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.     

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.     

On the Space of Sequences of p-Bounded Variation and Related Matrix Mappings

The difference sequence spaces (), c(), and c ₀() were studied by Kzmaz. The main purpose of the present paper is to introduce the space bv _p consisting of all sequences whose differences are in the space _p , and to fill up the gap in the existing literature. Moreover, it is proved that the space bv _p is the BK-space including the space _p . We also show that the spaces bv _p and _p are linearly isomorphic for 1 p . Furthermore, the basis and the -, -, and -duals of the space bv _p are determined and some inclusion relations are given. The last section of the paper is devoted to theorems on the characterization of the matrix classes (bv _p : ), (bv : _p ), and (bv _p : ₁), and the characterizations of some other matrix classes are obtained by means of a suitable relation.     

Finitely Valued f-Modules, An Addendum

In an -group M with an appropriate operator set it is shown that the -value set (M) can be embedded in the value set (M). This embedding is an isomorphism if and only if each convex -subgroup is an -subgroup. If (M) has a.c.c. and M is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets ₁ and ₂ and the corresponding -value sets and . If R is a unital -ring, then each unital -module over R is an f-module and has exactly when R is an f-ring in which 1 is a strong order unit.     

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).     

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.     

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.     

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.     

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.     

О лагранжевом интерполировании с узлами в корнях многочленов сонина-маркова

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Brownian charges around loops

Summary Let (X _t ⁿ ) be a Poisson sequence of independent Brownian motions in ^d ,d3; Let be a compact oriented submanifold of d, of dimensiond–2 and volume ; let t be the sum of the windings of (X _s ⁿ , 0st) around ; then t/t converges in law towards a Cauchy variable of parameter /2. A similar result is valid when the winding is replaced by the integral of a harmonic 1-form in ^d .     

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}.     

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.     

Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Mu–u and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions = we require =C ₀[0, 1] C ₂([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .     

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}.     

The Dual of the martingale Hardy space ℋΦ with general Young function Φwith general Young function Φ

[10], ¹ . [4] K _q-, , 1p2 _pp K _q, q=p/p–1)(q=+, p=1 K = ). — p ⊃<<. , , , — , K . , ( ) , q .     

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.     

L'indice de l'idéal de Stickelberger ℓ-adique

Sans résuméNotations un nombre premier 2. - v la valuation -adique telle quev ()=1 - _n le groupe des raciensn ^e de l'unité - f _k le conducteur d'un corps abélienk - f le conducteur d'un caractère de Dirichlet      

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.     

Sets with fixed congruence in a steiner system

In this paper we study particular sets of a Steiner systemS. More precisely, we study the setsA such that ¦A ¦ d modh for all lines ofS, withd andh integers satisfyingd 0,h 2.Dedicated to Professor M. Scafati Tallini on the occasion of her sixtyfifth birthday