Proximal methods in reflexive Banach spaces without monotonicity

We introduce the concept of hypomonotone point-to-set operators in Banach spaces, with respect to a regularizing function. This notion coincides with the one given by Rockafellar and Wets in Hilbertian spaces, when the regularizing function is the square of the norm. We study the associated proximal mapping, which leads to a hybrid proximal-extragradient and proximal-projection methods for nonmonotone operators in reflexive Banach spaces. These methods allow for inexact solution of the proximal subproblems with relative error criteria. We then consider the notion of local hypomonotonicity and propose localized versions of the algorithms, which are locally convergent.     

On PAC extensions and scaled trace forms

Any non-degenerate quadratic form over a Hilbertian field (e.g., a number field) is isomorphic to a scaled trace form. In this work we extend this result to more general fields, in particular, prosolvable and prime-to-p extensions of a Hilbertian field. The proofs are based on the theory of PAC extensions.     

Power Series Methods and Multisummability

We recall the notion of a power series summability method and show, using results of W. Balser and A. Beck , that existence of a power series method which is stronger than multisummability would be equivalent to existence of an entire function with certain particular properties. We then prove that such entire functions cannot exist. In contrast to this negative result, we show that an iteration of power series methods is indeed stronger than multisummability.     

On non-Archimedean hilbertian spaces

We consider a special class of non-Archimedean Banach spaces, called Hilbertian, for which every one-dimensional linear subspace has an orthogonal complement. We prove that all immediate extensions of c_o, contained in l^∞, are Hilbertian. In this way we construct examples of Hilbertian spaces over a non-spherically complete valued field without an orthogonal base.     

Constrained second order optimization on non-archimedean fields

Constrained optimization on non-Archimedean fields is presented. We formalize the notion of a tangent plane to the surface defined by the constraints making use of an implicit function Theorem similar to its real counterpart. Then we derive necessary and sufficient conditions of second order for the existence of a local minimizer of a function subject to a set of equality and inequality constraints, based on a concept of continuity and differentiability that is stronger than the conventional one.     

Local rotundity structure of Cesàro-Orlicz sequence spaces

Some criteria for extreme points and strong U-points in Cesàro-Orlicz spaces are given. In consequence we find a Cesàro-Orlicz sequence space different from c₀ which has no extreme points. Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. Various examples presented in this paper show that there are some differences between criteria for extreme points and strong U-points in Orlicz spaces and in Cesàro-Orlicz spaces. We also show that the uniqueness of the local best approximation needs the notion of SU-point, that is, the notion of the extreme point is not strong enough here.     

Note on g-Hilbertian fields and nonrational curves¶with the Hilbert property

In a recent paper, Fried and Jarden prove the existence, for all integers g, of non-Hilbertian fields K which cannot be covered by a finite number of sets of the form ϕ (X(K)), where X is a curve of genus ≤g and ϕ is a rational function on X of degree ≥ 2. (If no bound is given on the genus we recover the notion of Hilbertian field.) This generalizes the case g=0, obtained previously by Corvaja and Zannier with a more elementary method. By a suitable modification of that method, we give here a new proof of the result of Fried and Jarden which avoids the use of deep group theoretical results. By a somewhat related construction we give an example of a curve X/Q of any prescribed genus and a Hilbertian field K⊂ˉQ such that X/K has the Hilbert property, i.e. the set of rational points X(K) is not thin. Received: 10 March 1998 / Revised version: 20 April 1998     

Hilbert volume in metric spaces. Part 1

We introduce a notion of Hilbertian n-volume in metric spaces with Besicovitch-type inequalities built-in into the definitions. The present Part 1 of the article is, for the most part, dedicated to the reformulation of known results in our terms with proofs being reduced to (almost) pure tautologies. If there is any novelty in the paper, this is in forging certain terminology which, ultimately, may turn useful in an Alexandrov kind of approach to singular spaces with positive scalar curvature [Gromov M., Plateau-hedra, Scalar Curvature and Dirac Billiards, in preparation].     

On a Refined Stark Conjecture for Function Fields

We prove that a refinement of Stark's Conjecture formulated by Rubin in Ann. Inst Fourier 4 (1996) is true up to primes dividing the order of the Galois group, for finite, Abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions, a statement stronger than Rubin's holds true.     

Sous-dualités et noyaux (reproduisants) associés

A way to study non-Gaussian measures is to generalize Schwartz's theory of Hilbertian subspaces to the non-Hilbertian case. We initiate here a new theory based upon a given duality rather than an Hilbertian structure. Hence, we develop the concept of subduality of a locally convex vector space and of its associated kernel. We show in particular that to any subduality is associated a unique kernel whose image is dense in the subduality. The image of a subdality under a weakly continuous linear application is also given, which enables the definition of a vector space structure over the set of subdualities given a equivalence relation. We then exhibit a canonical representative. Finally, we study the particular case of subdualities of $\text{K}{}^{\mathrm{\Omega }}$ endowed with the product topology. To cite this article: X. Mary et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Parametric estimation for Gaussian fields indexed by graphs

In this paper, using spectral theory of Hilbertian operators, we study a special class of Gaussian processes indexed by graphs. We extend Whittle maximum likelihood estimation of the parameters for the corresponding spectral density and show their asymptotic optimality.     

On alternating and symmetric groups as Galois groups

Fix an integern≧3. We show that the alternating groupA _n appears as Galois group over any Hilbertian field of characteristic different from 2. In characteristic 2, we prove the same whenn is odd. We show that any quadratic extension of Hilbertian fields of characteristic different from 2 can be embedded in anS _n-extension (i.e. a Galois extension with the symmetric groupS _n as Galois group). Forn≠6, it will follow thatA _n has the so-called GAR-property over any field of characteristic different from 2. Finally, we show that any polynomialf=X ⁿ+…+a₁X+a₀ with coefficients in a Hilbertian fieldK whose characteristic doesn’t dividen(n-1) can be changed into anS _n-polynomialf ^* (i.e the Galois group off ^* overK Gal(f ^*, K), isS _n) by a suitable replacement of the last two coefficienta ₀ anda ₁. These results are all shown using the Newton polygon. The author acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2000-00114, GTEM.     

On VC-minimal fields and dp-smallness

In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.     

Hardy type spaces associated with compact unitary groups

We investigate Hilbertian Hardy type spaces of complex analytic functions of infinite many variables, associated with compact unitary groups and the corresponding invariant Haar’s measures. For such analytic functions we establish a Cauchy type integral formula and describe natural domains. Also we show some relations between constructed spaces of analytic functions and the symmetric Fock space.     

On singular formal deformations

We discuss the notion of singular formal deformation in algebraic setup. Such deformations show up in both finite and infinite dimensional structures. It turns out that there is a stronger version of singular deformation—called essentially singular—which arises from a singular curve in the base of the versal deformation.     

Robust locally testable codes and products of codes

We continue the investigation of locally testable codes, i.e., error‐correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations. We give two general results on local testability: First, motivated by the recently proposed notion of robust probabilistically checkable proofs, we introduce the notion of robust local testability of codes. We relate this notion to a product of codes introduced by Tanner and show a very simple composition lemma for this notion. Next, we show that codes built by tensor products can be tested robustly and somewhat locally by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing low‐degree multivariate polynomials (which are a special case of tensor codes). Combining these two results gives us a generic construction of codes of inverse polynomial rate that are testable with poly‐logarithmically many queries. We note that these locally testable tensor codes can be obtained from any linear error correcting code with good distance. Previous results on local testability, albeit much stronger quantitatively, rely heavily on algebraic properties of the underlying codes. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

On Minimal Subspaces in Tensor Representations

In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means of a tensor given in a typical representation format (Tucker, hierarchical, or tensor train). We show that this result holds in a tensor Banach space with a norm stronger than the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples using topological tensor products of standard Sobolev spaces are given.     

Commuting foliations

The aim of this paper is to extend the notion of commutativity of vector fields to the category of singular foliations using Nambu structures, i.e., integrable multi-vector fields. We will classify the relationship between singular foliations and Nambu structures and show some basic results about commuting Nambu structures.     

Generalized Gabidulin codes over fields of any characteristic

We generalize Gabidulin codes to a large family of fields, non necessarily finite, possibly with characteristic zero. We consider a general field extension and any automorphism in the Galois group of the extension. This setting enables one to give several definitions of metrics related to the rank-metric, yet potentially different. We provide sufficient conditions on the given automorphism to ensure that the associated rank metrics are indeed all equal and proper, in coherence with the usual definition from linearized polynomials over finite fields. Under these conditions, we generalize the notion of Gabidulin codes. We also present an algorithm for decoding errors and erasures, whose complexity is given in terms of arithmetic operations. Over infinite fields the notion of code alphabet is essential, and more issues appear that in the finite field case. We first focus on codes over integer rings and study their associated decoding problem. But even if the code alphabet is small, we have to deal with the growth of intermediate values. A classical solution to this problem is to perform the computations modulo a prime ideal. For this, we need study the reduction of generalized Gabidulin codes modulo an ideal. We show that the codes obtained by reduction are the classical Gabidulin codes over finite fields. As a consequence, under some conditions, decoding generalized Gabidulin codes over integer rings can be reduced to decoding Gabidulin codes over a finite field.     

On biorthogonal systems associated to orthogonal polynomials

We consider biorthogonal systems of functions associated to derivatives of orthogonal polynomials in the case of general weights. For Freud polynomials, it is proved that the derivatives of any orders of them constitute Hilbertian bases in the space of weighted square integrable functions.