Matrix exceptional Laguerre polynomials

We give an analog of exceptional polynomials in the matrix-valued setting by considering suitable factorizations of a given second-order differential operator and performing Darboux transformations. Orthogonality and density of the exceptional sequence are discussed in detail. We give an example of matrix-valued exceptional Laguerre polynomials of arbitrary size.     

Density of multivariate homogeneous polynomials on star like domains

The famous Weierstrass theorem asserts that every continuous function on a compact set in ${\mathbb{R}}^d$ can be uniformly approximated by algebraic polynomials. A related interesting problem consists in studying the same question for the important subclass of homogeneous polynomials containing only monomials of the same degree. The corresponding conjecture claims that every continuous function on the boundary of convex 0-symmetric bodies can be uniformly approximated by pairs of homogeneous polynomials. The main objective of the present paper is to review the recent progress on this conjecture and provide a new unified treatment of the same problem on non convex star like domains. It will be shown that the boundary of every 0-symmetric non convex star like domain contains an exceptional zero set so that a continuous function can be uniformly approximated on the boundary of the domain by a sum of two homogeneous polynomials if and only if the function vanishes on this zero set. Thus the Weierstrass type approximation problem for homogeneous polynomials on non convex star like domains amounts to the study of these exceptional zero sets. We will also present an extension of a theorem of Varjú which describes the exceptional zero sets for intersections of star like domains. These results combined with certain transformations of the underlying region will lead to the discovery of some new classes of convex and non convex domains for which the Weierstrass type approximation result holds for homogeneous polynomials.     

Extensions of homogeneous polynomials on c 0(l 2 i )

We show that a 2-homogeneous polynomial on the complex Banach space c ₀ l ₂ ⁱ ) is norm attaining if and only if it is finite (i.e, depends only on finite coordinates). As the consequence, we show that there exists a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on c ₀(l ₂ ⁱ ). The second author was supported by FAPESP, Brazil, Research Grant 01/04220-8.     

The number of convex polyominoes and the generating function of Jacobi polynomials

Lin and Chang gave a generating function of convex polyominoes with an m+1 by n+1 minimal bounding rectangle. Gessel showed that their result implies that the number of such polyominoes is

     

齐性等参子流形的多项式表示

本文首先讨论了Ａｋ，Ｂｋ和Ｄｋ群不变多项式，然后利用对称空产迷向表示的矩阵形式给出了齐性等参子流形的多项式表示。     

Characterizations of strictly convex sets by the uniqueness of support points

Abstract

This short paper characterizes strictly convex sets by the uniqueness of support points (such points are called unique support points or exposed points) under appropriate assumptions. A class of so-called regular sets, for which every extreme point is a unique support point, is introduced. Closed strictly convex sets and their intersections with some other sets are shown to belong to this class. The obtained characterizations are then applied to set-valued maps and to the separation of a convex set and a strictly convex set. Under suitable assumptions, so-called set-valued maps with path property are characterized by strictly convex images of the considered set-valued map.     

On the extensions of homogeneous polynomials

We investigate the problem of the uniqueness of the extension of -homogeneous polynomials in Banach spaces. We show in particular that in a nonreflexive Banach space that admits contractive projection of finite rank of at least dimension 2, for every there exists an -homogeneous polynomial on that has infinitely many extensions to . We also prove that under some geometric conditions imposed on the norm of a complex Banach lattice , for instance when satisfies an upper -estimate with constant one for some , any -homogeneous polynomial on attaining its norm at with a finite rank band projection , has a unique extension to its bidual . We apply these results in a class of Orlicz sequence spaces.

     

微分多项式的例外集

设ｆ为任一超越整函数，为ｆ的任一微分多项式；本文证明了在满足｜ａ＿ｎ－ａ＿ｍ｜＞ε｜ａ＿ｎ｜（ｎ≠ｍ）和的无穷个圆盘的并集之外取任意有限非零复数无穷次。     

Explicit formula of holomorphic automorphism group on complex homogeneous bounded domains

We known that the maximal connected holomorphic automorphism group Aut (D)(0) is a semi-direct product of the triangle group T(D) and the maximal connected isotropic subgroup Iso(D)(0) of a fixed point in the complex homogeneous bounded domain D and any complex homogeneous bounded domain is holomorphic isomorphic to a normal Siegel domain D(VN,F). In this paper, we give the explicit formula of any holomorphic automorphism in T(D(VN, F)) and Iso(D(VN,F))(0), where G(0) is the unit connected component of the Lie group G.     

Zero sets of bivariate Hermite polynomials

We establish various properties for the zero sets of three families of bivariate Hermite polynomials. Special emphasis is given to those bivariate orthogonal polynomials introduced by Hermite by means of a Rodrigues type formula related to a general positive definite quadratic form. For this family we prove that the zero set of the polynomial of total degree n+m$n+m$ consists of exactly n+m$n+m$ disjoint branches and possesses n+m$n+m$ asymptotes. A natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property. The results show that the properties of the zero sets, considered as affine algebraic curves in R²${\mathbb{R}}^2$, are completely different for the three families analyzed.     

The Virtual Poincaré Polynomials of Homogeneous Spaces

We factor the virtual Poincaré polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t^2u (t²–1)^r Q_G/H(t²) for a polynomial Q_G/H with nonnegative integer coefficients. Moreover, we show that Q_G/H(t²) divides the virtual Poincaré polynomial of every regular embedding of G/H, if H is connected.     

An Isolated Umbilical Point of the Graph of a Homogeneous Polynomial

Suppose that the origin o of R ³ is an isolated umbilical point of the graph of a homogeneous polynomial in two real variables of degree k3. Then we see that the index of o is an element of the set 1–k/2+i ^[k/2] _i=0. Moreover, we see that each element of 1–k/2+i ^[k/2] _i=0 may be the index of o on the graph of a suitable homogeneous polynomial of degree k.