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Andrius Kulikauskas

  • m a t h 4 w i s d o m - g m a i l
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  • My work is in the Public Domain for all to share freely.

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  • 读物 书 影片 维基百科

Introduction E9F5FC

Questions FFFFC0

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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

  • Consider the parallelism of Sheffer polynomials A(t)e^{xu(t)} and Hermitian matrix H e^{iH} where H is analogous to generating function and i is analogous to x.
  • Relate space builder interpretation of Sheffer polynomials with their expansion in terms of {$(q_{k0} + xq_{k1})$}.

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.

  • Give examples of generating functions of orthogonal polynomials that are not Sheffer sequences, such as the Chebyshev polynomials and the Jacobi polynomials.
  • Chebyshev polynomials of the first kind {$\sum_{n=0}^{\infty}P_n(x)\frac{t^n}{n!} = e^{tx}\textrm{cosh}(t\sqrt{x^2-1})$} we could write {$\frac{1}{2}e^{tu}+\frac{1}{2}e^{tv}$} where {$u=x+y$} and {$v=x-y$} and {$y=\sqrt{x^2–1}$} and {$x=\sqrt{y^2+1}$}.
  • Wick's theorem, quantum field theory and Feynman diagrams.

Physics

  • Why is the information encoded in the coefficients as opposed to the roots of the polynomial?
  • Note that each power of x involves a crossing of the curve.
  • The notion of space-time wrapper in the context - contributed by orthogonality.
  • The idea of a two frame physics.

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 calculus

Math Overflow: What's Up With Wick's Theorem?

Suggested by Tom Copeland

  • "Boson Normal Ordering via Substitutions and Sheffer-type Polynomials" by Blasiak, Horzela, Penson, Duchamp, and Solomon;
  • "Normal ordering problem and the extensions of the Striling grammar" by Ma, Mansour, and Schork;
  • "Combinatorial Models of Creation-Annihilation" by Blasiak and Flajolet;
  • the book Commutation Relations, Normal Ordering, and Stirling Numbers by Mansour and Schork with an extensive bibliography.

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.
MIT. Philippe Rigollet. Statistics for Applications.

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?

  • NEF-QVF have conjugate prior distributions on μ in the Pearson system of distributions (also called the Pearson distribution although the Pearson system of distributions is actually a family of distributions rather than a single distribution.) Examples of conjugate prior distributions of NEF-QVF distributions are the normal, gamma, reciprocal gamma, beta, F-, and t- distributions. Again, these conjugate priors are not all NEF-QVF.

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/Martingale_(probability_theory) A martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values.
  • https://en.wikipedia.org/wiki/Lévy_process A stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Lévy process may thus be viewed as the continuous-time analog of a random walk. The most well known examples of Lévy processes are the Wiener process, often called the Brownian motion process, and the Poisson process. Further important examples include the Gamma process, the Pascal process, and the Meixner process. Aside from Brownian motion with drift, all other proper (that is, not deterministic) Lévy processes have discontinuous paths.

https://en.wikipedia.org/wiki/Natural_exponential_family

  • normal distribution with known variance
  • Poisson distribution
  • gamma distribution with known shape parameter α (or k depending on notation set used)
  • binomial distribution with known number of trials, n
  • negative binomial distribution with known r

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).

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Physical Interpretation

Think of combinatorial quantum physics operator (raising and lowering operator) as a derived functor.

  • Ibraheem F. Al-Yousefm Moayad Ekhwan, Hocine Bahloul, Hocine Bahlouli, Abdulaziz Alhaidari. Quantum Mechanics Based on Energy Polynomials. We use a recently proposed formulation of quantum mechanics based, not on potential functions but rather, on orthogonal energy polynomials. In this context, the most important building block of a quantum mechanical system, which is the wavefunction at a given energy, is expressed as pointwise convergent series of square integrable functions in configuration space. The expansion coefficients of the series are orthogonal polynomials in the energy; they contain all physical information about the system. No reference is made at all to the usual potential function. We consider, in this new formulation, few representative problems at the level of undergraduate students who took at least two courses in quantum mechanics and are familiar with the basics of orthogonal polynomials. The objective is to demonstrate the viability of this formulation of quantum mechanics and its power in generating rich energy spectra illustrating the physical significance of these energy polynomials in the description of a quantum system. To assist students, partial solutions are given in an appendix as tables and figures.

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).

  • Ergodic theory relates the representations of the fivesome in terms of time and space.
  • Investigation of how a conceptual framework for decision making in space and time, which I call the fivesome, can help us interpret the mathematics of quantum physics. I am studying the information
  • Fivefold classification of orthogonal polynomials that are important for
  • Space has 3 dimensions external to the fivesome (5+3=0)(outside the division) and time has 1 dimension internal to the fivesome (the slack inside the division).
  • Quanta magazine. How We Can Make Sense of Chaos. The present is a homoclinic point. It orbit approaches a fixed point in the future and in the past. Then you have a horseshoe and chaos. But what is the fixed point?
  • Is the associativity diagram for monoidal categories an example of the fivesome?

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.

  • Energy is good will. Always show good will but as little as possible.
  • Relate particle clocks to the symmetric functions of eigenvalues such as the forgotten symmetric functions.

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This page was last changed on August 25, 2023, at 06:08 AM