Approximation to the Sobolev classesW q r of functions of several variables by bilinear forms inL p for 2≤q≤p≤∞

We study the approximation of functions of several variables by bilinear forms that are the pairwise products of functions of fewer variables. The order of approximation of Sobolev classesW _q ^r by bilinear forms inL _p for 2≤q≤p≤∞ is found. Translated by N. K. Kulman Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 18–34, July, 1997.     

Reconstruction of 3-submanifolds of large codimension in euclidean spaces from their gauss image

We present necessary and sufficient conditions for a regular three-dimensional manifold in the Grassmannian manifoldG(m, m+3) to be the Gauss image of a regular 3-submanifold in (m+3)-dimensional Euclidean space form>4. Translated fromMatemalicheskie Zametki, Vol. 62, No. 5, pp. 694–699, November, 1997. Translated by S. K. Lando     

Approximation of functions of boundedp-variation by polynomials in terms of the faber-schauder system

In this paper the best polynomial approximation in terms of the system of Faber-Schauder functions in the spaceC _p [0, 1] is studied. The constant in the estimate of Jackson’s inequality for the best approximation in the metric ofC _p [0, 1] and the estimate of the modulus of continuity ω_1−1/p are refined. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 363–371, September, 1997. Translated by N. K. Kulman     

Construction of the extremal function for a functional on the classH ω (n)

For a functional on the classH _ω ⁽ⁿ⁾ ,n≥3, we construct the extremal function on which the upper bound obtained by A. I. Stepanets is attained. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 519–529, April, 1997. Translated by N. K. Kulman     

Curvature identities for normal manifolds of killing type

We present two curvature identities and study the corresponding classesR ₁ andR ₂ of normal manifolds of Killing type. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 351–362, September, 1997. Translated by S. S. Anisov     

Calculation of a “nongeometric” S*-wave by using complex rays

By means of complex rays an explicit analytic expression is obtained for an S^*-wave which does not satisfy the classical geometric optics of elastic waves. Bibliography:11 titles. To dear Olga Alexandrovna Ladyzhenskaya on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 17–26. Translated by N. S. Zabavnikova.     

Strengthening theC t -closing lemma for dynamical systems and foliations on the torus

We prove a strengthenedC ^r -closing lemma (r≥1) for wandering chain recurrent trajectories of flows without equilibrium states on the two-dimensional torus and for wandering chain recurrent orbits of a diffeomorphism of the circle. The strengthenedC ^r -closing lemma (r≥1) is proved for a special class of infinitely smooth actions of the integer lattice ℤ ^k on the circle. The result is applied to foliations of codimension one with trivial holonomy group on the three-dimensional torus. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 323–331, March, 1997. Translated by S. K. Lando     

Classification of ordered nonsingular configurations of at most seven lines of ℝP 3 up to rigid isotopy

A projective m-configuration is a collection of m nonoriented pairwise disjoint lines in ℝP ³. An isotopy consisting of projective m-configurations is called a rigid isotopy. In the paper, the rigid isotopy classification of ordered projective m-configurations is obtained for m≤7. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 269–285. Translated by V. F. Mazurovskii.     

Some problems concerning universality in families ofn-Dimensional rational spaces

In this paper some problems concerning universality in families of n-dimensional rational spaces are given. Bibliography:17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 98–102. Translated by S. D. Iliadis.     

Scattering on a line. I. Existence of wave operators and asymptotic completeness in the periodic case

Existence of wave operators for scattering on a line in R³ is proved. It is shown that the completeness of have operators is violated in the case of a line with periodic boundary conditions. However, the asymptotic completeness and unitary property of wave operators are preserved. Bibliography:11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 136–141, 1990. Translated by Ya. V. Kurylev.     

On the measure of algebraic independence of values of derivatives of ap-adic modular function

The measure of the relative transcendence for the values of the derivatives of ap-adic Weyl-Lutz modular functionJ(q) at algebraic points is estimated. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 431–440, March, 1997. Translated by O. V. Sipacheva     

A library of matlab functions for polynomial matrices

This paper initiates the work pertaining to the preparation of a library of MATLAB functions. This library involves programs for personal computers, which realize methods and algorithms for solving spectral problems for polynomial one-parameter matrices, as well as some spectral problems for two-parameter polynomial matrices. Bibliography:4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 53–80. Translated by N. B. Lebedinskaya.     

Dirac operators on the quantum group SU(2) and the quantum sphere

The problem of constructing the Dirac operators on the quantum groupSU(2) and the quantum sphere S_qμ ² are discussed. In both cases, the constructions presented have the sameSU _q(2)-invariant form and are directly connected with the corresponding Laplace operators. Bibliography:16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 49–65. Translated by P. N. Bibikov.     

The problem of compensation

Let W₁, W₂ be stationary connected Wiener processes. We investigate the possibility of regular representation for an element of the linear space generated by the processes W₁, W₂. Biliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 144–152. Translated by V. Sudakov.     

Pitt's theorem for the Lorentz and Orlicz sequence spaces

LetL(X, Y) be the Banach space of all continuous linear operators fromX toY, and letK(X, Y) be the subspace of compact operators. Some versions of the classical Pitt theorem (ifp>q, thenK(l _p, l_q)=L(l_p, l_q)) for subspaces of Lorentz and Orlicz sequence spaces are established. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 18–25, January, 1997. Translated by V. N. Dubrovsky     

Integral representations of functions and embeddings of sobolev spaces on cuspidal domains

Some classes of cuspidal domainsG ⊂ ℝ ⁿ are introduced, and embeddings of the formW _p ^(l) (G)↪L_q(G),l ∈ ℕ, for sobolev spaces are established. To this end, estimates of some integral operators are needed. These operators cannot be estimated via Riesz potentials or their anisotropic analogs. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 201–219, February, 1997. Translated by V. E. Nazaikinskii     

Projective graph theory and configurations of lines

The spaces of disjoint configurations of k-dimensional subspaces in ℝP ^2k+1 (for example, lines in ℝP ³) are studied. These spaces are modeled by various simplicial schemes, and the homology groups of the latter are computed in certain cases. We use the fact that every configuration can be assigned a so-called projective graph, which is a class of graphs with respect to a certain equivalence relation. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 309–322. Translated by N. Yu. Netsvetaev.     

An additive divisor problem with a growing number of factors

Let τk(n) be the number of representations ofn as the product ofk positive factors, τ(n)=τ(n). The asymptotics of Σ _n≤x τ _k (n)τ(n+1) for 80k ¹⁰ (lnlnx)³≤lnx is shown to be uniform with respect tok. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 391–406, March, 1997. Translated by N. K. Kulman     

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     

Electronic and magnetic states in the two-band Hubbard model

Electronic and magnetic states are considered in the two-band Hubbard model. Bibliography:5 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 240–249. Translated by B. Bekker.