Finite rank operators in Jacobson radical \mathcal{R}_{\mathcal{N} \otimes \mathcal{M}}

In this paper we investigate finite rank operators in the Jacobson radical of Alg( ), where are nests. Based on the concrete characterizations of rank one operators in Alg( ) and , we obtain that each finite rank operator in can be written as a finite sum of rank one operators in and the weak closure of equals Alg( ) if and only if at least one of is continuous.     

Transformation of measures generated by jump Lévy processes

Let ξ(t), t ∈ [0, 1], be a jump Lévy process. We denote by the law of ξ in the Skorokhod space [0, 1]. Under some nondegeneracy condition on the Lévy measure Λ of the process, we construct a group of -preserving transformations of the space [0, 1]. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 175–188.     

Invariant and quasi-invariant transformations of measures that correspond to stable processes with independent increments

Let ξ(t), t ∈ [0, 1], be a strictly stable process with an index of stability α ∈ (0, 2). By we denote the law of ξ in the Skorokhod space [0,1]. For arbitrary strictly stable process ξ, we construct a semigroup of transformations of [0, 1] for which the measure is quasi-invariant. For strictly stable processes with positive and negative jumps, we construct a group of transformations of [0, 1] for which the measure is quasi-invariant. In the symmetric case, this group is a group of invariant transformations. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 135–149.     

The Spectrum of the C*-Algebra of Pseudodifferential Boundary Value Problems on a Manifold with Smooth Edges

The C ^*-algebra generated by the operators of pseudodifferential boundary value problems on a manifold with smooth closed disjoint edges and boundary is studied. The operators act in the space L ₂( ) L ₂( ). The goal of this paper is to describe all (up to an equivalence) irreducible representations of the algebra Bibliography: 12 titles.     

Furstenberg family and chaos

A Furstenberg family F is a family,consisting of some subsets of the set of positive integers,which is hereditary upwards,i.e.A■B and A∈F imply B∈F.For a given system (i.e.,a pair of a complete metric space and a continuous self-map of the space) and for a Furstenberg family F, the definition of F-scrambled pairs of points in the space has been given,which brings the well-known scrambled pairs in Li-Yorke sense and the scrambled pairs in distribution sense to be F-scrambled pairs corresponding respectively to suitable Furstenberg family F.In the present paper we explore the basic properties of the set of F-scrambled pairs of a system.The generically F-chaotic system and the generically strongly F-chaotic system are defined.A criterion for a generically strongly F-chaotic system is showed.     

Multipolytopes and convex chains

For a simple complete multipolytope in ℝⁿ, Hattori and Masuda defined a locally constant function on ℝⁿ minus the union of hyperplanes associated with , which agrees with the density function of an equivariant complex line bundle over a Duistermaat-Heckman measure when arises from a moment map of a torus manifold. We improve the definition of and construct a convex chain on ℝⁿ. The well-definiteness of this convex chain is equivalent to the semicompleteness of the multipolytope . Generalizations of the Pukhlikov-Khovanskii formula and an Ehrhart polynomial for a simple lattice multipolytope are given as corollaries. The constructed correspondence ⨑ub;simple semicomplete multipolytopes⫂ub; →; ⨑ub;convex chains⫂ub; is surjective but not injective. We will study its “kernel.”     

Isometries and direct decompositions of pseudo MV-algebras

In the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) there exists an internal direct decomposition of with commutative such that and for each x ∈ A. On the other hand, if is an internal direct decomposition of a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) with commutative, then the mapping g: A → A defined by is an isometry in and .      

Extremal Decomposition Problems in the Space of Riemann Surfaces

An extension of a theorem on extremal decomposition of a Riemann surface is obtained. The problem of extremal decomposition is extended from the case of a Riemann surface with a prescribed set of distinguished points to the case of the Teichmüller space corresponding to under quasiconformal homeomorphisms f. For the functional of our problem on extremal decomposition of a surface , we consider a function expressing the dependence of the extremal value of on a point . Differentiation formulas for the function are derived. These formulas are different and depend on the genus g of the surface . The case where the function is pluriharmonic is considered. Bibliography: 8 titles.     

Transversals of Families of Translations of a Planar Convex Compact

Let be a family of translations of a convex compact set such that every two elements of have a common point. Then there exist three points such that each element of contains one of these points. This answers in the affirmative an old question by Grünbaum. Bibliography: 13 titles.     

Categories with homotopy determine a closed model structure

In a category with homotopy (Definition 1.1), one can define a natural concept of (co)fibrations and weak equivalences (Sec. 2) such that some properties of a closed model category hold. If is a complete and cocomplete category (with respect to finite limits) with simplicial homotopy (Definition 1.3), one achieves a full closed model structure in (Theorem 4.6). For each category with homotopy , there exists a category with simplicial homotopy (see Sec. 5) (the simplicial envelope of ). If is already a complete category with simplicial homotopy, then is Quillen equivalent to (Theorem 5.9). __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 41, Topology and Its Applications, 2006.     

On the range of a generalized derivation

A generalized derivation , is defined by the formula , where and is the Banach algebra of bounded linear operators in a Hilbert space . Sufficient conditions under which and are given. Bibliography: 8 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 111–119.     

A note on a theorem of Daboussi

The following assertion is proved. Let be the set of integers the number of the prime power of which is . Let be the size of . Then for each irrational , uniformly in , \begin{equation*} \frac{1}{\pi_k(x)} \bigg|\sum _{\alul{n\le x}{n\in\cN_k}} f(n) e^{2\pi in\alpha}\bigg|\to 0, \end{equation*} where is an arbitrary multiplicative function with , is a positive constant. This revised version was published online in June 2006 with corrections to the Cover Date.     

A local duality theorem for categories of modules

Let Λ be an associative ring with identity and let be the category of left unitary Λ-modules. A subcateqory of the category is said to be small if the pairwise nonisomorphic objects of form a set. The main result of this paper consists of the fact that for every small full subcategory , there exists a ring Γ such that is dual to a small full subcategory of the category . Some applications of this result are indicated. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 66–73.     

On estimation and detection of a function of infinitely many variables

We observe an unknown function of infinitely many variables f = f(t), t = (t₁, ..., t_n, ... ) ∈, [0, 1]^∞, in the Gaussian white noise of level ε > 0. We suppose that in each variable there exists a 1-periodical σ-smooth extension of the function f(t) to IR ^∞. Taking a quantity σ > 0 and a positive sequence a = {a_k}, we consider the set that consists of functions f such that . We consider the cases a_k = k^α and a_k = exp(λk), α > 0, λ > 0. We would like to estimate a function f ∈ or to test the null hypothesis H₀: f = 0 against the alternatives f ∈ , where the set consists of functions f ∈ such that ∥f∥₂ ≥ r. In the estimation problem, we obtain the asymptotics (as ε → 0) of the minimax quadratic risk. In the detection problem, we study the sharp asymptotics of minimax separation rates f _ɛ ^* that provide distiguishability in the problems. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 91–113.     

Normality-Type Properties and Covariant Functors

Topological properties similar to normality are considered in subspaces of products and powers of topological spaces, of spaces of closed subsets, and of spaces having the form (X), where is a normal functor. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 2, pp. 57–98, 2003.     

Metrics in the sphere of a C*-module

Given a unital C*-algebra and a right C*-module over , we consider the problem of finding short smooth curves in the sphere = {x ∈ : 〈x, x〉 = 1}. Curves in are measured considering the Finsler metric which consists of the norm of at each tangent space of . The initial value problem is solved, for the case when is a von Neumann algebra and is selfdual: for any element x ₀ ∈ and any tangent vector ν at x ₀, there exists a curve γ(t) = e ^tZ (x ₀), Z ∈ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x ₀ and (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x ₀, x ₁ ∈ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f ₀ the selfadjoint projection I − x ₀ ⊗ x ₀, if the algebra f ₀ f ₀ is finite dimensional, then there exists a curve γ joining x ₀ and x ₁, which is minimizing along its path.      

Quantum duality in quantum deformations

In accordance with the quantum duality principle, the twisted algebra is equivalent to the quantum group and has two preferred bases: one inherited from the universal enveloping algebra and the other generated by coordinate functions of the dual Lie group . We show howthe transformation can be explicitly obtained for any simple Lie algebra and a factorable chain of extended Jordanian twists. In the algebra , we introduce a natural vector grading , compatible with the adjoint representation of the algebra. Passing to the dual-group coordinates allows essentially simplifying the costructure of the deformed Hopf algebra , considered as a quantum group . The transformation can be used to construct new solutions of the twist equations. We construct a parameterized family of extended Jordanian deformations and study it in terms of ; we find new realizations of the parabolic twist. Dedicated to the birthday of my teacher, Yurii Novozhilov __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 112–125, July, 2006.     

Property ∏σ and tensor products of operator algebras

In this paper, we introduce Property ∏σ of operator algebras and prove that nest subalgebras and the finite-width CSL subalgebras of arbitrary von Neumann algebras have Property ∏σ.Finally, we show that the tensor product formula alg ML1-(×)algNL2 = algM-(×)N(L1 (×) L2) holds for any two finite-width CSLs L1 and L2 in arbitrary von Neumann algebras M and N, respectively.     

A local wavelet basis for an irregular grid

The spaces of -splines are proved to be embedded for an arbitrary grid refinement; the direct (wavelet) decomposition for chains of embedded spaces of -splines on a sequence of refined irregular grids is discussed; a wavelet basis of functions with compact supports is constructed; formulas of decomposition and reconstruction are provided. Simple solutions of certain interpolation problems in the spaces considered are suggested. Examples of the spline spaces are presented. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 84–110.