A Note on Hitting Maximum and Maximal Cliques With a Stable Set

It was recently proved that any graph satisfying contains a stable set hitting every maximum clique. In this note, we prove that the same is true for graphs satisfying unless the graph is the strong product of an odd hole and .We also provide a counterexample to a recent conjecture on the existence of a stable set hitting every sufficiently large maximal clique.     

Maximal Strings in the Crystal Graph of Spin Representations of the Symmetric and Alternating Groups

We define a block-reduced version of the crystal graph of spin representations of the symmetric and alternating groups, and separate it into layers, each obtained by translating the previous layer and, possibly, adding new defect zero blocks. We demonstrate that each layer has weight-preserving central symmetry, and study the sequence of weights occurring in the maximal strings.

The Broué conjecture, that a block with abelian defect group is derived equivalent to its Brauer correspondent, has been proven for blocks of cyclic defect group and verified for many other blocks. This article is part of a study of the spin block case.     

The Structure of the Reduced Product of Preinjective Modules Over a Hereditary Right Pure Semisimple Ring

Abstract

For a monoid S , a (left) S -act is a nonempty set B together with a mapping S ×B→B sending (s, b) to sb such that S (tb)?=?lpar;st)b and 1b ?=?b for all S , t?∈?S and B ?∈?B. Right S -acts A can also be defined, and a tensor product A ??? _s B (a set)can be defined that has the customary universal property with respect to balanced maps from A?×?B into arbitrary sets. Over the past three decades, an extensive theory of flatness properties has been developed (involving free and projective acts, and flat acts of various sorts, defined in terms of when the tensor product functor has certain preservation properties). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by M. Kilp et al. (New York: Walter de Gruyter, 2000). To date, there have been only a few attempts to generalize this material to ordered monoids acting on partially ordered sets ( S -posets). The present paper is devoted to such a generalization. A unique decomposition theorem for S -posets is given, based on strongly convex, indecomposable S -subposets, and a structure theorem for projective S -posets is given. A criterion for when two elements of the tensor product of S -posets given, which is then applied to investigate several flatness properties.     

Vaught's Conjecture and Group Rings

We discuss Vaught's conjecture for complete theories of modules over some group rings over the integers and other Dedekind domains, and over pullback rings of Dedekind domains.     

A character theoretic proof of the homological sylow theorem

ABSTRACT

Motivated by the works of Cuntz and Quillen on cyclic homology and algebra extensions, we study higher traces on group rings.     

Witt equivalence of local and global fields of characteristic 2

ABSTRACT

In this note it is proved that certain level sets of some real proper polynomial maps are nothing but spheres. As an application of this, we provide new proofs of Theorems 1.1, 1.2 and of the fundamental theorem of algebra. In addition, we show that every strictly convex (concave) polynomial map is proper. The latter implies that every real polynomial map g(x): R ⁿ  → R ⁿ , whose Jacobian matrix is symmetric and has nonzero eigenvalues of the same sign, is a homeomorphism of R ⁿ onto R ⁿ .     

Torsion Units in the Integral Group Ring of the Alternating Group of Degree 6

It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring ?G of a finite group G conjugates to a group element within the rational group algebra ?G. We investigate the Zassenhaus Conjecture (ZC) and a conjecture by W. Kimmerle about prime graph in the normalized unit group of ?A₆.