transform, part of an #ebook on #DSP. The formatting is exemplary. #math
on 02016-09-23#Shalizi explains ergodicicty. #math
on 02016-08-13John Baez did some beautiful #visualization of roots of integer-coefficient polynomials. #math
on 02016-08-12“A Single-Scale Nomogram”, from Richard K. Guy, The Mathematical Gazette, Vol. 37, No. 319 (Feb. 1953), p. 39, DOI: 10.2307/3609499. Using an #elliptic-curve and a straightedge to multiply, divide, and extract square roots. #Analog computation. The explanation of why it works uses elementary algebra. #nomography #math
on 02016-08-12using #neural-networks to evaluate #math expressed in #handwriting on a chalkboard with #computer-vision. Linked videos. #toread
on 02016-08-02discussion thread on #CS theory, on unusual models of computation like the Rotary Element, Turing machines with tapes over arbitrary groups, Iota, Jot, etc. #math
on 02016-08-01Dan Piponi on monoids and monads. #math #toread
on 02016-08-01Telling if numbers are divisible by different constants by looking at their hexadecimal representation. #math
on 02016-07-22Multilinear interpolation is a generalization of linear interpolation. #math
on 02016-07-21"Singular" is a #computer-algebra system for polynomial computations, with special emphasis on commutative and non-commutative algebra, algebraic geometry, and singularity theory. It is free and open-source under the GNU General Public Licence. #math
on 02016-06-28A talk about Fourier analysis and graphical algebra. #explorable-explanations #math
on 02016-03-29the #ebook of the differential equations textbook that Freeman Dyson taught himself from over Christmas break, 14 hours a day. #math
on 02015-11-18Chebfun is a #Matlab/#Octave library for computations on continuous functions as if they were vectors, originally by way of approximations using Chebyshev polynomials. This example does interpolation with splines. #math
on 02015-11-16Takens’s 1981 “embedding theorem” in #math shows that the full state of (holomorphic?) chaotic systems can be derived from the time series of any one of their variables.
on 02015-10-19A holomorphic #math function is the same thing as a conformal function of the complex plane, and it turns out that an awful lot of complex functions are holomorphic or mostly holomorphic.
on 02015-10-19is a polyhedron made of 120 identical scalene triangles. It's what you get if you cut up the faces of a Platonic dodecahedron with five geodesics each, or the faces of a Platonic icosahedron with three geodesics each. In fact, it's made of 15 great circles. Would be ideal for a Dymaxion map. #math
on 02015-09-15The table of contents of Carl de Boor’s 1978 ebook on the #math of #splines.
on 02015-09-05"Fast B-Spline Transforms" #splines #math
on 02015-09-04“Cardinal spline filters: Stability and convergence to the ideal sinc interpolator”, "Aldroubi 1992", shows that polynomial spline interpolants (?) converge to the sinc function rapidly (?). #splines #math
on 02015-09-04a #pdf #ebook “Linear Predictive Coding and the Internet Protocol”, by Robert M. Gray, explaining the history of #LPC #speech-codecs in #DSP and the history of the development of IP, especially UDP. Covers all the #math of LPC. #protocols
on 02015-09-04Creating and Rendering "Convolution Surfaces", McCormack and Sherstyuk 1997. #3D #rendering #math #paper.
on 02015-09-01oh hey, “The most commonly used #splines are cubic spline, i.e., of order 3—in particular, cubic B-spline, which is equivalent to C2 continuous composite Bézier curves.” #math
on 02015-09-01Iterating a #math function half a time.
on 02015-08-29