初等模糊拟阵与基本截片模糊拟阵的若干性质

对两种初等模糊拟阵和基本截片模糊拟阵的定义进行了比较,研究了它们之间的关系.研究了初等模糊拟阵的若干性质,得到了初等模糊拟阵和基本截片模糊拟阵为闭正则模糊拟阵等结论,给出了初等模糊拟阵的等价刻画以及初等模糊拟阵与其截拟阵之间的关系.     

On Random Representable Matroids

Results are obtained on the likely connectivity properties and sizes of circuits in the column dependence matroid of a random r × n matrix over a finite field, for large r and n. In a sense made precise in the paper, it is shown to be highly probable that when n is less than r such a matroid is the free matroid on n points, while if n exceeds r it is a connected matroid of rank r. Moreover, the connectivity can be strengthened under additional hypotheses on the growth of n and r, using the notion of vertical connectivity; and the values of k for which circuits of size k exist can be determined in terms of n and r.     

On Large Circuits in Matroids

M be a matroid with a maximum-sized circuit C of size at least four. This paper proves that, for k∈{2,3}, if M is k-connected, then every element of M is contained in a circuit of size at least . Even when M is 3-connected and binary, the presence of a large circuit in M does not guarantee that M has a large circuit containing a nominated pair of elements. However, when M is 3-connected and graphic, it will be shown that every pair of distinct elements is contained in a circuit of at least . Examples will be given to show that these results are best-possible and some related results will also be proved. Received: March 12, 1998 Final version received: October 23, 1998     

On Matroids and Orlik-Solomon Algebras

In this paper we axiomatize combinatorics of arrangements of affine hyperplanes, which is a generalization of matroids, called quasi-matroids. We show that quasi-matroids are equivalent to pointed matroids. On the other hand, the Orlik-Solomon (OS) algebra of a quasimatroid can be constructed. We prove that the OS algebra of a quasi-matroid is isomorphic to the direct image of the OS algebra of a matroid by the linear derivation.AMS Subject Classification: 03B35, 13D03, 52C35.     

On Crapo's Beta Invariant for Matroids

In this paper we characterize the matroids for which Crapo's beta invariant does not exceed 4. In addition, we relate the beta invariant of a matroid to its chromatic number and to its connectivity.     

On the Generation of Oriented Matroids

We provide a multiple purpose algorithm for generating oriented matroids. An application disproves a conjecture of Grünbaum that every closed triangulated orientable 2-manifold can be embedded geometrically in R ³ , i.e., with flat triangles and without self-intersections. We can show in particular that there exists an infinite class of orientable triangulated closed 2-manifolds for each genus g \geq 6 that cannot be embedded geometrically in Euclidean 3-space. Our algorithm is interesting in its own right as a tool for many investigations in which oriented matroids play a key role. Received January 7, 1999, and in final form July 16, 1999.     

On the Integral Representation of Spherical k-Regular Functions in Clifford Analysis

We study the null solutions of iterated applications of the spherical (Atiyah-Singer) Dirac operator on locally defined polynomial forms on the unit sphere of ; functions valued in the universal Clifford algebra , here called spherical k-regular functions. We construct the kernel functions, get the integral representation formula and Cauchy integral formula of spherical k-regular functions, and as applications, the weak solutions of higher order inhomogeneous spherical (Atiyah-Singer) Dirac equations . We obtain, in particular, the weak solution of an inhomogeneous spherical Poisson equation Δ _s g = f. This work was partially supported by NNSF of China (No.10471107) and RFDP of Higher Education (No.20060486001).     

Symplectic Matroids

A symplectic matroid is a collection B of k-element subsets of J = {1, 2, ..., n, 1*, 2*, ...; n*}, each of which contains not both of i and i* for every i n, and which has the additional property that for any linear ordering of J such that i j implies j* i* and i j* implies j i* for all i, j n, B has a member which dominates element-wise every other member of B. Symplectic matroids are a special case of Coxeter matroids, namely the case where the Coxeter group is the hyperoctahedral group, the group of symmetries of the n-cube. In this paper we develop the basic properties of symplectic matroids in a largely self-contained and elementary fashion. Many of these results are analogous to results for ordinary matroids (which are Coxeter matroids for the symmetric group), yet most are not generalizable to arbitrary Coxeter matroids. For example, representable symplectic matroids arise from totally isotropic subspaces of a symplectic space very similarly to the way in which representable ordinary matroids arise from a subspace of a vector space. We also examine Lagrangian matroids, which are the special case of symplectic matroids where k = n, and which are equivalent to Bouchet's symmetric matroids or 2-matroids.     

On the Number of Bases of Bicircular Matroids

Let t(G) be the number of spanning trees of a connected graph G, and let b(G) be the number of bases of the bicircular matroid B(G). In this paper we obtain bounds relating b(G) and t(G), and study in detail the case where G is a complete graph K_n or a complete bipartite graph K_n,m.Received April 25, 2003