# Showing papers in "Communications on Pure and Applied Mathematics in 1988"

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Bell Labs

^{1}TL;DR: This work construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity, by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction.

Abstract: We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction. The construction then follows from a synthesis of these different approaches.

8,350 citations

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1,539 citations

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TL;DR: Soit Ω un ensemble ouvert borne regulier et connexe de R N, N≥3. On considere u:Ω→R telle que −Δu=u (N+2)/(N−2) dans Ω, u>0 dans ǫ, u=0 sur ∂Ω as discussed by the authors.

Abstract: Soit Ω un ensemble ouvert borne regulier et connexe de R N , N≥3. On considere u:Ω→R telle que −Δu=u (N+2)/(N−2) dans Ω, u>0 dans Ω, u=0 sur ∂Ω. On note par Hd(Ω; Z 2 ) l'homologie de diemnsion d de Ω a coefficients Z 2 . S'il existe un entier positif d tel que Hd(Ω, Z 2 )¬=0, alors l'equation a une solution

694 citations

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TL;DR: In this paper, the authors define a subset of the path space whose trajectories are given by the solutions of the Cauchy-Riemann equation with respect to a suitable almost complex structure on a symplectic manifold.

Abstract: The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this function, we define on a subset of the path space the flow whose trajectories are given by the solutions of the Cauchy-Riemann equation with respect to a suitable almost complex structure on P. In particular, we prove compactness and transversality results for the set of bounded trajectories.

396 citations

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ETH Zurich

^{1}TL;DR: In this paper, it was shown that estimates for the pressure do not play an essential role in partial regularity results for the Navier-Stokes equations, and that the regularity of Scheffer, Caffarelli, Kohn, and Nirenberg is not essential.

Abstract: Looking at the regularity results of Scheffer, respectively, Caffarelli, Kohn and Nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the Navier-Stokes equations.

337 citations

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TL;DR: Pour toutes donnees initiales d'energie finie, le modele sigma O(k) a des solutions faibles globales en chaque dimension d'espace-temps as discussed by the authors.

Abstract: Pour toutes donnees initiales d'energie finie, le modele sigma O(k) a des solutions faibles globales en chaque dimension d'espace-temps

299 citations

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TL;DR: On demontre un theoreme d'existence general pour des solutions distribution positives, a comportement singulier prescript, de l'equation scalaire semilineaire provenant de la deformation conforme des metriques de Riemann as discussed by the authors.

Abstract: On demontre un theoreme d'existence general pour des solutions distribution positives, a comportement singulier prescript, de l'equation scalaire semilineaire provenant de la deformation conforme des metriques de Riemann

263 citations

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TL;DR: In this paper, the boundary values of a continuous isotropic conductivity can be recovered from voltage and current measurements at the boundary using microlocal analysis, and sharp estimates to establish the continuous dependence of the boundary value of the conductivity on the voltage to current maps.

Abstract: We use the methods of microlocal analysis to give a new proof of a theorem of Kohn and Vogelius, showing that the boundary values of a continuous isotropic conductivity can be recovered from voltage and current measurements at the boundary. Moreover, we prove sharp estimates to establish the continuous dependence of the boundary values of the conductivity on the voltage to current maps.

213 citations

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TL;DR: The notion of a Morse index of a function on a finite-dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold.

Abstract: The notion of a Morse index of a function on a finite-dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold. In this paper, we define for any pair of critical points of a a relative Morse index, which corresponds to the difference of the two Morse indices in finite dimensions. It is based on the spectral flow of the Hessian of a and can be identified with a topological invariant recently defined by Viterbo, and with the dimension of the space of trajectories between the two critical points.

207 citations

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TL;DR: In this paper, the authors consider the problem of retrouver le coefficient a de l'equation elliptique div(a⊇u)=0 dans un domaine Ω avec la condition aux limites u=φ sur ∂Ω quand ∂u/∂N est donnee pour tout φ (regulier).

Abstract: On considere le probleme de retrouver le coefficient a de l'equation elliptique div(a⊇u)=0 dans un domaine Ω avec la condition aux limites u=φ sur ∂Ω quand ∂u/∂N est donnee pour tout φ (regulier). On demontre l'unicite pour a=a 0 +χ(Ω*)b discontinu, ou χ(Ω*) est la fonction indicateur d'un ensemble ouvert Ω*⊂Ω, a o est une fonction C 2 (Ω) donnee et b est une fonction C 2 (Ω*) inconnue

198 citations

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TL;DR: In this paper, a resultat general sur des bornes inferieures for des indices de Morse de points critiques obtenus par des principes de min-max is presented.

Abstract: On donne un resultat general sur des bornes inferieures pour des indices de Morse de points critiques obtenus par des principes de min-max. En combinant cette information avec une inegalite semiclassique on obtient des estimations pointues sur la croissance de certaines valeurs critiques, a partir desquelles on deduit de nouveaux resultats de multiplicite pour des solutions d'equations elliptiques semi-lineaires d'ordre 2

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TL;DR: In this article, the authors present an invariance d'echelle for le calcul numerique des solutions a singularites explosives des equations d'evolution non lineaires, i.e.

Abstract: On presente un algorithme a invariance d'echelle pour le calcul numerique des solutions a singularites explosives des equations d'evolution non lineaires

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Bell Labs

^{1}TL;DR: In this article, the decomposition of arbitrary f in L2(R) in terms of the family of functions cp,(x) = w-'/*exp{ - iimnab + iman - i(x - nb)2}, with a, b > 0.

Abstract: We look at the decomposition of arbitrary f in L2(R) in terms of the family of functions cp,,(x) = w-'/*exp{ - iimnab + iman - i(x - nb)2}, with a, b > 0. We derive bounds and explicit formulas for the minimal expansion coefficients in the case where ab = 2w/N, N an integer 2 2. Transported to the Hilbert space F of entire functions introduced by V. Bargmann, these results are expressed as inequalities of the form

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TL;DR: On etudie le probleme de Dirichlet pour une fonction u dans un domaine borne Ω de R n a frontiere lisse strictement convexe ∂Ω.

Abstract: On etudie le probleme de Dirichlet pour une fonction u dans un domaine borne Ω de R n a frontiere lisse strictement convexe ∂Ω. En un point x de Ω les courbures principales K=(K 1 , …, K n ) du graphe (x,u(x)) satisfont une relation f(K 1 , …, K n )=ψ(x)>0 ou ψ est une fonction positive lisse donnee sur Ω. De plus u doit satisfaire la condition aux limites de Dirichlet u=0 sur ∂Ω. Sous certaines conditions, on demontre qu'il existe une solution u unique admissible lisse dans Ω

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TL;DR: In this article, a new technique for proving instability of bound states for Hamiltonian systems is introduced, which is based on the variational structure of the problem and the difference between the number of negative eigenvalues of two selfadjoint operators.

Abstract: In this paper I am introducing a new technique for proving instability of bound states for Hamiltonian systems. There are already two disparate types of instability results in the literature. The approach developed by Strauss-Shatah [20] gave an instability criterion coming from the variational structure of the problem; on the other hand, Jones' approach [11] produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques. It turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results. Finally in order to demonstrate how this method works I apply the instability criterion in some specific examples.

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TL;DR: In this article, the half space boundary value problem for the Boltzmann equation with an incoming distribution was studied and the boundary layer arising in the kinetic theory of gases as the mean free path tends to zero.

Abstract: In the first part of this paper, we study the half space boundary value problem for the Boltzmann equation with an incoming distribution, obtained when considering the boundary layer arising in the kinetic theory of gases as the mean free path tends to zero. We linearize it about a drifting Maxwellian and prove that, as conjectured by Cercignani [4], the problem is well-posed when the drift velocity u exceeds the sound speed c, but that one (respectively four, five) additional conditions must be imposed when 0 < u < c (respectively - c < u < 0 and u < - c). In the second part, we show that the well-posedness and the asymptotic behavior results for kinetic layers equations with prescribed incoming flux can be extended to more general and realistic boundary conditions.

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TL;DR: Etude du comportement oscillatoire et de la convergence faible des solutions des approximations aux differences dispersives de u t +(1/2 u 2 ) x = 0 as mentioned in this paper.

Abstract: Etude du comportement oscillatoire et de la convergence faible des solutions des approximations aux differences dispersives de u t +(1/2 u 2 ) x =0

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TL;DR: It is established that, even though n2 real numbers possibly of high precision are required to define it, connectionist models do not possess any basic properties different from those of other (nonuniform) highly parallel hardware models.

Abstract: We show that an arbitrary “connectionist” model of n neutrons, defined by an n X n real matrix, can be simulated by a system of O(n3log n) Boolean gates with an O(log n) time slow-down factor. This establishes that, even though n2 real numbers possibly of high precision are required to define it, connectionist models do not possess any basic properties different from those of other (nonuniform) highly parallel hardware models.

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TL;DR: Recently, recent progress in the study of nonlinear hyperbolic wave interactions has revealed a surprising range of new mathematical phenomena and structures as discussed by the authors, such as Mach triple point formation, shock wave diffraction patterns and Riemann problems in one and higher dimensions.

Abstract: Nonlinearities in wave equations lead to focusing and defocusing of solutions. Focusing causes sharply defined wave fronts. The interaction of such sharply defined wave fronts and more generally of nonlinear hyperbolic waves is of fundamental importance and includes such phenomena as Mach triple point formation, shock wave diffraction patterns and the study of Riemann problems in one and higher dimensions.
Recent progress in the study of nonlinear hyperbolic wave interactions has revealed a surprising range of new mathematical phenomena and structures. This mathematical theory should be useful in the design of improved computational algorithms and in part was motivated by such considerations. It is also of considerable interest for its own sake as new mathematical phenomena as well as in terms of the direct insight it provides into physical phenomena.
Within the subject matter and point of view adopted here, we have attempted to present a broad and, we hope, a representative account of recent progress.

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TL;DR: In this article, the authors deduit a inegalite matricielle a partir d'une inegalitite differentielle, and use this result to deduire des inegalites for the solutions d'equations de Riccati matricielles.

Abstract: On deduit une inegalite matricielle a partir d'une inegalite differentielle. On utilise ce resultat pour deduire des inegalites pour les solutions d'equations de Riccati matricielles

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TL;DR: In this article, it was shown that an n-ple impulse solution resembling the superposition of n unstable solitary impulses has at most 2n - 1, and at least n, unstable modes: exactly n unstable modes corresponding to the amplitudes and the rest of them corresponding to spacings.

Abstract: We study McKean's caricature of a nerve conduction equation where H is the Heaviside function. It is proved that an n-ple impulse solution resembling the superposition of n unstable solitary impulses has at most 2n - 1, and at least n, unstable modes: exactly n unstable modes corresponding to the amplitudes and the rest of them corresponding to the spacings. The n amplitude modes always exist. We prove also that for an n-ple impulse solution resembling the superposition of n stable solitary impulses, there are at most n - 1 unstable modes and all of them are of spacing type.

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TL;DR: In this article, the Critical State Theory of Soil Mechanics (CSTOM) was used to examine partial differential equations for frictional materials flowing via plastic yield, including the equations given by the critical state theory of soil mechanics.

Abstract: This paper examines partial differential equations for frictional materials flowing via plastic yield, including the equations given by the Critical State Theory of Soil Mechanics. In particular, the material density is considered as a dependent variable. In previous work we demonstrated that two-dimensional plastic flow may be ill posed due to an instability along two rays in Fourier transform space. In this paper, we show that in three dimensions the equations are linearly well posed provided all three strain rates are nonzero.

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TL;DR: On decrit une nouvelle methode pour demontrer l'eclatement en temps fini pour une classe d'equations paraboliques semi-lineaires.

Abstract: On decrit une nouvelle methode pour demontrer l'eclatement en temps fini pour une classe d'equations paraboliques semi-lineaires

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TL;DR: In this paper, the authors et al. study the problem of finding the initial solution of a problem with respect to a set of initial variables, and propose a solution for each variable.

Abstract: On etudie la limite pour e→0 des solutions du probleme aux valeurs limites et initiales: u t e+(ue 2 /2) x =(eu xx e)/2; ue(x, 0)=u 0 (x); ue(0, t)=u b (t), dans D={(x, t): x≥0, t≥0}. On suppose que u 0 (x)∈L 1 (0, ∞) et u b ∈C(0, ∞)

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TL;DR: In this paper, the existence of weak solutions to the incompressible Euler equations for vortex sheet initial data is an open problem, and two approaches to this problem are the smoothing of vortex sheets by approximated identifies (which provides a sequence of smooth initial conditions, converging to vortex sheet starting conditions, for which classical solutions exist) and viscous smoothing.

Abstract: : The existence of weak solutions to the incompressible Euler equations for vortex sheet initial data is an open problem. Two approaches to this problem are the smoothing of vortex sheets by approximated identifies (which provides a sequence of smooth initial conditions, converging to vortex sheet initial conditions, for which classical solutions exist) and viscous smoothing of vortex sheets. These ideas are among the principal motivations for the recent series of papers in which the nature of the limiting behavior of sequences of solutions of Euler's equations, are examined. The concept of generalized Young measure-valued solution of the Euler equations is introduced. In two dimensions, approximate solution sequences always have subsequences which converge, in an appropriate sense, to non-oscillatory generalized Young measure-valued solutions of Euler's equations. Infinitesimal vortices of zero circulation were described in and named phantom vortices. The vortices in our example are somewhat different in that they have two length scales. The two length scales are important, for one cannot pack phantom vortices on successively finer lattices, as above, keeping the vorticity absolutely summable, without at the same time obtaining strong convergence in L-sq. Reprints.

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TL;DR: Etude des procedures de decision for les sous langages elementaires de la theorie des ensembles is described in this article, where a sous-classe restreinte des △ 0 -formules de la theory.

Abstract: Etude des procedures de decision pour les sous langages elementaires de la theorie des ensembles. Insolubilite du probleme de decision pour une sous-classe restreinte des △ 0 -formules de la theorie des ensembles. Etablissement de bornes

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TL;DR: In this paper, the application of scalar conservation laws to semiconductor device fabrication is described and conditions on the boundary values that characterize physically correct solutions are derived, and the analogue of the Riemann problem for these problems is analyzed and solved.

Abstract: The application of the theory of scalar conservation laws to semiconductor device fabrication is described. This application is the source of a Stefan problem and another moving boundary problem for a class of such equations. The analogue of the Riemann problem for these problems is analyzed and solved. Conditions on the boundary values that characterize physically correct solutions are derived.