We construct a new class of representations of the canonical commutation relations, which generalizes previously known classes. We perturb the infinitesimal generator of the initial Fock representation (in other words, the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize perturbations which lead to representations of the canonical commutation relations. We provide conditions entailing the irreducibility of such representations, show explicitly that our class of representations subsumes previously studied classes, and give necessary and sufficient conditions for our representations to be unitarily equivalent, or just quasi-equivalent, with Fock, coherent or quasifree representations. 1991 Mathematics Subject Classification: 81S05, 46L60, 81T05.

In recent years the theory of structured ring spectra (formerly known as A$_{\infty}$- and E$_{\infty}$-ring spectra) has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products. Now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra. In this paper we provide a general method for constructing model category structures for categories of ring, algebra, and module spectra. This provides the necessary input for obtaining model categories of symmetric ring spectra, functors with smash product, $\Gamma$-rings, and diagram ring spectra. Algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra. 1991 Mathematics Subject Classification: primary 55U35; secondary 18D10.

The Laplacian on a metric tree is $\Delta u = u''$ on its edges, with the appropriate compatibility conditions at the vertices. We study the eigenvalue problem on a rooted tree $\Gamma$: $$-\lambda \Delta u = V u \quad \text{on } \Gamma, \qquad u(o)=0.$$ Here $V \ge 0$ is a given ‘weight function’ on $\Gamma$, and $o$ is the root of $\Gamma$. The eigenvalues for such a problem decay no faster than $\lambda_n = O(n^{-2})$, this last case being typical for one-dimensional problems. We obtain estimates for the eigenvalues in the classes $l_p$, with $p > \frac12$, and their weak analogues $l_{p,\infty}$ with $p \ge \frac12$. The results for $p<1$ and $p>1$ are of different character. The case $p<1$ is studied in more detail. To analyse this case, we use two methods; in both of them the problem is reduced to a family of one-dimensional problems. One of the methods is based upon a useful orthogonal decomposition of the Dirichlet space on $\Gamma$. For a particular class of trees and weights, this method leads to the complete spectral analysis of the problem. We illustrate this in several examples, where we were able to obtain the asymptotics of the eigenvalues. 1991 Mathematics Subject Classification: primary 47E05, 05C05; secondary 34L15, 34L20.

We prove a rigid analytic version of Hironaka's Local Flattening Theorem, and establish a framework in which to study rigid subanalytic sets in the style of Denef and van den Dries.

1991 Mathematics Subject Classification: 32P05, 32B20, 12J25, 03C10.

We set up a singularity-theoretic framework for classifying one-parameter steady-state bifurcations with hidden symmetries. This framework also permits a non-trivial linearization at the bifurcation point. Many problems can be reduced to this situation; for instance, the bifurcation of steady or periodic solutions to certain elliptic partial differential equations with Neumann or Dirichlet boundary conditions. We formulate an appropriate equivalence relation with its associated tangent spaces, so that the usual methods of singularity theory become applicable. We also present an alternative method for computing those matrix-valued germs that appear in the equivalence relations employed in the classification of equivariant bifurcation problems. This result is motivated by hidden symmetries appearing in a class of partial differential equations defined on an $N$-dimensional rectangle under Neumann boundary conditions.

1991 Mathematics Subject Classification: 58C27, 58F14.

Asymptotic expansions are obtained for finite-dimensional symmetric stable distributions. They are used to prove the existence of continuous transition probability densities of stable and stable-like jump-diffusions, and to construct local multiplicative asymptotics and global two-sided estimates for these densities. As a consequence, the distribution of the first passage times for stable jump-diffusions is estimated and the integral test for the limsup behaviour of their sample paths as $t\to 0$ is provided. 1991 Mathematics Subject Classification: 60E07, 60G17, 60J35, 47D07.

A general selection theorem for a Vietoris-like hyperspace topology is obtained and, as a result, several known selection theorems for both `topology'- and `metric'-generated hyperspace topologies are improved from a common point of view. Applications are presented in characterizing classes of metrizable spaces by continuous selections for their hyperspaces.

1991 Mathematics Subject Classification: primary 54B20, 54C65; secondary 54E45, 54D65, 54E40, 54C10, 54G12.