10,380 research outputs found
A positive radial product formula for the Dunkl kernel
It is an open conjecture that generalized Bessel functions associated with
root systems have a positive product formula for non-negative multiplicity
parameters of the associated Dunkl operators. In this paper, a partial result
towards this conjecture is proven, namely a positive radial product formula for
the non-symmetric counterpart of the generalized Bessel function, the Dunkl
kernel. Radial hereby means that one of the factors in the product formula is
replaced by its mean over a sphere. The key to this product formula is a
positivity result for the Dunkl-type spherical mean operator. It can also be
interpreted in the sense that the Dunkl-type generalized translation of radial
functions is positivity-preserving. As an application, we construct Dunkl-type
homogeneous Markov processes associated with radial probability distributions.Comment: 25 page
An observer's view on the future of asteroseismology
Scientific research is a continuous process, and the speed of future progress
can be estimated by the pace of finding explanations for previous research
questions. In this observers based view of stellar pulsation and
asteroseismology, we start with the earliest observations of variable stars and
the techniques used to observe them. The earliest variable stars were large
amplitude, radial pulsators but were followed by other classes of pulsating
stars. As the field matured, we outline some cornerstones of research into
pulsating star research with an emphasis on changes in observational
techniques. Improvements from photographs, to photometry, CCDs, and space
telescopes allowed researchers to separate out pulsating stars from other stars
with light variations, recognize radial and nonradial pulsation courtesy of
increased measurement precision, and then use nonradial pulsations to look
inside the stars, which cannot be done any other way. We follow several
highlighted problems to show that even with excellent space data, there still
may not be quick theoretical explanations. As the result of technical changes,
the structure of international organizations devoted to pulsating stars has
changed, and an increasing number of conferences specialized to space missions
or themes are held. Although there are still many unsolved problems, such as
mode identification in non-asymptotic pulsating stars, the large amount of data
with unprecedented precision provided by space missions (MOST, CoRoT, Kepler)
and upcoming missions allow us to use asteroseismology to its full potential.
However, the enormous flow of data will require new techniques to extract the
science before the next missions. The future of asteroseismology will be
successful if we learn from the past and improve with improved techniques,
space missions, and a properly educated new generation.Comment: Review appeared in "Frontiers in Astronomy and Space Science" special
issue Future of Asteroseismolog
Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators
Based on the theory of Dunkl operators, this paper presents a general concept
of multivariable Hermite polynomials and Hermite functions which are associated
with finite reflection groups on \b R^N. The definition and properties of
these generalized Hermite systems extend naturally those of their classical
counterparts; partial derivatives and the usual exponential kernel are here
replaced by Dunkl operators and the generalized exponential kernel K of the
Dunkl transform. In case of the symmetric group , our setting includes the
polynomial eigenfunctions of certain Calogero-Sutherland type operators. The
second part of this paper is devoted to the heat equation associated with
Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem
is governed by a positive one-parameter semigroup; this is assured by a maximum
principle for the generalized Laplacian. The explicit solution to the Cauchy
problem involves again the kernel K, which is, on the way, proven to be
nonnegative for real arguments.Comment: 24 pages, AMS-LaTe
Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC
In this paper, we derive explicit product formulas and positive convolution
structures for three continuous classes of Heckman-Opdam hypergeometric
functions of type . For specific discrete series of multiplicities these
hypergeometric functions occur as the spherical functions of non-compact
Grassmann manifolds over one of the (skew) fields We write the product formula of these spherical
functions in an explicit form which allows analytic continuation with respect
to the parameters. In each of the three cases, we obtain a series of hypergroup
algebras which include the commutative convolution algebras of -biinvariant
functions on
Factors affecting the public judgement of consumer protection authorities
The article first introduces the state-provided institutions for consumer protection in Hungary, then examines and systematises those factors having a significant impact on the public judgement of the work of consumer protection authorities. The study is based on the presumption that the public judgement of the work of consumer protection institutions is affected by many factors, and that these factors can be grouped and counted. After examining the main fields of activities of consumer protection institutions and the main factors and aspects affecting the judgement of the public, a model for the measurement of satisfaction is proposed
- …