Spectral multiplicity for powers of weakly mixing automorphisms

We study the behavior of maximal multiplicities $mm (R^n)$ for the powers of a weakly mixing automorphism $R$. For some special infinite set $A$ we show the existence of a weakly mixing rank-one automorphism $R$ such that $mm (R^n)=n$ and $mm(R^{n+1}) =1$ for all $n\in A$. Moreover, the cardinality $cardm(R^n)$ of the set of spectral multiplicities for $R^n$ is not bounded. We have $cardm(R^{n+1})=1$ and $cardm(R^n)=2^{m(n)}$, $m(n)\to\infty$, $n\in A$. We also construct another weakly mixing automorphism $R$ with the following properties: $mm(R^{n}) =n$ for $n=1,2,3,..., 2009, 2010$ but $mm(T^{2011}) =1$, all powers $(R^{n})$ have homogeneous spectrum, and the set of limit points of the sequence $\{\frac{mm (R^n)}{n} : n\in \N \}$ is infinite

Typology of export specialization of the russian regions

© 2014, Mediterranean Center of Social and Educational Research. All rights reserved. The work carries out the primary analysis of the mass economic-geographical information which includes: 1) the stage of systematization using the methods of typing and zoning, 2) the analysis of the existing sectorial and regional characteristics of the foreign trade activity of the Russian Federation. Balances of foreign trade operations of all organizations as parts of the subjects of the Russian Federation that carry out these operations in the period from 2000 to 2010 within major product groups, are in the basis of the typing. Combining the indicators of foreign trade in seven groups is according to state statistics: food supplies, supply of fuel and energy components, petrochemical products, the supply of timber and related forest products, supply of products of the metallurgical and machine-building industries, as well as the group that unites all the other types of products. In the result of the research of the branch structure of the regions’ export we managed to allocate 8 types of subjects of the Russian Federation, where the regions without prevalent specialization belong to the 8th type

MHD alpha^2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator

It is shown that the alpha^2-dynamo of Magnetohydrodynamics, the hydrodynamic Squire equation as well as an interpolation model of PT-symmetric Quantum Mechanics are closely related as spectral problems in Krein spaces. For the alpha^2-dynamo and the PT-symmetric model the strong similarities are demonstrated with the help of a 2x2 operator matrix representation, whereas the Squire equation is re-interpreted as a rescaled and Wick-rotated PT-symmetric problem. Based on recent results on the Squire equation the spectrum of the PT-symmetric interpolation model is analyzed in detail and the Herbst limit is described as spectral singularity.Comment: 21 pages, LaTeX2e, 10 figures, minor improvements, references added, to appear in J. Math. Phy

Group measure space decomposition of II_1 factors and W*-superrigidity

We prove a "unique crossed product decomposition" result for group measure space II_1 factors arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups \Gamma in a fairly large family G, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T_n denotes the group of upper triangular matrices in PSL(n,Z), then any free, mixing p.m.p. action of the amalgamated free product of PSL(n,Z) with itself over T_n, is W*-superrigid, i.e. any isomorphism between L^\infty(X) \rtimes \Gamma and an arbitrary group measure space factor L^\infty(Y) \rtimes \Lambda, comes from a conjugacy of the actions. We also prove that for many groups \Gamma in the family G, the Bernoulli actions of \Gamma are W*-superrigid.Comment: Final version. Some extra details have been added to improve the expositio

Generic representations of abelian groups and extreme amenability

If $G$ is a Polish group and $\Gamma$ is a countable group, denote by \Hom(\Gamma, G) the space of all homomorphisms $\Gamma \to G$. We study properties of the group \cl{\pi(\Gamma)} for the generic \pi \in \Hom(\Gamma, G), when $\Gamma$ is abelian and $G$ is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on $\Gamma$, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic \cl{\pi(\Gamma)}; in the other two, we show that the generic \cl{\pi(\Gamma)} is extremely amenable. We also show that if $\Gamma$ is torsion-free, the centralizer of the generic $\pi$ is as small as possible, extending a result of King from ergodic theory.Comment: Version

Cybercities: Mediated Public Open Spaces - A Matter of Interaction and Interfaces.

In the near past, sources of information about public open spaces were: people, the place itself and historical archives. Accordingly, the information could be obtained by interviewing the visitors, by reading some poorly equipped signs on monuments or by research in libraries. Today, a new source appeared: The place itself covers its own information by the mean of the growing of the ICT (Information Communication Technologies). In addition, the information can be personalised in a way each people can access it individually. Ten years ago, a left-over newspaper on a park bench was a compact piece of information. Today, the newspaper resides on a smartphone in our pockets. In the future, the park bench will still be there, but dramatically changed to an IoT (Internet of things) object, bringing information to the people. Therefore, there is the need to re-think the park bench as an interface. A simple, fundamental point is: the quality of the interface rules the quality of the information. With a special focus on the latter, this chapter discusses how the classical model of the city is enhanced with the senseable city concept and how digital information influences, adopts, transforms and re-configures different objects in urban areas