2,827 research outputs found
Isonemal prefabrics with only parallel axes of symmetry
Isonemal weaving designs, introduced into mathematical literature by
Gr\"unbaum and Shephard, were classified into thirty-nine infinite sets and a
small number of exceptions by Richard Roth. This paper refines Roth's taxonomy
for the first ten of these families in order to solve three problems, which
designs exist in various sizes, which prefabrics can be doubled and remain
isonemal, and which can be halved and remain isonemal.Comment: 25 page
The Rahman Polynomials Are Bispectral
In a very recent paper, M. Rahman introduced a remarkable family of
polynomials in two variables as the eigenfunctions of the transition matrix for
a nontrivial Markov chain due to M. Hoare and M. Rahman. I indicate here that
these polynomials are bispectral. This should be just one of the many
remarkable properties enjoyed by these polynomials. For several challenges,
including finding a general proof of some of the facts displayed here the
reader should look at the last section of this paper.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Some Noncommutative Matrix Algebras Arising in the Bispectral Problem
I revisit the so called "bispectral problem" introduced in a joint paper with
Hans Duistermaat a long time ago, allowing now for the differential operators
to have matrix coefficients and for the eigenfunctions, and one of the
eigenvalues, to be matrix valued too. In the last example we go beyond this and
allow both eigenvalues to be matrix valued
A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application
The one variable Krawtchouk polynomials, a special case of the
function did appear in the spectral representation of the transition kernel for
a Markov chain studied a long time ago by M. Hoare and M. Rahman. A
multivariable extension of this Markov chain was considered in a later paper by
these authors where a certain two variable extension of the Appel
function shows up in the spectral analysis of the corresponding transition
kernel. Independently of any probabilistic consideration a certain
multivariable version of the Gelfand-Aomoto hypergeometric function was
considered in papers by H. Mizukawa and H. Tanaka. These authors and others
such as P. Iliev and P. Tertwilliger treat the two-dimensional version of the
Hoare-Rahman work from a Lie-theoretic point of view. P. Iliev then treats the
general -dimensional case. All of these authors proved several properties of
these functions. Here we show that these functions play a crucial role in the
spectral analysis of the transition kernel that comes from pushing the work of
Hoare-Rahman to the multivariable case. The methods employed here to prove this
as well as several properties of these functions are completely different to
those used by the authors mentioned above
The CMV bispectral problem
A classical result due to Bochner classifies the orthogonal polynomials on
the real line which are common eigenfunctions of a second order linear
differential operator. We settle a natural version of the Bochner problem on
the unit circle which answers a similar question concerning orthogonal Laurent
polynomials and can be formulated as a bispectral problem involving CMV
matrices. We solve this CMV bispectral problem in great generality proving
that, except the Lebesgue measure, no other one on the unit circle yields a
sequence of orthogonal Laurent polynomials which are eigenfunctions of a linear
differential operator of arbitrary order. Actually, we prove that this is the
case even if such an eigenfunction condition is imposed up to finitely many
orthogonal Laurent polynomials.Comment: 25 pages, final version, to appear in International Mathematics
Research Notice
A note on the invariant distribution of a quasi-birth-and-death process
The aim of this paper is to give an explicit formula of the invariant
distribution of a quasi-birth-and-death process in terms of the block entries
of the transition probability matrix using a matrix-valued orthogonal
polynomials approach. We will show that the invariant distribution can be
computed using the squared norms of the corresponding matrix-valued orthogonal
polynomials, no matter if they are or not diagonal matrices. We will give an
example where the squared norms are not diagonal matrices, but nevertheless we
can compute its invariant distribution
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