Imbedding problem with given localizations and ramification restrictions

For a solvable imbedding problem (L/K,G,A) of a number field with Abelian kernel A and a finite set of points S of the field K, a finite set T of points of the field K, is found which has the following property: for any solvable imbedding problem with given localizations, corresponding to the problem (L/K,G,A), there exists a solution which is unramified outside of SuT.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 57, pp. 85–99, 1976.     

Matroids with sylvester property

Let M, be a matroid of rankn, n being a non-negative integer. Let the matroid have the property that any two distinct cells ofM are contained in a circuit of size 3. Then we prove that |M| 2 ⁿ –1. We also establish that, up to isomorphism, the matroid corresponding to the non-nulln-tuples of the elements of GF(2) is the unique extremal matroid.     

Matroids with few non-common bases

In [On Mills's conjecture on matroids with many common bases, Discrete Math. 240 (2001) 271-276], Lemos proved a conjecture of Mills [On matroids with many common bases, Discrete Math. 203 (1999) 195-205]: for two (k+1)-connected matroids whose symmetric difference between their collections of bases has size at most k, there is a matroid that is obtained from one of these matroids by relaxing n₁ circuit-hyperplanes and from the other by relaxing n₂ circuit-hyperplanes, where n₁ and n₂ are non-negative integers such that n₁+n₂≤k. In [Matroids with many common bases, Discrete Math. 270 (2003) 193-205], Lemos proved a similar result, where the hypothesis of the matroids being k-connected is replaced by the weaker hypothesis of being vertically k-connected. In this paper, we extend these results.     

Matroids with many common bases

Lemos (Discrete Math. 240 (2001) 271–276) proved a conjecture of Mills (Discrete Math. 203 (1999) 195–205): for two (k+1)-connected matroids whose symmetric difference between their collections of bases has size at most k, there is a matroid that is obtained from one of these matroids by relaxing n₁ circuit-hyperplanes and from the other by relaxing n₂ circuit-hyperplanes, where n₁ and n₂ are non-negative integers such that n₁+n₂k. In this paper, we prove a similar result, where the hypothesis of the matroids being k-connected is replaced by the weaker hypothesis of being vertically k-connected.     

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.     

Matroids and Graphs with Few Non-Essential Elements

An essential element of a 3-connected matroid M is one for which neither the deletion nor the contraction is 3-connected. Tutte's Wheels and Whirls Theorem proves that the only 3-connected matroids in which every element is essential are the wheels and whirls. In an earlier paper, the authors showed that a 3-connected matroid with at least one non-essential element has at least two such elements. This paper completely determines all 3-connected matroids with exactly two non-essential elements. Furthermore, it is proved that every 3-connected matroid M for which no single-element contraction is 3-connected can be constructed from a similar such matroid whose rank equals the rank in M of the set of elements e for which the deletion M\e is 3-connected.     

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.     

Basis-Transitive Matroids

We consider the problem of classifying all finite basis-transitive matroids and reduce it to the classification of the finite basis-transitive and point-primitive simple matroids (or geometric lattices, or dimensional linear spaces). Our main result shows how a basis- and point-transitive simple matroid is decomposed into a so-called supersum. In particular each block of imprimitivity bears the structure of two closely related simple matroids, and the set of blocks of imprimitivity bears the structure of a point- and basis-transitive matroid.     

Lagrangian Matroids and Cohomology

We prove that bigtriangleup bigtriangleup -matroids associated with maps on compact closed surfaces are representable, with the space of representation provided by cohomology of the surface with punctured points.     

Almost-Graphic Matroids

A nongraphic matroid M is said to be almost-graphic if, for all elements e, either M\e or M/e is graphic. We determine completely the class of almost-graphic matroids, thereby answering a question posed by Oxley in his book “Matroid Theory.” A nonregular matroid is said to be almost-regular if, for all elements e, either M\e or M/e is regular. An element e for which both M\e and M/e are regular is called a regular element. We also determine the almost-regular matroids with at least one regular element.     

Matroids and linking systems

With the help of the concept of a linking system, theorems relating matroids with bipartite graphs and directed graphs are deduced. In this way natural generalizations of theorems of Edmonds & Fulkerson, Perfect, Pym. Rado, Brualdi and Mason are obtained. Furthermore some other properties of these linking systems are investigated.     

Uniform Families and Count Matroids

Given an r-graph G on [n], we are allowed to add consecutively new edges to it provided that every time a new r-graph with at least l edges and at most m vertices appears. Suppose we have been able to add all edges. What is the minimal number of edges in the original graph? For all values of parameters, we present an example of G which we conjecture to be extremal and establish the validity of our conjecture for a range of parameters. Our proof utilises count matroids which is a new family of matroids naturally extending that of White and Whiteley. We characterise, in certain cases, the extremal graphs. In particular, we answer a question by Erdős, Füredi and Tuza. Received: May 6, 1998 Final version received: September 1, 1999     

Mutation Polynomials and Oriented Matroids

Several polynomials are of use in various enumeration problems concerning objects in oriented matroids. Chief among these is the Radon catalog. We continue to study these, as well as the total polynomials of uniform oriented matroids, by considering the effect on them of mutations of the uniform oriented matroid. The notion of a ``mutation polynomial' is introduced to facilitate the study. The affine spans of the Radon catalogs and the total polynomials in the appropriate rational vector spaces of polynomials are determined, and bases for the Z -modules generated by the mutation polynomials are found. The Radon polynomials associated with alternating oriented matroids are described; it is conjectured that a certain extremal property, like that held by cyclic polytopes among simplicial polytopes, is possessed by them. Received November 20, 1998, and in revised form August 21, 1999. Online publication May 19, 2000.     

Matroids without adjoint

The purpose of this note is to give an example of a rank-4 matroid which not only shows that Levi's intersection property is not a sufficient condition for the existence of an adjoint but also seems to have an interesting structure of the lattice of flats.     

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.