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.     

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 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}.     

Polarizations on Abelian Varieties and Self-dual ℓ-adic Representations of Inertia Groups

It is well known that every finite subgroup of GL _d (Q ) is conjugate to a subgroup of GL _d (Z ). However, this does not remain true if we replace general linear groups by symplectic groups. We say that G is a group of inertia type of G is a finite group which has a normal Sylow-p-subgroup with cyclic quotient. We show that if >d+1, and G is a subgroup of Sp _2d (Q ) of inertia type, then G is conjugate in GL _2d (Q ) to a subgroup of Sp _2d (Z ). We give examples which show that the bound is sharp. We apply these results to construct, for every odd prime , isogeny classes of Abelian varieties all of whose polarizations have degree divisible by ². We prove similar results for Euler characteristic of invertible sheaves on Abelian varieties over fields of positive characteristic.     

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     

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.     

Dominated, diagonal polynomials on ℓ p spaces

We show that the r-dominated polynomials on _p(2 p ) are integral on ₁, and give examples proving that the converse is not true. We characterize when the 2-homogeneous, diagonal polynomials on _p(1 < p ) are r-dominated. We prove that, unlike the linear case, there are nuclear polynomials which are not 1-dominated.Received: 6 June 2004; revised: 28 September 2004     

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.     

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.     

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      

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.     

On the order of approximation to functions ofH p (R) (0<p≦1) by certain means of Fourier integrals

H ^P (R ₊ ² ) — R ₊ ² ={zC: Imz>0} ^p (R) — H ^p (R ₊ ² ). P ^k (f,x) — ë- — ,W ^k (f,x) — — R ^k, (f,x) — f H (R) (. §1,1)–3)); _k (, f) _p — - . , fH ^p (R) 0<p1,kN; (1+)^–1<p1, 0<<,kN.     

Unique <Emphasis Type="Italic">a</Emphasis>-closure for some ℓ-groups of rational valued functions

Usually, an abelian -group, even an archimedean -group, has a relatively large infinity of distinct a-closures. Here, we find a reasonably large class with unique and perfectly describable a-closure, the class of archimedean -groups with weak unit which are -convex. ( is the group of rationals.) Any C(X, ) is -convex and its unique a-closure is the Alexandroff algebra of functions on X defined from the clopen sets; this is sometimes C(X).     

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.     

Relèvement selon une isogénie de systèmes ℓ-adiques de représentations galoisiennes associés aux motifs

Résumé Soit :G _E H(Q ) un système de représentations galoisiennes -adiques associées à un motif sur un corps de nombresE et à valeurs dans un groupe algébriqueH. SoitHH une isogénie centrale telle que la structure de Hodge complexe se relève àH. Nous prouvons, au moins dans certains cas, que, après restriction à une extension finieE deE, le système de représentations galoisiennes se relève àH.

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Let :G _E H(Q ) be a system of -adic Galois representations associated to a motive over a number fieldE with values in the algebraic groupH. LetHH be a central isogeny such that the complex Hodge structure lifts toH. The main result is that, under some convenient hypothesis, and after restriction to a finite extension, the system of Galois representations lifts toH.

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Oblatum 1-VIII-1993 @ 10-X-1994     

Matrix Transformations Between Some Vector-Valued Sequence Spaces

In this paper, we give necessary and sufficient conditions for infinite matrices mapping from the Nakano vector-valued sequence space (X, p) into any BK-space, and by using this result, we obtain the matrix characterizations from (X, p) into the sequence spaces (Y), c₀(Y, q), c(Y), _s(Y), E_r(Y), and F_r(Y), where p = (p_k) and q = (q_k) are bounded sequences of positive real numbers such that p_k 1 for all k N, r 0, and s 1.AMS Subject Classification (2000): 46A45     

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.     

The interpolation ofℓ r sequences by hp functions

The sequence spaceH ^P (z)={{f (z_h)}:f H ^p} is defined for a fixed sequence Z={z_k} of different points of the open unit disk and the Hardy class HP of analytic functions in the disk. For an arbitrary p[1, ) is constructed a point sequence Z= {z_k} such that ¹h ^p(z), but ^r h^p (Z) for r > 1. It follows from a well-known result of L. Carleson that the inclusions ^r h (Z) for all r[1,] are equivalent.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 503–508, April, 1977.