拟阵Q6与W4可F-线性表示的构造

对拟阵Q6与W4可F-线性表示的构造进行了研究.用E(G)在R上的链群F0(G,R)表示G的圈拟阵M(G);用松弛拟阵M的极小圈超平面X的方法得到拟阵M'.得到主要结果为:(i)用链群表示了M(K4),M(W4);(ii)用松弛极小圈超平面的方法从M(K4)构造了Q6,从M(W4)构造了W4,找出了W4可线性表示的所有...     

在1/2Z~+中的(F-Z_1)-可流拟阵(M-Z_1)/Z_2

研究在1/2Z⁺中的F-可流拟阵的幼阵的可流性.首先给出F-可流拟阵的充要条件及在1/2Z+中的F-可流拟阵的幼阵的可流性.首先给出F-可流拟阵的充要条件及在1/2Z⁺中是F-可流拟阵的定义.证明了辅助命题:若拟阵是无环元的,则它的每个元素都恰在k个余极小圈之中;对满足一定的条件的极小圈集合,成立最小极小圈集合的等式.设映射p′在幼阵中满足1/2Z+中是F-可流拟阵的定义.证明了辅助命题:若拟阵是无环元的,则它的每个元素都恰在k个余极小圈之中;对满足一定的条件的极小圈集合,成立最小极小圈集合的等式.设映射p′在幼阵中满足1/2Z⁺中(F-Z_1)-可流拟阵的不等式,由p′定义p.证明p在拟阵中满足同样的不等式.由映射Φ满足是12/Z+中(F-Z_1)-可流拟阵的不等式,由p′定义p.证明p在拟阵中满足同样的不等式.由映射Φ满足是12/Z⁺中F-可流拟阵的等式,可找到最小属于幼阵的极小圈,定义Φ′(C′)则可证明Φ′满足在1/2Z+中F-可流拟阵的等式,可找到最小属于幼阵的极小圈,定义Φ′(C′)则可证明Φ′满足在1/2Z⁺中是(F-Z_1)-可流的等式.即由在1/2Z+中是(F-Z_1)-可流的等式.即由在1/2Z⁺中F-可流拟阵的充要条件,证明了幼阵在1/2Z+中F-可流拟阵的充要条件,证明了幼阵在1/2Z⁺中是(F-Z_1)-可流的.     

双圈拟阵

Sim■es Pereira于1992年提出双圈拟阵.本文讨论了(i)双圈拟阵及其秩函数;(ii)次模函数在双圈拟阵中的应用;(iii)双圈拟阵B(G)的横贯拟阵.主要结果:1°由圈矩阵Bf=[I,Bf12]和圈秩的概念,推出M(f0)为双圈拟阵;2°证明了双圈拟阵B(G)等于由子集族{Av∶v∈V(G)},e与v在G中相关联}所确定的横贯拟阵;3°用不同于Matthews(1977)的方法证明了(iii).     

超拟阵的独立集公理、基公理和圈公理

Dunstan等在1972年首先提出了超拟阵的概念,用以将定义拟阵的承载集合从有限集推广到偏序集.Barnabei等在1998年研究了另一种偏序集上的拟阵结构,即偏序集拟阵.由有限分配格和有限偏序集之间的对应关系可知,偏序集拟阵就是分配格上的超拟阵.本文研究超拟阵的公理系统,建立模格上的超拟阵的独立元公理,证明模格上超拟阵的中间基性质和基的交换性质并用这两个性质分别刻画了模超拟阵.最后指出了Barnabei等给出的分配超拟阵圈公理中的一个错误,重新提出并证明分配超拟阵的圈消去性质并建立了分配超拟阵的圈公理.作为圈消去性质的一个应用,本文证明了分配超拟阵中覆盖基的元素包含唯一的圈.     

关于模糊拟阵的模糊秩研究

本文受正规模糊拟阵启发,定义了普通模糊拟阵的正规模糊基概念;然后利用基子集套方法,证明了闭模糊拟阵存在正规模糊基,在同一模糊拟阵中的正规模糊基的模糊势相等,正规模糊基的模糊势是同一模糊拟阵中的模糊基的最大模糊势等性质。通过这些性质,给出了用正规模糊基描述的闭正规模糊拟阵的充要条件。还利用这些性质,得到计算正规模糊基模糊势的公式;最后拓展普通拟阵的秩定义了一般模糊拟阵的模糊秩。通过模糊拟阵的闭包概念,证明了模糊拟阵的模糊秩等于正规模糊基的模糊势,并得到计算模糊拟阵模糊秩的公式。同时,详细讨论了模糊拟阵模糊秩的许多性质,还对利用模糊拟阵模糊秩研究模糊拟阵做了一点尝试。模糊秩是模糊拟阵的固有特征之一,通过模糊秩来研究模糊拟阵,或者从模糊拟阵来讨论模糊秩都有大量工作可以做。     

关于模糊拟阵的圈子集套

本文主要采用通过导出拟阵来研究模糊拟阵的方法,探讨模糊拟阵模糊圈的性质和构造。这种方法的基本原理是两条:闭模糊拟阵可以由其导出拟阵序列和基本序列唯一确定,而模糊圈可以被分解为导出拟阵的圈和独立子集套。借助这种方法,本文主要做了三方面工作:一是讨论了模糊拟阵的模糊圈集和导出拟阵圈集之间的关系。比如模糊圈、初等模糊圈和最大初等模糊圈与导出拟阵圈之间的关系等;二是基于模糊圈和导出拟阵圈之间的关系,定义了导出拟阵圈函数和导出拟阵圈子集套两个概念。然后,详细研究了利用这两个概念来构造模糊圈的方法。同时,分析了在圈子集套和数列满足什么条件时,这种方法有效;三是分别用导出拟阵圈和圈子集套给出了准模糊图拟阵和精细模糊拟阵的充要条件。     

关于模糊横贯拟阵表示的初步研究

本文首先讨论了模糊子集族的全体模糊部分横贯能够形成模糊横贯拟阵的条件,得到两个充要条件。第一个充要条件使用截短模糊子集族来进行描述;第二个充要条件是通过部分横贯的指标集单射兼容这个概念来描述。然后研究了模糊横贯拟阵的表示形式,得到了三个结论。一是任何模糊横贯拟阵都有"精简表示",而且构造了从一个表示寻找精简表示的方法;二是模糊横贯拟阵的表示的模糊子集个数不小于导出拟阵的最大秩;三是一致模糊横贯拟阵的任何表示都包含"最小表示",也给出了求这个"最小表示"的方法。     

双曲空间中旋转对称的极小超曲面

本文给出了双曲空间Hⁿ中旋转对称的极小超曲面的微分方程,对这类超曲面进行了分类,它们是Hⁿ中的超平面或广义悬链面,而每个广义悬链面被夹在两个平行的超平面之间,且以这两个超平面为渐近平面.     

解非线性单调方程组的投影自调比对称秩1拟牛顿法

本文给出了求解非线性单调方程组的两个自调比对称秩1牛顿法,即投影SSR1法和投影有限储存SSR1法.这两个算法将自调比对称秩1校正参数进行了一个简单的修改并采用了保守策略.在非线性单调函数满足李普希茨连续的条件下,证明了算法的全局收敛性,并与相同类型的BFGS法进行了初步的数值比较试验,试验结果表明自调比对称秩1类投影算法求解非线性单调方程组与相同类型的BFGS数值结果相当.     

理想对称模

本文引进了理想对称模的概念,给出了理想对称模的系列等价刻画,用理想对称模给出了环R为理想对称环的若干等价条件,证明了对于环R的满足右Ore条件正则元的集S,如果S-挠自由R-模M是理想对称模,则M关于S的右分式模也是理想对称的,推广了理想对称环的相应结果.     

ContributionsOn binary k-paving matroids and Reed-Muller codes

A matroid M of rank r k is k-paving if all of its circuits have cardinality exceeding r − k. In this paper, we develop some basic results concerning k-paving matroids and their connections with codes. Also, we determine all binary 2-paving matroids.     

On properties of almost all matroids

We give several results about the asymptotic behaviour of matroids. Specifically, almost all matroids are simple and cosimple and, indeed, are 3-connected. This verifies a strengthening of a conjecture of Mayhew, Newman, Welsh, and Whittle. We prove several quantitative results including giving bounds on the rank, a bound on the number of bases, the number of circuits, and the maximum circuit size of almost all matroids.     

On the matroids in which all hyperplanes are binary

In this paper, it is shown that, for a minor-closed class of matroids, the class of matroids in which every hyperplane is in is itself minor-closed and has, as its excluded minors, the matroids U_1.1 N such that N is an excluded minor for . This result is applied to the class of matroids of the title, and several alternative characterizations of the last class are given.     

Characterizing 3‐connected planar graphs and graphic matroids

A well‐known result of Tutte states that a 3‐connected graph G is planar if and only if every edge of G is contained in exactly two induced non‐separating circuits. Bixby and Cunningham generalized Tutte's result to binary matroids. We generalize both of these results and give new characterizations of both 3‐connected planar graphs and 3‐connected graphic matroids. Our main result determines when a natural necessary condition for a binary matroid to be graphic is also sufficient. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 165–174, 2010     

On matroids with few circuits containing a pair of elements

In this note, we construct all the matroids that have a pair of elements belonging to just one of its circuits. We use this result to establish that, with two small exceptions, wheels and whirls are the only 3-connected matroids having a pair of elements contained in exactly two of its circuits.     

Tutte-martin polynomials and orienting vectors of isotropic systems

Isotropic systems are structures which unify some properties of 4-regular graphs and of pairs of dual binary matroids. In this paper we unify the properties of the symmetric Tutte polynomials (i.e. with equal variables) of binary matroids and of the Martin polynomials of 4-regular graphs. For this purpose we introduce the orienting vectors of an isotropic system in order to generalize the eulerian orientations of 4-regular graphs.     

Extension spaces of oriented matroids

We study the space of all extensions of a real hyperplane arrangement by a new pseudohyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the “Generalized Baues Problem” of Billera, Kapranov, and Sturmfels, via the Bohne-Dress theorem on zonotopal tilings. We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not known whether the extension space is connected for all realizable oriented matroids (hyperplane arrangements). We show that the subspace of realizable extensions is always connected but not necessarily spherical. Nonrealizable oriented matroids of rank 4 with disconnected extension spaces were recently constructed by Mnëv and Richter-Gebert.     

Characterizing binary simplical matroids

In an earlier paper we defined a class of matroids whose circuit are combinatorial generalizations of simple polytopes; these matroids are the binary analogue of the simplical geometrics of Crapo and Rota. Here we find necessary and sufficient conditions for a matroid to be isomorphic to such a binary simplical matroid.     

On finite matroids with two more hyperplanes than points

One of the most interesting results about finite matroids of finite rank and generalized projective spaces is the result of Basterfield, Kelly and Green (1968/1970) (J.G. Basterfield, L.M. Kelly, A characterization of sets of n points which determine n hyperplanes, in: Proceedings of the Cambridge Philosophical Society, vol. 64, 1968, pp. 585-588; C. Greene, A rank inequality for finite geometric lattices, J. Combin Theory 9 (1970) 357-364) affirming that any matroid contains at least as many hyperplanes as points, with equality in the case of generalized projective spaces. Consequently, the goal is to characterize and classify all matroids containing more hyperplanes than points. In 1996, I obtained the classification of all finite matroids containing one more hyperplane than points. In this paper a complete classification of finite matroids with two more hyperplanes than points is obtained. Moreover, a partial contribution to the classification of those matroids containing a certain number of hyperplanes more than points is presented.     

Matroid Enumeration for Incidence Geometry

Matroids are combinatorial abstractions for point configurations and hyperplane arrangements, which are fundamental objects in discrete geometry. Matroids merely encode incidence information of geometric configurations such as collinearity or coplanarity, but they are still enough to describe many problems in discrete geometry, which are called incidence problems. We investigate two kinds of incidence problem, the points–lines–planes conjecture and the so-called Sylvester–Gallai type problems derived from the Sylvester–Gallai theorem, by developing a new algorithm for the enumeration of non-isomorphic matroids. We confirm the conjectures of Welsh–Seymour on ≤11 points in ℝ³ and that of Motzkin on ≤12 lines in ℝ², extending previous results. With respect to matroids, this algorithm succeeds to enumerate a complete list of the isomorph-free rank 4 matroids on 10 elements. When geometric configurations corresponding to specific matroids are of interest in some incidence problems, they should be analyzed on oriented matroids. Using an encoding of oriented matroid axioms as a boolean satisfiability (SAT) problem, we also enumerate oriented matroids from the matroids of rank 3 on n≤12 elements and rank 4 on n≤9 elements. We further list several new minimal non-orientable matroids.