Subgroups and Hulls of Specker Lattice-Ordered Groups

In this article, it will be shown that every -subgroup of a Specker -group has singular elements and that the class of -groups that are -subgroups of Specker -group form a torsion class. Methods of adjoining units and bases to Specker -groups are then studied with respect to the generalized Boolean algebra of singular elements, as is the strongly projectable hull of a Specker -group.     

Fields with vanishing K2. Torsion in H1(X,K2) and Ch2(x)

This paper describes fields F of nonzero characteristic with the property that for all finite extensions E/F K₂E=0. We consider a somewhat wider class of fields which includes finite and separably closed fields. For smooth projective varieties X over such a field we show that the groups H¹(X, K₂){} and H²(X_et, (2)), NH³(X_et, (2)) and Ch²(X){} are isomorphic. These results are applied to describe the groups SK₁ of a smooth affine curve over such a field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 108–118, 1982.     

Application of fields formed by the Gauss periods to the investigation of cyclic diophantine equations

The question of the nonsolvability of the equation in rational integersx ₀,x ₁,...,x _t,z satisfying certain additional conditions is investigated. Two cases are considered: 1) is a regular prime number and 0 < t < – 1; 2) is an irregular prime number, =fe+1 (f is prime), > c₀ (f, t) and does not divide the Bernoulli numbers .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 67, pp. 201–222, 1977.     

The Szlenk index and local ℓ<Subscript>1</Subscript>-indices

We introduce two new local ₁-indices of the same type as the Bourgain ₁-index; the ⁺₁-index and the ⁺₁-weakly null index. We show that the ⁺₁-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain ₁. The ⁺₁-weakly null index has the same form as the Bourgain ₁-index: if it is countable it must take values for some <₁. The different ₁-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its ₁-index, via the ⁺₁-weakly null index. We show that I(C(^{))=^1++1}.     

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.     

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.     

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}.     

Uniqueness of the Unconditional Basis of ?1 (?p) and ?p (?1), 0 < p < 1

We prove that the quasi-Banach spaces ₁ (_p) and _p (₁), 0 < p < 1 have a unique unconditional basis up to permutation     

A unifying theory of a posteriori finite element error control

Summary Residual-based a posteriori error estimates are derived within a unified setting for lowest-order conforming, nonconforming, and mixed finite element schemes. The various residuals are identified for all techniques and problems as the operator norm |||| of a linear functional of the formin the variable of a Sobolev space V. The main assumption is that the first-order finite element space is included in the kernel Ker of . As a consequence, any residual estimator that is a computable bound of |||| can be used within the proposed frame without further analysis for nonconforming or mixed FE schemes. Applications are given for the Laplace, Stokes, and Navier-Lamè equations.Supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin.     

Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k

A new distribution called a generalized binomial distribution of order k is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer less than the specified length of the runs, an enumeration scheme called -overlapping way of counting is defined. Let k and be positive integers satisfying < k. Based on independent Bernoulli trials, it is shown that the number of (– 1)-overlapping occurrences of success-run of length k until the n-th overlapping occurrence of success-run of length follows the generalized binomial distribution of order (k–). In particular, the number of non-overlapping occurrences of success-run of length k until the n-th success follows the generalized binomial distribution of order (k– 1). The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order k is also given.     

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.     

The Archimedean ℓ-group tensor product

We introduce a construction (inZ F-set theory) for the Archimedean -group tensor product. We relate this tensor product to the existing ones in the theory of Archimedean vector lattices and -groups.     

Central Limit Theorems for dependent variables. I

Summary This paper gives a flexible approach to proving the Central Limit Theorem (C.L.T.) for triangular arrays of dependent random variables (r.v.s) which satisfy a weak mixing condition called -mixing. Roughly speaking, an array of real r.v.s is said to be -mixing if linear combinations of its past and future are asymptotically independent. All the usual mixing conditions (such as strong mixing, absolute regularity, uniform mixing, -mixing and -mixing) are special cases of -mixing. Linear processes are shown to be -mixing under weak conditions. The main result makes no assumption of stationarity. A secondary result generalises a C.L.T. that Rosenblatt gave for strong mixing samples which are nearly second order stationary.     

A characterization of 2-trivial Banach spaces with unconditional basis

Let X be a Banach space with an unconditional basis such that each operator from X into ² is 2-absolutely summing. Then X is isomorphic either to c_o or to ¹ or to c_o¹.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 76–87, 1987.The author is grateful to I. A. Komarchev for a series of useful marks and for the permission to publish the proof of Lemma 1.     

Classification of simple sℓ(2)-Modules and finite-dimensionality of the module of extensions of simple sℓ(2)-modules

A classification of simple s (2)-modules is carried out, and the finite-dimensionality of Ext_s(2)(M), where M and N are simple s(2)-modules is established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1174–1180, September, 1990.     

The Eta Invariant and the Connective K-Theory of the Classifying Space for Cyclic 2 Groups

We use the eta invariant to study the connective K-theory groups ko _m (B ) of the classifying space for the cyclic group where - 2 2.     

Problem of the extremal decomposition of a closed plane

One considers the problem of the maximum of the product of powers of conformal radii of nonoverlapping domains in the following formulation. Let A=a₁, ..., a_n and B=(b₁, ..., b_m be systems of distinguished points in ¯C and let ={₁,..., _m} be a system of positive numbers. ByU(D,b ) we denote the reduced modulus of the simply connected domain D relative to the pointb D. Find the maximum of the sum in the familyD of all systems of nonoverlapping simply connected domains D_j, j=1, ..., m, satisfying the following condition: the domain D_j does not contain points b_i B, different from b_j, and some collection A_j, for each domain, of points from A, _j=1 ^m A _j =A. The solution of this problem is obtained by the simultaneous use of the method of variation and of the method of the moduli of families of curves and is given by Theorem 1 of the present paper. As consequences of Theorem 1 one obtains Theorems 2 and 3, strengthening the corresponding results of a previous paper of the author.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 149–154, 1985.     

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.     

Application of spherical functions to a certain problem in the theory of quadratic forms

The problem regarding the number of integral points on multidimensional ellipsoids is investigated with the aid of modular forms. In the paper we consider the simplest special case of the following problem: one considers a multidimensional sphere and as a domain on it one selects a cap. The precise result is formulated in the following manner: let r(n) be the number of the representations of n by a sum of squares, 0n₁, 2 n; the expression for K(A), 4, is given in the paper. It is also shown that one can refine somewhat the results on the distribution of integral points on multidimensional ellipsoids, obtained by A. V. Malyshev by the circular method, remaining within the framework of the same methods.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 38–45, 1985.We express our gratitude to M. A. skopina for a consultatin regarding the theory of multiple fourier series     

Right-ordered and free lattice-ordered groups

The free -groups over nilpotent groups are considered. The upper bound of -solvable length is found for an -group over a nilpotent group of finite rank; the lower bound of solvable length is given for a free -group over a finitely generated nilpotent group.Translated fromAlgebra i Logika, Vol. 34, No. 1, pp. 33–40, January–February, 1995.Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-1524.