User:Michael Hardy/section

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On topics not within my professional qualifications[edit]

John Gillespie Magee, Junior, self-evidence, tidal resonance, Woods Hole Oceanographic Institution, list of religious topics (Many others have contributed to that one.) Fellowship of Reason, list of optical topics, iridescence, sylvanshine, Foundation for the Advancement of Art, Elisha Otis, Roman Republic (19th century), effect of sun angle on climate, Marlon Knauer, Shyamala Rajender v. University of Minnesota, Karlis Kaufmanis

On probability, statistics, probabilists, and statisticians[edit]

list of statistical topics, Ladislaus Bortkiewicz, Gauss–Markov theorem, Completeness (statistics), Sufficiency (statistics), logit, Kolmogorov's zero-one law, martingale, margin of error, zeta distribution, Zipf's law, confidence interval, Bruno de Finetti, Maxwell's theorem, law of total probability, law of total expectation, law of total variance, Galton–Watson process, coherence (philosophical gambling strategy), cumulant, canonical correlation, Wishart distribution, Rao–Blackwell theorem, Student's t-distribution, Wiener process, errors and residuals in statistics, Seymour Geisser, prediction interval, rankit, continuity correction, Studentized residual, Studentized range, Cramér–Rao inequality

Francis Ysidro Edgeworth (a stub; if it's a long article when you read this, then someone else has contributed),

ancillary statistic, empirical Bayes method, Herbert Robbins, memorylessness, factorial moment, rule of succession, conditional independence, pairwise independence, prior probability distribution, infinite divisibility, a term used in physics, probability theory, and several other disciplines, Schrödinger method, Vandermonde's identity, an inequality on location and scale parameters, compound Poisson distribution,

In estimation of covariance matrices, I describe what seems to me to be a surprisingly subtle and elegant application of linear algebra. I have no idea who originated it; I seem to recall that it is in Morris Eaton's book on multivariate statistics, and I suppose it is in lots of others. In that argument you find out why it is sometimes better to view a scalar as the trace of a 1×1 matrix than as a mere scalar and then to apply certain matrix decompositions to it.

Lévy process, Wigner semicircle distribution, multinomial distribution, Ewens's sampling formula, imputation (statistics), law of total cumulance, copula (statistics) (incorporating some material from Sklar's theorem, which was created by User:Oo64eva and is now a redirect page), normally distributed and uncorrelated does not imply independent, list of stochastic processes topics, method of moments (statistics), method of moments (probability theory)

Eduard Helly, David Blackwell (both stub articles; if they're more than stubs when you see them, then more recent edits have been done)

le Cam's theorem, McCullagh's parametrization of the Cauchy distributions, Lilliefors test, empirical distribution function--still a stub article at this time, Poisson regression, Shapiro–Wilk test, location-scale family, Behrens–Fisher problem (now still a stub), Hausdorff moment problem, Hamburger moment problem, Stieltjes moment problem, Chinese restaurant process, indecomposable distribution, Wick product, Robbins lemma, Fisher's inequality, Chernoff's distribution (a stub, but it clearly gives the definition), Lack-of-fit sum of squares, Bhatia–Davis inequality, Samuelson's inequality, Popoviciu's inequality on variances, Raikov's theorem, An Essay towards solving a Problem in the Doctrine of Chances

On other mathematical topics[edit]

inequality of arithmetic and geometric means, incidence algebra, Euler characteristic, stereographic projection, common logarithm, Dirichlet kernel,

exponential growth (a concept that "laymen" take to mean very fast growth, but which has a technical definition that need not imply great rapidity)

empty product This explains why, when you multiply no numbers at all, you get 1, and why 00 is almost always 1, and should be taken to be 1 for the purposes of set theory, combinatorics, probability, and power series.

binomial type, Sheffer sequence, umbral calculus, Bell numbers, Hermite polynomials, Chebyshev polynomials, Bernoulli polynomials, Gian-Carlo Rota

Archimedes Palimpsest This one mentions ancient history, mathematics, physics, engineering, an art museum, a federal lawsuit, and a very old hierarchical religious organization, in a very short space, without undue cramming;

How Archimedes used infinitesimals, Archimedean property, Eric Temple Bell, Arthur Cayley, Riemann–Stieltjes integral, bounded variation, Ferdinand von Lindemann, Charles Hermite,

orthogonal polynomials (Do not move that article to "orthogonal polynomial" under a delusion that that would conform to the convention of titling an article "dog" rather than "dogs". That would be absurd. There is no such thing as an orthogonal polynomial; there is such a thing as orthogonal polynomials.),

pointwise convergence, Bernstein polynomial, George Boolos, Cantor's theorem, Löwenheim–Skolem theorem, second-order logic (much expanded by others since I started it)

Cantor's first uncountability proof. This proof shows that the set of all real numbers is uncountable, but this proof is not a diagonal argument! This article has since been sweepingly revised by user:RJGray, who added material on a dispute over whether the proof is constructive and material derived from letters between Cantor and colleagues.

inclusion-exclusion principle, linearly ordered group, Boolean prime ideal theorem, uniform norm, Galton–Watson process, coherence (philosophical gambling strategy), dominated convergence theorem, Robertson–Seymour theorem, double integral (not the same as an "iterated integral"; see the article), Fubini's theorem, mathematical logic, Girard Desargues, Desargues' theorem, parallelogram law, Hamel basis, König's theorem, Schur complement (that article needs more work), combinatorial species (I left this one a stubby article that was barely a definition and two or three more-or-less obvious examples; AlexG has since added an account of operations on combinatorial species and lots of essential facts), A simple proof that 22/7 exceeds pi, Putnam Competition, Stone's representation theorem for Boolean algebras, Separation of variables, moment (a disambiguation page), list of mathematical examples (still in its infancy) Möbius transform, cross-ratio, Morera's theorem, Mahler's theorem, Cauchy principal value, Pincherle derivative

Radius of convergence -- This article includes an example of the fact that complex numbers are sometimes simpler than real numbers; they allow us to quickly find the radius of convergence of a power series in which the coefficients are Bernoulli numbers. (As you see from the previous sentence, I firmly believe in splitting infinitives on occasion.) Faà di Bruno's formula,

I moved the anonymously written "absolutely continuous" page to absolute continuity and rewrote it from scratch, including both absolute continuity of real functions, and absolute continuity of measures and the Radon–Nykodym theorem.

An infinitely differentiable function that is not analytic -- Although this is merely the usual example, I explained (albeit tersely, so far) its relevance to Schwartz's theory of generalized functions: One can construct test functions (i.e., infinitely differentiable functions with bounded support) with prescribed behavior on an interval. The existence of such functions must be known before we can confidently say that Schwartz's whole theory is not vacuous.

Moreau's necklace-counting function, cyclotomic identity, proof that holomorphic functions are analytic, branch point, Bell polynomials, Pedoe's inequality, Hadwiger's theorem (a stub; if it's a long article when you read this, then someone else has contributed), defect (geometry) (a fairly terse article so far....), Dandelin spheres, matrix exponential (others have added material to that one), noncrossing partition, uses of trigonometry, Bohr–Mollerup theorem, characterization (mathematics), list of Boolean algebra topics, germ (mathematics), Lissajous curve -- someone has since edited that one by adding illustrations, Cauchy product, Gibbs phenomenon -- a number of people have worked on this one, Schrödinger method, Vandermonde's identity, list of Fourier analysis topics, Muirhead's inequality, Stanley's reciprocity theorem, exponential formula, method of distinguished element, Stirling transform, osculating circle, list of circle topics, list of knot theory topics, Cantor's back-and-forth method, list of set theory topics, rencontres numbers, Bernstein's theorem, multiplicities of entries in Pascal's triangle, Dobinski's formula, Brahmagupta–Fibonacci identity (originally titled Fibonacci's identity; that is now a disambiguation page), alternating permutation, spread polynomials, Faulhaber's formula, Appell sequence, caloric polynomial, Johan Frederik Steffensen---stubby so far..., Lester's theorem, Monge's theorem, polar sine, Hermite–Hadamard inequality, Steiner inellipse, Marden's theorem, (Lommel–Weber function After I created this, it was found that three articles on this topic existed under different titles, which then got merged, so this page now redirects to another.), Regiomontanus' angle maximization problem, Bloch's theorem (complex variables), List of topics named after Bernhard Riemann, Cavalieri's principle, Napkin ring problem (on a startling result in elementary geometry; the natural way to prove this is by Cavalieri's principle, but some authors perversely persist in relying only on other methods), Lune of Hippocrates, Mollweide's formula, Bonse's inequality, Landau's algorithm (a stub article that identifies the purpose and the creator of the algorithm but doesn't describe it; hopefully others, if not I, will expand it), List of topics named after Fibonacci, List of topics named after Augustin-Louis Cauchy, Harold Edwards (mathematician), nLab, Erdős–Kac theorem, Triangular array (currently stubby, but with a fairly good list of examples), List of topics named after Karl Weierstrass, Weierstrass substitution, Gerald J. Toomer — I created this one in somewhat stubby form after I noticed that dozens of Wikipedia articles mention this person; before creating it I put links to it into dozens of articles, Ptolemy's table of chords, Haynsworth inertia additivity formula, Patrick Billingsley, Integral of the secant function, Tikhonov's theorem (dynamical systems), Śleszyński–Pringsheim theorem, Tree of primitive Pythagorean triples, Hermite's cotangent identity, Kummer's theorem, Niven's theorem, Sendov's conjecture