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

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

New remarks on the Cosmological Argument

We present a formal analysis of the Cosmological Argument in its two main forms: that due to Aquinas, and the revised version of the Kalam Cosmological Argument more recently advocated by William Lane Craig. We formulate these two arguments in such a way that each conclusion follows in first-order logic from the corresponding assumptions. Our analysis shows that the conclusion which follows for Aquinas is considerably weaker than what his aims demand. With formalizations that are logically valid in hand, we reinterpret the natural language versions of the premises and conclusions in terms of concepts of causality consistent with (and used in) recent work in cosmology done by physicists. In brief: the Kalam argument commits the fallacy of equivocation in a way that seems beyond repair; two of the premises adopted by Aquinas seem dubious when the terms `cause' and `causality' are interpreted in the context of contemporary empirical science. Thus, while there are no problems with whether the conclusions follow logically from their assumptions, the Kalam argument is not viable, and the Aquinas argument does not imply a caused origination of the universe. The assumptions of the latter are at best less than obvious relative to recent work in the sciences. We conclude with mention of a new argument that makes some positive modifications to an alternative variation on Aquinas by Le Poidevin, which nonetheless seems rather weak.Comment: 12 pages, accepted for publication in International Journal for Philosophy of Religio

Spectral Difference Equations Satisfied by KP Soliton Wavefunctions

The Baker-Akhiezer (wave) functions corresponding to soliton solutions of the KP hierarchy are shown to satisfy eigenvalue equations for a commutative ring of translational operators in the spectral parameter. In the rational limit, these translational operators converge to the differential operators in the spectral parameter previously discussed as part of the theory of "bispectrality". Consequently, these translational operators can be seen as demonstrating a form of bispectrality for the non-rational solitons as well.Comment: to appear in "Inverse Problems

A superintegrable finite oscillator in two dimensions with SU(2) symmetry

A superintegrable finite model of the quantum isotropic oscillator in two dimensions is introduced. It is defined on a uniform lattice of triangular shape. The constants of the motion for the model form an SU(2) symmetry algebra. It is found that the dynamical difference eigenvalue equation can be written in terms of creation and annihilation operators. The wavefunctions of the Hamiltonian are expressed in terms of two known families of bivariate Krawtchouk polynomials; those of Rahman and those of Tratnik. These polynomials form bases for SU(2) irreducible representations. It is further shown that the pair of eigenvalue equations for each of these families are related to each other by an SU(2) automorphism. A finite model of the anisotropic oscillator that has wavefunctions expressed in terms of the same Rahman polynomials is also introduced. In the continuum limit, when the number of grid points goes to infinity, standard two-dimensional harmonic oscillators are obtained. The analysis provides the $N\rightarrow \infty$ limit of the bivariate Krawtchouk polynomials as a product of one-variable Hermite polynomials

Some comments on the inverse problem of pure point diffraction

In a recent paper, Lenz and Moody (arXiv:1111.3617) presented a method for constructing families of real solutions to the inverse problem for a given pure point diffraction measure. Applying their technique and discussing some possible extensions, we present, in a non-technical manner, some examples of homometric structures.Comment: 6 pages, contribution to Aperiodic 201