908 research outputs found
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 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
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