On the structure of incoming and outgoing subspaces for a wave equations in ℝ n

We investigate the structure of incoming and outgoing subspaces in the Lax-Phillips scheme for the classic wave equation in ℝ ⁿ . Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp. 708–712, May, 1999.     

On Jackson's theorem in the space ℓ2(ℤ 2 n )

Estimates of Jackson's constants in the space ℓ₂(ℤ ₂ ⁿ ) are given for the case of approximation by sums of subspaces on which irreducible representations of the isometry group of (ℤ ₂ ⁿ ) act and for the case in which the modulus of continuity is defined using generalized translations. Coding theory results on efficiency estimates for binaryd-codes with respect to the Hamming distance are used. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 390–405, September, 1996.     

Irreducible characters of the group S n that are semiproportional on A n

Previously, we dubbed the conjecture that the alternating group A_n has no semiproportional irreducible characters for any natural n [1]. This conjecture was then shown to be equivalent to the following [3]. Let α and β be partitions of a number n such that their corresponding characters χ^α and χ^β in the group S_n are semiproportional on A_n. Then one of the partitions α or β is self-associated. Here, we describe all pairs (α, β) of partitions satisfying the hypothesis and the conclusion of the latter conjecture. Supported by RFBR (grant No. 07-01-00148) and by RFBR-NSFC (grant No. 05-01-39000). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 135–156, March–April, 2008.     

Uncomplemented subspaces of Lorentz spaces

In the paper we construct a system of bounded functions which generates an uncomplemented subspace in the Lorentz space Λ(α) for all α∈(0,1). Lower bounds of the norms of the projector onto such subspaces are obtained. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 57–65, July, 2000.     

Irreducible characters with equal roots in the groups Sn and An

We show that treating of (non-trivial) pairs of irreducible characters of the group S_n sharing the same set of roots on one of the sets A_n and S_n \ A_n is divided into three parts. This, in particular, implies that any pair of such characters χ^α and χ^β (α and β are respective partitions of a number n) possesses the following property: lengths d(α) and d(β) of principal diagonals of Young diagrams for α and β differ by at most 1. Supported by RFBR grant No. 04-01-00463 and by RFBR-NSFC grant No. 05-01-39000. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 3–25, January–February, 2007.     

The algebraic structure ofH-dissipative operators in a finite-dimensional space

We study properties of Jordan representations ofH-dissipative operators in a finite-dimensional indefiniteH-space. An algebraic proof is given of the fact that such operators always have maximal semidefinite invariant subspaces. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 163–169, February, 1998. The authors axe grateful to Professor A. A. Shkalikov for useful discussions. The research of the first author was supported by INTAS under grant No. 93-0249. The research of the second author was supported by the International Science Foundation and the Russian Government under grant No. NZP300.     

Large hyperbolic lattices

For a fundamental group of a compact orientable manifold, a condition is specified that is sufficient to guarantee the presence of a “virtual” epimorphism onto a free non-Abelian group. A consequence is deriving a strong Tits alternative. An arbitrary noncompact finitely generated discrete subgroup in PO(3, 1) either is large or is virtually Abelian. An application is provided to the problem of uniform exponential growth for lattices in a 3-dimensional hyperbolic space and of growth of Betti numbers for lattices in a hyperbolic n-dimensional space, where n is an odd number. Supported by RFBR (project No. 08-01-00067), by DFG grant Gr 627-11, and by Forschergruppe “Spektrale Analysis, asymptotical Verteilungen und stochastische Dynamiken,” Billfold University. (G. A. Noskov) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 174–189, March–April, 2009.     

The property of being equationally Noetherian for some soluble groups

Let B be a class of groups A which are soluble, equationally Noetherian, and have a central series A = A₁ ⩾ A₂ ⩾ … A_n ⩾ … such that ⋂A_n = 1 and all factors A_n/A_n+1 are torsion-free groups; D is a direct product of finitely many cyclic groups of infinite or prime orders. We prove that the wreath product D ≀ A is an equationally Noetherian group. As a consequence we show that free soluble groups of arbitrary derived lengths and ranks are equationally Noetherian. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 46–59, January–February, 2007.     

Widths of classes fromL 2[0, 2π] and minimization of exact constants in Jackson-type inequalities

The problem of minimizing the constant in Jackson-type inequalities with respect to all subspaces of dimensionN, i.e., with respect to the entire set of approximating subspaces of dimensionN, is solved. It is shown that the constant in question is equal to the value of different widths of classL ₂ (δ, τ, β, γ, θ). Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 816–820, June, 1999.     

Asymptotic growth of averaged Dehn functions for nilpotent groups

It is proved that in any finite representation of any finitely generated nilpotent group of nilpotency class l ⩾ 1, the averaged Dehn function σ(n) is subasymptotic w.r.t. the function n^l+1. As a consequence it is stated that in every finite representation of a free nilpotent group of nilpotency class l of finite rank r ⩾ 2, the Dehn function σ(n) is Gromov subasymptotic. Supported by RFBR grant No. 04-01-00489. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 60–74, January–February, 2007.     

Algebraic sets in metabelian groups

The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases: (1) G = F_n is a free metabelian group of rank n; (2) G = W_n,k is a wreath product of free Abelian groups of ranks n and k. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 503–513, July–August, 2007.     

Quasiconformal immersions of Riemannian manifolds and a picard type theorem

We study singularities of quasiconformal immersions of Riemannian manifolds and show that the phenomenon of compulsory continuation holds in dimensionn≥3. In particlar, this result in a stronger version of the Picard theorem—one without omitted values. Supported by RFBR grant No. 99-01-01179. Moscow State University, Department of mechanics and mathematics Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 3, pp. 37–48, July–September, 2000. Translated by V. A. Zorich     

Universally equivalent solvable groups

Every group that is finitely presented in the varietyA ⁿ of solvable groups. and is universally equivalent to a free group F_r(A ⁿ) in this variety, is embedded in the Cartesian degree of F₂(A ⁿ). All subgroups on a set of two generators in that Cartesian degree which are universally equivalent to F₂(A ⁿ) are determined. Free solvable and nilpotent groups are proved universally equivalent. Supported by RFFR grant No. 99-01-00567, and through the RP “Universities of Russia. Fundamental Research.” Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 227–240, March–April, 2000.     

Infinite commensurable hyperbolic groups are bi-Lipschitz equivalent

It is proved that commensurable hyperbolic groups are bi-Lipschitz equivalent. Therefore, subgroups of finite index in an arbitrary hyperbolic group also share this property. In addition, it is shown that any two separated nets Γ¹ and Γ² in the hyperbolic space Hⁿ of dimension n≥2 are bi-Lipschitz-equivalent. These results answer the questions posed in [1]. Supported by RFFR grant No. 96-01-01781. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 259–272, May–June, 1997.     

On the definability of endomorphisms of a free group of the variety\mathfrak{A}\mathfrak{M} by a finite set of valuesby a finite set of values

In the paper we find out in what cases any endomorphism of a free metabelian group of rankn is uniquely determined by its values on finitely many elements of the group. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 916–920, December, 1997 Translated by A. I. Shtern     

Finite-codimensional Chebyshev subspaces in the complex spaceC(Q)

We consider finite-condimensional Chebyshev subspaces in the complex spaceC(Q), whereQ is a compact Hausdorff space, and prove analogs of some theorems established earlier for the real case by Garkavi and Brown (in particular, we characterize such subspaces). It is shown that if the real spaceC(Q) contains finite-codimensional Chebyshev subspaces, then the same is true of the complex spaceC(Q) (with the sameQ). Translated fromMatermaticheskie Zametki, Vol. 62, No. 2, pp. 178–191, August, 1997. Translated by V. E. Nazaikinskii     

Intermediate predicate logic without the beth property

It is shown that a logic J _fd ^* characterized by all Kripke frames the domains of all nonmaximal worlds of which are finite lacks the Beth property. The logic is the first example of an intermediate superintuitionistic logic without the Beth property. The interpolation and the Beth properties are also proved missing in all predicate superintuitionistic logics which contain J _fd ^* and are contained in a logic characterized by frames of the form〈N _n , ≤,{Dk}k∈N n〉. Supported by the Russian Foundation for Humanities, grant No. 97-03-04089. Translated fromAlgebra i Logika, Vol. 37, No. 1, pp. 107–117, January–February, 1998.     

Gauge-invariant discrete models of Yang-Mills equations

Two discrete models of Yang-Mill equations are constructed in the space ℝⁿ for some matrix-valued Lie group. A gauge-invariant discrete model is examined. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 742–754, May, 1997. Translated by O. V. Sipacheva     

Irreducible Algebraic Sets in Metabelian Groups

We present the construction for a u-product G₁ ○ G₂ of two u-groups G₁ and G₂, and prove that G₁ ○ G₂ is also a u-group and that every u-group, which contains G₁ and G₂ as subgroups and is generated by these, is a homomorphic image of G₁ ○ G₂. It is stated that if G is a u-group then the coordinate group of an affine space Gⁿ is equal to G ○ F_n, where F_n is a free metabelian group of rank n. Irreducible algebraic sets in G are treated for the case where G is a free metabelian group or wreath product of two free Abelian groups of finite ranks. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 601–621, September–October, 2005. Supported by RFBR grant No. 05-01-00292, by FP “Universities of Russia” grant No. 04.01.053, and by RF Ministry of Education grant No. E00-1.0-12.