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.     

Asymptotic formulas for the enumerator of trees with a given number of hanging or internal vertices

Let t(r, n) be the number of trees with n vertices of which r are hanging and q are internal (r=n−9). For a fixed r or q we prove the validity of the asymptotic formulas (r > 2)t(r, n)≈t/r|(r−2)| 2^2−r n ^2r−4 (n→∞)t(n−q, n)≈1/q|(q−1)|q ^q−2 n ^q−1 (n→∞) In the derivation of these formulas we do not use the expression for the enumerator of the trees with respect to the number of hanging vertices. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 65–70, January, 1977.     

q-Numbers of quantum groups, Fibonacci numbers, and orthogonal polynomials

We obtain algebraic relations (identities) for q-numbers that do not contain q ^α-factors. We derive a formula that expresses any q-number [x] in terms of the q-number [2]. We establish the relationship between the q-numbers [n] and the Fibonacci numbers, Chebyshev polynomials, and other special functions. The sums of combinations of q-numbers, in particular, the sums of their powers, are calculated. Linear and bilinear generating functions are found for “natural” q-numbers. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1055–1063, August, 1998.     

Rate of escape on the lamplighter tree

Suppose we are given a homogeneous tree {ie173-01} of degree q ≥ 3, where at each vertex sits a lamp, which can be switched on or off. This structure can be described by the wreath product (ℤ/2)≀Γ, where Γ = * _i=1^qℤ/2 is the free product group of q factors ℤ/2. We consider a transient random walk on a Cayley graph of (ℤ/2) ≀Γ, for which we want to compute lower and upper bounds for the rate of escape, that is, the speed at which the random walk flees to infinity. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.     

Comparison of approximation properties of generalized polynomials and splines

We establish that, for p ∈ [2, ∞), q = 1 or p = ∞, q ∈ [ 1, 2], the classes W _p^rof functions of many variables defined by restrictions on the L _p-norms of mixed derivatives of order r = (r ₁, r ₂, ..., r _m) are better approximated in the L _q-metric by periodic generalized splines than by generalized trigonometric polynomials. In these cases, the best approximations of the Sobolev classes of functions of one variable by trigonometric polynomials and by periodic splines coincide. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1011–1020, August, 1998.     

Bernstein widths of embeddings of Lebesgue spaces

Let (K, μ) be a measurable space with μ(K)=1. Let Ip,q: L^p (K, μ)→L^q (K, μ) be the embedding operator. The Bernstein widths of I_{p, q} are considered. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 166–169. Translated by S. V. Kislyakov.     

An estimate of the error for eigen functions in the eigenfunction problem for the operator of the linear elasticity theory

We consider an eigenfunction problem for a system of Lamé equations in a three-dimensional parallelepiped in the case of a mixed boundary-value condition on the boundary. By using Steklov averaging operators, the approximation error is given in divergent form. The accuracy of the difference scheme is studied for generalized solutions from the spaceW ₂ ³(Ω). An O(|h|^1.5)-estimate is obtained for eigenfunctions in the grid metric ofW ₂ ¹(ω). Bibliography: 5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 100–109.     

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     

On the Minimum Length of some Linear Codes of Dimension 5

In this paper, we shall prove that the minimum length n_q(5,d) is equal to g_q(5,d) +1 for q⁴−2q²−2q+1≤ d≤ q⁴ − 2q² − q and 2q⁴ − 2q³ − q² − 2q+1 ≤ d ≤ 2q⁴−2q³−q²−q, where g_q(5,d) means the Griesmer bound . Communicated by: J.D. Key     

Bounds for exponential sums modulo <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>

In this paper we consider exponential sums over subgroups G ⊂ ℤ _q ^* . Using Stepanov’s method, we obtain nontrivial bounds for exponential sums in the case where q is a square of a prime number. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 81–94, 2005.     

Calibration relations for nonpolynomial splines

Nonpolynomial (X, A, ϕ)-splines of the third order and the special case of B _ϕ-splines of class C² are studied. For such splines calibration relations are obtained, owing to which the coordinate splines on the original grid is represented in terms of the coordinate splines on a refined grid. A nonlinear mapping (ℝ⁴)⁹ ↦ ℝ⁴ and locally orthogonal chains of vectors are used for this purpose. Bibliography: 22 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 39–54.     

Trigonometric widths of the classes B p,θ r of functions of many variables in the space L q

We obtain estimates exact in order for the trigonometric widths of the Besov classes B _p,θ^r of periodic functions of many variables in the space L _q for 1 ≤ p ≤ 2 < q < p/(p - 1). Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1089–1097, August, 1998.     

Orthogonal graphs of characteristic 2 and their automorphisms

The (singular) orthogonal graph O(2ν + δ,q) over a field with q elements and of characteristic 2 (where ν 1, and δ = 0,1 or 2) is introduced. When ν = 1, O(2 · 1,q), O(2 · 1 + 1,q) and O(2 · 1 + 2,q) are complete graphs with 2, q + 1 and q2 + 1 vertices, respectively. When ν 2, O(2ν + δ,q) is strongly regular and its parameters are computed. O(2ν + 1,q) is isomorphic to the symplectic graph Sp(2ν,q). The chromatic number of O(2ν + δ,q) except when δ = 0 and ν is odd is computed and the group of graph automo...     

The autotopism group of ap-primitive semifield plane of orderq 4

Let κ be a semifield plane of order q⁴ with kernel K≅GF(q²), where q=p^r, p is prime. Previously, Johnson determined the form of p-primitive semifield planes of order q⁴, q=p^r, and Cordero specified the form of autotopisms and proved the solvability of an autotopism group for the particular case q=p. The goal of the present article is to give an explication of the form of autotopisms and prove the solvability of an autotopism group in the general case. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 334–344, May–June, 1996.     

The Hering plane and associated designs

Absract We associate an affine planeA _ƒ with any automorphism ƒ of the additive group of the fieldF=GF(q), whereq is odd,F ^*=Δ ∪−Δ, and Δ=x ^ƒ x |x εF ^*. We compute the ternar of the planeA _ƒ. A simple construction of the Hering plane in the caseq=27,x ^ƒ=x−Trx=−x ³−x ⁹ and two designs associated with it are described in detail. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 873–881, December, 1996.     

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.     

A Characterization of the Complement of a Hyperbolic Quadric in PG (3, q), for q Odd

The incidence structure NQ⁺(3, q) has points the points not on a non-degenerate hyperbolic quadric Q⁺(3, q) in PG(3, q), and its lines are the lines of PG(3, q) not containing a point of Q⁺(3, q). It is easy to show that NQ⁺(3, q) is a partial linear space of order (q, q(q−1)/2). If q is odd, then moreover NQ⁺(3, q) satisfies the property that for each non-incident point line pair (x,L), there are either (q−1)/2 or (q+1)/2 points incident with L that are collinear with x. A partial linear space of order (s, t) satisfying this property is called a ((q−1)/2,(q+1)/2)-geometry. In this paper, we will prove the following characterization of NQ⁺(3,q). Let S be a ((q−1)/2,(q+1)/2)-geometry fully embedded in PG(n, q), for q odd and q>3. Then S = NQ⁺(3, q).     

On the sharpness of a theorem of B. segre

The theorem of B. Segre mentioned in the title states that a complete arc of PG(2,q),q even which is not a hyperoval consists of at mostq−√q+1 points. In the first part of our paper we prove this theorem to be sharp forq=s ² by constructing completeq−√q+1-arcs. Our construction is based on the cyclic partition of PG(2,q) into disjoint Baer-subplanes. (See Bruck [1]). In his paper [5] Kestenband constructed a class of (q−√q+1)-arcs but he did not prove their completeness. In the second part of our paper we discuss the connections between Kestenband’s and our constructions. We prove that these constructions result in isomorphic (q−√q+1)-arcs. The proof of this isomorphism is based on the existence of a traceorthogonal normal basis in GF(q ³) over GF(q), and on a representation of GF(q)³ in GF(q ³)³ indicated in Jamison [4].     

Parabolic permutation representations of the groups2 F 4(q) and3 D 4(q 3)

In the paper, the ranks, degrees, subdegrees, and double centralizers of permutation representations of the bounded groups² F ₄(q) and³ D ₄(q ³) with respect to parabolic maximal subgroups of nonminimal index are found. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 69–76, January, 2000.     

Tightening Turyn’s bound for Hadamard difference sets

This work examines the existence of (4q ²,2q ²−q,q ²−q) difference sets, for q=p ^f , where p is a prime and f is a positive integer. Suppose that G is a group of order 4q ² which has a normal subgroup K of order q such that G/K ≅ C _q ×C ₂×C ₂, where C _q ,C ₂ are the cyclic groups of order q and 2 respectively. Under the assumption that p is greater than or equal to 5, this work shows that G does not admit (4q ²,2q ²−q,q ²−q) difference sets.