On the maximal dimension of a completely entangled subspace for finite level quantum systems

LetH _ibe a finite dimensional complex Hilbert space of dimensiond _i associated with a finite level quantum system A_i for i = 1, 2, ...,k. A subspaceS ⊂ is said to becompletely entangled if it has no non-zero product vector of the formu ₁⊗u ₂ ⊗ ... ⊗u _k with u_i inH _i for each i. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that

A ∞-simplicial objects andA ∞-topological groups

The notion ofA _∞-topological group is introduced. It is proved that, if a space is obtained by deformation retraction of a topological groupG, then it has the structure of anA _∞-topological group, and theA _∞-homotopy equivalence holds. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 913–919, December, 1999.     

Systems with nonextendable convergence of quasipolynomials

The system , where Λ={λ _n } is the set of zeros (of multiplicitiesm _n ) of the Fourier transform

Problems similar to the additive divisor problem

For multiplicative functions ƒ(n), let the following conditions be satisfied: ƒ(n)≥0 ƒ(p ^r)≤A ^r,A>0, and for anyε>0 there exist constants ,α>0 such that and Σ _p≤x ƒ(p) lnp≥αx. For such functions, the following relation is proved:

THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS

Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aw of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr(Aw)=1/d.tr(Aρ).It is proved that the set of all C^*-algebras of sections of locally trivial C^*-algebra bundles over S^2 with fibres Aω has a group sturcture,denoted by π1^s(Aut(Aω)),which is isomorphic to Zif Ed>1 and {0} if d>1.Let Bcd be a cd-homogeneous C^*-algebra over S^2×T^2 of which no non-trivial matrix algebra can be factored out.The spherical noncommutative torus Sρ^cd is defined by twisting C^*(T2×Z^m-2) in Bcd ×C^*(Z^m-3) by a totally skew multiplier ρ on T^2×Z^m-2。It is shown that Sρ^cd×Mρ∞ is isomorphic to C(S^2)×C^*(T^2×Z^m-2,ρ）× Mcd(C)×Mρ∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.     

Trigonometric series of classesL^p (\mathbb{T}), p \in ]1;\infty [ and their conservative meansand their conservative means

Suppose that a lower triangular matrix μ:[μ _m ⁽ⁿ⁾ ] defines a conservative summation method for series, i.e.,

A boundary value problem for Poisson’s two-media equation inL p /2 -spaces

In the paper we study a binding boundary value problem for two media for Poisson's equation μΔu=f(x) with solutions in the class , 1<p<∞, with the corresponding seminorm, where

Weighted estimates for multilinear commutators of Marcinkiewicz integral

Let μ be the n-dimensional Marcinkiewicz integral and μb the multilinear commutator of μ. In this paper, the following weighted inequalities are proved for ω ∈ A∞ and 0 〈 p 〈 ∞,
｜｜μ（f）｜｜LP（ω）≤C｜Mf｜LP（ω） and ｜｜μb（f）｜｜LP（ω）≤C｜｜ML（log L）^1/r f｜｜LP（ω）.
The weighted weak L（log L）^1/r -type estimate is also established when p=1 and ω∈A1.     

Almost semirecursive sets

LetA be a subset of , and leta∉A. The setA is said to be almost semirecursive, if there is a two-place general recursive functionf such thatf(x, y)ε{x, y, a}∧({x, y}⊆A⇌f(x, y)εA) for all . Among other facts, it is proved that ifA and are almost semirecursive sets, thenA is a semirecursive set, and that there exists a wsr^*-set that is neither a wsr-nor an almost semirecursive set. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 188–193, August, 1999.     

On the equivalence of differential operators of infinite order in analytic spaces

It is shown that in the spacesA _R (0 <R ⩽ ∞) of all functions which are single-valued and analytic in the disk |z| < R with the topology of compact convergence, the differential operator of infinite order with constant coefficients is equivalent to the operator Dⁿ (n is a fixed natural number) if and only if and |ϕ _n | = 1 for R < ∞ or ϕ _n ≠ 0 for R = ∞. Also the equivalence of two shift operators in the space A_∞ is investigated. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 33–37, January, 1977.     

Exponential Stability of Semigroups Related to Operator Models in Mechanics

In this paper, we consider equations of the form , where is a function with values in the Hilbert space , the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in . The linear operator generating the C ₀-semigroup in the energy space is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.     

Analytic solutions of the Borel problem

LetF ⊂ ℂ be a dense-in-itself set that has a nonempty connected interior and contains the origin, and let be the space of infinitely differentiable complex-valued functions onF. For some classes of such setsF, we prove that for an arbitrary sequence of complex numbers there exists a functionf ε withf ⁽ⁿ⁾(0)=d _n,n=0, 1, 2, ..., and study the analyticity properties off. The functionf is constructed in the form of various function series, namely, a power series, a series of simple fractions, and an exponential series. Analytic solutions of the multidimensional Borel problem are also considered. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 525–538, April, 2000.     

On the twisted endomorphism algebra of a vector bundle on a curve

Let X be a smooth projective curve, L∈Pic (X) and E a vector bundle on X. Here we study the bigraded algebra , mainly when E is a general stable bundle on X. We give informations on the degrees of the minimal generators and study a generalization of the notion of projective normality. Entrata in Redazione il 19 gennaio 2000.     

On the Approximability by Finite p-Groups of Free Products of Groups with Normal Amalgamation

A sufficient condition for the residual p-finiteness (approximability by the class of finite p-groups) of a free product G = (A * B; H) of groups A and B with a normal amalgamated subgroup H is obtained. This condition is used to prove that if A and B are extensions of residually -groups by -groups, where stands for the class of finitely generated torsion-free nilpotent groups, and if H is a normal p′-isolated polycyclic subgroup, then the group G is residually p-finite (i.e., residually -group), provided the quotient group G/H ^p H′ is residually p-finite.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 125–131.Original Russian Text Copyright © 2005 by E. V. Sokolov.     

Sharp Jackson-Stechkin inequality inL2 for multidimensional spheres

In this paper we prove the Jackson-Stechkin inequalityE _n−1(f)<ω _n (f, 2τ _n ,λ),n≥1,m≥5,r≥1, f ∈L2( ),f ≢ const, which is sharp for eachn=2, 3, ...; hereE _n−1 (f) is the best approximation of a functionf by spherical polynomials of degree ≤n−1, ω _n (f, τ) is theτth modulus of continuity off based on the translations ,t ∈ ℝ,x ∈ , , is the measure of the unit Euclidean sphere , , andτ _n ,λ is the first positive zero of the Gegenbauer cosine polynomial (cost). Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 333–355, September, 1996. The present paper was discussed at Ural State University in a seminar headed by Professor Arestov. The author is grateful to Professor Arestov and Associate Professor Popov for useful conversations. This research was supported by the State Commission for Higher Education of the Russian Federation under grant No. 2-16-5-31 and by the Russian Foundation for Basic Research under grant No. 93-011-196.     

Realising Smashing Localisations as Morphisms of DG Algebras

For a smashing localisation L of the derived category of a differential graded (dg) algebra A we construct a dg algebra A _L and a morphism of dg algebras A→A _L that induces the canonical map in cohomology. As a first application we obtain a localisations of a dg algebra A with graded commutative homology at a prime ideal in the homology H ^* A, namely a morphism of dg algebras. As a second application we can use results of Keller to “model” every smashing localisation of compactly generated algebraic triangulated categories by a morphism of dg algebras.      

On the representation of differentiations by Hamiltonians,operating from an algebra of local observable spin systems

Let and be algebras of local and quasilocal observable spin systems corresponding to the group Zr, be a differentiation invariant with respect to displacements. The question of representation of D in the form of formal Hamiltonian formed by the displacements of an elementx ε is considered. It is shown that such a representation exists if the condition holds, where means an element obtained from the elements [T_kX,a] by some r-multiple process of summation. Translated from Matematicheskii Zametki, Vol. 21, No. 1, pp. 93–98, January, 1977.     

Irreducibility of the punctual quotient scheme of a surface

It is shown that the punctual quotient schemeQ _l ^r parametrizing all zero-dimensional quotients of lengthl and supported at some fixed point O∈A ² in the plane is irreducible.