Approximate Isomorphisms in <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

This paper is a survey on the Hyers–Ulam–Rassias stability of the following Cauchy–Jensen functional equation in C ^*-algebras:

Improved random graph isomorphism

Canonical labeling of a graph consists of assigning a unique label to each vertex such that the labels are invariant under isomorphism. Such a labeling can be used to solve the graph isomorphism problem. We give a simple, linear time, high probability algorithm for the canonical labeling of a G(n,p) random graph for p[ω(ln⁴n/nlnlnn),1−ω(ln⁴n/nlnlnn)]. Our result covers a gap in the range of p in which no algorithm was known to work with high probability. Together with a previous result by Bollobás, the random graph isomorphism problem can be solved efficiently for p[Θ(lnn/n),1−Θ(lnn/n)].     

Foundational Isomorphisms in Continuous Differentiation Theory

We provide an elementary proof of existence for the Foundational Isomorphism in each of the categories of convergence spaces, compactly generated topological spaces and sequential convergence spaces. This isomorphism embodies the germ of differentiation and its inverse the germ of integration.     

On the measure theoretic structure of compact groups

If is a compact group with , we show the following results:

(i)

There exist direct products of compact metric groups and continuous open surjections with respect to Haar measure; and

(ii)

the Haar measure on is Baire and at the same time Jordan isomorphic to the Haar measure on a direct product of compact Lie groups.

Applications of the above results in measure theory are given.

     

New designs by changing … the signs

We introduce an equivalence relation between difference families which declare two difference families to be similar if one can be obtained from the other by changing the signs of some base blocks. By means of an algorithm, written in PHP language and published at http://www.designtheory.it/tni, we investigate the problem of establishing how many non-isomorphic difference families are contained in a given similarity class.     

Projective push-forwards in the Witt theory of algebraic varieties

We define push-forwards along projective morphisms in the Witt theory of smooth quasi-projective varieties over a field. We prove that they have standard properties such as functoriality, compatibility with pull-backs and projection formulas.     

Neighborhood hypergraphs of bipartite graphs

Matrix symmetrization and several related problems have an extensive literature, with a recurring ambiguity regarding their complexity and relation to graph isomorphism. We present a short survey of these problems to clarify their status. In particular, we recall results from the literature showing that matrix symmetrization is in fact NP‐hard; furthermore, it is equivalent with the problem of recognizing whether a hypergraph can be realized as the neighborhood hypergraph of a graph. There are several variants of the latter problem corresponding to the concepts of open, closed, or mixed neighborhoods. While all these variants are NP‐hard in general, one of them restricted to the bipartite graphs is known to be equivalent with graph isomorphism. Extending this result, we consider several other variants of the bipartite neighborhood recognition problem and show that they all are either polynomial‐time solvable, or equivalent with graph isomorphism. Also, we study uniqueness of neighborhood realizations of hypergraphs and show that, in general, for all variants of the problem, a realization may be not unique. However, we prove uniqueness in two special cases: for the open and closed neighborhood hypergraphs of the bipartite graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 69–95, 2008     

Conjugation in Representations of the Zassenhaus Algebra

Let F be an algebracially closed field of characteristic p > 2, and L be the p ⁿ -dimensional Zassenhaus algebra with the maximal invariant subalgebra L ₀ and the standard filtration {L _i }| ^pn−2 _{i =−1}. Then the number of isomorphism classes of simple L-modules is equal to that of simple L ₀-modules, corresponding to an arbitrary character of L except when its height is biggest. As to the number corresponding to the exception there was an earlier result saying that it is not bigger than p ⁿ . Received May 10, 1999, Accepted December 8, 1999     

二型序数与良序二型类的同构

证明了二型序数具有如下性质对任一二型类良序结构［A,R］,存在惟一的二型序数B,使得［B,E］与［A,R］同构,特别地对良序聚会［