On an application of infinitely divisible distributions to quadrature problems

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A distribution of the number of ℓ-steps in a random binary sequence subject to some constraints

There are investigated the joint distribution of random variables _kn(₁),..., _kn(_s), and distributions of some functionals of _kn(), for n. Here _kn(), 1ln–1 is the number of -steps in a binary sequence (b.s.), selected randomly and equiprobably from the totality of all n-dimensional b.s. that have a prescribed number of ones and k 1-steps. By an -step of a b.s. we understand a configuration of the form 1...0, where the ellipsis stands for an ( –1)-dimensional b.s.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1186–1193, September, 1991.     

On the equivalence of some rearrangements of the two-parameter Haar system

h ₁, L _p .

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This research was supported in part by MTA-NSF grants INT-8400708 and 8620153.     

Images of two-dimensional motivic Galois representations

Let M be a two-dimensional motive which is pure of weight w over a number field K and let (: G_K Aut(H(M) )) be the system of the -adic realizations. Choose G_K-invariant -lattices T of H(M) and let (:G_K GL (T))be the corresponding system of integral representations. Then either for almost all primes (G_K) consist of all the elements of GL(T) with determinant in ( ^*)^–w or the system () is associated to algebraic Hecke characters. We also can prove an adelic version of our results.Mathematics Subject Classification (2000): 11F80     

Extensions of distance reducing mappings to piecewise congruent mappings on ℓm

In this note I will show any distance reducing mapping f: M ⁿ, where M is a finite subset of ^m (m n), can be extended to a piecewise conqruent mapping f: ^{m n}.     

Finite Representability of ℓp in Quotients of Banach Spaces

A typical result of the paper states that if X is a Banach space with a basis and for some 1pq, the spaces _p and _q are finitely block representable in every block subspace of X, then every block subspace of X admits a block quotient Z such that for every r[p,q], the space _r is finitely block representable in Z. Results of a similar nature are also established for ^N _p-block-sequences and asymptotic spaces.     

On flags in a topological projective plane

In this paper we study some properties of the flag space () of a topological projective plane . The line set, sayL, of (), splits into two subfamiliesL ₀ andL ₁ which an non-trivial partitions of the point set of (). For any two distinct lines , ' belonging to one of the two subfamilies, there is exactly one line in the other subfamily meeting botl and '; turns out to be a continuous mapping of (-,'–. Further, the intersection o: two lines, having exactly one point in common, is a continuous and open mapping. Conversely, every irreducible semilinear space, satisfying the above properties, is isomor phic to the flag space of a topological projective plane.This paper has been prepared from some results of a research project supported by N.A.T.O.     

Generalization of some results concerning Walsh series and the dyadic derivative

, , , . , . , , x(0,1),x2^–j ,j=1,2,..., 2 ⁿ . , ka _k 0 k k. , (0, 1) , , , , . , .     

Tests of Normality Based on Transformed Empirical Processes

The present work introduces four families of tests of normality. The tests in two of them are of the Kolmogorov–Smirnov type, and the tests of the other two are of the Cramér–von Mises type. One family of each type is focused to detect alternatives of skewness and the other one is designed to be specially sensitive to changes in kurtosis. The tests in each family depend on a parameter : for each integer , the test statistic involves the computation on the standardized sample points of the Hermite polynomials up to degree +3. The resulting tests are consistent against all alternative distributions such that at least one of their moments up to order +3 differ from the corresponding moment of the normal distribution with the same mean and variance. Therefore, a sequence of tests for samples of size n and = (n) is consistent against any nonnormal alternative, when lim _n (n)=. The performance of the proposed tests compares favorably with Shapiro–Wilk and Anderson–Darling omnibus tests, LaRiccia's focused tests, and Kallenberg and Ledwina data driven smooth tests, and the statistics can be easily computed. Hints for their computation are provided.     

Majorization versus power majorization

, . , [1]. 42.     

Net determinant over a Bezoutian local ring

Let be any D-net of ideals of order n over a commutative local Bezoutian ring R and denote by G() the corresponding net subgroup in the general linear group of degree n over R (RZhMat, 1977, 2A280). We give an explicit computation of the factor group G()/E(), where E() is the subgroup generated by all elementary transvections in G().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 5–13, 1982.     

Generalized ornstein-uhlenbeck process having a characteristic operator with polynomial coefficients

Summary Let be a weighted Schwartz's space of rapidly decreasing functions, the dual space and (t) a perturbed diffusion operator with polynomial coefficients from into itself. It is proven that (t) generates the Kolmogorov evolution operator from into itself via stochastic method. As applications, we construct a unique solution of a Langevin's equation on : whereW(t) is a Brownian motion and ^*(t) is the adjoint of (t) and show a central limit theorem for interacting multiplicative diffusions.     

Convergence of simultaneous Padé approximants to two dilogarithms

. .     

Constructing Representations of the Nonstandardly Deformed Algebra sℓξ(2)

The generators of the algebra s(2), which result from the nonstandard (Jordanian) deformation of the algebra s(2), are realized in the form of finite-difference operators acting in a function space. This allows realizing arbitrary-dimensional representations of s(2) in the polynomial space that are in one-to-one correspondence with usual matrices of an appropriate dimension. We discuss using the suggested realization to construct and investigate the universal R-matrix invariant with respect to the action of the algebra s(2).     

The Residue of p(N) Modulo Small Primes

For primes we obtain a simple formula for p(N) (mod ) as a weighted sum over -square affine partitions of N. When {3,5,7,11}, the weights are explicit divisor functions. The Ramanujan congruences modulo 5, 7, 11, 25, 49, and 121 follow immediately from these formulae.     

On the classical logarithmic barrier function method for a class of smooth convex programming problems

In this paper, we describe a natural implementation of the classical logarithmic barrier function method for smooth convex programming. It is assumed that the objective and constraint functions fulfill the so-called relative Lipschitz condition, with Lipschitz constantM>0.In our method, we do line searches along the Newton direction with respect to the strictly convex logarithmic barrier function if we are far away from the central trajectory. If we are sufficiently close to this path, with respect to a certain metric, we reduce the barrier parameter. We prove that the number of iterations required by the algorithm to converge to an -optimal solution isO((1+M ²) log) orO((1+M ²)nlog), depending on the updating scheme for the lower bound.on leave from Eötvös University, Budapest, Hungary.     

Hypercentral Series and Paired Intersections of Sylow Subgroups of Chevalley Groups

Let G(k) be the Chevalley group of normal type associated with a root system G = , or of twisted type G = ^m,m = 2,3, over a field K. Its root subgroups X_s, for all possible s G⁺, generate a maximal unipotent subgroup U = UG(k) if p = charK < 0, U is a Sylow p-subgroup of G(K). We examine G and K for which there exists a paired intersection U U⁹, g G(K), which is not conjugate in G(K) to a normal subgroup of U. If K is a finite field, this is equivalent to a condition that the normalizer of U U⁹ in G(K)has a p-multiple index. Put p() = max(r,r)/(s,s) | r,s . We prove a statement (Theorem 1) saying the following. Let G(K) be a Chevalley group of Lie rank greater than 1 over a finite field K of characteristic p and U be its Sylow p-subgroup equal to UG(K); also, either G = and p() is distinct from p and 1, or G(K) is a twisted group. Then G(K) contains a monomial element n such that the normalizer U of Uⁿ in G(K) has a p-multiple index. Let K be an associative commutative ring with unity and (K,J) be a congruence subgroup of the Chevalley group (K) modulo a nilpotent ideal J. We examine an hypercentral series 1 Z₁ Z₂ ... Z_c-1 of the group U(K) (K,J). Theorem 2 shows that under an extra restriction on the quotient (J_t : J) of ideals, central series are related via Z_i = T_c-iC, 1 i < c, where C is a subgroup of central diagonal elements. Such a connection exists, in particular, if K = Z_pm and J = (p^d), 1 d < m, d| m.     

Stable solvability of Padé difference schemes for parabolic equations in Hölder spaces

We study Padé difference scheme for the approximate solution of the Cauchy problem for parabolic equations generated by the Padé fractions R_j,, of exponential approximation. We establish an estimate of the coerciveness of the difference schemes for j=–2, –1, or even j= in a smaller space than C₀ (E).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1466–1476, November, 1992.     

The Eta Invariant and the Real Connective K-Theory of the Classifying Space for Quaternion Groups

We express the real connective K-theory groups o_4k–1(B Q ) ofthe quaternion group Q of order = 2 ^j 8 in terms of therepresentation theory of Q by showing o_4k–1(B Q ) = Sp(S ^4k+3/Q )where is any fixed point free representation of Q in U(2k + 2).     

Approximation of functions of several variables by trigonometric polynomials with given number of harmonics,and estimates of ε-entropy

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