Théorème de Whitehead stratifié et invariants de nœuds

By considering homotopies that preserve the stratification, one obtains a natural notion of homotopy for stratified spaces. In this short note, we introduce invariants of stratified homotopy, the stratified homotopy groups. We show that they satisfy a stratified version of Whitehead's theorem. As an example, we introduce a complete knot invariant defined via the stratified homotopy groups.     

Théorie spectrale de la propagation des ondes acoustiques dans un milieu stratifié perturbé

In this article the spectral analysis of the self adjoint operator governing the propagation acoustic waves in a perturbed stratified medium is given. Both a medium whose sound speed is a short range perturbation of that of a stratified medium and a stratified medium exterior to a compact set are considered. The basic tool is a division theorem for the self adjoint operator governing the propagation of acoustic waves in a pure stratified medium.     

Relative arithmetic

In nonstandard mathematics, the predicate ‘x is standard’ is fundamental. Recently, ‘relative’ or ‘stratified’ nonstandard theories have been developed in which this predicate is replaced with ‘x is y ‐standard’. Thus, objects are not (non)standard in an absolute sense, but (non)standard relative to other objects and there is a whole stratified universe of ‘levels’ or ‘degrees’ of standardness. Here, we study stratified nonstandard arithmetic and the related transfer principle. Using the latter, we obtain the ‘reduction theorem’ which states that arithmetical formulas can be reduced to equivalent bounded formulas. Surprisingly, the reduction theorem is also equivalent to the transfer principle. As applications, we obtain a truth definition for arithmetical sentences and we formalize Nelson's notion of impredicativity (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

THE AMBIGUOUS TYPE THEORY IS HEREDITARILY UNDECIDABLE

The ambiguous type theory ATT is introduced in [1] in order to obtain a new proof of Specker's theorem on typical ambiguity [3]. In the present paper we imbed the recursively undecidable theory TR of a single transitive-reflexive relation in an arbitrary stratified theory T such that T ? ATT and so reduce the problem of derivability in TR to the same problem for T. From this it follows that ATT is hereditarily undecidable, i.e. ATT has no decidable subtheories.     

The Limiting Absorption Principle for Elastic Wave Propagation Problems in Perturbed Stratified Media ℝ3

We consider the self-adjoint operator governing the propagation of elastic waves in perturbed stratified media ℝ³ with free boundary–interface conditions. In this paper we establish the limiting absorption principle for this self-adjoint operator in appropriate Hilbert space. The proof of the limiting absorption principle is based on the division theorem which is proved by means of eigenfunction expansions for the self-adjoint operator governing the propagation of elastic waves in unperturbed stratified media ℝ³.     

Comparison between two complexes on a singular space

The aim of this Note is twofold. In the first step we study the Witten deformation for stratified spaces X and radial Morse functions on them and prove a spectral gap theorem for the Witten Laplacian. In the second step we focus on spaces with isolated conic singularities, where we construct a geometric complex associated to the Morse function and give two comparison results.     

On topological invariants of stratified maps with non-Witt target

The Cappell-Shaneson decomposition theorem for self-dual sheaves asserts that on a space with only even-codimensional strata any self-dual sheaf is cobordant to an orthogonal sum of twisted intersection chain sheaves associated to the various strata. In sharp contrast to this result, we prove that on a space with only odd-codimensional strata (not necessarily Witt), any self-dual sheaf is cobordant to an intersection chain sheaf associated to the top stratum: the strata of odd codimension do not contribute terms. As a consequence, we obtain formulae for the pushforward of characteristic classes under a stratified map whose target need not satisfy the Witt space condition. To prove these results, we introduce a new category of superperverse sheaves, which we show to be abelian. Finally, we apply the results to the study of desingularization of non-Witt spaces and exhibit a singular space which admits a PL resolution in the sense of M. Kato, but no resolution by a stratified map.

     

Global inversion and covering maps on length spaces

In order to obtain global inversion theorems for mappings between length metric spaces, we investigate sufficient conditions for a local homeomorphism to be a covering map in this context. We also provide an estimate of the domain of invertibility of a local homeomorphism around a point, in terms of a kind of lower scalar derivative. As a consequence, we obtain an invertibility result using an analog of the Hadamard integral condition in the frame of length spaces. Some applications are given to the case of local diffeomorphisms between Banach-Finsler manifolds. Finally, we derive a global inversion theorem for mappings between stratified groups.     

A Kuran Type Regularity Criterion for Sub-Laplacians

The aim of this paper is to show the strong connection between regularity of bounded open set boundary points and quasi-boundedness, on the same set, of the fundamental solution of stratified Lie group sub-Laplacians. In the euclidean case the theorem was proved by Kuran (J Lond Math Soc 2(19):301–311, 1979). We later give two examples using some direct consequences of main theorem.     

Weak Solvability of the Dirichlet Problem on Stratified Sets

The Dirichlet problem is posed for an analog of the Beltrami--Laplace operator on sets consisting of manifolds of various dimensions regularly adjacent to one another (stratified sets). A special system of notions permits one to prove analogs of Green's integral identities and the Poincaré inequality for Sobolev type spaces. The weak solvability of the Dirichlet problem for this operator, as well as for an analog of the biharmonic operator, is proved on the basis of the Riesz theorem on the representation of linear functionals.     

Existence and qualitative theory for stratified solitary water waves

This paper considers two-dimensional gravity solitary waves moving through a body of density stratified water lying below vacuum. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the water and vacuum is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity field and density function, there exists a continuous curve of such solutions that includes large-amplitude surface waves. Furthermore, following this solution curve, one encounters waves that come arbitrarily close to possessing points of horizontal stagnation.We also provide a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the wave speed from above and below, some of which are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores in this physical regime; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry.     

Théorème d'unicité pour un problème d'ondes élastiques

In this Note we prove a uniqueness theorem for the an elastic waves problem (in frequency domain). The propagation domain is a stratified half-space with a vertical borehole. We impose radiation conditions at infinity which ensure uniqueness of the solution. To cite this article: L. Alem, L. Chorfi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Inverse sounding problem using stabilization of the magnetic field of a loop

We consider the inverse sounding problem for a stratified medium using the variable magnetic field of a loop. A uniqueness theorem and a theorem ensuring stable determination of the integral conductivity of the medium are proved. An algorithm is proposed for the solution of the inverse problem based on a transformation from the time domain to the frequency domain. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 87–95.     

关于二层海中的二阶波浪绕射问题*

在本文中,我们用二层海模型,探讨了层化海洋中任意三维物体的二阶波浪绕射问题,给出了多色波场中二阶波浪散射势边值问题的数学提法以及基于一个弱的远场辐射条件下解的表式。同时,利用Green定理,并通过引入一个辅助势函数,我们导出了结构所受二阶波浪荷载的积分表式。结果表明,海水的层化特性对结构物所受之二阶差频波浪荷载可能具有显著的影响。     

Typical ambiguity and elementary equivalence

A sentence of the usual language of set theory is said to be stratified if it is obtained by “erasing” type indices in a sentence of the language of Russell's Simple Theory of Types. In this paper we give an alternative presentation of a proof the ambiguity theorem stating that any provable stratified sentence has a stratified proof. To this end, we introduce a new set of ambiguity axioms, inspired by Fraïssé's characterization of elementary equivalence; these axioms can be naturally used to give different proofs of the ambiguity theorem (semantic or syntactic, classical or intuitionistic). MSC: 03B15, 03F50, 03F55.     

Compactification of moduli space of harmonic mappings

We introduce the notion of harmonic nodal maps from the stratified Riemann surfaces into any compact Riemannian manifolds and prove that the space of the energy minimizing nodal maps is sequentially compact. We also give an existence result for the energy minimizing nodal maps. As an application, we obtain a general existence theorem for minimal surfaces with arbitrary genus in any compact Riemannian manifolds. Received: 1 April 1997; revised: 15 April 1998.     

Sampling of Paley-Wiener functions on stratified groups

We consider a generalization of entire functions of spherical exponential type on stratified groups. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a stratified group is uniquely determined by its values on some discrete subgroups. The main result of the article is reconstruction formula of spectral entire functions from their values on discrete subgroups using Lagrangian splines.     

Stone''''s representation theorem in fuzzy topology

In this paper, a complete solution to the problem of Stone's repesentation theorem in fuzzy topology is given for a class of completely distributive lattices. Precisely, it is proved that if L is a frame such that 0 ∈ L is a prime or 1 ∈ L is a coprime, then the category of distributive lattices is dually equivalent to the category of coherent L-locales and that if L is moreover completely distributive, then the category of distributive lattices is dually equivalent to the category of coherent stratified L-topological spaces.     

The Hauptsatz for Stratified Comprehension: A Semantic Proof

We prove the cut-elimination theorem, Gentzen's Hauptsatz, for the system for stratified comprehension, i. e. Quine's NF minus extensionality. Mathematics Subject Classification: 03B15, 03F05.     

Local and global methods in representations of Hecke algebras

This paper aims at developing a “local-global” approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. We first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. We prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary (the “good” prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells.