Some ℵ0-bounded subsets of stone-čech compactifications

We characterize the maximalm-bounded extension of an arbitrary completely regular Hausdorff spaceX. The other principal results are:Theorem. LetX be a locally compact, σ-compact non-compact space with no more than 2^ℵ0 zero-sets. Then assuming the continuum hypothesis,βX − X can be written as the union of 2^2ℵ0 pairwise disjoint, dense ℵ₀-bounded subspaces.Theorem. LetX be a locally compact, σ-compact metric space without isolated points. Then both the set of remote points ofβX and the complement of this set inβX −X are ℵ₀-bounded.     

Singular parts of pluriharmonic measures

A measure μ defined on the complex sphere S is called pluriharmonic if its Poisson integral is a pluriharmonic function (in the unit ball of ℂⁿ). A probability measure ρ is called representing if ∫_Sfdp=f(0) for all f in the ball algebra. It is shown that singular parts of pluriharmonic measures and representing measures are mutually singular. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 54–58.     

Totally dissipative measures for the shift and conformal σ-finite measures for the stable holonomies

In this paper we investigate some results of ergodic theory with infinite measures for a subshift of finite type. We give an explicit way to construct σ-finite measures which are quasi-invariant by the stable holonomy and equivalent to the conditional measures of some σ-invariant measure. These σ-invariant measures are totally dissipative, σ-finite but satisfy a Birkhoff Ergodic-like Theorem. The constructions are done for the symbolic case, but can be extended for uniformly hyperbolic flows or diffeomorphisms.     

Conservative markov processes on a topological space

A Markov operator preservingC(X) is known to induce a decomposition of the locally compact spaceX to conservative and dissipative parts. Two notions of ergodicity are defined and the existence of subprocesses is studied. A sufficient condition for the existence of a conservative subprocess is given, and then the process is assumed to be conservative. When it has no subprocesses, sufficient conditions for the existence of a σ-finite invariant measure are given, and are extended to continuous-time processes. When the invariant measure is unique, ratio limit theorems are proved for the discrete and continuous time processes. Examples show that some combinations of conservative processes are not necessarily conservative. This paper is a part of the authors’s Ph.D. thesis prepared at the Hebrew University under the direction of Professor S. R. Foguel, to whom the author is grateful for his helpful advice and kind encouragement.     

Variational problem of the two-phase medium elasticity theory for the zero surface tension coefficient

In this article, we give a proof of the existence theorem for an equilibrium state for the surface tension coefficient σ=0 and investigate the behavior of the equilibrium state for small σ. Bibliography: 4 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1900, pp. 208–225. Translated by S. Yu. Pilyugin.     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

Distribution of the supremum of random processes from quasi-BanachK σ-spaces

We study random processes from quasi-BanachK _σ-spaces of random variables whose domain of definition is not necessarily a compact set. We establish conditions under which the supremum of a properly normalized process belongs to the same space as the process itself. We also obtain estimates for the norm of this supremum. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 7, pp. 918–930, July, 1999.     

Extension of the Stieltjes moment sequence to the left and related problems of the spectral theory of inhomogeneous string

For an inhomogeneous string with known mass distribution (the total mass is assumed to be infinite), known finite length, and unknown spectral measure dσ(t), we construct an analogous string with spectral measure dσ(t)/t. This enables one to determine the moments of all non-negative orders for the measure dσ(t). The mechanical interpretation of Stieltjes’ investigation of the problem of moments proposed by Krein enables one to solve the problem of finding the moments of negative orders for the Stieltjes moment sequence that has a unique solution. This problem is equivalent to the problem of determining the asymptotic behavior of the associated Stieltjes function near zero on the basis of its known asymptotic behavior at infinity. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 6, pp. 815–825, June, 2007.     

A Note on the characterization of Shannon-Wiener’s measure of information

Summary In [1] the author treated a characterization problem of the ShannonWiener measure of information for continuous probability distributions defined over an abstract measure space (R, S, m), wherem is a σ-finite measure over a σ-field S of subsets ofR, whose rangeM(S) is such thatM(S)=[0, ∞]. This condition on the range of the basic measure, however, can slightly be altered such thatM(S)=[0,1], and this modification is useful for characterization of the Kullback-Leibler mean information. In the present paper, it is shown that the characterization procedure of [1] can be applicable to continuous probability distributions defined on a finite measure space.     

τ-Completeness of topological groups

An analog of weak τ-completeness of uniform spaces, namely, the notion of τ-completeness of topological groups is introduced. The corresponding results for rather wide classes of topological groups such as τ-balanced and τ-bounded groups are obtained. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 103–114.     

On the existence ofσ-finite invariant measures for operators

Several necessary and sufficient conditions are given for the existence of aσ-finite invariant measure for a positive operator onL _∞. They are ofσ-type: the entire space is an increasing union of setsX _k each of which is well-behaved. To the Memory of Shlomo Horowitz Research in part supported by the National Science Foundation (U.S.A.).     

Strongly 1-Bounded Von Neumann Algebras

Suppose F is a finite tuple of selfadjoint elements in a tracial von Neumann algebra M. For α > 0, F is α-bounded if where is the free packing α-entropy of F introduced in [J3]. M is said to be strongly 1-bounded if M has a 1-bounded finite tuple of selfadjoint generators F such that there exists an with . It is shown that if M is strongly 1-bounded, then any finite tuple of selfadjoint generators G for M is 1-bounded and δ₀(G) ≤ 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and δ₀ is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II ₁-factors which have property Γ, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of . If M and N are strongly 1-bounded and M ∩ N is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II ₁-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded. Received: November 2005, Revision: March 2006, Accepted: March 2006     

The inverse spectral problem for finite-rank Hankel operators

The following theorem is proved. Let Λ be a divisor of n points of the unit disk and let σ₁, σ₂,...σ _n be a finite sequence of nonzero complex numbers. Then there exists a Hankel operator Γ of rank n such that the divisor of the poles of its symbol is Λ and the eigenvalues of Γ (counted with the multiplicities) are σ₁, σ₂,...σ _n Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 5–15     

Finite-dimensional Jordan algebras admitting the structure of a Jordan bialgebra

We define the concepts of a triangular and a quasitriangular Jordan bialgebras. It is proved that a finite-dimensional Jordan algebra J over an algebraically closed field Φ admits the structure of a quasitriangular Jordan bialgebra with nonzero comultiplication, provided that J is not a direct sum of fields, algebras H(Φ₂) and H(Φ₃), null extensions of Φ, and of algebras with zero multiplication. Supported by RFFR grant No. 98-01-01142. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 40–67, January–February, 1999.     

A dyadic endomorphism which is Bernoulli but not standard

Any measure preserving endomorphism generates both a decreasing sequence of σ-algebras and an invertible extension. In this paper we exhibit a dyadic measure preserving endomorphism (X,T,μ) such that the decreasing sequence of σ-algebras that it generates is not isomorphic to the standard decreasing sequence of σ-algebras. However the invertible extension is isomorphic to the Bernoulli two shift.     

Some remarks on estimation of parameters of uniform distribution

The best (in the sense of quadratic risk) unbiased estimators are constructed for the function f(x)=σ(2x/(n+1)−1)+μ from a sample of size n from the uniform distribution over [μ−σ, μ+σ] with unknown μ and σ. The best unbiased estimator for σ with μ being known is also presented. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 36–39, Perm, 1991.     

Measures on diffeomorphism groups for non-archimedean manifolds: Group representations and their applications

Nondegenerate σ-additive measures with ranges in ℝ and ℚ_q (q≠p are prime numbers) that are quasi-invariant and pseudodifferentiable with respect to dense subgroups G′ are constructed on diffeomorphism and homeomorphism groups G for separable non-Archimedean Banach manifolds M over a local fieldK,K ⊃ ℚ_q, where ℚ_q is the field of p-adic numbers. These measures and the associated irreducible representations are used in the non-Archimedean gravitation theory. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 381–396, June, 1999.     

p-Energia in spazi metrici generalizzati

Given a setS and a function σ:S x S→[0, +∞[ such that σ(x, x)=0, we define the generalizedp-energy of a curve γ: [a, b]→S, so that, ifS is a Hilbert space and σ(x, y)=‖x−y‖ we obtain . Sufficient conditions for the equality of the defined energies are also given. Moreover the case in whichS is a set of measurable parts of ℝⁿ and σ_r is a family of functions used in order to study the generalized minimizing motions, is discussed.

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Conferenza tenuta il 30 ottobre 1995     

δ-Derivations of simple finite-dimensional Jordan superalgebras

We describe non-trivial δ-derivations of semisimple finite-dimensional Jordan algebras over an algebraically closed field of characteristic not 2, and of simple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic 0. For these classes of algebras and superalgebras, non-zero δ-derivations are shown to be missing for δ ≠ 0, 1/2, 1, and we give a complete account of 1/2-derivations. Supported by RFBR grant No. 05-01-00230 and by RF Ministry of Education and Science grant No. 11617. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 585–605, September–October, 2007.     

Spectral theory of group representations and their nonstandard hull

We construct the nonstandard hull of a not necessarily bounded strongly continuous representationU of the locally compact semigroupS on a Banach spaceE. Then we apply our results to the theory of the spectrum σ (U) ofU, mainly in cases whereS is an abelian group, e.g.S=R. First of all we obtain generalizations to the unbounded case of results known for the bounded one. Secondly we introduce the notion of the Riesz part R σ(U) of σ(U) and characterize those representations satisfying σ(U)=R σ(U). We illustrate the theory developed so far by applications to representations on Banach lattices. Dedicated to Prof. Dr. H. H. Schaefer on the occasion of his 60th birthday