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See: Fivesome, Orthogonal polynomials, Quantum physics, Hermite polynomials, Probability distribution Investigation: Derive and interpret, combinatorially and physically, the fivefold classification of orthogonal Sheffer polynomials. Orthogonal Sheffer polynomials
Three conditions: monic, orthogonal (quadratic), Sheffer (exponential) Think of combinatorial quantum physics operator (raising and lowering operator) as a derived functor. 5 notions of independency
What are the transition matrices between orthogonal polynomials?
Frédéric Chapoton. Ramanujan-Bernoulli numbers as moments of Racah polynomials For almost two years I have been studying quantum physics with my old friend John Harland who is passionate about it. We both took courses in quantum mechanics in college and later met at the University of California at San Diego where he got his PhD in math doing functional analysis and I got mine doing algebraic combinatorics. Recently, I thought that I should learn quantum physics better as a source of inuition about Lie theory, which I think is central to the way that mathematics unfolds, especially through affine, projective, conformal and symplectic geometries. I was glad to join John in studying "Introduction to Quantum Mechanics" by David Griffiths, which is an excellent textbook for learning to calculate as physicists do. Relearning this, as a combinatorialist, I noticed the orthogonal polynomials in solutions of the Schroedinger equation and I became curious to learn what structures they encode.
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Lin Jiu. Research. relates Bernoulli and Euler polynomials, and also Euler and Meixner-Pollaczek polynomials. What are zonal polynomials? Moments of Classical Orthogonal Polynomials Rota, Kahaner, Odlyzko. Finite Operator Calculus. About combinatorics of Sheffer polynomials. Bernouli polynomials - umbral calculusMath Overflow: What's Up With Wick's Theorem? Suggested by Tom Copeland
http://users.dimi.uniud.it/~giacomo.dellariccia/Table%20of%20contents/He2006.pdf The Generalized Stirling Numbers, Sheffer-type Polynomials and Expansion Theorems. Tian-Xiao Generalized linear model is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value.
Canonical link function - distinguishes the essence of the NEF-QVFs.
Morris and Lock (2014), "Starting with a solitary member distribution of an NEF, all possible distributions within that NEF can be generated via ﬁve operations: using linear functions (translations and re-scalings), convolution and division (division being the inverse of convolution), and exponential generation..." (Statistics Stack Exchange) NEF with QVF What is the relation between the Pearson distribution and the natural exponential families with quadratic variance functions?
Meixner classified all the orthogonal Sheffer sequences: there are only Hermite, Laguerre, Charlier, Meixner, and Meixner–Pollaczek. In some sense Krawtchouk should be on this list too, but they are a finite sequence. These six families correspond to the NEF-QVFs and are martingale polynomials for certain Lévy processes.
https://en.wikipedia.org/wiki/Natural_exponential_family
These five examples – Poisson, binomial, negative binomial, normal, and gamma – are a special subset of NEF, called NEF with quadratic variance function (NEF-QVF) because the variance can be written as a quadratic function of the mean. Given a positive semidefinite inner product on the vector space of all polynomials, we have a notion of orthogonality. Then the orthogonal polynomials can be obtained from the monomials {$1,x,x^2,x^3\dots$} by the Gram-Schmidt process. Tom Copeland: Btw, the refinement of the Stirling polynomials of the second kind is OEIS A036040, which is the coproduct of the Faa di Bruno Hopf algebra (see the Figueroa et al. paper in the entry and Zeidler in his book QFT I, p. 859 onward ). In some contexts, a more convenient/enlightening basis for composition with the famous associahedra polynomials for the antipode / compositional inverse is the set of refined Lah partition polynomials of A130561. Both of these, of course, are related to Scherk-Graves-Lie infinigins that are umbralizations of infinigins for the coarser polynomials. Rota et al paper on Finite Operator Calculus notes connection between umbral calculus (Sheffer polynomials) and the Hopf algebra (for polynomials). 读物
Physical Interpretation Think of combinatorial quantum physics operator (raising and lowering operator) as a derived functor.
Nima Arkani-Hamed: Fluctuations (at short distances) based on a magnitude of discrepancy of 1 in {$10^120$} (at long distances). But this could be given by the addition of one discrete node to the existing {$10^120$} nodes, depending on whether or not it is included, whether the center is expressed as a node or not. Are the particle clocks - the carving up of space for an observer - related to the existence of mass and interaction with the Higgs boson? Trying to verify that a space is empty yields pairs of particles and anti-particles. The closer you look, the more chance that you will find exotic things. How does this relate to carving up space into what does not happen? Expansion of space is related to the growth of discrete space, as with the Sheffer polynomials. In the generating functions for the various Sheffer polynomials, why is it that the Hermite and Meixner-Pollaczek polynomials have f=0 but the other ones have {$f\neq 0$}? Does this have a physical meeting regarding the collapse of the wave function, the lack of some distance between the clocks? Also, Hermite and Meixner-Pollaczek both allow for sequences of odd and even polynomials in terms of {$x^{\frac{1}{2}}$}? And both have moments in terms of combinatorial objects that are pairs (like involutions, secant numbers).
Why is least squares the best fit?
Linear regression is focused on the effects "y". The generalized linear model (and the fivesome) relates that to the causes "x_1", "x_2", etc. There can then be a cause of an effect and likewise and effect of a cause and also a critical point. The critical point is where we have symmetry so that we can flip around x and y, cause and effect.
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