Volume 18, Issue 1

We prove the linear independence of the L -functions, and of their derivatives of any order, in a large class 𝒞 defined axiomatically. Such a class contains in particular the Selberg class as well as the Artin and the automorphic L -functions. Moreover, 𝒞 is a multiplicative group, and hence our result also proves the linear independence of the inverses of such L -functions.

A stochastic dynamics ( X ( t )) t ≥0 of a classical continuous system is a stochastic process which takes values in the space Г of all locally finite subsets (configurations) in ℝ d and which has a Gibbs measure μ as an invariant measure. We assume that μ corresponds to a symmetric pair potential ϕ( x − y ). An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics—the so-called gradient stochastic dynamics, or interacting Brownian particles—has been investigated. By using the theory of Dirichlet forms from [Ma Z.-M., Röckner M.: An Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer-Verlag, Berlin 1992], we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form ℰ μ Г on L 2 (Г; μ), and under general conditions on the potential ϕ, prove its closability. For a potential ϕ having a “weak” singularity at zero, we also write down an explicit form of the generator of ℰ μ Г on the set of smooth cylinder functions. We then show that, for any Dirichlet form ℰ μ Г , there exists a diffusion process that is properly associated with it. Finally, in a way parallel to [ Grothaus M., Kondratiev Yu. G., Lytvynov E., Röckner M.: Scaling limit of stochastic dynamics in classical continuous systems. Ann. Prob. 31 (2003), 1494-1532 ], we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in C([0, ∞), 𝒟′), where 𝒟′ is the dual space of 𝒟:= C 0 ∞ (ℝ d ).

We show that every finite-dimensional p -adic Lie group of class C k (where k ∈ ℕ ∪ {∞}) admits a C k -compatible analytic Lie group structure. We also construct an exponential map for every k + 1 times strictly differentiable ( SC k +1 ) ultrametric p -adic Banach-Lie group, which is an SC 1 -diffeomorphism and admits Taylor expansions of all finite orders ≤ k .

Mixed modules of finite torsion-free rank over a discrete valuation domain which are either pi-Butler modules or homomorphic images of a finite direct sum of purely indecomposable modules are characterized.

A maximal chain in a finite lattice L is called smooth if any two intervals of the same length are isomorphic. We call a lattice L totally smooth if all maximal chains of L are smooth. The main concept in this paper is that of a generalized smooth group, that is, a group G such that [ G / H ] is totally smooth for every subgroup H of G of prime order. The purpose of this paper is to determine the structure of generalized smooth groups.

For an abelian group A we investigate the question as to when End( A ) is a maximal subring of the near-ring of 0-preserving mappings on A . We find this is closely related to the question of when the abelian group A is E -locally cyclic. Several classes of E -locally cyclic abelian groups are determined.

An affine isometry of ℝ 3 , with hyperbolic linear part, admits a measure of signed Lorentzian displacement along an invariant line, called its Margulis invariant . In order for a group generated by hyperbolic isometries to act properly on ℝ 3 , the sign of the Margulis invariant must be constant over the group. One might ask whether positivity of the Margulis invariant over some finite generating set implies that the group acts properly on ℝ 3 . We show that in general no so such set exists, in the case when the hyperbolic structure determined by the linear holonomy representation in SO(2, 1) 0 corresponds to a punctured torus. This contrasts with the case of a pair of pants, where it suffices to check the sign of the Margulis invariant for a certain triple of generators.

Let X be a pointed connected simplicial set with loop group G . The linearisation map in K -theory as defined by Waldhausen uses G -equivariant spaces. This paper gives an alternative description using presheaves of sets and abelian groups on the simplex category of X . In other words, the linearisation map is defined in terms of X only, avoiding the use of the less geometric loop group. The paper also includes a comparison of categorical finiteness with the more geometric notion of finite CW objects in cofibrantly generated model categories. The application to the linearisation map employs a model structure on the category of abelian group objects of retractive spaces over X .

Let q be a power of an odd prime. We prove that the mod-2 cohomologies of and as algebras over the mod-2 Steenrod algebra, together with the associated Bockstein spectral sequence, determine the homotopy types of respectively and .