Zeros of orthogonal Laurent polynomials and solutions of strong Stieltjes moment problems

The strong Stieltjes moment problem for a bisequence consists of finding positive measures μ with support in [0,∞) such that

     

On some semilattice structures for production technologies

Tracing back from Charnes et al. [9] many approaches have been proposed to extend the DEA production model to non-convex technologies. The FDH method were introduced by Deprins et al. [13] and it only assumes a free disposal assumption of the technology. This paper, continues further an earlier work by Briec and Horvath [7]. Among other things, a new class of semilattice production technologies is introduced. Duality results as well as computational issues are proposed.     

On the packing chromatic number of some lattices

For a positive integer k, a k-packing in a graph G is a subset A of vertices such that the distance between any two distinct vertices from A is more than k. The packing chromatic number of G is the smallest integer m such that the vertex set of G can be partitioned as V₁,V₂,…,V_m where V_i is an i-packing for each i. It is proved that the planar triangular lattice T and the three-dimensional integer lattice Z³ do not have finite packing chromatic numbers.     

Colorings of k-balanced matrices and integer decomposition property of related polyhedra

We show that a class of polyhedra, arising from certain 0,1 matrices introduced by Truemper and Chandrasekaran, has the integer decomposition property. This is accomplished by proving certain coloring properties of these matrices.     

Uniform asymptotics of some q-orthogonal polynomials

Uniform asymptotic formulas are obtained for the Stieltjes-Wigert polynomial, the q⁻¹-Hermite polynomial and the q-Laguerre polynomial as the degree of the polynomial tends to infinity. In these formulas, the q-Airy polynomial, defined by truncating the q-Airy function, plays a significant role. While the standard Airy function, used frequently in the uniform asymptotic formulas for classical orthogonal polynomials, behaves like the exponential function on one side and the trigonometric functions on the other side of an extreme zero, the q-Airy polynomial behaves like the q-Airy function on one side and the q-Theta function on the other side. The last two special functions are involved in the local asymptotic formulas of the q-orthogonal polynomials. It seems therefore reasonable to expect that the q-Airy polynomial will play an important role in the asymptotic theory of the q-orthogonal polynomials.     

Conservation laws for some compacton equations using the multiplier approach

This paper is an application of the variational derivative method to the derivation of the conservation laws for partial differential equations. The conservation laws for (1+1) dimensional compacton k(2,2) and compacton k(3,3) equations are studied via multiplier approach. Also the conservation laws for (2+1) dimensional compacton Zk(2,2) equation are established by first computing the multipliers.     

Hilbert series of invariants, constant terms and Kostka-Foulkes polynomials

A problem that arose in the study of the mass of the neutrino led us to the evaluation of a constant term with a variety of ramifications into several areas from Invariant Theory, Representation Theory, the Theory of Symmetric Functions and Combinatorics. A significant by-product of our evaluation is the construction of a trigraded Cohen Macaulay basis for the Invariants under an action of SL_n(C) on a space of 2n+n² variables.     

Quantum Brownian motion and generalizations of Hermite polynomials

We introduce new families of orthogonal polynomials H_D,n, motivated by the non-equilibrium evolution of a quantum Brownian particle (qBp). The H_D,n’s generalize non-trivially the standard Hermite polynomials, employed for classical Brownian motion. We treat several models (labelled by D) for a non-equilibrium qBp, by means of the Wigner function W, in the presence of a “heat bath” at thermal equilibrium, with and without ab initio friction. For long times (for a suitable class of initial conditions), the non-equilibrium Wigner function W should approach, in some sense, the (time-independent) equilibrium Wigner function W_eq,D, which describes the thermal equilibrium of the qBp with the “heat bath” and plays a central role. W_eq,D is chosen to be the weight function which orthogonalizes the H_D,n’s. New results on W_eq,D and on the H_D,n’s are reported. We justify the key role of the H_D,n’s as follows. Using the H_D,n’s, moments W_eq,D,n and W_n are introduced for W_eq,D and W, respectively. At equilibrium, all moments W_eq,D,n except the lowest one (W_eq,D,0) vanish identically. Off-equilibrium, one expects that, for long times (for suitable initial conditions): (i) all non-equilibrium moments W_n (except the lowest moment W₀), will approach zero, while (ii) the lowest non-equilibrium moment W₀ will tend to W_eq,D,0(≠0). To complete the justification, we outline how the approximate long-time non-equilibrium theories determined by W₀ for the different models (D) yield Smoluchowski equations and irreversible evolutions of the qBp towards thermal equilibrium.     

Upper triangular operator matrices with the single-valued extension property

If A=(Aij)_1?i,j?n∈B(X) is an upper triangular Banach space operator such that AiiAij=AijAjj for all 1?i?j?n, then A has SVEP or satisfies (Dunford's) condition (C) or (Bishop's) property (β) or (the decomposition) property (δ) if and only if Aii, 1?i?n, has the corresponding property.     

On some iterated weighted spaces

It is proved that the Hörmander and spaces (Ω₁⊂Rn, Ω₂⊂Rm open sets, 1?p<∞, ki Beurling-Björck weights, k=k₁⊗k₂) are isomorphic whereas the iterated spaces and are not if 1<p≠q<∞. A similar result for weighted Lp-spaces of entire analytic functions is also obtained. Finally a result on iterated Besov spaces is given: and are not isomorphic when 1<q≠2<∞.     

Operator matrices on topological vector spaces

A series of basic summability results are established for matrices of linear and some nonlinear operators between topological vector spaces.     

Partition and composition matrices

This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another.We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel.We show that composition matrices on X are in one-to-one correspondence with (2+2)-free posets on X. Also, composition matrices whose rows satisfy a column-ordering relation are shown to be in one-to-one correspondence with parking functions. Finally, we show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on {1,…,n}.     

Exact asymptotics of the characteristic polynomial of the symmetric Pascal matrix

We have obtained the exact asymptotics of the determinant . Inverse symbolic computing methods were used to obtain exact analytical expressions for all terms up to relative order L⁻¹⁴ to the leading term. This determinant is known to give weighted enumerations of cyclically symmetric plane partitions, weighted enumerations of certain families of vicious walkers and it has been conjectured to be proportional to the one point function of the O(1) loop model on a cylinder of circumference L. We apply our result to the loop model and give exact expressions for the asymptotics of the average of the number of loops surrounding a point and the fluctuation in this number. For the related bond percolation model at the critical point, we give exact expressions for the asymptotics of the probability that a point is on a cluster that wraps around a cylinder of even circumference and the probability that a point is on a cluster spanning a cylinder of odd circumference.     

On constants in the Füredi-Hajnal and the Stanley-Wilf conjecture

For a given permutation matrix P, let fP(n) be the maximum number of 1-entries in an n×n(0,1)-matrix avoiding P and let SP(n) be the set of all n×n permutation matrices avoiding P. The Füredi-Hajnal conjecture asserts that cP:=limn→∞fP(n)/n is finite, while the Stanley-Wilf conjecture asserts that is finite.In 2004, Marcus and Tardos proved the Füredi-Hajnal conjecture, which together with the reduction introduced by Klazar in 2000 proves the Stanley-Wilf conjecture.We focus on the values of the Stanley-Wilf limit (sP) and the Füredi-Hajnal limit (cP). We improve the reduction and obtain which decreases the general upper bound on sP from sP?constconstO(klog(k)) to sP?constO(klog(k)) for any k×k permutation matrix P. In the opposite direction, we show .For a lower bound, we present for each k a k×k permutation matrix satisfying cP=Ω(k²).     

Scattering matrices of regular coverings of graphs

We give a decomposition formula for the determinant on the bond scattering matrix of a regular covering of G. Furthermore, we define an L-function of G, and give a determinant expression of it. As a corollary, we express the determinant on the bond scattering matrix of a regular covering of G by means of its L-functions.     

The Andrews-Stanley partition function and Al-Salam-Chihara polynomials

For any partition λ let ω(λ) denote the four parameter weight

ω(λ)=a^{∑_i≥1⌈λ_2i−1/2⌉}b^{∑_i≥1⌊λ_2i−1/2⌋}c^{∑_i≥1⌈λ_2i/2⌉}d^{∑_i≥1⌊λ_2i/2⌋},     

On a new class of Finsler metrics

In this paper, the geometric meaning of (α,β)-norms is made clear. On this basis, a new class of Finsler metrics called general (α,β)-metrics are introduced, which are defined by a Riemannian metric and a 1-form. These metrics not only generalize (α,β)-metrics naturally, but also include some metrics structured by R. Bryant. The spray coefficients formula of some kinds of general (α,β)-metrics is given and the projective flatness is also discussed.     

The intersection of essential approximate point spectra of operator matrices

When A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the infinite-dimensional separable Hilbert space H⊕K of the form . In this paper, it is shown that there exists some operator C∈B(K,H) such that MC is upper semi-Fredholm and ind(MC)?0 if and only if there exists some left invertible operator C∈B(K,H) such that MC is upper semi-Fredholm and ind(MC)?0. A necessary and sufficient condition for MC to be upper semi-Fredholm and ind(MC)?0 for some C∈Inv(K,H) is given, where Inv(K,H) denotes the set of all the invertible operators of B(K,H). In addition, we give a necessary and sufficient condition for MC to be upper semi-Fredholm and ind(MC)?0 for all C∈Inv(K,H).     

On the topology of the Kasparov groups and its applications

In this paper we establish a direct connection between stable approximate unitary equivalence for *-homomorphisms and the topology of the KK-groups which avoids entirely C^*-algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this topology coincides with both Pimsner's topology and the Brown-Salinas topology. We study the generalized Rørdam group , and prove that if a separable exact residually finite dimensional C^*-algebra satisfies the universal coefficient theorem in KK-theory, then it embeds in the UHF algebra of type 2^∞. In particular such an embedding exists for the C^*-algebra of a second countable amenable locally compact maximally almost periodic group.