Brown–Peterson Cohomology from Morava K-Theory

We give some structure to the Brown–Peterson cohomology (or its p-completion) of a wide class of spaces. The class of spaces are those with Morava K-theory even-dimensional. We can say that the Brown–Peterson cohomology is even-dimensional (concentrated in even degrees) and is flat as a BP*-module for the category of finitely presented BP*(BP)-modules. At first glance this would seem to be a very restricted class of spaces but the world abounds with naturally occurring examples: Eilenberg-Mac Lane spaces, loops of finite Postnikov systems, classifying spaces of most finite groups whose Morava K-theory is known (including the symmetric groups), QS²ⁿ, BO(n), MO(n), BO, Im J, etc. We finish with an explicit algebraic construction of the Brown–Peterson cohomology of a product of Eilenberg–Mac Lane spaces and a general Künneth isomorphism applicable to our situation.     

Representations of a class of lattice type vertex algebras

In this paper we study the representation theory for certain “half lattice vertex algebras.” In particular we construct a large class of irreducible modules for these vertex algebras. We also discuss how the representation theory of these vertex algebras are related to the representation theory of some associative algebras.     

Book review

Topics in Matrix Analysis, Roger A. Horn and Charles R. Johnson, Cambridge University Press, New York, 1991, viii+607 pages. ISBN 0-521-30587-X     

Centralizers in symmetric inverse semigroups: Structure and order

The representation [5] of the centralizerC(x) of a permutationx in (a symmetric inverse semigroup)C _n involves direct products of wreath products. Indeed, this semigroup case extends its group theory counterpart. Here, the last case (forx nilpotent) is addressed: A quotient of a wreath product is introduced and used to obtain a representation of the correspondingC(x). It follows that, for anyx∈C _n ,C(x) can be imbedded in a direct product of wreath products with a quotient of a wreath product. A formula for calculating the order ofC(x) is given. The independent parameters in the formula are precisely those that define the path structure ofx∈C _n . Part of this research was supported by a Mary Washington College faculty development grant.     

Ultrafast dynamics of polybutadiene probed by optically heterodyne-detected optical-Kerr-effect spectroscopy

The ultrafast dynamics of polybutadiene have been studied using ultrafast optical-Kerr-effect spectroscopy. The data are compared with measurements on 1,3- and 1,4-pentadiene. The two diene derivatives have quite distinct subpicosecond dynamics, with an important contribution from an intramolecular torsional mode in the 1,4-derivative. The main part of the polymer spectral density can be assigned, by analogy with the data for 1,4-pentadiene, to intramolecular torsional motion about carbon–carbon single bonds. The picosecond diffusive orientational relaxation times of the dienes are not well described by simple hydrodynamics.