Quadratisation de Certaines Structures de Poisson

On sait associer à certaines structures de Poisson surRⁿ, de 1-jet nul en 0, des actions de R² sur Rⁿ, donnéespar le ‘rotationnel’ de leur partie quadratiqueet un autre champ de vecteurs. Lorsque ces actions sont ‘nonrésonantes’ et ‘hyperboliques’, onmontre que ces structures sont ‘quadratisables’,en ce sens qu'il existe des coordonnées dans lesquelles,elles sont quadratiques. Dans le cas de la dimension 3, nosrésultats mènent à la ‘non-dégénérescence’générique des structures de Poisson quadratiquesà rotationnels inversibles. We can associate with some Poisson structures defined on Rⁿwith a zero 1-jet at zero, actions from R² on Rⁿ, given by the‘curl’ of their quadratic part and another vectorfield. Assuming that those actions are ‘hyperbolics’and without ‘resonances’, we give a normal formfor those structures. On R³, we prove that every quadratic Poissonstructure with invertible curl, is generically ‘non degenerate’.     

Monodromy and Connectedness

The purpose of this note is initially to present an elementarybut surprising connectedness principle pertaining to the intersectionof a fixed subvariety X of some ambient space Z with anothersubvariety Y which is ‘mobile’ (in the sense ofbeing movable, rather than actually moving). It is via thismobility that monodromy enters the picture, permitting the crucialpassage from ‘relative’ or total-space irreducibilityto ‘absolute’ or fibrewise connectedness (and sometimesirreducibility). A general form of this principle is given inTheorem 2 below. 1991 Mathematics Subject Classification 14C99,15N05.     

Locally Nilpotent p-Groups whose Proper Subgroups are Hypercentral or Nilpotent-by-Chernikov

Let G be a group and P be a property of groups. If every propersubgroup of G satisfies P but G itself does not satisfy it,then G is called a minimal non-P group. In this work we studylocally nilpotent minimal non-P groups, where P stands for ‘hypercentral’or ‘nilpotent-by-Chernikov’. In the first case weshow that if G is a minimal non-hypercentral Fitting group inwhich every proper subgroup is solvable, then G is solvable(see Theorem 1.1 below). This result generalizes [3, Theorem1]. In the second case we show that if every proper subgroupof G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov(see Theorem 1.3 below). This settles a question which was consideredin [1–3, 10]. Recently in [9], the non-periodic case ofthe above question has been settled but the same work containsan assertion without proof about the periodic case. The main results of this paper are given below (see also [13]).     

On the Hecke Equivariance of the Borcherds Isomorphism

The Borcherds isomorphism is proved to be Hecke equivariantif one considers multiplicative Hecke operators acting on theintegral weight meromorphic modular forms. This answers a partof a question of Borcherds (see ‘Automorphic forms onO_{s+2, 2}(R) and infinite products’, Invent. Math. 120 (1995)161–213, 17.10), using his suggestion to define the multiplicativeHecke operators. 2000 Mathematics Subject Classification 11F37.     

The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings

Motivated in part by the first author's work [23] on the Weyl-Berryconjecture for the vibrations of ‘fractal drums’(that is, ‘drums with fractal boundary’), M. L.Lapidus and C. Pomerance [31] have studied a direct spectralproblem for the vibrations of ‘fractal strings’(that is, one-dimensional ‘fractal drums’) and establishedin the process some unexpected connections with the Riemannzeta-function = (s) in the ‘critical interval’0 < s < 1. In this paper we show, in particular, thatthe converse of their theorem (suitably interpreted as a naturalinverse spectral problem for fractal strings, with boundaryof Minkowski fractal dimension D (0,1)) is not true in the‘midfractal’ case when D = , but that it is true for all other D in the criticalinterval (0,1) if and only if the Riemann hypothesis is true.We thus obtain a new characterization of the Riemann hypothesisby means of an inverse spectral problem. (Actually, we provethe following stronger result: for a given D (0,1), the aboveinverse spectral problem is equivalent to the ‘partialRiemann hypothesis’ for D, according to which = (s)does not have any zero on the vertical line Re s = D.) Therefore,in some very precise sense, our work shows that the question(à la Marc Kac) "Can one hear the shape of a fractalstring?" – now interpreted as a suitable converse (namely,the above inverse problem) – is intimately connected withthe existence of zeros of = (s) in the critical strip 0 <Res < 1, and hence to the Riemann hypothesis.     

A safeguarded dual weighted residual method

Andreas Veeser ^{The dual weighted residual (DWR) method yields reliable a posteriorierror bounds for linear output functionals provided that theerror incurred by the numerical approximation of the dual solutionis negligible. In that case, its performance is generally superiorthan that of global ‘energy norm’ error estimatorswhich are ‘unconditionally’ reliable. We presenta simple numerical example for which neglecting the approximationerror leads to severe underestimation of the functional error,thus showing that the DWR method may be unreliable. We proposea remedy that preserves the original performance, namely a DWRmethod safeguarded by additional asymptotically higher ordera posteriori terms. In particular, the enhanced estimator isunconditionally reliable and asymptotically coincides with theoriginal DWR method. These properties are illustrated via theaforementioned example.}     

Interfacial instabilities and fingering formation in Hele-Shaw flow

The interfacial instability of Hele-Shaw flow has been a crucialissue for the understanding of the pattern formation of viscousfingers in a Hele-Shaw cell. By using a unified asymptotic approach,we derive two different types of instability mechanisms for‘slightly’ time-dependent finger solutions; namely,(i) the global-trapped-wave (GTW) instability; and (ii) thezero-frequency (null-f) instability. On the basis of these instabilitymechanisms, the selection of viscous finger formation is clarified;the apparent contradiction between the previous linearstabilityanalysis by Tanveer (1987, Phys. Fluid 30, 1589) and othersand the numerical simulations by DeGregoria & Schwartz (1986,J. Fluid Mech. 164, 383)and the experimental evidence is reconciled.     

Reduced-basis output bound methods for parabolic problems

^** Email: rovas{at}uiuc.edu^*** Email: luc_machiels{at}mckinsey.com^**** Corresponding author. Email: maday{at}ann.jussieu.fr In this paper, we extend reduced-basis output bound methodsdeveloped earlier for elliptic problems, to problems describedby ‘parameterized parabolic’ partial differentialequations. The essential new ingredient and the novelty of thispaper consist in the presence of time in the formulation andsolution of the problem. First, without assuming a time discretization,a reduced-basis procedure is presented to ‘efficiently’compute accurate approximations to the solution of the parabolicproblem and ‘relevant’ outputs of interest. In addition,we develop an error estimation procedure to ‘a posteriorivalidate’ the accuracy of our output predictions. Second,using the discontinuous Galerkin method for the temporal discretization,the reduced-basis method and the output bound procedure areanalysed for the semi-discrete case. In both cases the reduced-basisis constructed by taking ‘snapshots’ of the solutionboth in time and in the parameters: in that sense the methodis close to Proper Orthogonal Decomposition (POD).     

Representations of compact linear operators in Banach spaces and nonlinear eigenvalue problems

Let X and Y be reflexive Banach spaces with strictly convexduals, and let T be a compact linear map from X to Y. It isshown that a certain nonlinear equation, involving T and itsadjoint, has a normalised solution (an ‘eigenvector’)corresponding to an ‘eigenvalue’, and that the sameis true for each member of a countable family of similar equationsinvolving the restrictions of T to certain subspaces of X. Theaction of T can be described in terms of these ‘eigenvectors’.There are applications to the p-Laplacian, the p-biharmonicoperator and integral operators of Hardy type.     

Transformations Between Sturm-Liouville Problems with Eigenvalue Dependent and Independent Boundary Conditions

Explicit relationships are given connecting ‘almost’isospectral Sturm-Liouville problems with eigen-value dependent,and independent, boundary conditions, respectively. Applicationis made to various direct and inverse spectral questions. 2000Mathematics Subject Classification 34B24, 34L05, 47E05.     

On weakly eutactic forms

We make precise some properties of the Hermite function in relationwith the Morse theory introduced by Avner Ash in his papers‘On eutactic forms’,Canad. J. Math. 29 (1977) 1040–1054and ‘On the existence of eutactic forms’,Bull. LondonMath. Soc. 12 (1980) 192–196, and with the cellular decompositionof the space of positive definite quadratic forms. We also establisha link between Ash's and Bavard's mass formulae.     

Are Operads Algebraic Theories?

I exhibit a pair of non-symmetric operads that, while not themselvesisomorphic, induce isomorphic monads. The existence of sucha pair implies that if ‘algebraic theory’ is understoodas meaning ‘monad’ then operads cannot be regardedas algebraic theories of a special kind. 2000 Mathematics SubjectClassification 18D50 (primary), 18C10, 18C15 (secondary).     

Ordered C*-Modules

In this first part of a study of ordered operator spaces, wedevelop the basic theory of ‘ordered C^*-bimodules’.A crucial role is played by ‘open tripotents’, aJB^*-triple variant of Akemann's notion of open projection. 2000Mathematics Subject Classification 46L08, 47L07 (primary), 46L07,47B60, 47L05 (secondary).     

Necessary and Sufficient Tauberian Conditions, Under Which Convergence Follows From Summability (C, 1)

Let (s_k: k = 0, 1, ...) be a sequence of real numbers whichis summable (C, 1) to a finite limit. We prove that (s_k) isconvergent if and only if the following two conditions are satisfied: where _n denotes the integer part of the productn. Both conditions are clearly satisfied if (s_k) is slowly decreasingin the sense of R. Schmidt and G. H. Hardy. The symmetric counterparts of the conditions above are thosewhen ‘limsup’ and ‘liminf’ are interchangedon the left-hand sides, while the inequality sign ‘ ’is changed for the opposite ‘ ’ in them. Next, let (s_k) be a sequence of complex numbers which is summable(C, 1) to a finite limit. We prove that (s_k) is convergent ifand only if one of the following conditions is satisfied: We also prove a general Tauberian theorem forsequences in ordered linear spaces.     

Uniqueness cases in odd-type groups of finite Morley rank

There is a longstanding conjecture, due to Gregory Cherlin andBoris Zilber, that all simple groups of finite Morley rank aresimple algebraic groups. One of the major theorems in the areais Borovik's trichotomy theorem. The ‘trichotomy’here is a case division of the generic minimal counterexampleswithin odd type, that is, groups with a large and divisibleSylow° 2-subgroup. The so-called ‘uniqueness case’in the trichotomy theorem is the existence of a proper 2-generatedcore. It is our aim to drive the presence of a proper 2-generatedcore to a contradiction, and hence bind the complexity of theSylow° 2-subgroup of a minimal counterexample to the Cherlin–Zilberconjecture. This paper shows that the group in question is aminimal connected simple group and has a strongly embedded subgroup,a far stronger uniqueness case. As a corollary, a tame counterexampleto the Cherlin–Zilber conjecture has Prüfer rankat most two.     

Generalising Group Algebras

We generalise group algebras to other algebraic objects withbounded Hilbert space representation theory; the generalisedgroup algebras are called ‘host’ algebras. The mainproperty of a host algebra is that its representation theoryshould be isomorphic (in the sense of the Gelfand–Raikovtheorem) to a specified subset of representations of the algebraicobject. Here we obtain both existence and uniqueness theoremsfor host algebras as well as general structure theorems forhost algebras. Abstractly, this solves the question of whena set of Hilbert space representations is isomorphic to therepresentation theory of a C*-algebra. To make contact withharmonic analysis, we consider general convolution algebrasassociated to representation sets, and consider conditions fora convolution algebra to be a host algebra.     

On Manifolds Whose Tangent Bundle is Big and 1-Ample

A line bundle over a complex projective variety is called bigand 1-ample if a large multiple of it is generated by globalsections and a morphism induced by the evaluation of the spanningsections is generically finite and has at most 1-dimensionalfibers. A vector bundle is called big and 1-ample if the relativehyperplane line bundle over its projectivisation is big and1-ample. The main theorem of the present paper asserts that any complexprojective manifold of dimension 4 or more, whose tangent bundleis big and 1-ample, is equal either to a projective space orto a smooth quadric. Since big and 1-ample bundles are ‘almost’ample, the present result is yet another extension of the celebratedMori paper ‘Projective manifolds with ample tangent bundles’(Ann. of Math. 110 (1979) 593–606). The proof of the theorem applies results about contractionsof complex symplectic manifolds and of manifolds whose tangentbundles are numerically effective. In the appendix we re-provethese results. 2000 Mathematics Subject Classification 14E30,14J40, 14J45, 14J50.     

The Kissing Numbers of Convex Bodies -- A Brief Survey

Nearly four hundred years ago, the cubic close-packing of equalspheres in R³ was discovered by Kepler [21], in which each spheretouches 12 others. In 1694, Gregory and Newton discussed thefollowing thirteen spheres problem. Can a rigid material spherebe brought into contact with 13 other such spheres of the samesize? Gregory believed ‘yes’, while Newton thought‘no’. 1991 Mathematics Subject Classication 11H31,52C17.