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A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (16 August 2016) | Viewed by 23964

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Prof. Dr. Charles F. Dunkl

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Guest Editor

Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
Interests: harmonic analysis; representation theory; special functions of several variables; applications to mathematical physics, especially exactly solvable systems of quantum mechanics
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Special Issue Information

Dear Colleagues,

The concept of symmetry has been fundamental and studied for millennia. The ancient geometers already knew the five regular solids. For a long time, symmetry was a part of the discipline of geometry, but in more recent times it has become very important in analysis, mathematical physics, and of course, group theory. Symmetry is a key tool in analyzing functions of several variables. For example, the harmonic homogeneous polynomials, which are invariant under the group of rotations fixing the North Pole on the unit sphere in are essentially the same as Gegenbauer polynomials of index N/2−1. By now, this idea has been vastly generalized, for example, to interpreting Jacobi polynomials of several variables (defined on a simplex and orthogonal with respect to a Dirichlet measure) as harmonic polynomials with certain subgroup invariance properties.

In mathematical physics there are the quantum-mechanical models of Calogero–Moser–Sutherland type: N identical particles with 1/r² interaction and possibly an external potential, of which wavefunctions involve Jack polynomials. The symmetric group occurs naturally in any system of identical particles, where the properties are invariant under the interchange of two particles. Recent developments have extended this to “supermodels” and introduced super polynomials with bosonic (commuting) and fermionic (anticommuting) variables. These are involved in the open question of whether SUSY (supersymmetry) manifests in the real world. Another application of orthogonal polynomials is as wavefunctions of isotropic quantum harmonic oscillators. Lie and quadratic algebras are being used to provide more insight into the Askey tableau, a scheme for organizing the classical polynomials of hypergeometric type.

Symmetry appears in algebraic combinatorics, for example in association schemes and distance-regular graphs. These structures are analyzed with the help of orthogonal polynomials, which arise as eigenfunctions of an associated Laplacian operator.

Symmetry in orthogonal polynomials also appears when the domain is a symmetric shape and the weight function is invariant under a group generated by reflections: for example orthogonal polynomials on a regular hexagon find an application in wave-front analysis for hexagonal mirror segments in large astronomical telescopes. Another example is the analysis of trigonometric polynomials, which are periodic on a lattice (or tesselation of space by regular polytopes).

In this Special Issue we aim to present the newest developments in the interaction of symmetry and orthogonal polynomials, in areas such as quantum physics, combinatorics, and classical analysis problems dealing with convergence of polynomial expansions. In the situations discussed above, much has been discovered, nevertheless, more needs to be done, in more precise formulas, approximation theorems about expansions in orthogonal polynomials of several variables, dependence on the parameters of a weight function, vanishing properties of specific polynomials (such as Jack and Macdonald), and so on.

Prof. Charles F. Dunkl
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

weight functions invariant under reflection groups
Calogero-Moser-Sutherland models
Jack polynomials
orthogonal polynomials in several variables of classical type
polynomials periodic on a lattice

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Published Papers (6 papers)

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277 KiB

Open AccessFeature PaperArticle

Planar Harmonic and Monogenic Polynomials of Type A

by Charles F. Dunkl

Symmetry 2016, 8(10), 108; https://doi.org/10.3390/sym8100108 - 21 Oct 2016

Viewed by 3439

Abstract

Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on

R^{N}

. The Dunkl operators are denoted by

T_{j}

for

1 \leq j \leq N

, and the [...] Read more.

Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on

R^{N}

. The Dunkl operators are denoted by

T_{j}

for

1 \leq j \leq N

, and the Laplacian

Δ_{κ} = \sum_{j = 1}^{N} T_{j}^{2}

. This paper finds the homogeneous harmonic polynomials annihilated by all

T_{j}

for

j > 2

. The structure constants with respect to the Gaussian and sphere inner products are computed. These harmonic polynomials are used to produce monogenic polynomials, those annihilated by a Dirac-type operator. Full article

(This article belongs to the Special Issue Symmetry in Orthogonal Polynomials)

1565 KiB

Open AccessArticle

The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

by Luca Bagnato, Mario Faliva and Maria Grazia Zoia

Symmetry 2016, 8(8), 77; https://doi.org/10.3390/sym8080077 - 5 Aug 2016

Cited by 1 | Viewed by 3770

Abstract

This paper carries out an investigation of the orthogonal-polynomial approach to reshaping symmetric distributions to fit in with data requirements so as to cover the multivariate case. With this objective in mind, reference is made to the class of spherical distributions, given that they provide a natural multivariate generalization of univariate even densities. After showing how to tailor a spherical distribution via orthogonal polynomials to better comply with kurtosis requirements, we provide operational conditions for the positiveness of the resulting multivariate Gram–Charlier-like expansion, together with its kurtosis range. Finally, the approach proposed here is applied to some selected spherical distributions. Full article

(This article belongs to the Special Issue Symmetry in Orthogonal Polynomials)

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1490 KiB

Open AccessArticle

Cubature Formulas of Multivariate Polynomials Arising from Symmetric Orbit Functions

by Jiří Hrivnák, Lenka Motlochová and Jiří Patera

Symmetry 2016, 8(7), 63; https://doi.org/10.3390/sym8070063 - 14 Jul 2016

Cited by 17 | Viewed by 4260

Abstract

The paper develops applications of symmetric orbit functions, known from irreducible representations of simple Lie groups, in numerical analysis. It is shown that these functions have remarkable properties which yield to cubature formulas, approximating a weighted integral of any function by a weighted finite sum of function values, in connection with any simple Lie group. The cubature formulas are specialized for simple Lie groups of rank two. An optimal approximation of any function by multivariate polynomials arising from symmetric orbit functions is discussed. Full article

(This article belongs to the Special Issue Symmetry in Orthogonal Polynomials)

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Figure 1

285 KiB

Open AccessArticle

Three New Classes of Solvable N-Body Problems of Goldfish Type with Many Arbitrary Coupling Constants

by Francesco Calogero

Symmetry 2016, 8(7), 53; https://doi.org/10.3390/sym8070053 - 24 Jun 2016

Cited by 9 | Viewed by 3409

Abstract

Three new classes of N-body problems of goldfish type are identified, with N an arbitrary positive integer (

N \geq 2

). These models are characterized by nonlinear Newtonian (“accelerations equal forces”) equations of motion describing N equal point-particles moving in the [...] Read more.

Three new classes of N-body problems of goldfish type are identified, with N an arbitrary positive integer (

N \geq 2

). These models are characterized by nonlinear Newtonian (“accelerations equal forces”) equations of motion describing N equal point-particles moving in the complex z-plane. These highly nonlinear equations feature many arbitrary coupling constants, yet they can be solved by algebraic operations. Some of these N-body problems are isochronous, their generic solutions being all completely periodic with an overall period T independent of the initial data (but quite a few of these solutions are actually periodic with smaller periods

T / p

with p a positive integer); other models are isochronous for an open region of initial data, while the motions for other initial data are not periodic, featuring instead scattering phenomena with some of the particles incoming from, or escaping to, infinity in the remote past or future. Full article

(This article belongs to the Special Issue Symmetry in Orthogonal Polynomials)

260 KiB

Open AccessArticle

On a Reduction Formula for a Kind of Double q-Integrals

by Zhi-Guo Liu

Symmetry 2016, 8(6), 44; https://doi.org/10.3390/sym8060044 - 8 Jun 2016

Cited by 8 | Viewed by 4056

Abstract

Using the q-integral representation of Sears’ nonterminating extension of the q-Saalschütz summation, we derive a reduction formula for a kind of double q-integrals. This reduction formula is used to derive a curious double q-integral formula, and also allows us to prove a general q-beta integral formula including the Askey–Wilson integral formula as a special case. Using this double q-integral formula and the theory of q-partial differential equations, we derive a general q-beta integral formula, which includes the Nassrallah–Rahman integral as a special case. Our evaluation does not require the orthogonality relation for the q-Hermite polynomials and the Askey–Wilson integral formula. Full article

(This article belongs to the Special Issue Symmetry in Orthogonal Polynomials)

314 KiB

Open AccessArticle

Multivariate Krawtchouk Polynomials and Composition Birth and Death Processes

by Robert Griffiths

Symmetry 2016, 8(5), 33; https://doi.org/10.3390/sym8050033 - 9 May 2016

Cited by 7 | Viewed by 3799

Abstract

This paper defines the multivariate Krawtchouk polynomials, orthogonal on the multinomial distribution, and summarizes their properties as a review. The multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets of functions defined on each of N multinomial trials. The dual multivariate Krawtchouk polynomials, which also have a polynomial structure, are seen to occur naturally as spectral orthogonal polynomials in a Karlin and McGregor spectral representation of transition functions in a composition birth and death process. In this Markov composition process in continuous time, there are N independent and identically distributed birth and death processes each with support

0, 1, \dots

. The state space in the composition process is the number of processes in the different states

0, 1, \dots

. Dealing with the spectral representation requires new extensions of the multivariate Krawtchouk polynomials to orthogonal polynomials on a multinomial distribution with a countable infinity of states. Full article

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