Functional Analysis and Its Applications最新文献

We construct natural symbolic representations of intrinsically ergodic, but not necessarily expansive, principal algebraic actions of countably infinite amenable groups and use these representations to find explicit generating partitions (up to null-sets) for such actions.

The article studies bundle towers (M^{n+1}to M^{n}to dots to S^1), (geqslant 1), with fiber (S^1), where (M^n = L^n!/Gamma^n) are compact smooth nilmanifolds and (L^nthickapprox mathbb{R}^n) is a group of polynomial transformations of the line (mathbb{R}^1). The focus is on the well-known problem of calculating cohomology rings with rational coefficients of manifolds (M^n). Using the canonical bigradation in the de Rham complex of manifolds (M^n), we introduce the concept of polynomial Eulerian characteristic and calculate it for these manifolds.

In a paper of Croot, Lev and Pach and a later paper of Ellenberg and Gijswijt, it was proved that for a group (G=G_0^n), where (G_0ne {1,-1}^m) is a fixed finite Abelian group and (n) is large, any subset (Asubset G) without 3-progressions (triples (x), (y), (z) of different elements with (xy=z^2)) contains at most (|G|^{1-c}) elements, where (c>0) is a constant depending only on (G_0). This is known to be false when (G) is, say, a large cyclic group. The aim of this note is to show that the algebraic property corresponding to this difference is the following: in the first case, a group algebra (mathbb{F}[G]) over a suitable field (mathbb{F}) contains a subspace (X) with codimension at most (|X|^{1-c}) such that (X^3=0). We discuss which bounds are obtained for finite Abelian (p)-groups and for some matrix (p)-groups: the Heisenberg group over (mathbb{F}_p) and the unitriangular group over (mathbb{F}_p). We also show how the method allows us to generalize the results of [14] and [12].

We associate a noncommutative curve to a periodic, bipartite, planar dimer model with polygonal boundary. It determines the inverse Kasteleyn matrix and hence all correlations. It may be seen as a quantization of the limit shape construction of Kenyon and the author. We also discuss various directions in which this correspondence may be generalized.

In this paper, we compare the Stone–Čech compactification (beta mathcal{P}(X)) of the space (mathcal{P}(X)) of Radon probability measures on a Tychonoff space (X), equipped with the weak topology, with the space (mathcal{P}(beta X)) of Radon probability measures on the Stone–Čech compactification (beta X) of the space (X). It is shown that for any noncompact metric space (X), the compactification (beta mathcal{P}(X)) does not coincide with (mathcal{P}(beta X)). We discuss the case of more general Tychonoff spaces and also the case of the Samuel compactification, for which the coincidence holds.

For a metric space (M), we prove existence of continuous maps ({M_n}^{infty}_{n=1}) associating to each compact set (K subset M), a probability measure (M_n(K)) with (operatorname{supp}(M_n(K)) = K) in such a way that the set ({M_n(K)}^{infty}_{n=1}) is dense in the space of probability measures on (K).

We answer a question posed by Vershik regarding connections between quasi-similarity of dynamical systems and Kolmogorov entropy. We prove that all Bernoulli actions of a given countably infinite group are quasi-similar to each other. The existence of non-Bernoulli actions in the same quasi-similarity class is an open problem. A notion opposite to quasi-similarity is that of disjointness (or independence) of actions. Pinsker proved that a deterministic action is independent from an action with completely positive entropy. Using joinings, we obtain the following generalization of Pinsker’s theorem: an action with zero (P)-entropy (an invariant defined by Kirillov and Kushnirenko) and an action with completely positive (P)-entropy are disjoint.

A simple convex approach and block techniques are used to obtain new sharpened versions of numerical radius inequalities for Hilbert space operators. These include comparisons of norms of operators, their Cartesian parts, their numerical radii, and the numerical radius of the product of two operators.

In this paper, we aim to prove an index theorem for linear relations and apply it to study the invertibility and the essential invertibility of certain upper triangular block relation matrices.