Facets and volume of Gorenstein Fano polytopes

It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension d is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension . Furthermore, investigation on combinatorial properties, especially, Ehrhart polynomials and volume of these new polytopes will be achieved. Finally, some curious examples of Gorenstein Fano polytopes will be discovered.     

Coxeter cones and their h-vectors

Coxeter cones are formed by intersecting the nonnegative sides of a collection of root hyperplanes in some root system. They are shellable subcomplexes of the Coxeter complex, and their h-vectors record the distribution of descents among their chambers. We identify a natural class of “graded” Coxeter cones with the property that their h-vectors are symmetric and unimodal, thereby generalizing recent theorems of Reiner-Welker and Brändén about the Eulerian polynomials of graded partially ordered sets.     

On h-convexity

We introduce a class of h-convex functions which generalize convex, s-convex, Godunova-Levin functions and P-functions. Namely, the h-convex function is defined as a non-negative function which satisfies f(αx+(1−α)y)?h(α)f(x)+h(1−α)f(y), where h is a non-negative function, α∈(0,1) and x,y∈J. Some properties of h-convex functions are discussed. Also, the Schur-type inequality is given.     

h-Stability for linear dynamic equations on time scales

We study the h-stability for linear dynamic equations on time scales and their perturbations by using the Bihari type inequality on time scales and the unified time scale quadratic Lyapunov functions.     

On properties of h-homogeneous spaces of first category

A metric space X is called h-homogeneous if and each nonempty open-closed subset of X is homeomorphic to X. We describe how to assign an h-homogeneous space of first category and of weight k to any strongly zero-dimensional metric space of weight ?k. We investigate the properties of such spaces. We show that if Q is the space of rational numbers and Y is a strongly zero-dimensional metric space, then Q×Yω is an h-homogeneous space and F×Q×Yω is homeomorphic to Q×Yω for any Fσ-subset F of Q×Yω. L. Keldysh proved that any two canonical elements of the Borel class α are homeomorphic. The last theorem is generalized for the nonseparable case.     

Unimodal sequences of Gorenstein ideals of codimension 4

We present Gorenstein ideals of condimension 4 which have unimodal Hilbert functions. The present studies were supported (in part) by the Basic Science Research Institute Program, BSRI 97-1423, Ministry of Education, Korea     

Socles of Buchsbaum modules, complexes and posets

The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel's conjecture for the maximum value of the Euler characteristic of a 2k-dimensional simplicial manifold on n vertices as well as Kalai's conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension, number of vertices, and the first Betti number.     

Coupling of h and p finite elements: Application to free vibration analysis of plates with curvilinear plan-forms

This paper presents a method for coupling isoparametric cubic quadrilateral h-elements and straight sided serendipity quadrilateral p-elements. The p-elements are used to model the interior of the domain while the h-elements are used to describe accurately the curved boundaries. At a common side shared by a p-element and an arbitrary number of h-elements, the field variables are minimized in the least square sense with respect to the degrees-of-freedom of the h-elements. This leads to a set of equations which relate the degrees-of-freedom of the coupled elements on the shared side. The method is applied to the calculation of frequencies for plates with curvilinear plan-forms. The effects of shear deformation and rotary inertia are taken into account. The frequencies are obtained for a sectorial plate with simply supported radial edges and free circular edge, an annular sectorial plate with simply supported radial edges and clamped circular edges, and a circular plate with one concentric ring support. Furthermore, new accurate frequencies are given for a fully clamped square plate with a corner cut-out. Constant meshes are used and convergence is sought by increasing progressively the degree p of the interpolating polynomial. The fast convergence and high accuracy of the method are validated through convergence and comparison studies.     

<Emphasis Type="Italic">f</Emphasis>-Vectors of barycentric subdivisions

For a simplicial complex or more generally Boolean cell complex Δ we study the behavior of the f- and h-vector under barycentric subdivision. We show that if Δ has a non-negative h-vector then the h-polynomial of its barycentric subdivision has only simple and real zeros. As a consequence this implies a strong version of the Charney–Davis conjecture for spheres that are the subdivision of a Boolean cell complex or the subdivision of the boundary complex of a simple polytope. For a general (d − 1)-dimensional simplicial complex Δ the h-polynomial of its n-th iterated subdivision shows convergent behavior. More precisely, we show that among the zeros of this h-polynomial there is one converging to infinity and the other d − 1 converge to a set of d − 1 real numbers which only depends on d. F. Brenti and V. Welker are partially supported by EU Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272 and the program on “Algebraic Combinatorics” at the Mittag-Leffler Institut in Spring 2005.     

Weighted Ehrhart theory and orbifold cohomology

We introduce the notion of a weighted δ-vector of a lattice polytope. Although the definition is motivated by motivic integration, we study weighted δ-vectors from a combinatorial perspective. We present a version of Ehrhart Reciprocity and prove a change of variables formula. We deduce a new geometric interpretation of the coefficients of the Ehrhart δ-vector. More specifically, they are sums of dimensions of orbifold cohomology groups of a toric stack.     

Monomial Ideals and the Gorenstein Liaison Class of a Complete Intersection

In an earlier work, the authors described a mechanism for lifting monomial ideals to reduced unions of linear varieties. When the monomial ideal is Cohen–Macaulay (including Artinian), the corresponding union of linear varieties is arithmetically Cohen–Macaulay. The first main result of this paper is that if the monomial ideal is Artinian then the corresponding union is in the Gorenstein linkage class of a complete intersection (glicci). This technique has some interesting consequences. For instance, given any (d + 1)-times differentiable O-sequence H, there is a nondegenerate arithmetically Cohen–Macaulay reduced union of linear varieties with Hilbert function H which is glicci. In other words, any Hilbert function that occurs for arithmetically Cohen–Macaulay schemes in fact occurs among the glicci schemes. This is not true for licci schemes. Modifying our technique, the second main result is that any Cohen–Macaulay Borel-fixed monomial ideal is glicci. As a consequence, all arithmetically Cohen–Macaulay subschemes of projective space are glicci up to flat deformation.     

The g-theorem matrices are totally nonnegative

The g-theorem proved by Billera, Lee, and Stanley states that a sequence is the g-vector of a simplicial polytope if and only if it is an M-sequence. For any d-dimensional simplicial polytope the face vector is gMd where Md is a certain matrix whose entries are sums of binomial coefficients. Björner found refined lower and upper bound theorems by showing that the (2×2)-minors of Md are nonnegative. He conjectured that all minors of Md are nonnegative and that is the result of this note.     

Newton polytopes,increments, and roots of systems of matrix functions for finite-dimensional representations

The asymptotic root distribution is computed for systems of matrix functions associated with finite-dimensional holomorphic representations of a Lie group. This distribution can be expressed via the increments of the representations involved. If the group is reductive, then the number of equations in the system can be arbitrary, from 1 to the dimension of the group. In this case, the computation results are stated in the language of convex geometry. These computations imply the previously known formulas for the density of the solution variety of a system of exponential equations as well as for the number of solutions of a polynomial system and, more generally, of a system formed by matrix functions of representations of a complex reductive Lie group.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 22–35, 2004Original Russian Text Copyright © by B. Ya. KazarnovskiiSupported by INTAS grant 00-259 and NWO grant 047.008.005.     

Lattice polytopes of degree 2

A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly i>0 interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimensions. In particular, there is only a finite number of quadratic polynomials with fixed leading coefficient being the h^∗-polynomial of a lattice polytope.     

Linear relations for p-harmonic functions

We discuss the p-harmonicity of the linear combination of p-harmonic functions in the Euclidean space and on a tree. If p≠2, the p-harmonicity is non-linear, i.e., the linear combination of p-harmonic functions need not be p-harmonic. In spite of this non-linear nature, we find some p-harmonic functions whose linear combinations become p-harmonic.     

The Murnaghan-Nakayama rule for k-Schur functions

We prove the Murnaghan-Nakayama rule for k-Schur functions of Lapointe and Morse, that is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene.     

Combinatorial interpretations of the q-Faulhaber and q-Salié coefficients

Recently, Guo and Zeng discovered two families of polynomials featuring in a q-analogue of Faulhaber's formula for the sums of powers and a q-analogue of Gessel-Viennot's formula involving Salié's coefficients for the alternating sums of powers. In this paper, we show that these are polynomials with symmetric, nonnegative integral coefficients by refining Gessel-Viennot's combinatorial interpretations.     

Examples of proper k-ball contractive retractions in F-normed spaces

Assume X is an infinite dimensional F-normed space and let r be a positive number such that the closed ball Br(X) of radius r is properly contained in X. The main aim of this paper is to give examples of regular F-normed ideal spaces in which there is a 1-ball or a (1+ε)-ball contractive retraction of Br(X) onto its boundary with positive lower Hausdorff measure of noncompactness. The examples are based on the abstract results of the paper, obtained under suitable hypotheses on X.