6,087 research outputs found
Towards a q-analogue of the Kibble--Slepian formula in 3 dimensions
We study a generalization of the Kibble-Slepian (KS) expansion formula in 3
dimensions. The generalization is obtained by replacing the Hermite polynomials
by the q-Hermite ones. If such a replacement would lead to non-negativity for
all allowed values of parameters and for all values of variables ranging over
certain Cartesian product of compact intervals then we would deal with a
generalization of the 3 dimensional Normal distribution. We show that this is
not the case. We indicate some values of the parameters and some compact set in
R^{3} of positive measure, such that the values of the extension of KS formula
are on this set negative. Nevertheless we indicate other applications of so
generalized KS formula. Namely we use it to sum certain kernels built of the
Al-Salam-Chihara polynomials for the cases that were not considered by other
authors. One of such kernels sums up to the Askey-Wilson density disclosing its
new, interesting properties. In particular we are able to obtain a
generalization of the 2 dimensional Poisson-Mehler formula. As a corollary we
indicate some new interesting properties of the Askey-Wilson polynomials with
complex parameters. We also pose several open questions
Mixed Ehrhart polynomials
For lattice polytopes , Bihan (2014)
introduced the discrete mixed volume in analogy
to the classical mixed volume. In this note we initiate the study of the
associated mixed Ehrhart polynomial . We study properties of this polynomial and we
give interpretations for some of its coefficients in terms of (discrete) mixed
volumes. Bihan (2014) showed that the discrete mixed volume is always
non-negative. Our investigations yield simpler proofs for certain special
cases. We also introduce and study the associated mixed -vector. We show
that for large enough dilates the corresponding mixed
-polynomial has only real roots and as a consequence the mixed
-vector becomes non-negative.Comment: 12 page
A Converse Sum of Squares Lyapunov Result with a Degree Bound
Sum of Squares programming has been used extensively over the past decade for
the stability analysis of nonlinear systems but several questions remain
unanswered. In this paper, we show that exponential stability of a polynomial
vector field on a bounded set implies the existence of a Lyapunov function
which is a sum-of-squares of polynomials. In particular, the main result states
that if a system is exponentially stable on a bounded nonempty set, then there
exists an SOS Lyapunov function which is exponentially decreasing on that
bounded set. The proof is constructive and uses the Picard iteration. A bound
on the degree of this converse Lyapunov function is also given. This result
implies that semidefinite programming can be used to answer the question of
stability of a polynomial vector field with a bound on complexity
Positivity results for the Hecke algebras of non-crystallographic finite Coxeter groups
This paper is a report on a computer check of some positivity properties of
the Hecke algebra in type H4, including the non-negativity of the coefficients
of the structure constants in the Kazhdan-Lusztig basis. This answers a
long-standing question of Lusztig's. The same algorithm, carried out by hand,
also allows us to deal with the case of dihedral Coxeter groups.Comment: septembre 2005; 13 pages; cet article s'accompagne d'un logicie
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