We show that over a field k of characteristic zero an affine (n — 1)- space embedded as a closed subvariety in affine n-space and homogeneous for a codimension two linear torus action on is defined by the vanishing of a variable.

An extension of the Naimark dilation theorem [N], [SzF2] to positive-definite functions on free semigroups is given. This is used to extend the operatorial trigonometric moment problem [A] to a non-commutative setting and to characterize the classes 𝐶ρ (ρ > 0) of all n-tuples of operators that have a p-isometric dilation (see [SzF2] for the case n = 1). It is also shown that 𝐶ρ ⊂ 𝐶ρ′ and 𝐶ρ ≠ 𝐶ρ′ for 0 < ρ < ρ′ < ∞.

The von Neumann inequality [vN], [Po2] is extended to the classes 𝐶p. This is used to prove that any element in 𝐶ρ is simultaneously similar to an element in 𝐶1.

Let X be a Banach space and (fn)n be a bounded sequence in L1(X). We prove a complemented version of the celebrated Talagrand's dichotomy, i.e., we show that if (en)n denotes the unit vector basis of c0, there exists a sequence gn ∈ conv(fn, fn+1,...) such that for almost every ω, either the sequence (gn(ω) ⊗ en) is weakly Cauchy in or it is equivalent to the unit vector basis of ℓ1. We then get a criterion for a bounded sequence to contain a subsequence equivalent to a complemented copy of ℓ1 in L1(X). As an application, we show that for a Banach space X, the space L1(X) has Pełczyńiski's property (V*) if and only if X does.

The aim of this paper is to derive the expansion of the following class of multi-dimensional integrals

with respect to the large parameter λ when Ω is a subset of ℝn, a > 0, w is a strictly positive and bounded function on Σ and fp means an integration in the finite part sense of Hadamard (see Section 2). This is performed for weak assumptions bearing on pseudofunction K and by extending to higher dimensional cases the tools developed in the one-dimensional context. The range of applications of the proposed results is outlined by the exhibition of several examples.

A new approach to the analysis of the well-posedness of difference parabolic problems is proposed, which is based on weaker assumptions than in earlier works. The results are applied to the study of multi-dimensional difference parabolic problems in mesh Lebesgue spaces.

Generalized Riesz products similar to the type which arise as the spectral measure for a rank-one transformation are studied. A condition for the mutual singularity of two such measures is given. As an application, a probability space of transformations is presented in which almost all transformations are singular with respect to Lebesgue measure.

Our main result implies that for any choice 1 ≤ m ≤ n ≤ p of integers there exist finitary algebras A1 and A2 that generate the same variety, and such that the initial k-segments of their centralizer clones coincide exactly when k ≤ m, are isomorphic exactly when k ≤ n and are elementarily equivalent exactly when k ≤ p. The proof uses the existence and properties of disciplined topological spaces which we introduce and investigate here.

For amenable Lie groups of NC-type the heat kernel satisfies pt ~ t-a. We find the exact value of a ≥ 0.

Let A be a finite dimensional associative K-algebra with an identity over an algebraically closed field K, d a natural number, and modA(d) the affine variety of d-dimensional A-modules. The general linear group Gld(K) acts on modA(d) by conjugation, and the orbits correspond to the isomorphism classes of d-dimensional modules. For M and N in modA(d), N is called a degeneration of M, if TV belongs to the closure of the orbit of M. This defines a partial order ≤deg on modA(d). There has been a work [1], [10], [11], [21] connecting ≤deg with other partial orders ≤ext and ≤deg on modA(d) defined in terms of extensions and homomorphisms. In particular, it is known that these partial orders coincide in the case A is representation-finite and its Auslander-Reiten quiver is directed. We study degenerations of modules from the additive categories given by connected components of the Auslander-Reiten quiver of A having oriented cycles. We show that the partial orders ≤ext, ≤deg and < coincide on modules from the additive categories of quasi-tubes [24], and describe minimal degenerations of such modules. Moreover, we show that M ≤degN does not imply M ≤ext N for some indecomposable modules M and N lying in coils in the sense of [4].

We provide an isomorphism between the Grothendieck ring of modules of the twisted quantum double of a finite group, and a product of centres of twisted group algebras of centralizer subgroups. It follows that this Grothendieck ring is semisimple. Another consequence is a formula for the characters of this ring in terms of representations of twisted group algebras of centralizer subgroups.