44,968 research outputs found
A class of symmetric -orthogonal polynomials with four free parameters
By using a generalization of Sturm-Liouville problems in -difference
spaces, a class of symmetric -orthogonal polynomials with four free
parameters is introduced. The standard properties of these polynomials, such as
a second order -difference equation, the explicit form of the polynomials in
terms of basic hypergeometric series, a three term recurrence relation and a
general orthogonality relation are presented. Some particular examples are then
studied in detail.Comment: arXiv admin note: substantial text overlap with arXiv:1306.644
A basic class of symmetric orthogonal polynomials of a discrete variable
By using a generalization of Sturm-Liouville problems in discrete spaces, a
basic class of symmetric orthogonal polynomials of a discrete variable with
four free parameters, which generalizes all classical discrete symmetric
orthogonal polynomials, is introduced. The standard properties of these
polynomials, such as a second order difference equation, an explicit form for
the polynomials, a three term recurrence relation and an orthogonality relation
are presented. It is shown that two hypergeometric orthogonal sequences with 20
different weight functions can be extracted from this class. Moreover, moments
corresponding to these weight functions can be explicitly computed. Finally, a
particular example containing all classical discrete symmetric orthogonal
polynomials is studied in detail
A symmetric generalization of Sturm-Liouville problems in -difference spaces
Classical Sturm-Liouville problems of -difference variables are extended
for symmetric discrete functions such that the corresponding solutions preserve
the orthogonality property. Some illustrative examples are given in this sense.Comment: 11 page
Five dimensional gauge theories and vertex operators
We study supersymmetric gauge theories in five dimensions, using their
relation to the K-theory of the moduli spaces of torsion free sheaves. In the
spirit of the BPS/CFT correspondence the partition function and the expectation
values of the chiral, BPS protected observables are given by the matrix
elements and more generally by the correlation functions in some q-deformed
conformal field theory in two dimensions. We show that the coupling of the
gauge theory to the bi-fundamental matter hypermultiplet inserts a particular
vertex operator in this theory. In this way we get a generalization of the main
result of \cite{CO} to -theory. The theory of interpolating Macdonald
polynomials is an important tool in our construction.Comment: 33 page
On the generalization of McVittie's model for an inhomogeneity in a cosmological spacetime
McVittie's spacetime is a spherically symmetric solution to Einstein's
equation with an energy-momentum tensor of a perfect fluid. It describes the
external field of a single quasi-isolated object with vanishing electric charge
and angular momentum in an environment that asymptotically tends to a
Friedmann--Lemaitre--Robertson--Walker universe. We critically discuss some
recently proposed generalizations of this solution, in which radial matter
accretion as well as heat currents are allowed. We clarify the hitherto
unexplained constraints between these two generalizing aspects as being due to
a geometric property, here called spatial Ricci-isotropy, which forces
solutions covered by the McVittie ansatz to be rather special. We also clarify
other aspects of these solutions, like whether they include geometries which
are in the same conformal equivalence class as the exterior Schwarzschild
solution, which leads us to contradict some of the statements in the recent
literature.Comment: LaTeX (RevTeX), 15 pages, no figur
Cosmological models (Carg\`{e}se lectures 1998)
The aim of this set of lectures is a systematic presentation of a 1+3
covariant approach to studying the geometry, dynamics, and observational
properties of relativistic cosmological models. In giving (i) the basic 1+3
covariant relations for a cosmological fluid, the present lectures cover some
of the same ground as a previous set of Carg\`{e}se lectures \cite{ell73}, but
they then go on to give (ii) the full set of corresponding tetrad equations,
(iii) a classification of cosmological models with exact symmetries, (iv) a
brief discussion of some of the most useful exact models and their
observational properties, and (v) an introduction to the gauge-invariant and
1+3 covariant perturbation theory of
almost-Friedmann-Lema\^{\i}tre-Robertson-Walker universes, with a fluid
description for the matter and a kinetic theory description of the radiation.Comment: 87 pages. Fully hyperlinked. Published in proceedings of the NATO
Advanced Study Institute on Theoretical and Observational Cosmology,
Carg\`{e}se, France, August 17-29, 1998 / edited by Marc Lachi\`{e}ze-Rey.
Boston: Kluwer Academic, 1999. NATO science series. Series C, Mathematical
and physical sciences, vol. 541, p.1-11
Do orthogonal polynomials dream of symmetric curves?
The complex or non-hermitian orthogonal polynomials with analytic weights are
ubiquitous in several areas such as approximation theory, random matrix models,
theoretical physics and in numerical analysis, to mention a few. Due to the
freedom in the choice of the integration contour for such polynomials, the
location of their zeros is a priori not clear. Nevertheless, numerical
experiments, such as those presented in this paper, show that the zeros not
simply cluster somewhere on the plane, but persistently choose to align on
certain curves, and in a very regular fashion.
The problem of the limit zero distribution for the non-hermitian orthogonal
polynomials is one of the central aspects of their theory. Several important
results in this direction have been obtained, especially in the last 30 years,
and describing them is one of the goals of the first parts of this paper.
However, the general theory is far from being complete, and many natural
questions remain unanswered or have only a partial explanation.
Thus, the second motivation of this paper is to discuss some "mysterious"
configurations of zeros of polynomials, defined by an orthogonality condition
with respect to a sum of exponential functions on the plane, that appeared as a
results of our numerical experiments. In this apparently simple situation the
zeros of these orthogonal polynomials may exhibit different behaviors: for some
of them we state the rigorous results, while other are presented as conjectures
(apparently, within a reach of modern techniques). Finally, there are cases for
which it is not yet clear how to explain our numerical results, and where we
cannot go beyond an empirical discussion.Comment: 36 pages, 11 figures. Based on a plenary talk of the first author at
FoCM 2014 Conference. To appear in Foundations of Computational Mathematic
Theory of Band Warping and its Effects on Thermoelectronic Transport Properties
Optical and transport properties of materials depend heavily upon features of
electronic band structures in proximity to energy extrema in the Brillouin zone
(BZ). Such features are generally described in terms of multi-dimensional
quadratic expansions and corresponding definitions of effective masses.
Multi-dimensional expansions, however, are permissible only under strict
conditions that are typically violated by degenerate bands and even some
non-degenerate bands. Suggestive terms such as "band warping" or "corrugated
energy surfaces" have been used to refer to such situations and ad hoc methods
have been developed to treat them. While numerical calculations may reflect
such features, a complete theory of band warping has not been developed. We
develop a generally applicable theory, based on radial expansions, and a
corresponding definition of angular effective mass. Our theory also accounts
for effects of band non-parabolicity and anisotropy, which hitherto have not
been precisely distinguished from, if not utterly confused with, band warping.
Based on our theory, we develop precise procedures to evaluate band warping
quantitatively. As a benchmark demonstration, we analyze the warping features
of valence bands in silicon using first-principles calculations and we compare
those with previous semi-empirical models. We use our theory and angular
effective masses to generalize derivations of tensorial transport coefficients
for cases of either single or multiple electronic bands, with either
quadratically expansible or warped energy surfaces. From that theory we
discover the formal existence at critical points of transport-equivalent
ellipsoidal bands that yield identical results from the standpoint of any
transport property. Additionally, we illustrate the drastic effects that band
warping can induce on thermoelectric properties using multi-band models.Comment: 23 pages, 5 figures. To appear in Physical Review
Exact Evolution of Discrete Relativistic Cosmological Models
We study the effects of inhomogeneities on the evolution of the Universe, by
considering a range of cosmological models with discretized matter content.
This is done using exact and fully relativistic methods that exploit the
symmetries in and about submanifolds of spacetimes that themselves possess no
continuous global symmetries. These methods allow us to follow the evolution of
our models throughout their entire history, far beyond what has previously been
possible. We find that while some space-like curves collapse to anisotropic
singularities in finite time, others remain non-singular forever. The resulting
picture is of a cosmological spacetime in which some behaviour remains close to
Friedmann-like, while other behaviours deviate radically. In particular, we
find that large-scale acceleration is possible without any violation of the
energy conditions.Comment: 46 pages, 23 figure
Meixner matrix ensembles
We construct a family of matrix ensembles that fits Anshelevich's regression
postulates for "Meixner laws on matrices". We show that the Laplace transform
of a general n by n Meixner matrix ensemble satisfies a system of partial
differential equations which is explicitly solvable for n=2. We rely on these
solutions to identify the six types of 2 by 2 Meixner matrix ensembles
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