Let

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be a presentation for the group G where F = 〈a1, …, an〉 is a free group. Put A = {a1, …, an a−11, …, a−1n} and call n the degree of the presentation. For g ∈ G, c ∈ A we let W(k, g, c) be the set of freely reduced words w ∈ F having length k, which end (on the right) in c and which represent the element g ∈ G. Let w(k, g, c) = |W(k, g, c)|. The function

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will be called the cogrowth function for this presentation. Let

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A countably infinite first-order structure M is ω-categorical if every countable N which has the same first-order theory as M is isomorphic to M. By a theorem due independently to Engeler[8], Ryll–Nardzewski[14] and Svenonius[19], this is equivalent to the condition that Aut M has finitely many orbits on Mn for all positive integers n. It is well known that ω-categorical algebraic structures are often intimately related to Boolean algebras. As examples, we mention Theorem 1·3, which is used later, and also the following theorem of J. S. Wilson (proved in [3]), which partly motivates our work.

The classical cobar construction ΩC for a coalgebra C (first introduced by Adams[1]) is an important algebraic concept motivated by the singular chain complex of a loop space ΩX. If X is a 1-reduced simplicial set with realization |X|, Adams proved that there is a natural isomorphism of homology groups

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where C(X) si the coalgebra given by th chain complex on X abd tge Alexander-Whitney diagonal. Here the homology has coefficients in an abelian group A. The purpose of this paper is the extension of this result to the case of twisted coefficients given by π1Ω|X|−modules A, with π1Ω|X| = H2X.

A β-ring is supplied with operations βH where Hruns over the conjugacy classes of subgroups of the symmetric groups Sn. In an earlier paper we introduced a second set of operations λH and we show here that the two sets are related by the isomorphism

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We then consider the operations βH and λH as combinatorial species, in the sense of Joyal, and express their molecular decomposition as a finite sum of products of the exponential species with molecular species of degree at most n. We give combinatorial interpretations for βSn-structures and λSn-structures and derive various species isomorphisms.

For E a subset of ℝn and 0 [les ] m [les ] n we define a ‘family of dimensions’ DimmE, closely related to the packing dimension of E, with the property that the orthogonal projection of E onto almost all m-dimensional subspaces has packing dimension DimmE. In particular the packing dimension of almost all such projections must be equal. We obtain similar results for the packing dimension of the projections of measures. We are led to think of DimmE for m ∈ [0, n] as a ‘dimension profile’ that reflects a variety of geometrical properties of E, and we characterize the dimension profiles that are obtainable in this way.

For a Radon probability measure μ on ℝn we can use the Hausdorff dimension dimH and the packing dimension dimp to define lower indices

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and

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For two such measures μ and ν intersection measures μ ∩ f# ν are obtained by slicing a certain product measure on ℝn × ℝn using diagonal n-planes, and by projecting the result onto ℝn. Here f is a similarity on ℝn and f# ν is the image of the measure ν under f. Our main result gives a lower bound for the packing dimension of these intersection measures under the condition dimH μ+dimH ν > n and some additional assumptions.

Given an analytic path of flat connections on a principal U(N)-bundle over a closed odd-dimensional manifold M, how can we calculate the spectral flow of the corresponding path Dt of signature operators on an associated vector bundle E → M? In this paper we give an algebraic-topological answer to this question in terms of cohomology, cup products and higher Massey products.

Let S be an immersed compact surface into a Riemannian manifold M. We denote by H and K the mean curvature vector field of S and the sectional curvature function of M with respect to the tangent space of S and define

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1. In the paper [Kai] it was observed that if A is a Banach algebra (over R or C) then the dual space is not only an A-A-bimodule, but is also injective as a left (or right) A-module. Furthermore, if M is a left (or right) Banach module over the unital Banach algebra A, then there is a natural bilinear map, there denoted TrA, from M × M′ to A′, defined by

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(or 〈TrA(m, m′), a〉 = 〈m′, ma〉). The map TrA can be extended to the projective tensor product M[otimesas]M′, which is also an A–A–bimodule.It is easy to see that the map TrA is a bimodule homomorphism, so that the image is an A–A–submodule of A′. This module was denoted EA (M) in [Kai] and is in general not closed as a subspace of M′. It does, however, have a natural norm (as a quotient space of M[otimesas]M′) and the unit ball can be used to define a new norm ∥a∥new = sup {|〈e, a〉 | e∈ the unit ball of EA(M)} on A, and it is easy to see that this new norm is simply the operator norm of a as an operator on M. The conclusion is that if a′∈A′ is not only continuous with respect to the norm ∥a∥L(M) (which is of course in general smaller thatn the given norm on A) but also with respect to the weak topology on A given by the set of all functionals of the form (0·1), then a′ has a representation of the form

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There are several ways of shortening a curve on a surface in order to establish the existence of closed geodesics. These methods are often useful in the study of much more general spaces, e.g. locally compact length spaces (see [1] chapter 10). Recently, J. Hass and P. Scott in [6] have introduced a new curve flow, the polygonal flow, which takes into consideration the complexity of the resulting geodesic. They construct the polygonal flow on surfaces so that the number of self-intersection points of a curve does not increase throughout the flow. In this paper we are concerned with ideal polyhedra with finitely many cells of dimension 2. These spaces consist of finite ideal hyperbolic triangles glued together by isometries along their sides.

This paper presents the asymptotic expansion, with respect to the large real parameter λ, of the singular Fourier integral

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where 0 < b [les ] + ∞, the symbol fp designates an integration in the finite part sense of Hadamard and complex function K(x, u) belongs to a specific set of pseudo-functions which may present a singular behaviour of logarithmic nature at the endpoints.