Möbius Polynomials of Face Posets of Convex Polytopes

The Möbius polynomial is an invariant of ranked posets, closely related to the Möbius function. In this paper, we study the Möbius polynomial of face posets of convex polytopes. We present formulas for computing the Möbius polynomial of the face poset of a pyramid or a prism over an existing polytope, or of the gluing of two or more polytopes in terms of the Möbius polynomials of the original polytopes. We also present general formulas for calculating Möbius polynomials of face posets of simplicial polytopes and of Eulerian posets in terms of their f-vectors and some additional constraints.     

Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials

The asymptotic theory is developed for polynomial sequences that are generated by the three-term higher-order recurrence

Uniform Spacing of Zeros of Orthogonal Polynomials

It is shown that for doubling weights, the zeros of the associated orthogonal polynomials are uniformly spaced in the sense that if cos θ _m,k , θ _m,k ∈[0,π] are the zeros of the m-th orthogonal polynomial associated with w, then θ _m,k −θ _m,k+1∼1/m. It is also shown that for doubling weights, neighboring Cotes numbers are of the same order. Finally, it is proved that these two properties are actually equivalent to the doubling property of the weight function.     

The Analysis and the Representation of Balanced Complex Polytopes in 2D

In this paper, we deepen the theoretical study of the geometric structure of a balanced complex polytope (b.c.p.), which is the generalization of a real centrally symmetric polytope to the complex space. We also propose a constructive algorithm for the representation of its facets in terms of their associated linear functionals. The b.c.p.s are used, for example, as a tool for the computation of the joint spectral radius of families of matrices. For the representation of real polytopes, there exist well-known algorithms such as, for example, the Beneath–Beyond method. Our purpose is to modify and adapt this method to the complex case by exploiting the geometric features of the b.c.p. However, due to the significant increase in the difficulty of the problem when passing from the real to the complex case, in this paper, we confine ourselves to examine the two-dimensional case. We also propose an algorithm for the computation of the norm the unit ball of which is a b.c.p. This work was supported by INdAM-GNCS.     

Ehrhart h ?-Vectors of Hypersimplices

We consider the Ehrhart h ^?-vector for the hypersimplex. It is well-known that the sum of the $h_{i}^{*}$ is the normalized volume which equals the Eulerian numbers. The main result is a proof of a conjecture by R.?Stanley which gives an interpretation of the $h^{*}_{i}$ coefficients in terms of descents and exceedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes.     

Hölder Continuity of Roots of Complex and p-adic Polynomials

Let f be a polynomial with coefficients in an algebraically closed, valued field. We show a refinement of the principle of continuity of roots, namely, that each root α of f is locally Hölder continuous of order 1/μ as a function of the coefficients of f, where μ is the root multiplicity of α. This is derived as a consequence of a principle that could be called continuity of factors, namely, that if f = gh is a factorisation with (g, h) = 1, then the coefficients of g and h are locally Lipschitz continuous as functions of the coefficients of f. The proofs are elementary and of an algebraic nature.     

Mixed Type Multiple Orthogonal Polynomials for Two Nikishin Systems

We study the logarithmic and ratio asymptotics of linear forms constructed from a Nikishin system which satisfy orthogonality conditions with respect to a system of measures generated by a second Nikishin system. This construction combines type I and type II multiple orthogonal polynomials. The logarithmic asymptotics of the linear forms is expressed in terms of the extremal solution of an associated vector valued equilibrium problem for the logarithmic potential. The ratio asymptotics is described by means of a conformal representation of an appropriate Riemann surface of genus zero onto the extended complex plane.     

Weighted Poisson Cells as Models for Random Convex Polytopes

We introduce a parametric family for random convex polytopes in ? ^d which allows for an easy generation of samples for further use, e.g., as random particles in materials modelling and simulation. The basic idea consists in weighting the Poisson cell, which is the typical cell of the stationary and isotropic Poisson hyperplane tessellation, by suitable geometric characteristics. Since this approach results in an exponential family, parameters can be efficiently estimated by maximum likelihood. This work has been motivated by the desire for a flexible model for random convex particles as can be found in many composite materials such as concrete or refractory castables.     

On the Local Time of Random Walks Associated with Gegenbauer Polynomials

The local time of random walks associated with Gegenbauer polynomials \(P_{n}^{(\alpha)}(x)\), x∈[?1,1], is studied in the recurrent case: \(\alpha\in [-\frac{1}{2},0]\). When α is nonzero, the limit distribution is given in terms of a Mittag-Leffler distribution. The proof is based on a local limit theorem for the random walk associated with Gegenbauer polynomials. As a by-product, we derive the limit distribution of the local time of some particular birth-and-death Markov chains on ?.     

Denjoy–Carleman Differentiable Perturbation of Polynomials and Unbounded Operators

Let \({t\mapsto A(t)}\) for \({t\in T}\) be a C ^M -mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here C ^M stands for C ^ω (real analytic), a quasianalytic or non-quasianalytic Denjoy–Carleman class, C ^∞, or a Hölder continuity class C ^0,α . The parameter domain T is either \({\mathbb R}\) or \({\mathbb R^n}\) or an infinite dimensional convenient vector space. We prove and review results on C ^M -dependence on t of the eigenvalues and eigenvectors of A(t).     

Primitive Roots and Linearized Polynomials

Let f(x) = sum from t=0 to n α_ix~i∈GF(p)[x],we associate it with a ploynomial f~*(x)=sum from i=0 to n α_ix~(p~i),f(x) and f~*(x)are called p-associates of each other. f~*(x) is called a p-ploynomial,customary to speak of linearized polynomial. Let f(x)=x~m- 1/g(x), m = m_1~r, q = p~m, g(x)∈GF(p)[x],r be the order of g(x). Cohen and the author observed that if m_1≥2, there alwaysexsists a primitive roots ζ∈GF(q) suck that f~*(ζ) = f~*(c), here f~*(c)≠0. In fact     

Approximation Properties of Julia Polynomials

Let G be a finite simply connected domain in the complex plane C, bounded by a rectifiable Jordan curve L, and let w = φ0 （z） be the Riemann conformal mapping of G onto D （0, r0） ：= {E-mail： ： ｜｜ 〈 r0}, normalized by the conditions φ0 （z0） = 0, φ＇0 （z0） = 1. In this work, the rate of approximation of φ0 by the polynomials, defined with the help of the solutions of some extremal problem, in a closed domain G is studied. This rate depends on the geometric properties of the boundary L.     

Explicit Forms of q-Deformed Levy-Meixner Polynomials and Their Generating Functions

The classical Levy-Meixner polynomials are distinguished through the special forms of their generating functions. In fact, they are completely determined by 4 parameters： c1, c2,γ and β. In this paper, for-1 〈q〈 1, we obtain a unified explicit form of q-deformed Levy-Meixner polynomials and their generating functions in term of c1, c2, γand β, which is shown to be a reasonable interpolation between classical case （q=1） and fermionic case （q=-1）.In particular, when q=0 it＇s also compatible with the free case.     

A Unified Construction of Generalized Classical Polynomials Associated with Operators of Calogero–Sutherland Type

In this paper we consider a large class of many-variable polynomials which contains generalizations of the classical Hermite, Laguerre, Jacobi and Bessel polynomials as special cases, and which occur as the polynomial part in the eigenfunctions of Calogero–Sutherland type operators and their deformations recently found and studied by Chalykh, Feigin, Sergeev, and Veselov. We present a unified and explicit construction of all these polynomials.     

On the Asymptotic Behavior of Faber Polynomials for Domains With Piecewise Analytic Boundary

Let φ(z) be an analytic function on a punctured neighborhood of ∞, where it has a simple pole. The nth Faber polynomial F _n (z) (n=0,1,2,…) associated with φ is the polynomial part of the Laurent expansion at ∞ of [φ(z)] ⁿ . Assuming that ψ (the inverse of φ) conformally maps |w|>1 onto a domain Ω bounded by a piecewise analytic curve without cusps pointing out of Ω, and under an additional assumption concerning the “Lehman expansion” of ψ about those points of |w|=1 mapped onto corners of ∂ Ω, we obtain asymptotic formulas for F _n that yield fine results on the limiting distribution of the zeros of Faber polynomials.      

Asymptotic Properties of Orthogonal Polynomials with Respect to a Non-discrete Jacobi-Sobolev Inner Product

Let {Q _n ^(α,β) (x)} _n=0 ^∞ denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product

$\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)d\mu_{\alpha,\beta}(x)+\lambda\int_{-1}^{1}f'(x)g'(x)d\nu_{\alpha,\beta}(x)$

where λ>0 and d μ _α,β(x)=(x?a)(1?x)^α?1(1+x)^β?1 dx, d ν _α,β(x)=(1?x) ^α (1+x) ^β dx with aα,β>0. Their inner strong asymptotics on (?1,1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Q _n ^(α,β) are obtained.

     

q-Hermite Polynomials and a q-Beta Integral

The main object of the present paper is to derive a formula for the q-Hahn.polynomials by using the q-Mehler formula about the q-Hermite polynomials.This result, together with other properties of the q-Hermite polynomials, provides an evalution of a q-beta integral.Both the celebrated Askey-Wilson integral and the Ismail-Stanton-Viennot integral are special cases of this integral.     

On the Existence of Optimal Unions of Subspaces for Data Modeling and Clustering

Given a set of vectors F={f ₁,…,f _m } in a Hilbert space H\mathcal {H}, and given a family C\mathcal {C} of closed subspaces of H\mathcal {H}, the subspace clustering problem consists in finding a union of subspaces in C\mathcal {C} that best approximates (is nearest to) the data F. This problem has applications to and connections with many areas of mathematics, computer science and engineering, such as Generalized Principal Component Analysis (GPCA), learning theory, compressed sensing, and sampling with finite rate of innovation. In this paper, we characterize families of subspaces C\mathcal {C} for which such a best approximation exists. In finite dimensions the characterization is in terms of the convex hull of an augmented set C⁺\mathcal {C}^{+}. In infinite dimensions, however, the characterization is in terms of a new but related notion; that of contact half-spaces. As an application, the existence of best approximations from π(G)-invariant families C\mathcal {C} of unitary representations of Abelian groups is derived.     

Asymptotic Form of Hermite-Pad? Polynomials

Permanent address: Department of Physics, Saint Louis University, St Louis, Missouri, U.S.A. A method is given which locates the area on which the zerosof Hermite-Pad? polynomials of large order lie. The method alsogives the density of these zeros along the ares, and so theasymptotic form of the polynomials for large order. Severalexamples which confirm and illustrate the method are given.These results are necessary for proving the convergence of schemesfor extracting information from power series based on Hermite-Pad?approximants (the so-called "quadratic" and "differential equation"or "integral" Pad? approximants).