文章 发现 ms@ms.lt +370 607 27 665 My work is in the Public Domain for all to share freely. 读物 书 影片 维基百科 Software Upload 数学笔记 What are the transition matrices between the natural bases of the orthogonal polynomials? What are the combinatorial interpretations for their entries? They are indexed by integers rather than partitions. What would be the natural bases for the Yoneda lemma and adjunctions? What space are they describing? What are they indexed by? How is adjunction related to the inversion of a functor? How is adjunction related to the inversion of a perspective? How are the natural orthogonal polynomials related to the natural bases of the symmetric functions? Hermite - pairs - monomial? Charlier - cylces - power? Laguerre - strict ordering - elementary Meixner-Pollaczek - alternatin permutation - Schur Meixner - weak ordering - homogeneous Leaving the forgotten as an extra basis. The 5 natural bases for {$L^2$} or the 6 natural bases for the symmetric functions are perhaps related to M-theory. What are natural bases for {$L^2$} ? The 5 Sheffer polynomials. What are natural bases for symmetric functions? There are 6. Are there four bases for the Yoneda lemma? What are the invariant subspaces of the shift operator? How does Viennot's "coherent transport" work as a functor? The Sheffer polynomials are perhaps characterized by the operator {$x$} and its action on {$P_n(x)$} whereas the classical orthogonal polynomials are perhaps characterized by the operator {$\frac{\textrm{d}}{\textrm{dx}}$} as with the Rodrigues formula. Interpret combinatorially the formula for an orthogonal polynomial as a determinant of moments for the given measure. Wolfram's cellular automata can be thought of as acting on the hierarchy of levels, as with raising and lowering operators. Is the Raudys classification some how related to the Pearson classification? In orthogonal polynomials, understand {$x P_n(x)$} as the basis for the recurrence relation. It is the basis for the definition of {$P_n(x)$} in terms of {$x$}. {$x$} stands for the gap, encodes the gap between levels above and below. There is a hierarchy of gaps. {$x$} is the raising operator which defines {$x p_n(x)$} in terms of {$p_{n+1}$}, {$p_n$}, {$p_{n-1}$}. {$\frac{\textrm{d}}{\textrm{dx}}$} is the lowering operator which defines {$y'=\frac{\textrm{d}y}{\textrm{dx}}$} in terms of {$y''$}, {$y'$}, {$y$}. We can think of {$x$} as moving from {$xP_n(x)$} to {$P_{n-1}$}. Which is why it does not vanish, why it does not go to zero by linearization. The middle term is defined with regard to the terms above and below. This is like Wildberger's mutation game defined for Dynkin diagrams that are a chain. The chain prganizes dimensions, higher and lower. We can consider triality as given by Dynkin diagrams with a threefold branching, the D-series and the E-series. Here we may have three different lowering operators but one raising operator. At the root we may have three lowering operators and three raising operators, thus two trinities, a sixfold structure. Sheffer classification gives 5 possibilities for synchronization. Consider {$\alpha$} and {$beta$} as vectors in the plane. Then we can have: Two points at the origin. A point at the origin and a vector. The same vector twice. Two nonzero vectors of different length on a line. Complex conjugates in the plane. Krautchouk polynomials are a family that consists of only finitely many polynomials. The higher polynomials are zero. This introduces the notion of finiteness. It may be a sixth possibility that expresses the finite human situation, morality, the sixsome. https://www.youtube.com/watch?v=zPtnEZeuzH0 Wildberger. Consider how the mutation games express how and why the four classical Lie groups get distinguished. How is this expressed in the Lie groups and in geometry? How does this relate to the propagation of signal that I had observed? And what is going on with the exceptional Lie algebras? https://www.youtube.com/watch?v=Qw5jonrLbPU&t=2868s Representation theory of Lie algebras and groups echarts apache - graph routines Adjoint functors may relate to chains of perspectives of human's view and God's view. The identity triangles for units and counits are in terms of FGF. This reminds me of my thoughts on bisecting a view. FGF also shows the ambiguity of F, that it can be thought of as acting on the outside, F(GF->I), or on the inside (FG->I)_F Pair polytope types to get six ways of structuring mathematics as in the house of knowledge. In what sense are the Krawtchouk polynomials possibly a sixth kind of orthgonal polynomial? They are a special case of the Mexiner polynomials of the first kind. They are only a finite sequence. Fourier transform transforms the shift operator into the differentiation operator and vice versa. So how does it relate shift invariance (relativity, as with Sheffer polynomials) and invariance with regard to differentiation (as with weight functions like {$e^x$})? The collapse of the wave function resets the clock of time for that system. And it diffuses by the phase factor from there. Self-adjoint operator means diagonalizable. What does that mean for adjunctions? Lie groups and algebras express (continuous) symmetry that I am discovering at various meta levels. In the classification of Sheffer polynomials, we have the choices: {$\alpha = 0$}, {$\alpha \neq 0$} {$\beta = 0$}, {$\beta = \alpha$}, {$\beta \neq \alpha$}, {$\beta = \overline{\alpha}$} So there are eight possibilities, but some of them are related. Wick's theorem? organizes creation and annihilation operators in very much the same way as the Hermite polynomials organize Zeng saplings. This suggests that momentum (the movement of a particle) can be thought of as the annihilation of a particle in one location and the creation of a particle in a different location. I am learning about Feynman diagrams? to see how they might relate to Zeng saplings. How are the combinatorics of the Hermite (involution - one dimension), Laguerre (rook polynomials - two dimension) and Jacobi polynomials related? Are the Jacobi polynomials related to three dimensions? X.Viennot. The cellular ansatz: bijective combinatorics and quadratic algebra Robinson-Schensted-Knuth correspondence, trees and tableau, PASEP, quadratic algebra. Razumov-Stroganov conjecture Combinatorial proof by Cantini, Sportielo Alternating Sign Matrices Motivated by study of {$\lambda$}-determinants. Schur function The Hankel determinant representations for the partition function and boundary correlation functions of the six-vertex model with domain wall boundary conditions are investigated by the methods of orthogonal polynomial theory. For specific values of the parameters of the model, corresponding to 1-, 2- and 3-enumerations of alternating sign matrices (ASMs), these polynomials specialize to classical ones (continuous Hahn, Meixner–Pollaczek and continuous dual Hahn, respectively). As a consequence, a unified and simplified treatment of ASM enumerations turns out to be possible, leading also to some new results such as the refined 3-enumerations of ASMs. Furthermore, the use of orthogonal polynomials allows us to express, for generic values of the parameters of the model, the partition function of the (partially) inhomogeneous model in terms of the one-point boundary correlation functions of the homogeneous one. Vertex model Six-vertex model Combinatorics and physics Christian Brouder. Quantum field theory meets Hopf algebra. Relates Feynman diagrams and semistandard Young tableaux. Gérard Henry Edmond Duchamp, H. Cheballah. Some Open Problems in Combinatorial Physics. Multiplicities in diag. Combinatorics of Riordan-Sheffer one-parameter groups. A corpus for combinatorial vector fields. Probabilistic study of approximate substitutions. Combinatorics, Physics and their Interactions Journal. Jacobi polynomials (such as the physicist Hermite polynomials) include a factor of 2x. These seem related to probability. But also for the geometry, as with the Legendre polynomials. Can we use the same combinatorics as Kim Zeng, but just alter by substitution for y=2x? and multiplying each polynomial by a fraction? Can we compare with Viennot's interpretation? Do they relate physics with probability? Spin gives the number of choices. Boson - odd number (includes zero), fermion even number (does not include zero). How does that relate to the kinds of classical Lie algebras? {$F_1$} photon chooses from one choice {$F_2$} electron chooses from two choices {$F_3$} spin one boson chooses from three choices {$F_5$} spin two graviton chooses from five choices (fivesome for space and time?) Do these relate to the divisions of everything? What happens with the eightsome? and ninesome? Attiyah: Limits (singularities, zeroes) exist in the ideal world but not the real world. Thus the house of knowledge for mathematics constructs the ideal world from the real world. Relativistic invariance of the Dirac operator. Delayed signals? (Kinks?) Attiyah: Retarded Dirac operator. Attiyah, Moore. A Shifted View of Fundamental Physics. Connes noncommutative geometry is based on cyclic permutations where ABC = CAB, and this yields cyclic homology for associative algebras. Connes noncommutative geometry is based on every point being actually a pair of points (on two sheets), on which 2x2 matrices operate, and yields the Standard Model. Are these pairs of points expressing the dual causality of the fivesome, links and kinks? How is Kim-Zeng linearization (generalized derangements) related to the representation of the symmetric group and general linear group? John: Unitary dynamics - self-adjoint operator {$H*=H$}. Unitary operator. Attiyah: Number theory (Connes - noncommutative algebra) is linked by Geometry (Penrose - twistors) to Physics (Witten - strings). Atiyah: K-theory is related to quantum theory, cohomology is related to classical theory. Dirac operator is the basis for spinors and for Bott periodicity. Dirac operator is, conceptually, the square root of the Laplacian. In what sense is a tableaux the square root of a matrix? Atiyah: What does octonionification mean for the magic square? {$\forall D\exists f\forall g(f=g)$} arithmetic hierarchy: derivative {$\forall \epsilon \exists \delta \forall x$} arithmetic hierarchy: universality: for all objects there exists a unique morphism Robert Harper - meta theory, metalogic Residuated Boolean algebra, residuated semilattice graphicallinearalgebra.net idris2 Quantum Field Theory Physics What is the connection between imaginary time (the dynamical evolution of the quantum world {$e^{-iHt/ħ}$}) and inverse temperature (the Boltzmann factor {$e^{-\beta H}$}). Partition function {$\textrm{tr}e^{-\beta H}$}. How to understand negative energy and negative probability? Think of negative probability as removing a possibility, by having it not happen, by having it happen in our mirror universe (presumed to exist by CPT symmetry). Given wave function {$x^ke^{-x^2}$}, the chance of being at x=0 is 0 unless it is the ground state. Orthogonal polynomials https://en.m.wikipedia.org/wiki/Sheffer_sequence is what i am interested in. Sturmian functions - harmonic oscillator Sturmian words - billiard ball bouncing between horizontal and vertical sides of table Sturmian functions https://arxiv.org/pdf/quant-ph/9809062.pdf This paper relates Sturmian words, Sturmian functions and Sturmian numbers. Relativistic quantum harmonic oscillator A.Poszwa. Relativistic Generalizations of the Quantum Harmonic Oscillator Spherical harmonics R. P. Martı́nez-y-Romero. Relativistic quantum mechanics of a Dirac oscillator 1999. Free article. Refers to spinor spherical harmonics. The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. While the standard spherical harmonics are a basis for the angular momentum operator, the spinor spherical harmonics are a basis for the total angular momentum operator (angular momentum plus spin). These functions are used in analytical solutions to Dirac equation in a radial potential. See: Greiner, Walter (6 December 2012). "9.3 Separation of the Variables for the Dirac Equation with Central Potential (minimally coupled)". Relativistic Quantum Mechanics: Wave Equations. Springer. ISBN 978-3-642-88082-7. Relativistic hydrogen atom R. P. Martı́nez-y-Romero. Relativistic hydrogen atom revisited American Journal of Physics 68, 1050 (2000). The hydrogen atom is solved using a simple method. We show that this system has an exact solution that can be written in terms of Laguerre polynomials of noninteger index, instead of the hypergeometric series. This point is important because Laguerre polynomials of integer index appear in the solution of the nonrelativistic hydrogen atom, giving students a more unified point of view for this system. A. D. Alhaidari, H. Bahlouli, M. E. H. Ismail. The Dirac-Coulomb Problem: a mathematical revisit Alhaidari researches orthogonal polynomials. Describes the relativistic hydrogen atom in terms of Pollaczek polynomials. Guillermo Palma, Ulrich Raff. The One Dimensional Hydrogen Atom Revisited Spacetime is made of events (the coordinates). But in quantum mechanics, an event is only when a wave function collapses. So time and space combine only when the wave function collapses. So relativity is only important for the actual collapse of the wave function. The wrapper {$e^{-x^2}$} may explain the quadratic nature of gravity. But how? Orthogonality measures self-duality. Two pyramids give the perspectives of position and momentum. How does adjunction relate to automata? https://varkor.github.io/blog/2020/11/25/announcing-quiver.html Quiver - for drawing commutative diagrams Penrose's three-cycle - being, doing, thinking. What would a three-cycle of functors look like, modeling Penrose's three worlds? Based on the case of the harmonic oscillator, setting units like h to 1, energy is simply the frequency. In general, is energy the frequency of conversion of potential energy to kinetic energy and back again, as with the harmonic oscillator? Enriched preorder category. Preorder is a boolean enriched category. Example of self-enriched category is Bool where we have the category with two objects and one morphism between them, and we map that morphism to 0 (false) or 1 (true) with 0<=1. HomSet is an enrichment of a category with a preorder of nonempty homsets. Diachronic category is preorder, synchronic category (linear time) is monoid defined on one element (all time, all actions at once). An object is a point in time. In the diachronic point of view, we move by morphisms from object to object. In the synchronic point of view we keep returning to the same point in time, so all morphisms act in parallel. Extensional vs. intensional view of functions. On Lambda Calculus. Enriched category is the "graded category" that I was seeking for in statistics, etc. Chess educator of the Year, Jerald Times https://www.youtube.com/watch?v=isAfKbPDeYc Wick's theorem is used extensively in quantum field theory to reduce arbitrary products of creation and annihilation operators to sums of products of pairs of these operators. A more general idea in probability theory is Isserlis' theorem. Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. Wick's contraction interprets the double factorial in terms of pairs of connections. In measuring an average quality, such as average position: You can't average the values you get upon repeatedly measuring one particle because it has taken on a particular state with the first measurement. You can't average the values you get by making measurements on an ensemble of particles having the same state because, in general, you don't know what that state is, and even if you did put the particles in such a state, you cannot absolutely maintain them in that state, but only up to a degree of certainty. Thus you can average the values that you get by making meaurements on an ensemble of particles that are in rather similar states. The axioms {$p \equiv -iħ\frac{\delta}{\delta x}$} and {$E \equiv iħ\frac{\delta}{\delta t}$} and then the energy equation {$E=K+V=\frac{p^2}{2m} + V(x)$} are theoretically more fundamental than Shroedinger's equation, which is empirical, and which is derived by plugging in. Thus Schroedinger's equation simply describes the circumstances for measurement by way of the wave function. The factor of 2 for momentum and the factor i for position are both half-way factors, square root factors. i is the square root of negative 1. The factor 2 arises because of differentiating {$p^2$}, perhaps twice? There are four dimensions of space-time and accordingly there are four dimensions of the four momentum where the energy {$\frac{E}{c}$} corresponds to time. The principle of least action says that given the two forms of energy, kinetic and potential, {$E=K+V$}, we shift from one to the other as little as possible, thus minimizing K-V. In this sense we can think of time as the dimension in which this conversion occurs, and thus time is like an expressly local dimension, of limited extent, as little as possible. Time is the distance traveled up the potential V. Does that mean that change takes place as slowly and gradually as possible? And can that limitation express the speed of light as a consequence of the principle of least action? And also the shape of the mother function. How do we rethink Schroedinger's equation for space time so that energy and momentum are both related in terms of derivatives of the same order? How do we rethink energy as {$-E=iħ\frac{\delta}{\delta t}$}, so that it is compatible with p? So we can write this as {$0=(-E)+K+V$} or {$(-V)=(-E)+K$}. How doe we rethink the operator {$V=-iħ\sum_{j}\frac{\delta}{\delta x_j}$} where multiplying by the potential is the same as acting by the partial derivatives. How can the second law of thermodynamics be described in terms of time which is defined as the distance traveled up a potential? The combinatorial configurations into cells and relationships are presumably the same regardless of which is assigned to position and which to momentum... and other, possible hybrid, physical qualities? Or does the Fourier transform introduce some subtle difference? So what is the meaning of these deeper configurations? And what does it mean when they signify the result of a measurement? 维基百科: Adjoint Notions of adjoint. Symmetry is the laws for the laws (of physics). Physicists know how to copy equations, how to manipulate them, how to solve them, how to set them up, but not how to read them mathematically, conceptually. Good physicists can read the physics and yet not the math. Josef Meixner. Orthogonale polynomsysteme mit besonderen Gestalt der erzeugenden Funktion. Journal of the London Mathematical Society. January, 1934. Consider the weights of the various families of orthogonal polynomials and their properties. For example, 1/2cosh(pi x/2) is, up to a dilation, its own Fourier transform. Are the other weights similarly related to geometric transformations? Dunce, dunce hat, Duns Scotus {${}_0F_1$} confluent hypergeometric limit function {${}_1F_1$} Kummer’s confluent hypergeometric function of the first kind {${}_2F_1$} Gauss’ hypergeometric function Art ideas: Mahjong tiles and game. Dominoes. Art ideas: The Tailors. Einstein with person's watch: "Pulse is steady." Measurement is a trivial translation of coordinate system from x to x-a and thus relates to the trivial self-entanglement of a tree. Roger Penrose, Jurekfest, 2019 4:20 "I think probably twister theory has now as little connection with spin networks as loop quantum gravity does with the original aims of spin network theory which was to produce a combinatorial approach to physics where you just ... calculate everything from integers. We've both drifted away from that. I don't know whether we're coming to it back to it or not. I sometimes hope so." What is the meaning and significance of the conic sections "dot" and "X" (two lines crossed)? For the conic sections, given the two cones, can we identify one with the inside and the other with the outside? Renato ́Alvarez-Nodarse. On some applications of Orthogonal Polynomials in Mathematical-Physics Alberto Grunbaum one time said (in a summer school in 2004):“Special functions are to mathematics what pipes are to a house: nobodywants to exhibit them openly but nothing works without them”. Danilo Latini, D.Riglioni. From ordinary to discrete quantum mechanics: The Charlier oscillator and its coalgebra symmetry The coalgebraic structure of the harmonic oscillator is used to underline possible connections between continuous and discrete superintegrable models which can be described in terms of SUSY discrete quantum mechanics. A set of 1-parameter algebraic transformations is introduced in order to generate a discrete representation for the coalgebraic harmonic oscillator. This set of transformations is shown to play a role in the generalization of classical orthogonal polynomials to the realm of discrete orthogonal polynomials in the Askey scheme. As an explicit example the connection between Hermite and Charlier oscillators, that share the same coalgebraic structure, is presented and a two-dimensional maximally superintegrable version of the Charlier oscillator is constructed. Richard Askey explained why hypergeometric functions appear so frequently in mathematical applications: "Riemann showed that the requirement that a differential equation have regular singular points at three given points and every other complex point is a regular point is so strong a restriction that the differential equation is the hypergeometric equation with the three singularities moved to the three given points. Differential equations with four or more singular points only infrequently have a solution which can be given explicitly as a series whose coefficients are known, or have an explicit integral representation. This partly explains why the classical hypergeometric function arises in many settings that seem to have nothing to do with each other. The differential equation they satisfy is the most general one of its kind that has solutions with many nice properties. Special functions: group theoretical aspects and applications. How do regular singular points and Riemann's differential equation relate to the eccentricities of the conics (0,1,infinity)? and to the Mobius transformations? How do Bayesian statistics and the usual statistics differ as regards the fivesome and causalities? Alain Connes about Bott periodicity and CTP. "Why Four Dimensions and the Standard Model Coupled to Gravity... Video 39:00 Ali H. Chamseddine, Alain Connes. Why the Standard Model Connes. Why Four Dimensions and the Standard ModelCoupled to Gravity - A Tentative Explanation Fromthe New Geometric Paradigm of NCG Alain Connes. Noncommutative Geometry and the Standard Model with Neutrino Mixing.] 6+4=2 modulo 8 When you move from K-theory to KO-theory, you add charge conjugation operator J. Kinks There are 5 conic sections plus 3 more: the point C (where the cones meet), two lines that cross in an X, and the line segment (which arises in the limiting case as the hyperbola passes over to an ellipse). Given any other point P on the lower cone, P not C, then we get conic figures by rotating the plane that contains C along the axis that is tangent to the cone. We can start with a circle, then it becomes an ellipse ever longer, until it reaches infinity and becomes a parabola, which then intersects the upper cone as a hyperbola, which then becomes a line as the hyperbola gets squeezed and infinitely narrow, but then the line drops down to a line segment from P to C, from where it opens up as an ellipse, which grows in width until it becomes the circle. So the line segment is a moment of ambiguity where it equals the line and yet the limit cases from either side are different. The line segment seems like a string or loop in string theory. Nurseries, Zeng saplings What would a kink look like for a tableaux? An observer carves up spacetime into kinks. Kinks express the need that a question be answered, that there be resolution, that tension be resolved, which is important in all conceptual languages. Kinks express the yearning for justice, that everything be set right, that there be no loose ends. Kinks express the obligations met by a pushdown automata, as with balanced parentheses. Random matrices https://arxiv.org/abs/1712.07903 https://link.springer.com/book/10.1007/978-94-009-0501-6 Orthogonal Polynomials, Theory and Practice, ed. Paul Nevai https://www.sciencedirect.com/science/article/pii/S0001870816310453?via%3Dihub Orthogonal polynomials through the invariant theory of binary forms, P.Petrullo, D.Senato, R.Simone A kink is a contraction (a line) as defined for Wick's theorem. What is a normal ordering? An abnormal ordering? A link? Reply to this question: Thomas asked, Why are there 3 + 1 dimensions of time and space? What could my philosophy say? The relevant structure would be the eightfold way: 1 + 3 + 3 + 1. It suggests that (external) space and (internal) time are two different ways of relating a relative framework with an absolute framework. For space, the absolute framework (God) is beyond the relative framework. There are three independent dimensions (being, doing, thinking) which relate the relative to the absolute. So the relative precedes the absolute. For time, the absolute framework (good) is within the relative framework (the three-cycle) and prior to it. Time is the dimension of slack within the three-cycle which goes on infinitely and for which forwards and backwards are qualitatively different directions. https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula used to establish the existence of Sheffer's polynomials. WorldCat Heyting algebra - notion of adjunction? meet vs. join? product vs. homset http://builds.openlogicproject.org/ WhatIf book https://en.wikipedia.org/wiki/Del_Pezzo_surface see especially from degree 2 to 9, is this an eightfold phenomenon? How is it related to https://en.wikipedia.org/wiki/M-theory ? How is it related to https://en.wikipedia.org/wiki/Veronese_surface and the conics? More formally, in the context of QFT, the S-matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the in-states and the out-states) in the Hilbert space of physical states. A multi-particle state is said to be free (non-interacting) if it transforms under Lorentz transformations as a tensor product, or direct product in physics parlance, of one-particle states as prescribed by equation (1) below. Asymptotically free then means that the state has this appearance in either the distant past or the distant future. (Distant time!) Petri Muinonen I'm reading your doctoral thesis as I have a physics model under development where particles are described as combinatoric aggregates of logarithmic poles (singularities in 4D). They form negative simplex vertices around center of positive ones (convention). See my Youtube channel petrimuinonen Neutrino oscillations under Logarithmic pole model v1.0 - a candidate for a Theory of Everything Petri: From your diary "This suggested to me that the collapse must occur when a coordinate system is introduced and thereby separates the observer from the observed. ... at a certain point ... which collapses the wave function.... Thus this would be the place to look for a relation between general relativity and quantum mechanics." This makes perfectly sense to me. My understanding is that we are talking about division algebra, quaternions, and a negative 4-distance expressed in quaternion terms (squared separation of 4D locations, outside light cone) collapses into a vectorial quaternion (corresponding to a purely spatial separation) There is infinite number of roots to that, all space-like. Will have to explain this better later! You might also want to check Julian Barbour's concept of shape dynamics Matematika iškyla vairuojant, valdant automobilį. Sukame vairą - ratą. Skaičiuojame bėgius. Važiuojame pirmyn ir atgal. Įsivaizduokime, kad valdome lėktuvą arba netgi erdvėlaivį sugebantį skristi į praeitį ir ateitį ar netgi visai kitokius matus, pavyzdžiui, nuotaikų pasaulius. Suktumėme ne ratą bet gaublį arba keturmatį ar netgi aštuonmatį gaublį. Kad važiuotumėme ne tik pirmyn ir atgal, bet aukštyn ir žemyn, į praeitį ir ateitį. Wenbo How can structure arise that maintains itself? Foucault "The Order of Things". Oulipo. Kūryba pagal savo taisykles. Schelling požiūris panašus į mano, anot Wenbo. Yu-Gi-Oh! Trading card game. Embodied agency (video game). Valios įgyvendinimas. Distinction between extrinsic symmetry (relative to an outside observer) and intrinsic symmetry (relative to itself). Askey-Wilson polynomials subsume the various orthogonal polynomials. They are defined by five parameters. They have a combinatorial interpretation in terms of PASEP and their moments are given by staircase tableaux. Arrows, structures, and functors_ the categorical imperative-Academic Press (1975) Logical Foundation of Cognition - Lawvere - 4. Tools for the Advancement of the Logic of Closed Categories Programmig in Martin-Loef's Type Theory - Bengt Noerdstrom et al
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