Eléments analytiques et filtres percés sur un ensemble infraconnexe

Summary Let K be a complete ultrametric algebraically closed field, let D be an infraconnected bounded closed set of K and let H(D) be the Banach algebra of the analytical elements on D. The properties of the elements f of H(D) are learnt introducing a function v(f, μ) continuous and affine on pieces in the intervals where it is bounded. We learn the elements f ε H(D) which are not a product of a polynomial with an invertible element. We introduce the notion of monotonous filter, related with the continuous multiplicative semi-norms of H(D) and we prove these such elements are annulated by a monotonous filter and more precisely, a pierced monotonous filter.

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Entrata in Redazione il 13 settembre 1975.     

Lubin-Hensel factorization for Laurent series

Summary Let K be a complete ultrametric algebraically closed field. Let D be a bounded closed strongly infraconnected set in K with no T-filter, and let H(D) be the Banach algebra of the analytic elements in D. Let r, r be functions from D toR with bounds a, b such that 0 (D,r,r) be the Banach algebra of the Laurent series with coefficients a_s in H(D) such that , provided with a suitable norm. In (D, r, r) we give a kind of Hensel Factorization for series whose dominating coefficients at r(x) and at r(x) conserve the same rank. We take advantage of this method to correcting a mistake that happened in our previous article on the Hensel Factorization for Taylor series.And Erratum to «Maximum principle for analytic elements and Lubin-Hensel's Theorem inH(D)Y»,135, pp. 265–278 of this Journal.     

Local Regularity of L∞ ∞-Refinable Function Vectors

This article discusses the local regularity of refinable function vectors associated with a dilation matrix M. Suppose that D is a complete set of representatives of Z^s/MZ^s. Under the assumptions that the self-affine tile T (M,D), associated with dilation matrix M and digit set D, has measure 1 and that the corresponding refinable function vector φ ∈ L_∞, we prove that there is a set H ? R^s of full measure such that the restriction φ|_H of φ on H has a positive Hölder exponent α(x) at every x ∈ H. Similar result holds for the derivatives of refinable function vectors provided that the dilation matrix is diagonalizable.     

Qualitative analysis of a class of competitive and cooperative systems

Hirsch[1,2] studied the limiting behavior of solutions of competitive or cooperative systems, and showed that ifL is an ω-limit set of a three-dimensional cooperative system, which contains no equilibrium, thenL is a nonattracting closed orbit. Smith^[3] considered a three-dimensional irreducible competitive system and showed that an ω-limit set containing no equilibrium must be a closed orbit which has a simple Floquet multiplier λ<1, and may be attracting. In this paper we carry out the qualitative analysis of a class of competitive and cooperative systems, and a generalization of the result of Levine^[4] is given. The stability problem of closed orbits raised in [5] and [6] is resolved.     

Hyperbolic sets for semilinear parabolic equations

In this note we considerC ^r semiflows on Banach spaces, roughly speakingC ^r flows defined only for positive values of time. Such semiflows arise as the “general solution” of a large class of partial differential equations that includes the Navier-Stokes equation. Our main result (Proposition B) is that under certain assumptions on the P.D.E. (satisfield by the Navier-Stokes equation) a hyperbolic set for the corresponding semiflow (hyperbolicity is defined following closely the finite dimensional case) is always ε-equivalent to a hyperbolic set for an ordinary differential equation that can be easily deduced from the P.D.E. As an example we consider the P.D.E. (0) $$\frac{{\partial u}}{{\partial t}} = - \Delta u + \varepsilon F(x,u,u')$$ where u:M → ? ^k andM is a closed smooth Riemannian manifold. Applying normal hyperbolicity techniques the phase portrait of (0) can be analyzed proving that every example of hyperbolic set for O.D.E. can appear as a hyperbolic set for the semiflow generated by (0).     

Polynomial growth and ideals in group algebras

Let G be a locally compact group with polynomial growth and symmetric group algebra L¹ (G). To every closed subset C of Prim^* (L¹(G)), there exists a smallest twosided closed ideal j (C) in L¹(G), whose hull is equal to C. If H is a closed normal subgroup of G, then H¹ is a set of synthesis in Prim^* (L¹(G)).     

Über die Gleichstetigkeit von Mengen von linearen Abbildungen und eine Klasse von topologischen linearen Räumen

In the first part of this paper we proof the following theorem: Let E and F be topological linear spaces, α an infinite cardinal number, and H a set of linear mappings from E into F such that every subset G of H with cardinality |G|≤α is equicontinuous. Then H is equicontinuous on every linear subspace of E which is the closed linear hull of a family (B_L;L∈I), |I|≤α, of precompact subsets of E. In the second part we introduce the class of all topological linear spaces E with the following property: A set H of linear mappings from E into a topological linear space is equicontinuous, if every countable subset of H is equicontinuous. We show that this class is closed with respect to forming topological products and linear final topologies.     

Geodesics Closed On The Projective Plane

Given a C ^∞ Riemannian metric g on P ² we prove that (, g) has constant curvature iff all geodesics are closed. Therefore is the first non-trivial example of a manifold such that the smooth Riemannian metrics which involve that all geodesics are closed are unique up to isometries and scaling. This remarkable phenomenon is not true on the 2-sphere, since there is a large set of C ^∞ metrics whose geodesics are all closed and have the same period 2π (called Zoll metrics), but no metric of this set can be obtained from another metric of this set via an isometry and scaling. As a corollary we conclude that all two-dimensional P-manifolds are SC-manifolds. Received: April 2007; Revision: September 2007; Accepted: September 2007     

Localizations over the Steenrod algebra The lost chapter

Let H be an unstable algebra over the Steenrod algebra, and let be a multiplicatively closed subset. inherits an action of the Steenrod algebra from H, which is, however, in general no longer unstable. In this note we consider the following three statements. (1)] H is Noetherian, (2) the integral closure, , of H in the localization with respect to S is Noetherian, (3) , where denotes the unstable part. If the set S contains only (nonzero) nonzero divisors and the algebras are reduced then If S contains zerodivisors, then only remains true, to show the converse is false we construct a counterexample. The implication is always true, while needs a bunch of technical assumptions to remain true. However, none of them can be removed: we illustrate this also with examples. Finally, as a technical tool, we characterize -finite algebras. Received February 8, 1999 / in final form August 16, 1999 / Published online July 20, 2000     

Shilov boundary and p-adic injective analytic functions

Let K be an algebraically closed field complete with respect to a dense ultrametric absolute value |.|. Let D be an infraconnected affinoid subset of K and let H(D) be the Banach algebra of analytic elements on D. Let f ∈ H(D) be injective in D and let f _* be the mapping defined on the multiplicative spectrum of H(D) that identifies with the set of circular filters on D. We show that f _* is injective and maps bijectively the Shilov boundary of H(D) onto this of H(f(D)). Thanks to this property we give a new proof of the equality $\left| {f(x) - f(y)} \right| = \left| {x - y} \right|\sqrt {\left| {f'(x)f'(y)} \right|} $ .     

Slice-Continuous Sets in Reflexive Banach Spaces: Some Complements

In this paper we study a class of closed convex sets introduced recently by Ernst et al. (J Funct Anal 223:179–203, 2005) and called by these authors slice-continuous sets. This class, which plays an important role in the strong separation of convex sets, coincides in ℝ ⁿ with the well known class of continuous sets defined by Gale and Klee in the 1960s. In this article we achieve, in the setting of reflexive Banach spaces, two new characterizations of slice-continuous sets, similar to those provided for continuous sets in ℝ ⁿ by Gale and Klee. Thus, we prove that a slice-continuous set is precisely a closed and convex set which does not possess neither boundary rays, nor flat asymptotes of any dimension. Moreover, a slice-continuous set may also be characterized as being a closed and convex set of non-void interior for which the support function is continuous except at the origin. Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

On finite conductor domains

An integral domain D is a FC domain if for all a, b in D, aDbD is finitely generated. Using a set of very general and useful lemmas, we show that an integrally closed FC domain is a Prüfer v-multiplication domain (PVMD). We use this result to improve some results which were originally proved for integrally closed FC domains (or for coherent domains) to results on PVMD's. Finally we provide examples of integrally closed integral domains which are not FC domains.     

The GL2 Main Conjecture for Elliptic Curves without Complex Multiplication

Let G be a compact p-adic Lie group, with no element of order p, and having a closed normal subgroup H such that G/H is isomorphic to Z_p. We prove the existence of a canonical Ore set S^* of non-zero divisors in the Iwasawa algebra Λ(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S^*, we are able to define a characteristic element for every finitely generated Λ(G)-module M which has the property that the quotient of M by its p-primary submodule is finitely generated over the Iwasawa algebra of H. We discuss the evaluation of this characteristic element at Artin representations of G, and its relation to the G-Euler characteristics of the twists of M by such representations. Finally, we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of Iwasawa theory for an elliptic curve E over Q, without complex multiplication, over the field F generated by the coordinates of all its p-power division points; here p is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over Q.     

Several theorems of combinatorial geometry

A set is said to be H-convex if it can be represented by an intersection of a family of closed half-spaces whose outer normals belong to a given subset of the set H of the unit sphereS ⁿ⁻¹⊂R. On the basis of Helly’s theorem for H-convex sets recently obtained by us, we prove in this note certain extensions of Blaschke’s theorem (on the radius of an inscribed sphere) and of several other well-known theorems of combinatorial geometry. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 117–124, January, 1977.     

Uniform duality in semi-infinite convex optimization

We study infinite sets of convex functional constraints, with possibly a set constraint, under general background hypotheses which require closed functions and a closed set, but otherwise do not require a Slater point. For example, when the set constraint is not present, only the consistency of the conditions is needed. We provide hypotheses, which are necessary as well as sufficient, for the overall set of constraints to have the property that there is no gap in Lagrangean duality for every convex objective function defined on ℝⁿ. The sums considered for our Lagrangean dual are those involving only finitely many nonzero multipliers. In particular, we recover the usual sufficient condition when only finitely many functional constraints are present. We show that a certain compactness condition in function space plays the role of finiteness, when there are an infinite number of functional constraints. The author's research has been partially supported by Grant ECS8001763 of the National Science Foundation.     

Monoid Domain Constructions of Antimatter Domains

An integral domain without irreducible elements is called an antimatter domain. We give some monoid domain constructions of antimatter domains. Among other things, we show that if D is a GCD domain with quotient field K that is algebraically closed, real closed, or perfect of characteristic p > 0, then the monoid domain D[X; ?⁺] is an antimatter GCD domain. We also show that a GCD domain D is antimatter if and only if P^?1 = D for each maximal t-ideal P of D.     

Doeblin's and Harris' theory of Markov processes

Our notation and definitions are taken from (Chung, K. L.: The general theory of Markov processes according to Doeblin. Z. Wahrscheinlichkeitstheorie und verw. Gebiete 2, 230–254 (1964)). A closed set H is called recurrent in the sense of Harris if there exists a σ-finite measure ? such that for E=H, ?(E) >0 implies Q(x, E)=1 for all tx?H. Theorem 1. Let X be absolutely essential and indecomposable. Then there exists a closed set B?X. such that B contains no acountable disjoint collection of perpetuable sets if and only if X=H+1 where H is recurrent in the sense of Harris and I is either inessential or improperly essential. Theorem 2. If there exists no uncountable disjoint collection of closed sets, then there exists a countable disjoint collection {D_n} _n=1 ^∞ of absolutely essential and indecomposable closed sets such that \(I = X - \sum\nolimits_{n = 1}^\infty {D_n } \) . Under the additional assumption that Suslin's Conjecture holds, Theorem 2 was proved by Jamison (Jamison, B.: A Result in Doeblin's Theory of Markov Chains implied by Suslin's Conjecture. Z. Wahrscheinlichkeitstheorie verw. Gebiete 24, 287–293 (1972)).     

A counterexample to the Bishop-Phelps Theorem in complex spaces

The Bishop-Phelps Theorem asserts that the set of functionals which attain the maximum value on a closed bounded convex subsetS of a real Banach spaceX is norm dense inX ^*. We show that this statement cannot be extended to general complex Banach spaces by constructing a closed bounded convex set with no support points.     

Approximate fixed points of nonexpansive mappings in unbounded sets

It follows from Banach’s fixed point theorem that every nonexpansive self-mapping of a bounded, closed and convex set in a Banach space has approximate fixed points. This is no longer true, in general, if the set is unbounded. Nevertheless, as we show in the present paper, there exists an open and everywhere dense set in the space of all nonexpansive self-mappings of any closed and convex (not necessarily bounded) set in a Banach space (endowed with the natural metric of uniform convergence on bounded subsets) such that all its elements have approximate fixed points.