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This paper introduces and analyzes a class of directionally reinforced random walks. The work is motivated by an elementary model for time and space correlations in ocean surface wave fields. We develop some basic properties of these walks. For instance, we investigate recurrence properties and give conditions under which the limiting continuous versions of the walks are Gaussian diffusion processes.

The weight lattice of a crystallographic root system is partially ordered by the rule that λ > μ if λ − μ is a nonnegative integer linear combination of positive roots. In this paper, we study the subposet formed by the dominant weights. In particular, we prove that λ covers μ in this partial order only if λ − μ belongs to a distinguished subset of the positive roots. Also, if the root system is irreducible, we prove that the Mobius function of the partial order takes on only the values {0, ±1, ±2}.

/quad A regular polytope is locally toroidal if its minimal sections which are not of spherical type are toroids. The locally toroidal polytopes of rank 4 have been extensively discussed, and their classification is now nearly complete. In this paper, the locally toroidal polytopes of higher rank are investigated. Again, an almost complete classification of these regular polytopes is obtained; as well as sporadic examples, there are several infinite families.

Let F denote a field of characteristic different from two. In this paper we describe the mod2 cohomology of a Galois group GF (called the W-group of F) which is known to essentially characterize the Witt ring WF of anisotropic quadratic modules over F. We show that H*(GF, F2) contains the mod2 Galois cohomology of F and that its structure will reflect important properties of the field. We construct a space XF endowed with an action of an elementary abelian group E such that the computation of the cohomology of GF reduces to calculating the equivariant cohomology H*E(XF, F2). For the case of a field which is not formally real this amounts to computing the cohomology of an explicit Euclidean space form, an object which is interesting in its own right. We provide a number of examples and a substantial combinatorial computation for the cohomology of the universal W-groups.

In this paper we present a self-contained and detailed exposition of the new renormalization group technique proposed in [1, 2]. Its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator, the partition function, or correlation functions. We apply this renormalization transformation to a Hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field, and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing.

The study of the action of the Steenrod algebra on the mod $p$ cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of Steenrod operations on the cohomology of Grassmannians, both in the Borel and the Schubert picture. We consider integral lifts of Steenrod operations, which lie in a certain Hopf algebra of differential operators. The latter has been considered recently as a realization of the Landweber-Novikov algebra in complex cobordism theory; it also has connections with the action of the Virasoro algebra on the boson Fock space. Our formulas for Steenrod operations are based on combinatorial methods which have not been used before in this area, namely Hammond operators and the combinatorics of Schur functions. We also discuss several applications of our formulas to the geometry of Grassmannians.

We consider algebras with one binary operation $\cdot$ and one generator ({\it monogenic}) and satisfying the left distributive law $a\cdot (b\cdot c)=(a\cdot b)\cdot (a\cdot c)$. One can define a sequence of finite left-distributive algebras $A_n$, and then take a limit to get an infinite monogenic left-distributive algebra~$A_\infty$. Results of Laver and Steel assuming a strong large cardinal axiom imply that $A_\infty$ is free; it is open whether the freeness of $A_\infty$ can be proved without the large cardinal assumption, or even in Peano arithmetic. The main result of this paper is the equivalence of this problem with the existence of a certain algebra of increasing functions on natural numbers, called an {\it embedding algebra}. Using this and results of the first author, we conclude that the freeness of $A_\infty$ is unprovable in primitive recursive arithmetic.