Basis of graded identities of the superalgebra M 1,2(F)

Denote by Mat _k,l (F) the algebraM _n (F) of matrices of order n = k + l with the grading (Mat _k,l ⁰ (F),Mat _k,l ¹ (F)), where Mat _k,l ⁰ (F) admits the basis $$ \{ e_{ij} ,i \leqslant k,j \leqslant k\} \cup \{ e_{ij} ,i > k,j > k\} $$ and Mat _k,l ¹ (F) admits the basis $$ \{ e_{ij} ,i \leqslant k,j > k\} \cup \{ e_{ij} ,i > k,j \geqslant k\} . $$ . Denote byM _k,l (F) the Grassmann envelope of the superalgebra Mat _k,l (F). In the paper, bases of the graded identities of the superalgebras Mat_1,2(F) and M _1,2(F) are described.     

Exact solution of the hypergraph Turán problem for k-uniform linear paths

A k-uniform linear path of length ?, denoted by ? _? ^(k) , is a family of k-sets {F ₁,...,F _? such that |F _i ∩ F _i+1|=1 for each i and F _i ∩ F _bj = \(\not 0\) whenever |i?j|>1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H, denoted by ex _k (n, H), is the maximum number of edges in a k-uniform hypergraph \(\mathcal{F}\) on n vertices that does not contain H as a subhypergraph. With an intensive use of the delta-system method, we determine ex _k (n, P _? ^(k) exactly for all fixed ? ≥1, k≥4, and sufficiently large n. We show that $ex_k (n,\mathbb{P}_{2t + 1}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} )$ . The only extremal family consists of all the k-sets in [n] that meet some fixed set of t vertices. We also show that $ex(n,\mathbb{P}_{2t + 2}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} ) + (_{k - 2}^{n - t - 2} )$ , and describe the unique extremal family. Stability results on these bounds and some related results are also established.     

Convolutions of random measures on compact groups

LetG be a compact group andM ₁(G) be the convolution semigroup of all Borel probability measures onG with the weak topology. We consider a stationary sequence {μ _n } _n=?∞ ^+∞ of random measures μ _n =μ _n (ω) inM ₁(G) and the convolutions $$v_{m,n} (\omega ) = \mu _m (\omega )* \cdots *\mu _{n - 1} (\omega ), m< n$$ and $$\alpha _n^{( + k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n,n + i} (\omega ),} \alpha _n^{( - k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n - i,n} (\omega )} $$ We describe the setsA _m ⁺ (ω) andA _n ⁺ (ω) of all limit points ofv _m,n(ω) asm→?∞ orn→+∞ and the setA _∞(ω) of its two-sided limit points for typical realizations of {μ _n (ω)} _n=?∞ ^+∞ . Using an appropriate random ergodic theorem we study the limit random measures ρ _n ^(±) (ω)=lim _k→∞ α _n ^(±k) (ω).     

The least degree of identities in the subspace M 1 (m,k) (F) of the matrix superalgebra M (m,k)(F)

We determine the least degree of identities in the subspace M ₁ ^(m,k) (F) of the matrix superalgebra M ^(m,k)(F) over a field F for arbitrary m and k. For the subspace M ₁ ^(m,k) (F) (k > 1) we obtain concrete minimal identities and generalize some results by Chang and Domokos.     

Random systems of equations in free abelian groups

We study the solvability of random systems of equations on the free abelian group ? ^m of rank m. Denote by SAT(? ^m , k, n) and \(SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)\) the sets of all systems of n equations of k unknowns in ? ^m satisfiable in ? ^m and ? ^m respectively. We prove that the asymptotic density \(\rho \left( {SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)} \right)\) of the set \(SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)\) equals 1 for n ≤ k and 0 for n > k. As regards, SAT(? ^m , k, n) for n < k, some new estimates are obtained for the lower and upper asymptotic densities and it is proved that they lie between (Π _j=k?n+1 ^k ζ(j))^?1 and \(\left( {\tfrac{{\zeta (k + m)}} {{\zeta (k)}}} \right)^n\) , where ξ(s) is the Riemann zeta function. For n ≤ k, a connection is established between the asymptotic density of SAT(? ^m , k, n) and the sums of inverse greater divisors over matrices of full rank. Starting from this result, we make a conjecture about the asymptotic density of SAT(? ^m , n, n). We prove that ρ(SAT(? ^m , k, n)) = 0 for n > k.     

Jackson-type inequalities and widths of function classes in L 2

The sharp Jackson-type inequalities obtained by Taikov in the space L ₂ and containing the best approximation and the modulus of continuity of first order are generalized to moduli of continuity of kth order (k = 2, 3, ... ). We also obtain exact values of the n-widths of the function classes F(k, r, Φ) and F _k ^r (h), which are a generalization of the classes F(1, r, Φ) and F _k ^r (h) studied by Taikov.     

ON possible Turán densities

The Turán density π(F) of a family F of k-graphs is the limit as n → ∞ of the maximum edge density of an F-free k-graph on n vertices. Let Π _∞ ^(k) consist of all possible Turán densities and let Π _fin ^(k) ? Π _∞ ^(k) be the set of Turán densities of finite k-graph families. Here we prove that Π _fin ^(k) contains every density obtained from an arbitrary finite construction by optimally blowing it up and using recursion inside the specified set of parts. As an application, we show that Π _fin ^(k) contains an irrational number for each k ≥ 3. Also, we show that Π _∞ ^(k) has cardinality of the continuum. In particular, Π _∞ ^(k) ≠ Π _fin ^(k) .     

On the exactness of certain inequalities in approximation theory

We prove the following: for every sequence {F_v}, F_v ? 0, F_v > 0 there exists a functionf such that

1. E_n(f)?F_n (n=0, 1, 2, ...) and
2. A_kn^?k? _v=1 ⁿ v^k?1 F_v?1?Ω_k (f, n^?1) (n=1, 2, ...).

     

Inversion complexity of self-correcting circuits for a certain sequence of boolean functions

The asymptotics L _k ^? (f ₂ ⁿ ) ?? n min{k+1, p} is obtained for the sequence of Boolean functions $f_2^n \left( {x_1 , \ldots ,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n}$ for any fixed k, p ?? 1 and growing n, here L _k ^? (f ₂ ⁿ ) is the inversion complexity of realization of the function f ₂ ⁿ by k-self-correcting circuits of functional elements in the basis B = {&, ?}, p is the weight of a reliable invertor.     

Interpolation in certain Hilbert spaces of analytic functions

In this article we study, for a Hilbert spaceB of analytic functions in the open unit disk, the dependence of the structure of the space of sequencesB(Z)={{f(z_k)} _k=1 ^∞ :f∈B} on the choice of the sequence Z={z_k} _k=1 ^∞ of distinct points of the unit disk [6].     

Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

Binomial coefficients, Catalan numbers and Lucas quotients

Let p be an odd prime and let a,m ∈ Z with a 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0kpa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.     

Pointwise estimates of potentials for higher-order capacities

In a domain D=Ω\E∈ R ⁿ , we consider a nonlinear higher-order elliptic equation such that the corresponding energy space is W _p ^m (D)?W _q ¹ (D), q>mp, and estimate a solution u(x) of this equation satisfying the condition u(x)?kf(x)∈W _p ^m (D)?W _q ¹ (D), where k∈R ¹, f(x)∈ C ₀ ^∞ (Ω), and f(x)=1 for x∈F. We establish a pointwise estimate for u(x) in terms of the higher-order capacity of the set F and the distance from the point x to the set F.     

Mehrfache gitterförmige Kreis- und Kugelanordnungen

LetG be a lattice inR ⁿ and letS ₁ ,S ₂ , ... be the family of unit spheres whose centres are the lattice-points ofG. This set is called ak-fold lattice packing (k-fold lattice covering) if each point ofR ⁿ lies in at most (at least)k of the open (closed) spheresS _i . Letd _k ⁿ be the density of the closestk-fold lattice packing and letD _k ⁿ be the density of the thinnestk-fold lattice covering ofR ⁿ . In the present paper we are considering the following problem: For which valuesn≧2 andk≧2 are the inequalitiesd _k ⁿ >kd ₁ ⁿ ,D _k ⁿ ₁ ⁿ valid?Theorem 1:For all pairs (n, k), n≧3, k≧2, with the exception of (3, k), (4, k), k=3, 5, 7, 9, 11 and (5, 3) we prove d _k ⁿ >kd ₁ ⁿ .Theorem 2:For each k≧3 is D _k ² ₁ ² . The proofs make use of the works ofBlundon, Danzer, Few andHeppes.     

Expansions in Series of Suitable Systems of Functions

The paper is devoted to the study of expansions of functions ?, which are holomorphic in a circle Ω around 0, in a series of the form where the given system of functions (f_k) _k=0 ^? has the property that each function f_k has a representation as a power series of the form with a ₀ ^(k) = 1. It is the question if and where an expansion of the form (*) holds. We present sufficient conditions for the sequence (a _n ^(k) ) _k,n=0 ^∞ , under which there exists a subset Ω₀ of Ω such that for arbitrary ? the expansion (*) holds on Ω₀ and the series converges absolutely uniformly on compact subsets of Ω₀. The obtained results can be applied to prove expansions of analytic functions in a series of a suitable system of m-fold products of Bessel functions. Expansions of this type have been treated by many authors in special cases, but we are able to state a general expansion theorem.     

On the convergence to infinity of Fourier series along dense subsequences of numbers

В статье доказываетс я Теорема.Какова бы ни была возрастающая последовательность натуральных чисел {H _k } _{k = 1} ^∞ c $$\mathop {\lim }\limits_{k \to \infty } \frac{{H_k }}{k} = + \infty$$ , существует функцияf∈L(0, 2π) такая, что для почт и всех x∈(0, 2π) можно найти возраст ающую последовательность номеров {n_k(x)} _k=1 ^∞ ,удовлетворяющую усл овиям 1) $$n_k (x) \leqq H_k , k = 1,2, ...,$$ 2) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t} (x)} (x,f) = + \infty ,$$ 3) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t - 1} (x)} (x,f) = - \infty$$ .     

The conjunction complexity asymptotic of self-correcting circuits for monotone symmetric functions with threshold 2

It is shown that the conjunction complexity L _k ^& (f ₂ ⁿ ) of monotone symmetric Boolean functions \(f_2^n (x_1 , \ldots ,x_n ) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n} x_i x_j\) realized by k-self-correcting circuits in the basis B = {&, ?} asymptotically equals to (k + 2)n for growing n providing the price of a reliable conjunctor is ≥ k + 2.     

On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces

We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = ?y″ + q(x)y with potentials from the Sobolev space W ₂ ^θ?1 , θ ≥ 0, including the nonclassical case θ ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Let s _2k (q) = λ _k ^1/2 (q) ? k, s _2k?1(q) = μ _k ^1/2 (q) ? k ? 1/2, where {λ _k } ₁ ^∞ and {μ _k } ₁ ^∞ are the sequences of eigenvalues of the operator L generated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spaces t ₂ ^θ such that the mapping F:W ₂ ^θ?1 →t ₂ ^θ defined by the equality F(q) = {s _n } ₁ ^∞ is well defined for all θ ≥ 0. The main result is as follows: for θ > 0, the mapping F is weakly nonlinear, i.e., can be expressed as F(q) = Uq + Φ(q), where U is the isomorphism of the spaces W ₂ ^θ?1 and t ₂ ^θ , and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥_τ ≤ C∥q∥_θ?1, where the exact value of τ = τ(θ) > θ ? 1 is given and the constant C depends only on the radius of the ball ∥q∥_θ? ≤ R, but is independent of the function q varying in this ball.     

О некоторых свойства х частичных сумм рядо в Фурье—Уолша—Пэли

In this paper we consider the behaviour of partial sums of Fourier—Walsh—Paley series on the group62-01. We prove the following theorems: Theorem 1. Let {n _k } _k ^=1/∞ be some increasing convex sequence of natural numbers such that $$\mathop {\lim sup}\limits_m m^{ - 1/2} \log n_m< \infty $$ . Then for anyf∈L ^∞(G) $$\left( {\frac{1}{m}\sum\limits_{j = 1}^m {|Sn_j (f;0)|^2 } } \right)^{1/2} \leqq C \cdot \left\| f \right\|_\infty $$ . Theorem 2. Let {n _k } _k ^=1/∞ be a lacunary sequence of natural numbers,n _k+1/n _k≧q>1. Then for anyfεL ^∞(G) $$\sum\limits_{j = 1}^m {|Sn_j (f;0)| \leqq C_q \cdot m^{1/2} \cdot \log n_m \cdot \left\| f \right\|_\infty } $$ . Theorems. Let µ _k =2 ^k +2 ^k-2+2 ^k-4+...+2^α ₀,α ₀=0,1. Then $$\begin{gathered} \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in L^\infty (G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = 0(m)^2 \} .} \hfill \\ \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = o(m)^2 \} = } \hfill \\ = \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} \hfill \\ \end{gathered} $$ . Theorem 4. {{S ₂ k(f: 0)} _k ^=1/∞ ,f∈L ^∞(G)}=m. $$\{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = c. \{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} = c_0 $$ .     

On positive linear interpolation operators

В этой работе мы даем о бобщение понятия нор мальной системы точек, введен ного Фейером [3]. Наше определ ение включает и случа й бесконечного интерв ала (0, ∞). Доказано, в частности, что систе ма точек 0<x ₁ ⁽ⁿ⁾ /(n)<... _n ⁽ⁿ⁾ <∞ является нормальной в смысле нашего определения тогда и т олько тогда, когда вып олняются оценки — фиксированное чис ло, 0≦?<1. Мы доказываем, что есл и точкиx _k ⁽ⁿ⁾ /(n) являются ну лями многочлена ЛагерраL _n ^(α) (x), то они образуют норма льную систему в том и т олько том случае, когда ?1<α≦0. Мы получаем, таким обр азом, положительный интерполяционный пр оцесс для каждой нормальной системы т очек и устанавливаем теорему сходимости для того с лучая, когда эти точки являются ну лямиL _n ^(α) (x) при — 1相似文献