Asymptotic behaviour of Gaussian random fields

Summary Let X={X(t), t ^N} be a centred Gaussian random field with covariance X(t)X(s)=r(t–s) continuous on ^N×^N and r(0)=1. Let (t,s)=((X(t)–X(s)) ²)^1/2; (t,s) is a pseudometric on ^N. Assume X is -separable. Let D ₁ be the unit cube in ^N and for 0<k, D _k= {x^N: k ^–1 xD₁}, Z(k)=sup{X(t),tD _k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦) as ¦t¦8 then Z(k)-(2Nlogk) ^1/20 as k a.s.     

Constraint decomposition algorithms in global optimization

Many global optimization problems can be formulated in the form min{c(x, y): x X, y Y, (x, y) Z, y G} where X, Y are polytopes in ^p , ⁿ , respectively, Z is a closed convex set in ^p+n, while G is the complement of an open convex set in ⁿ . The function c: ^p+n is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ⁿ . Computational experiments show that the resulting algorithms work well for problems with smalln.     

Problèmes elliptiques à frontière libre axi-symétriques: Estimation du diamètre de la section au moyen de la capacité

Résumé Le but de cet article est d'estimer la capacité d'un pétit segment [a(1–), a(1+)]×{z=0} du démi-plan := {x = (r, z)|r > 0, z } de ² par rapport à la capacité définie par la norme . Ce résultat est ensuite utilisé pour estimer le diamètre de la surface libre de certains problèmes elliptiques non linéaires.     

C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers

We prove that if aC ¹ smooth change of variable : generates a bounded composition operatorff° in the spaceA _p()=L ^p ,p2, then is linear (affine).We also prove that for a nonlinearC ¹ mapping , the norms of exponentialse ⁱ as Fourier multipliers inL ^p () tend to infinity (,||). In both results the condition C ¹ is sharp, it cannot be replaced by the Lipschitz condition.     

Wiener-Ditkin sets for certain beurling algebras

It is proved that closed subgroups of ⁿ are Wiener-Ditkin sets for the Beurling algebrasL ¹ ( ⁿ ), <1.     

The transposition of locally compact,connected translation planes

This note deals with the transposition of translation planes in the topological context. We show that a topological congruenceC of the real vector space ²ⁿ has the property that every hyperplane of ²ⁿ contains a component ofC. This makes it possible to define the transposeP of the topological translation planeP associated withC; it is proved that the translation planeP is topological also. The relationship between collineation groups and the relationship between coordinatizing quasifields ofP andP are also discussed.     

Asymptotic stability at infinity of planar vector fields

Let >0 andX be aC ¹ vector field on the plane such that: (i) for allq², Det(DX(q))>0; and (ii) for allp², with p, Trace(D(X(p))<0. IfX has a singularity and ² Trace(DX)dxdy is less than 0 (resp. greater or equal than 0), then the point at infinity of the Riemann sphere ²{} is a repellor (resp. an attractor) ofX.     

Conditions for separately subharmonic functions to be subharmonic

Letu be a function on ^m × ⁿ , wherem2 andn2, such thatu(x, .) is subharmonic on ⁿ for each fixedx in ^m andu(.,y) is subharmonic on ^m for each fixedy in ⁿ . We give a local integrability condition which ensures the subharmonicity ofu on ^m × ⁿ , and we show that this condition is close to being sharp. In particular, the local integrability of (log⁺ u ⁺) ^m+n–2+ is enough to secure the subharmonicity ofu if >0, but not if <0.     

On a class of Hankel operators: Fredholm properties and invertibility

A study is presented on the Fredholm properties and invertibility of a Hankel integral operator inL ₊ ² () with a kernel function inS whose Fourier transformK is a measurable essentially bounded function in . This study is based on the properties of a Wiener-Hopf operator with a matrix valued symbol naturally associated with the operator mentioned above. Further results are obtained for the case whereK PC(), and an application to a diffraction problem is presented.     

On Non-Negative Elliptic-Type Differential Inequalities in ℝ n , Vanishing Almost Everywhere on Some Open Subset in ℝ n

It is proved in this article that any generalized solution of a sufficiently general class of elliptic-type differential inequalities in  ⁿ that is non-negative almost everywhere in  ⁿ and vanishes almost everywhere on an open set ⁿ is trivial in  ⁿ .     

A functional inequality characterizing convex functions,conjugacy and a generalization of Hölder's and Minkowski's inequalities

Summary The main result says that, iff: _{+ +} satisfies the functional inequalityaf(s) + bf(t) f (as + bt) (s,t 0) for somea, b such that 0 <a < 1 <a + b, thenf(t) = f(1)t, (t 0). A relevant result for the reverse inequality is also discussed. Applying these results we determine the form of all functionsf: _k ⁺ ₊ satisying the above inequalities. This leads to a characterization of both concave and convex functions defined on ₊ ^k–1 , to a notion of conjugate functions and to a general inequality which contains Hölder's and Minkowski's inequalities as very special cases.     

Willmore Surfaces of ?4 and the Whitney Sphere

We make a contribution to the study of Willmore surfaces infour-dimensional Euclidean space ⁴ by making useof the identification of ⁴ with two-dimensionalcomplex Euclidean space ². We prove that theWhitney sphere is the only Willmore Lagrangian surface of genus zero in⁴ and establish some existence and uniquenessresults about Willmore Lagrangian tori in ^{4 2}.     

Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh

Summary We study here the discretisation of the nonlinear hyperbolic equationu _t +div(vf(u))=0 in ³ × ₊, with given initial conditionu(.,0)=u ₀(.) in ², wherev is a function from ² × ₊ to ² such that divv=0 andf is a given nondecreasing function from to . An explicit Euler scheme is used for the time discretisation of the equation, and a triangular mesh for the spatial discretisation. Under a usual stability condition, we prove the convergence of the solution given by an upstream finite volume scheme towards the unique entropy weak solution to the equation.     

Variations on the theme of repeated distances

We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ^{d, d}4. We also give asymptotically tight upper bounds on the total number of occurrences of the favourite distances fromn points in ^{d, d}4. Related results are proved for distances determined byn disjoint compact convex sets in ².At the time this paper was written, both authors were visiting the Technion — Israel Institute of Technology.     

On a Class of Convex Sets and Functions

Given a convex subset C of ⁿ, the set-valued mapping C (where 0C is, by convention, the recession cone of C) is increasing on ₊ if and only if C contains the origin, and decreasing on ₊ if and only if C is contained in its recession cone. This simple fact enables us to define a binary operation which combines a concave or convex function on ^m with a convex subset of ⁿ to produce a convex subset of ^n+m. This binary operation is the set theoretic counterpart of a functional operation introduced by the author. In this paper, we present a detailed study of the class of convex subsets which are contained in their recession cones, and we establish some remarkable properties of our binary operation.Mathematics Subject Classifications (2000) 26A51, 26B25, 26E25.     

Une caracterisation du type de la loi de Cauchy-conforme sur ℝ n

Résumé La loi de Cauchy-conforme est la mesure de probabilité sur ⁿ de densitéC/(1+X²)ⁿ. Le type d'une mesure sur ⁿ étant l'ensemble des mesures images de par les similitudes-translations de ⁿ et étant une mesure de probabilité sans atome, on démontre que le type de est invariant par les inversions de ⁿ si et seulement si est du type de la loi de Cauchy-conforme.

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The conformal Cauchy law is the probability on ⁿ with densityC/(1+X²)ⁿ. It is shown that for a non-atomic measure on ⁿ the following is true: its type is invariant under inversions of ⁿ if and only if it is the type of a conformal Cauchy law. (The type of a measure is defined as the set of its images under similarities and translations.)

     

Parametric and non-parametric minima

We prove that if uBV() is a local minimum for the non-parametric functional F(u) associated to a parametric integrand g××ⁿ⁺¹[0,+), then the hypograph of u is locally minimizing for the parametric functionalG associated to g, among the sets of finite perimeter in ×.     

Minimal Surfaces in a Cone

We prove the convex hull property for properly immersed minimal hypersurfaces in a cone of ⁿ . We deal with the existence of new barriers for the maximum principle application in noncompact truncated tetrahedral domains of ³, describing the space of such domainsadmitting barriers of this kind. Nonexistence results for nonflatminimal surfaces whose boundary lies in opposite faces of a tetrahedraldomain are obtained. Finally, new simple closed subsets of ³ whichhave the property of intersecting any properly immersed minimal surfaceare shown.     

On the problem of two linearized wells

We study the behaviour of sequences of elastic deformationsy ^{n n} whose gradients approach two linearized wells, and give an application to magnetostriction.This article was processed by the author using the style filepljour1m from Springer-Verlag.     

Generalized Elliptic Genus and Cobordism Class of Nonspin Real Grassmannians

We prove that Witten's generalized elliptic genusis rigid under certain group actions and constant(identically zero) on the real Grassmanniansr₄^2m+5, m 1, for appropriate vectorbundles.We also prove that all the Pontryagin numbers of theseGrassmannians are zero, i.e. they have cobordism class zero,which implies the vanishing of all genera.