Two Decompositions in Topological Combinatorics with Applications to Matroid Complexes

This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of M-shellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the rank-numbers of M-shellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex ear-decomposition, and, using results of Kalai and Stanley on -vectors of simplicial polytopes, we show that -vectors of pure rank- simplicial complexes that have this property satisfy and for . We then show that the abstract simplicial complex formed by the collection of independent sets of a matroid (or matroid complex) admits a special type of convex ear-decomposition called a PS ear-decomposition. This enables us to construct an associated M-shellable poset, whose set of rank-numbers is the -vector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi that the -vector of a matroid complex satisfies the above two sets of inequalities.

     

第一类Cartan—Egg或的Bergman核函数

以显式给出了第一类Cartan-Egg域的Bergman核函数及其全纯自同构群。     

一类Hartogs域的Bergman核函数的零点问题(英文)

We obtain the Bergman kernel for a new type of Hartogs domain.The corresponding LU Qi-Keng’s problem is considered.     

A Note on the Sticky Matroid Conjecture

A matroid is sticky if any two of its extensions by disjoint sets can be glued together along the common restriction (that is, they have an amalgam). The sticky matroid conjecture asserts that a matroid is sticky if and only if it is modular. Poljak and Turzík proved that no rank-3 matroid having two disjoint lines is sticky. We show that, for r ≥ 3, no rank−r matroid having two disjoint hyperplanes is sticky. These and earlier results show that the sticky matroid conjecture for finite matroids would follow from a positive resolution of the rank-4 case of a conjecture of Kantor.     

New classes of domains with explicit Bergman kernel

We introduce two classes of egg type domains, built on general bounded sym-metric domains, for which we obtain the Bergman kernel in explicit formulas. As an aux-iliary tool, we compute the integral of complex powers of the generic norm on a boundedsymmetric domains using the well-known integral of Selberg. This generalizes matrix in-tegrals of Hua and leads to a special polynomial with integer or half-integer coefficientsattached to each irreducible bounded symmetric domain.     

Matroid bases with cardinality constraints on the intersection

Mathematical Programming - Given two matroids $$\mathcal {M}_{1} = (E, \mathcal {B}_{1})$$ and $$\mathcal {M}_{2} = (E, \mathcal {B}_{2})$$ on a common ground set E with base sets $$\mathcal...     

Density of algebras generated by Toeplitz operators on Bergman spaces

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC ^*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA ²(Ω) for a wide class of plane domains Ω⊂C, and in Fock spacesA ²(C ^N),N≧1.     

An essay on Bergman completeness

We give first of all a new criterion for Bergman completeness in terms of the pluricomplex Green function. Among several applications, we prove in particular that every Stein subvariety in a complex manifold admits a Bergman complete Stein neighborhood basis, which improves a theorem of Siu. Secondly, we give for hyperbolic Riemann surfaces a sufficient condition for when the Bergman and Poincaré metrics are quasi-isometric. A consequence is an equivalent characterization of uniformly perfect planar domains in terms of growth rates of the Bergman kernel and metric. Finally, we provide a noncompact Bergman complete pseudoconvex manifold without nonconstant negative plurisubharmonic functions.     

Compact Operators on Bergman Spaces

We prove that a bounded operator S on L _a ^p for p > 1 is compact if and only if the Berezin transform of S vanishes on the boundary of the unit disk if S satisfies some integrable conditions. Some estimates about the norm and essential norm of Toeplitz operators with symbols in BT are obtained.     

Density of algebras generated by Toeplitz operators on Bergman spaces

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC ^*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA ²(Ω) for a wide class of plane domains Ω?C, and in Fock spacesA ²(C ^N),N≧1.     

The Bergman Kernels on W_Ⅲ

In this paper, we compute the Bergman kernel function on WIII.and RIII(q) denote the Cartan domain of the third class. Because domain WIII is neither homogeneous domain nor Reinhardt domain, we will use a new way to solve this problem. First, we give a holomorphic automorphism group, such that for any Zo, there exists an element of this group, which maps (W, Zo) into (W,O). Second, introduce the concept of semi-Reinhardt and discuss the complete orthonormal system of this domain.     

Area operator on Bergman spaces

We characterize the non-negative measuresμon the unit disk D for which the area operator Aμis bounded from Bergman space Aαp to Lq ((?)D).     

Harmonic Bergman Functions on Half-Spaces

We study harmonic Bergman functions on the upper half-space of . Among our main results are: The Bergman projection is bounded for the range ; certain nonorthogonal projections are bounded for the range ; the dual space of the Bergman -space is the harmonic Bloch space modulo constants; harmonic conjugation is bounded on the Bergman spaces for the range ; the Bergman norm is equivalent to a ``normal derivative norm' as well as to a ``tangential derivative norm'.

     

Toeplitz operators on Bergman spaces

Let ? be an element in \(H^\infty (D) + C(\overline D )\) such that ?^* is locally sectorial. In this paper it is shown that the Toeplitz operator defined on the Bergman spaceA ² (D) is Fredholm. Also, it is proved that ifS is an operator onA ²(D) which commutes with the Toeplitz operatorT _? whose symbol ? is a finite Blaschke product, thenS H ^∞ (D) is contained inH ^∞ (D). Moreover, some spectral properties of Toeplitz operators are discussed, and it is shown that the spectrum of a class of Toeplitz operators defined on the Bergman spaceA ² (D), is not connected.     

Operators on harmonic Bergman spaces

We study the commutator of the multiplication and harmonic Bergman projection, Hankel and Toeplitz operators on the harmonic Bergman spaces. The same type operators have been well studied on the analytic Bergman spaces. The main difficulty of this study is that the bounded harmonic function space is not an algebra! In this paper, we characterize theL ^p boundedness and compactness of these operators with harmonic symbols. Results about operators in Schatten classes, the cut-off phenomenon and general symbols are also included.Partially supported by a grant from the Research Grants Committee of the University of Alabama.     

关于广义Bergman空间的乘子

In this paper,we characterize the multipliers of generalized Bergman spaces Ap,q,αwith 00) in almost every case considered. The corollaries on multipliers of the spaces Ap,q,αextend some related results.     

有界域的Bergman核函数显式表示的最新进展

对多维复数空间的有界域，如何求出它的Bergman核函数的显表达式，是多复变研究中的一个重要方向。本文综述了迄今为止的所有重要结果以及方法上的进展，特别对新近引进的华罗域，综述了它们的Bergman核函数的显表达式及其计算方法上的创新。     

拟阵闭包算子诱导拓扑空间的分离性

拟阵是图论与代数的结合,而拓扑学是一门重要的数学基础学科.利用拟阵理论研究拓扑空间,从而将两者进行融合,通过拟阵的闭包算子定义拟阵拓扑空间,进一步利用拟阵闭包算子与拓扑闭包算子之间的关系,讨论拟阵拓扑空间的分离性.     

由上近似数诱导的拟阵及其特征

拟阵理论与粗糙集理论之间有很多相似之处,近年来,探讨这二者之间的联系成为一个研究热点.首先利用基于等价关系的上近似数诱导了一系列拟阵结构,然后刻画了这类拟阵的独立集、基集、秩函数以及闭包等,讨论了与其它拟阵之间的一些联系.     

Decomposing 40 billion integers by four tetrahedral numbers

Based upon a computer search performed on a massively parallel supercomputer, we found that any integer less than billion (B) but greater than can be written as a sum of four or fewer tetrahedral numbers. This result has established a new upper bound for a conjecture compared to an older one, B, obtained a year earlier. It also gives more accurate asymptotic forms for partitioning. All this improvement is a direct result of algorithmic advances in efficient memory and cpu utilizations. The heuristic complexity of the new algorithm is compared with that of the old, .