On Selmer groups of abelian varieties over ℓ-adic Lie extensions of global function fields

Let F be a global function field of characteristic p > 0 and A/F an abelian variety. Let K/F be an ?-adic Lie extension (? ≠ p) unramified outside a finite set of primes S and such that Gal(K/F) has no elements of order ?. We shall prove that, under certain conditions, Sel _A (K) _? ^∨ has no nontrivial pseudo-null submodule.     

Symmetric duality for mathematical programming in complex spaces with F-convexity

Under F-convexity, F-concavity/F-pseudoconvexity, F-pseudoconcavity, appropriate duality results for a pair of Wolfe and Mond-Weir type symmetric dual nonlinear programming problems in complex spaces are established. These results are then used to develop second order F-convexity, F-concavity, second order F-pseudoconvexity, F-pseudoconcavity, and appropriate second order symmetric dual nonlinear programming problems in complex spaces.     

Theory of cones

This survey deals with the aspects of archimedian partially ordered finite-dimensional real vector spaces and order preserving linear maps which do not involve spectral theory. The first section sketches some of the background of entrywise nonnegative matrices and of systems of inequalities which motivate much of the current investigations. The study of inequalities resulted in the definition of a polyhedral cone K and its face lattice F(K). In Section II.A the face lattice of a not necessarily polyhedral cone K in a vector space V is investigated. In particular the interplay between the lattice properties of $\text{F}$(K) and geometric properties of K is emphasized. Section II.B turns to the cones Π(K) in the space of linear maps on V. Recall that Π(K) is the cone of all order preserving linear maps. Of particular interest are the algebraic structure of Π(K) as a semiring and the nature of the group Aut(K) of nonsingular elements A?Π(K) for which A^-1?Π(K) as well. In a short final section the cone P_n of n×n positive semidefinite matrices is discussed. A characterization of the set of completely positive linear maps is stated. The proofs will appear in a forthcoming paper.     

Abhyankar places admit local uniformization in any characteristic

We prove that every place P of an algebraic function field F|K of arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue field FP over K is equal to the transcendence degree of F|K, and the extension FP|K is separable. We generalize this result to the case where P dominates a regular local Nagata ring R⊆K of Krull dimension dimR?2, assuming that the valued field (K,vP) is defectless, the factor group vPF/vPK is torsion-free and the extension of residue fields FP|KP is separable. The results also include a form of monomialization.     

Elementary Abelianp-extensions of algebraic function fields

LetK be a field of characteristicp>0 andF/K be an algebraic function field. We obtain several results on Galois extensionsE/F with an elementary Abelian Galois group of orderp ⁿ.

1. E can be generated overF by some elementy whose minimal polynomial has the specific formT ^pn?T?z.
2. A formula for the genus ofE is given.
3. IfK is finite, then the genus ofE grows much faster than the number of rational points (as [E∶F] → ∞).
4. We present a new example of a function fieldE/K whose gap numbers are nonclassical.

     

Approximation algorithms and hardness results for the clique packing problem

For a fixed family F of graphs, an F-packing in a graph G is a set of pairwise vertex-disjoint subgraphs of G, each isomorphic to an element of F. Finding an F-packing that maximizes the number of covered edges is a natural generalization of the maximum matching problem, which is just F={K₂}. In this paper we provide new approximation algorithms and hardness results for the K_r-packing problem where K_r={K₂,K₃,…,K_r}.We show that already for r=3 the K_r-packing problem is APX-complete, and, in fact, we show that it remains so even for graphs with maximum degree 4. On the positive side, we give an approximation algorithm with approximation ratio at most 2 for every fixed r. For r=3,4,5 we obtain better approximations. For r=3 we obtain a simple3/2-approximation, achieving a known ratio that follows from a more involved algorithm of Halldórsson. For r=4, we obtain a (3/2+?)-approximation, and for r=5 we obtain a (25/14+?)-approximation.     

Théorèmes d'existence pour des inclusions différentielles du second ordre

We prove several existence theorems for second order differential inclusions of the form $\text{x}\text{?}\text{(t)∈K(x(t))}$, $\text{x}\text{?}\text{(t)∈?N(K(x(t));}\text{x}\text{?}\text{(t))+F(t,}\text{x}\text{?}\text{(t)),}$ when K and F are set-valued mappings taking nonconvex and convex values, respectively, in a Hilbert space H. To cite this article: M. Bounkhel, D. Laouir-Azzam, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

What does a group algebra of a free group “know” about the group?

We describe solutions to the problem of elementary classification in the class of group algebras of free groups. We will show that unlike free groups, two group algebras of free groups over infinite fields are elementarily equivalent if and only if the groups are isomorphic and the fields are equivalent in the weak second order logic. We will show that the set of all free bases of a free group F is 0-definable in the group algebra $K(F)$ when K is an infinite field, the set of geodesics is definable, and many geometric properties of F are definable in $K(F)$. Therefore $K(F)$ “knows” some very important information about F. We will show that similar results hold for group algebras of limit groups.     

On a theorem of Helly

We consider a group of problems related to the well-known Helly theorem on the intersections of convex bodies. We introduce convex subsetsK(?) of a compact convex setK defined by the relation $$K(f) = co\left\{ {\frac{N}{{N + 1}}x + \frac{N}{{N + 1}}f(x)} \right\}{\text{ }}(x \in K \subset \mathbb{R}^N ),$$ where?: K→K are continuous mappings, and prove that the intersection ∩ _?∈F K(?) is not empty; hereF is the set of all continuous mappings?: K→K.     

Some results on the capitulation problem

Let K be an unramified abelian extension of a number field F with Galois group G. K corresponds to a subgroup H of the ideal class group of F. We study the subgroup J of ideal classes in H which become trivial in K. There is an epimorphism from the cohomology group H^?1(G, Cl_K) to J which is an isomorphism if G is cyclic; Cl_K is the ideal class group of K. Some results on the structure of J and Cl_K are obtained.     

The hilbert series of the polynomial identities for the tensor square of the grassmann algebra

LetE be the Grassmann (or exterior) algebra of an infinite-dimensional vector space over a fieldK of characteristic 0. We compute the Hilbert series of the relatively free algebraF _m(varE?_K E)=K(x _l,...,x _m) /(T(E?_K E)∩K〈x _l,...,x _m〉) of the variety of algebrasvarE?_KE, whereT(E?_K E) is the set of all polynomial identities forE?_K EE from the free associative algebraK〈x ₁,x₂…〉.     

A fixed point theorem for eventually nonexpansive semigroups of mappings

Let K be a subset of a Banach space X. A semigroup $\text{F}$ = {?_α ∥ α ∞ A} of Lipschitz mappings of K into itself is called eventually nonexpansive if the family of corresponding Lipschitz constants $\text{{k}{}_{\mathrm{\alpha }}\text{¦ α ? A}}$ satisfies the following condition: for every ? > 0, there is a γ?A such that k_β < 1 + ? whenever ?_β ? ?_γ$\text{F}$ = {?_gg?_α ¦ ?_α ? $\text{F}$}. It is shown that if K is a nonempty, closed, convex, and bounded subset of a uniformly convex Banach space, and if $\text{F}$:K → K is an eventually nonexpansive, commutative, linearly ordered semigroup of mappings, then $\text{F}$ has a common fixed point. This result generalizes a fixed point theorem by Goebel and Kirk.     

Domination by second countable spaces and Lindelöf Σ-property

Given a space M, a family of sets A of a space X is ordered by M if A={AK:K is a compact subset of M} and K⊂L implies AK⊂AL. We study the class M of spaces which have compact covers ordered by a second countable space. We prove that a space Cp(X) belongs to M if and only if it is a Lindelöf Σ-space. Under MA(ω₁), if X is compact and (X×X)\Δ has a compact cover ordered by a Polish space then X is metrizable; here Δ={(x,x):x∈X} is the diagonal of the space X. Besides, if X is a compact space of countable tightness and X²\Δ belongs to M then X is metrizable in ZFC.We also consider the class M^? of spaces X which have a compact cover F ordered by a second countable space with the additional property that, for every compact set P⊂X there exists F∈F with P⊂F. It is a ZFC result that if X is a compact space and (X×X)\Δ belongs to M^? then X is metrizable. We also establish that, under CH, if X is compact and Cp(X) belongs to M^? then X is countable.     

Existence results for a class of second order evolution inclusions and its corresponding first order evolution inclusions

We deal with an abstract second order nonlinear evolution inclusion with its principal part having a small parameter ?. We prove the existence of a weak solution when the nonlinearity F is convex as well as nonconvex valued. Then we study the asymptotic behavior of a sequence of solutions {u _? } when ? → 0. We prove that there exists a limit function u, and u is a solution of the corresponding first order evolution inclusion.     

L-functions of twisted Legendre curves

Let K be a global field of char p and let Fq be the algebraic closure of Fp in K. For an elliptic curve E/K with nonconstant j-invariant, the L-function L(T,E/K) is a polynomial in 1+T⋅Z[T]. For any N>1 invertible in K and finite subgroup T⊂E(K) of order N, we compute the mod N reduction of L(T,E/K) and determine an upper-bound for the order of vanishing at 1/q, the so-called analytic rank of E/K. We construct infinite families of curves of rank zero when q is an odd prime power such that for some odd prime ?. Our construction depends upon a construction of infinitely many twin-prime pairs (Λ,Λ−1) in Fq[Λ]×Fq[Λ]. We also construct infinitely many quadratic twists with minimal analytic rank, half of which have rank zero and half have (analytic) rank one. In both cases we bound the analytic rank by letting T≅Z/2⊕Z/2 and studying the mod-4 reduction of L(T,E/K).     

Products of basic doubly stochastic matrices over a field

It is shown that if n?3, then every doubly stochastic matrix of order n over a field F is a product of doubly stochastic matrices with exactly two nonzero off- diagonal entries if and only if the characteristic of F is not 2 and F has more than three elements. A number of related results are also obtained.     

On a test of equality of two-parameter exponential distributions

In this paper, an asymptotic expansion of the distribution of the statistic for testing the equality of p two-parameter exponential distributions is obtained upto the order n^?4 with the second term of the order n^?3 where n is the size of the sample drawn from the i th exponential population. The asymptotic expansion can therefore be used to obtain accurate approximations to the critical values of the test statistic even for comparatively small values of n. Also we have shown that F-tables can be used to test the hypothesis when the sample size is moderately large.     

Bernstein theorems and transformations of correlation measures in statistical physics

We study the class of endomorphisms of the cone of correlation functions generated by probability measures. We consider algebraic properties of the products (·, ?) and the maps K, K ^?1 which establish relationships between the properties of functions on the configuration space and the properties of the corresponding operators (matrices with Boolean indices): F(γ) → F?(γ) = {F(α?β)}α,β?γ. For the operators F?(γ) and F?(γ), we prove conditions which ensure that these operators are positive definite; the conditions are given in terms of complete or absolute monotonicity properties of the function F(γ).     

Approche logarithmique des noyaux étales sauvages des corps de nombres

We study the ?-part of the wild étale kernels WK2i(F) of an arbitary number field F for a given prime ? in connection with the logarithmic ?-class groups . From the logarithmic arithmetic we deduce rank formulas, periodicity and reflection theorems, triviality characterizations and various consequences.