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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

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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.

History

Roelof Koekoek. Inversion formulas involving orthogonal polynomials and some of their applications

Richard P. Stanley. A Survey of Alternating Permutations

Wilf. Generatingfunctionology.

Math Stack Exchange. Expected value of falling factorials from axioms of Poisson process.

The ordered Bell numbers for {$n=3$} can be organized like the root system {$G_2$} as pictured here.

Patrick Njionou Sadjang. Moments of Classical Orthogonal Polynomials.

How do the growth structures (causal trees) of orthogonal polynomials relate to group representations (such as pairs of Young tableaux - pairs of causal trees)? How are pairs of permutations (which are pairs of involutions) related to pairs of involutions (as with the Hermite polynomials)? Do the pairs of Hermite polynomials relate to the Robinson-Schensted algorithm?

The second type of matching polynomial has remarkable connections with orthogonal polynomials. For instance, if G = Km,n, the complete bipartite graph, then the second type of matching polynomial is related to the generalized Laguerre polynomial Lnα(x) by the identity:

  • {\displaystyle M_{K_{m,n}}(x)=n!L_{n}^(m-n)(x^{2}).\,}{\displaystyle M_{K_{m,n}}(x)=n!L_{n}^(m-n)(x^{2}).\,}

If G is the complete graph Kn, then MG(x) is an Hermite polynomial:

  • {\displaystyle M_{K_{n}}(x)=H_{n}(x),\,}{\displaystyle M_{K_{n}}(x)=H_{n}(x),\,}

where Hn(x) is the "probabilist's Hermite polynomial" (1) in the definition of Hermite polynomials. These facts were observed by Godsil (1981).

Lectures on Orthogonal Polynomials and Special Functions Ne per brangi knyga.

Rotation accords with orientation (of a simplex) accords with an imaginary number i or j. Orientation is related to permutation as with the linearization for orthogonal polynomials.

Robert Gilmore. Group Theory. XIV. Group Theory and Special Functions. Relates Lie groups and orthogonal polynomials.

Orthogonal Sheffer polynomials

  • Sheffer: Two particles have their own independent clocks but when they interact it synchronizes their clocks. Then when the synchronization collapses, one indicates the forward time and the other the backwards time so they are conjugates.
  • How does the combinatorics of orthogonality manifest itself? Orthogonality converts compartments into causal trees in the case of orthogonal Sheffer polynomials. Can we make combinatorial sense of the impositions of this orthogonality constraint? As with Favard's theorem or Meixner's classification?
  • Consider the generating function for (associated) Legendre polynomials and compare it with the generating function for orthogonal Sheffer polynomials.
  • Consider how forgetful functors take us through the five zones for the moments of orthogonal Sheffer polynomials. We start with ordered set partitions. We forget the order and get set partitions. They have an intrinsic order, starting with the largest (and on the inside starting with the smallest). So you should get a permutations. Every permutation is a pair of involutions. If you forget the one involution, then you should still have another involution. And what would you forget to get the alternating permutations?

Interpret the combinatorics of Associated Legendre polynomials, and substituting {$\cos\;\theta$} and {$\sin\;\theta$}, the related spherical harmonics. Consider how they integrate with the Laguerre polynomials. Understand how they describe the possible states of the hydrogen atom and the periodic table of elements.

Electron shells are given by twice the square {$2n^2$} which yields {$2,8,18,32,50...$} And the square number is understood as the sum of odd numbers: {$1+3+5+7+...$}. The {$2$} is the spin of the electron which we can think of as two sides of the square (as a sheet of paper). The shells (their odd number portions) are filled in a zig-zag pattern. This is based on the radial model of the hydrogen atom.

  • Relate these numbers to the Laguerre polynomials and express them as causal trees.

Meixner-Pollaczek polynomials have complex conjugate weights {$\alpha$} and {$\beta=\bar\alpha$} but actually they appear as link weight {$-\alpha -\bar\alpha$} which is real, and kink weight {$-\alpha\bar\alpha$} which is real. Note that the kink is the full portion (the amplitude) of {$\alpha$} whereas the link is twice the real portion. Thus the kink is a measure of entanglement, of the imaginary portion.

Sheffer polynomial coefficients can be expressed in terms of elementary symmetric functions. How does that relate to the particle clocks? And {$\alpha$} and {$\beta$}? And do those relate to the forgotten symmetric functions with the two different kinds of labels?

How do the colorings and derangements in the linearization coefficients of orthogonal Sheffer polynomials relate to John Baez's interest in colorings and derangements?

  • How does {$e^x$} for groupoids {$X$} relate to the connection between Lie algebras and Lie groups?

For Sheffer polynomials as such, in the implicate order, prior to orthogonality, there are no notions of moments or distribution or weight function.

Legendre polynomials appear when solving the Schrödinger equation in three dimensions for a central force.

Wave function arises when two systems interact. As given by orthogonal Sheffer polynomials.

Jacobi polynomials have a notion of combinatorial space that may be relevant for thermodynamics as it relateswo disjoint sets malping into their union.

Path integrals depend on the number of points in space, or the number of interactions. But my approach suggests that this number is actually given by the degree of x in the relevant polynomial.

Orthogonal Sheffer polynomial recurrence relation has 5 inputs. 3 are first-step, external, weighted, meaningful in the broader environment. 2 are later step, dependent on {$n$} or {$n(n-1)$}, meaningful internally.

The pair of causal trees - the squaring of the wave function - may be expressing sexual combination.

Integrating the square of the wave function with a definite integral - summing over the step function of the orthogonality equation for the Meixner polynomials - is simply a way of grouping together combinatorial objects, and if you like, giving a linear weight to them (if you are dealing with nonintegers). But you can group them in other ways as well - so this interpreation is superior to that of the wave function. It lets you think of measuring (grouping) units of information in different ways.

Combinatorics of the residue theorem can provide a combinatorial interpretation of the distributions arising from the moments of the orthogonal Sheffer polynomials. Here is an example of how to consider it combinatorially:

Potential energy source {$\frac{y}{2}$} (adding free cell) is in balance with kinetic energy source {$i\frac{\partial}{\partial y}$} (half-link)

Feinsilver. Lie algebras, Representations, and Analytic Semigroups through Dual Vector Fields. Group theory related to orthogonal Sheffer sequences.

Locality is the whole achievement of the continuum. Local means low overhead and the actual global time frame is even lower overhead. Locality arises with orthogonality, assumes measurement, observers, space time wrapper.

A sum of particle clocks is like a prism operator (in the proof for singular homology that homotopic maps induce the same homomorphism for the homology groups) but without the minus signs.

The wave equation - and waves in general - are expressions of analytic symmetry.

Think of {$\alpha$} and {$\beta$} as the steps in two frames that are centered on two events. If {$\alpha=\beta=0$}, then the two frames coincide and so the kinematics collapses, the edge statistics collapse.

Linear regression

  • In multiple regression the constant {$b_0$} acts like free space, the initial compartment. The random variables are like compartments.

Understand the analytic symmetry in this expression for a weight function: {$\omega(x)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\sum_{n=0}^{\infty}\omega(x)\sum_{k=0}^{\infty}\frac{(-1)^n(2\pi i \xi x)^{n+k}}{n!k!}dx d\xi$}.

  • Note that {$\sum_{k=0}^{n}\frac{1}{k!(n-k)!}=\frac{1}{n!}\sum_{k=0}^n\binom{n}{k}=\frac{2^n}{n!}$} and {$\sum_{k=0}^n(-1)^k\binom{n}{k}=(1-1)^n=0$}.
  • Why doesn't the formula equal {$0$}?
  • What happens if we differentiate the formula with regard to {$x$} or {$\xi$}?
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This page was last changed on November 15, 2024, at 12:53 AM