On 2-sylow intersections

Letz be an involution in the finite groupG and suppose thatz belongs to the center of a Sylow subgroup ofG. Ifz belongs to a unique Sylow subgroup ofG and ifG is not a trivial intersection group, thenG is not a simple group.     

On central 2-sylow intersections

In this paper we classify finite groups, in which a center of a Sylow 2-subgroup is contained in no more than six distinct Sylow 2-subgroups. This paper is part of the author’s Ph.D. thesis, done at Tel Aviv University under the supervision of Professor M. Herzog.     

On simple groups with a quaternion maximal 2-sylow intersection

Finite simple groupsG with a generalized quaternion maximal 2-Sylow intersectionV are determined under the assumption that [G: N _G (V)] is odd.     

On matroid intersections

This paper exploits and extends results of Edmonds, Cunningham, Cruse and McDiarmid on matroid intersections. Letr ₁ andr ₂ be rank functions of two matroids defined on the same setE. For everyS ⊂E, letr ₁₂(S) be the largest cardinality of a subset ofS independent in both matroids, 0≦k≦r ₁₂(E)−1. It is shown that, ifc is nonnegative and integral, there is ay: 2 ^E →Z ⁺ which maximizes and , subject toy≧0, ∀j∈E, .     

On almost complete intersections

Let R be a local ring such that R=S/I where S is a regular local ring and I is a prime ideal of height r. In this paper it is shown that if I is minimally generated by r+1 elements, then there exists an R-homomorphism : K_RR^r+1 such that is an injection and R^r+1/(K_R)I/I² where K_R:=Ext _S ^r (R,S) the canonical module of R. Moreover, in case where S is a locality over a perfect field k, it is also shown that if R is Cohen-Macaulay and I² is a primary ideal, then the homological dimension of the differential module _R/k is infinite.The author wishes to thank his colleague Mr.Y.Aoyama for valuable discussions in connection with this subject.     

On the regularity of products and intersections of complete intersections

This paper proves the formulae

for arbitrary monomial complete intersections and , and provides examples showing that these inequalities do not hold for general complete intersections.

     

On intersections of interval graphs

If one can associate with each vertex of a graph an interval of a line, so that two intervals intersect just when the corresponding vertices are joined by an edge, then one speaks of an interval graph.It is shown that any graph on v vertices is the intersection (“product”) of at most [$\text{1}\text{2}$v] interval graphs on the same vertex set.For v=2k, k factors are necessary for, and only for, the complete k-partite graph K_2,2,…,2.Some results for the hypergraph generalization of this question are also obtained.     

On set systems determined by intersections

The set systems determined by intersections are studied and a sufficient condotion for this property is given. For case of graphs a necessary and sufficient condition is established. Some connections to other results are discussed.     

On Landsberg's criterion for complete intersections

Partially supported by ISF grant MSC000     

On complete intersections with trivial canonical class

We prove birational boundedness results on complete intersections with trivial canonical class of base point free divisors in (some version of) Fano varieties. Our results imply in particular that Batyrev–Borisov toric construction produces only a bounded set of Hodge numbers in any given dimension, even as the codimension is allowed to grow.     

On the homotopy type of complete intersections

The homotopy classification problem for complete intersections is settled when the complex dimension is larger than the total degree.     

On the possible dimensions of subspace intersections

A problem concerning the dimensions of the intersections of a subspace in the direct sum of a finite number of the finite-dimensional vector spaces with the pairwise sums of direct summands, provided that the subspace intersection with these direct summands is zero has been discussed. The problem is naturally divided into two ones, namely, the existence and representability of the corresponding matroid. Necessary and sufficient conditions of the existence of a matroid with the specified ranks of some subsets of the base set have been given. Using these conditions, necessary conditions of the existence of a matroid with a base set composed of a finite series of pairwise disjoint sets of the full rank and the given ranks of their pairwise unions have been presented. A simple graphical representation of the latter conditions has also been considered. These conditions are also necessary for the subspace to exist. At the end of the paper, a conjecture that these conditions are sufficient has also been stated.     

On intersections of conjugacy classes and bruhat cells

For a connected semisimple algebraic group G over an algebraically closed field k and a fixed pair (B, B ^– ) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB ^– is nonempty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution m _C ∈ 2 W associated to C. We prove that the element m _C is the unique maximal length element in its conjugacy class in W, and we classify all such elements in W. For G = SL(n + 1; k), we describe m _C explicitly for every conjugacy class C, and when w ∈ W ≌ S_n+1 is an involution, we give an explicit answer to when C ∩ (BwB) is nonempty.