Radical Classes of Lattice-Ordered Groups vs. Classes of Compact Spaces

For a given class T of compact Hausdorff spaces, let Y(T) denote the class of -groups G such that for each gG, the Yosida space Y(g) of g belongs to T. Conversely, if R is a class of ;-groups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some gGR. The correspondences TY(T) and RT(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of -groups whose Yosida spaces are zero-dimensional. There is a thorough discussion of hyper-projectable -groups, followed by presentations on Y(e.d.), where e.d. denotes the class of compact extremally disconnected spaces, and, for each regular uncountable cardinal , the class Y(disc), where disc stands for the class of all compact -disconnected spaces. Sample results follow. Every strongly projectable -group lies in Y(e.d.). The -group G lies in Y(e.d.) if and only if for each gG Y(g) is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(disc). Finally, there is a discussion of Y(F), with F standing for the class of compact F-spaces. It is shown that an Archimedean -group G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G=P +Q .     

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.     

Hulls for Various Kinds of α-Completeness in Archimedean Lattice-Ordered Groups

Within Archimedean -groups, and with an infinite cardinal or , we consider X-hulls where X stands for any of the following classes of -groups: -projectable; laterally -complete; boundedly laterally -complete; conditionally -complete; combinations of the preceding, together with divisibility and/or relative uniform completeness. All these hulls exist, and may be obtained by iterated adjunction of the required extra elements, within the essential hull. When the -groups is relatively -complemented one step in the iteration suffices for several crucial properties. We derive from the above a considerable number of equations involving combinations of these hull operators.     

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     

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     

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.     

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     

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}.     

Fine Analysis of the Quasi-Orderings on the Power Set

We pursue the fine analysis of the quasi-orderings and on the power set of a quasi-ordering (Q,). We set X Y if every xX is majorized in by some yY, and X Y if every yY is minorized in by some xX. We show that both these quasi-orderings are -wqo if and only if the original quasi-ordering is ( )-wqo. For this holds also restricted to finite subsets, thus providing an example of a finitary operation on quasi-orderings which does not preserve wqo but preserves bqo.     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

ℓ1-Rigid Graphs

An ₁-graph is a graph whose nodes can be labeled by binary vectors in such a way that the Hamming distance between the binary addresses is, up to scale, the distance in the graph between the corresponding nodes. We show that many interesting graphs are ₁-rigid, i.e., that they admit an essentially unique such binary labeling.     

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 Cross-intersecting Families of Sets

A family of -element subsets and a family of k-element subsets of an n-element set are cross-intersecting if every set from has a nonempty intersection with every set from . We compare two previously established inequalities each related to the maximization of the product , and give a new and short proof for one of them. We also determine the maximum of for arbitrary positive weights ,_k.     

Admissibility of unbiased tests for a composite hypothesis with a restricted alternative

This paper discusses -admissiblility and d-admissiblity which are important concepts in studying the performance of statistical tests for composite hypotheses. A sufficient condition for -admissibility is presented. When =1/m, the Nomakuchi-Sakata test, which is uniformly more powerful than the likelihood ratio test for hypotheses min (₁, ₁) = 0 versus min (₁, ₁) > 0, is generalized for a class of distributions in an exponential family, and its unbiasedness and -admissibility are shown. Finally, the case of 1/m is discussed in brief.     

On the divisibility of homogeneous hypergraphs

We denote byK _k ,k, 2, the set of allk-uniform hypergraphsK which have the property that every element subset of the base ofK is a subset of one of the hyperedges ofK. So, the only element inK ₂ ² are the complete graphs. If is a subset ofK _k then there is exactly one homogeneous hypergraphH whose age is the set of all finite hypergraphs which do not embed any element of . We callH -free homogeneous graphsH _n have been shown to be indivisible, that is, for any partition ofH _n into two classes, oue of the classes embeds an isomorphic copy ofH _n . [5]. Here we will investigate this question of indivisibility in the more general context of-free homogeneous hypergraphs. We will derive a general necessary condition for a homogeneous structure to be indivisible and prove that all-free hypergraphs for K _k with 3 are indivisible. The-free hypergraphs with K _k ² satisfy a weaker form of indivisibility which was first shown by Henson [2] to hold forH _n . The general necessary condition for homogeneous structures to be indivisible will then be used to show that not all-free homogeneous hypergraphs are indivisible.This research has been supported by NSERC grant 69–1325.     

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.     

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.     

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.     

Regularity and Unit-regularity of Generalized Semigroups of Linear Transformations

Let V and W be vector spaces over a division ring D and L_D (V, W) the set of all linear transformations from V into W. For L_D(W, V), let (L_D (V, W), ) denote the semigroup L_D (V, W) with the operation * defined by * = for all , L_D(V, W). By a unit-regular semigroup we mean a semigroup S with identity having the property that for each a S, a = aua for some unit u S. The main purpose of this paper is to prove the following statements. The semigroup (L_D(V, W), ) is regular if and only if V = {0}, W = {0} or is an isomorphism from W onto V. The semigroup (L_D (V, W), ) is unit-regular if and only if (i) V = {0}, (ii) W = {0} or (iii) is an isomorphism from W onto V and dim_D V .     

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