Showing papers by "Courant Institute of Mathematical Sciences published in 1991"
TL;DR: It is shown that gravitational waves from astronomical sources have a nonlinear erect on laser interferometer detectors on Earth, an erect which has hitherto been neglected, but which is of the same order of magnitude as the linear erects.
Abstract: It is shown that gravitational waves from astronomical sources have a nonlinear effect on laser interferometer detectors on Earth, an effect which has hitherto been neglected, but which is of the same order of magnitude as the linear effects. The signature of the nonlinear effect is a permanent displacement of test mases after the passage of a wave train.
537 citations
TL;DR: In this article, the authors consider real C' functions F defined on a Banach space X and present a rather general deformation result for finding critical points of F. They use versions of Ekeland's Principle and deformation arguments.
Abstract: In the course of writing a chapter of a book we observed some simple facts dealing with the Palais SmaIe property and critical points of functions. Some of these facts turned out to be known, though not well-known, and we think it worthwhile to make them more available. In addition, we present some other recent results which we believe will prove to be useful-in particular, a result of Ghoussoub and Preiss; see [ 91, [ 81. There are two useful techniques used in obtaining critical points. One is Ekeland’s Principle (see below), the other is based on deformation arguments. We will use versions of both of them. In particular we present a rather general deformation result. Throughout this paper we consider real C’ functions F defined on a Banach space X. When looking for critical points of F it has become standard to assume the following “compactness condition”:
366 citations
TL;DR: In this paper, the authors presented an interpretation of effective resistance in electrical networks in terms of random walks on underlying graphs using simple and elegant proofs for some known results in random walks and electrical networks.
Abstract: In this article we present an interpretation ofeffective resistance in electrical networks in terms of random walks on underlying graphs Using this characterization we provide simple and elegant proofs for some known results in random walks and electrical networks We also interpret the Reciprocity theorem of electrical networks in terms of traversals in random walks The byproducts are (a) precise version of thetriangle inequality for effective resistances, and (b) an exact formula for the expectedone-way transit time between vertices
352 citations
TL;DR: In this paper, the authors studied coherent, energy-minimizing mixtures of two linearly elastic phases with identical elastic moduli and derived a formula for the "relaxed" or "macroscopic" energy of the system, by identifying microstructures that minimize the total energy when the volume fractions and the average strain are fixed.
Abstract: This paper studies coherent, energy-minimizing mixtures of two linearly elastic phases with identical elastic moduli. We derive a formula for the “relaxed” or “macroscopic” energy of the system, by identifying microstructures that minimize the total energy when the volume fractions and the average strain are fixed. If the stress-free strains of the two phases are incompatible then the relaxed energy is nonconvex, with “double-well structure”. An optimal microstructure always exists within the class of layered mixtures. The optimal microstructure is generally not unique, however; we show how to construct a large family of optimal, sequentially laminated microstructures in many circumstances. Our analysis provides a link between the work of Khachaturyan and Roitburd in the metallurgical literature and that of Ball, James, Pipkin, Lurie, and Cherkaev in the recent mathematical literature. We close by explaining why the corresponding problem for three or more phases is fundamentally more difficult.
352 citations
TL;DR: In this paper, the authors prove the hydrodynamic limit of Ginzburg-Landau models by considering relative entropy and its rate of change with respect to local Gibbs states.
Abstract: We prove the hydrodynamic limit of Ginzburg-Landau models by considering relative entropy and its rate of change with respect to local Gibbs states. This provides a new understanding of the role played by relative entropy in the hydrodynamics of interacting particle systems.
340 citations
01 Sep 1991
TL;DR: A special-purpose point clustering algorithm is described, and its application to automatic grid generation, a technique used to solve partial differential equations, is considered.
Abstract: A special-purpose point clustering algorithm is described, and its application to automatic grid generation, a technique used to solve partial differential equations, is considered. Extensions of techniques common in computer vision and pattern recognition literature are used to partition points into a set of enclosing rectangles. Examples from 2-D calculations are shown, but the algorithm generalizes readily to three dimensions. >
337 citations
TL;DR: In this article, the authors studied the convexity and star-shapedness of level sets of positive solutions to elliptic or parabolic equations and showed that these properties can be easily described and are accessible to variational and maximum principles in analysis.
Abstract: Some geometrical properties such as convexity and star-shapedness of level sets of positive solutions to elliptic or parabolic equations have been studied via various authors (see an excellent lecture set of notes by B. Kawohl, [ 81 and the references therein). One reason to choose star-shapedness and convexity, other than other properties, as geometrical properties, is that they can be easily described and are accessible to variational and maximum principles in analysis. Here we are interested in some general questions regarding level sets of solutions to elliptic and parabolic equations, such as the size (i.e., HausdorB measure of appropriate dimension) and the topology of these level sets and estimates on critical point sets, etc. The study of such problems was motivated by the study of moving defects in evolution problems of harmonic maps and liquid crystals (see [lo]). In [lo], I have studied a model for the evolution of nematic liquid crystals. The singular set of optical axes (i.e., defects) of liquid crystal in motion can be described precisely by the nodal set of solutions to certain parabolic equations. Recently, there were several rather interesting articles studying the nodal sets of eigenfunctions of Laplacians on a compact Riemannian manifold by Donnelly and Feffermann [ 2 1, [ 3 ] or, generally, solutions of second-order elliptic equations in Hardt and Simon [ 61. The present work can also be viewed as a natural extension of [2], [3], and [ 61. Our main result can be stated as:
271 citations
TL;DR: In this article, the authors studied the limiting behavior of u = U' as E + 0 and showed that the solutions of (1.1) are in a certain sense compact as functions of space-time.
264 citations
264 citations
TL;DR: In this article, the hydrodynamic behavior of asymmetric simple exclusions and zero range processes in several dimensions was studied and a nonlinear conservation law was derived for the time evolution of the macroscopic particle density.
Abstract: We study the hydrodynamic behavior of asymmetric simple exclusions and zero range processes in several dimensions. Under Euler scaling, a nonlinear conservation law is derived for the time evolution of the macroscopic particle density.
256 citations
TL;DR: In this paper, the authors established multiple solutions for a semilinear elliptic equation with superlinear nonlinearity without assuming any symmetry, and proved that these solutions can be obtained without any assumption of symmetry.
Abstract: In this note we establish multiple solutions for a semilinear elliptic equation with superlinear nonlinearility without assuming any symmetry.
TL;DR: A new version of the Lovasz Local lemma is used to prove the existence of Latin transversals in matrices where no symbol appears too often.
04 Dec 1991
TL;DR: It is proved that no on-line scheduling algorithm can guarantee a cumulative value greater than 1/4th the value obtainable by a clairvoyant scheduler, and it is shown that an online uniprocessor scheduling algorithm TD1 actually has a competitive factor of 1/ 4; this bound is thus shown to be tight.
Abstract: The authors study the performance of online algorithms in environments where no value is obtained for the partial execution of a request. They prove that no online scheduling algorithm can have a competitive factor greater than 0.25 times the optimal. They further refine this bound by considering the effect of the loading factor. Other models of task systems (for example, tasks systems consisting of many types of task requests), are considered. Similar upper bounds on the competitive factor that can be made by online scheduling algorithms in these environments are proved. It is shown that the performance bound of 0.25 is tight by means of a simple online uniprocessor scheduling algorithm has a competitive factor of 1/4. The authors extend the discussion to systems with dual processors. They show that the upper bound for the dual-processor online scheduling problem is 1/2 if all tasks have the same value density. This bound is tight if the tasks all also have zero laxity. >
TL;DR: In this article, the authors proved the existence of multidimensional traveling wave solutions of the bistable reaction-diffusion equation with periodic coefficients under the condition that these coefficients are close to constants.
Abstract: We prove the existence of multidimensional traveling wave solutions of the bistable reaction-diffusion equation with periodic coefficients under the condition that these coefficients are close to constants. In the case of one space dimension, we prove their asymptotic stability.
TL;DR: Results show that assimilating the profiler wind data leads to a large reduction of forecast/analysis error in heights as well as in winds, over the Profiler region and also downstream, when compared with the results of assimilated the radiosonde data alone.
Abstract: The behavior of forecast error covariances in a fairly realistic setting is demonstrated via a Kalman filter algorithm. It is used to assimilate simulated data from the existing radiosonde network, from the demonstration network of 31 Doppler wind profilers in the central U.S., and from hypothetical radiometers located at five of the profiler sites. Some theoretical justification of the hypothesis advanced by Phillips (1982), and the hypothesis is used to formulate the model error covariance matrix required by the Kalman filter. The results show that assimilating the profiler wind data leads to a large reduction of forecast/analysis error in heights as well as in winds, over the profiler region and also downstream, when compared with the results of assimilating the radiosonde data alone. The forecast error covariance matrices that the Kalman filter calculates to obtain this error reduction differ considerably from those prescribed by the optimal interpolation schemes that are employed for data assimilation at operational centers.
01 Oct 1991
TL;DR: It is found that analytical comparisons provide information of limited value, and that probabilistic comparisons overcome some of these limitations.
Abstract: A person testing a program has many methods to choose from, but little solid information about how these methods compare. Where analytic comparisons do exist, their significance is often in doubt. In this paper we examine various comparisons that have been used or proposed for test data selection and adequacy criteria. We characterize them by type and identify their strengths and weaknesses. We examine useful properties of comparisons and study the relationship between analytical and probabilistic comparisons. We find that analytical comparisons provide information of limited value, and that probabilistic comparisons overcome some of these limitations.
01 Jun 1991
TL;DR: Borders on the number of vertices on the upper envelope of a collection of Voronoi surfaces are derived, and efficient algorithms to calculate these vertices are provided.
Abstract: Given a setS ofsources (points or segments) in ℜ211C;d, we consider the surface in ℜ211C;d+1 that is the graph of the functiond(x)=minpeSρ(x, p) for some metricρ. This surface is closely related to the Voronoi diagram, Vor(S), ofS under the metricρ. The upper envelope of a set of theseVoronoi surfaces, each defined for a different set of sources, can be used to solve the problem of finding the minimum Hausdorff distance between two sets of points or line segments under translation. We derive bounds on the number of vertices on the upper envelope of a collection of Voronoi surfaces, and provide efficient algorithms to calculate these vertices. We then discuss applications of the methods to the problems of finding the minimum Hausdorff distance under translation, between sets of points and segments.
TL;DR: In this article, the existence, local uniqueness, and global regularity of minimizers were proved and the size of possible defects was also estimated for nematic liquid crystals with variable degree of orientation, which can be described by a map which minimizes certain degenerate variational integrals.
Abstract: Energy minimizing static configurations of nematic liquid crystals with variable degree of orientation can be described by a map which minimizes certain degenerate variational integrals. This mathematical model was recently posed by J. L. Erickson. In this paper, we shall prove the existence, local uniqueness, and global regularity of minimizers. We shall also estimate the size of possible defects.
TL;DR: The model is extended by incorporating electrogenic Na+/K+ exchange, and the predicted dependence of guanylate cyclase activity on [Ca2+] closely matches a cooperative inhibition equation suggested by the experimental results of Koch and Stryer (1988) oncyclase activity in bovine rods.
01 Mar 1991
TL;DR: In this paper, a randomized incremental algorithm for computing a single face in an arrangement of n line segments in the plane is presented, and the expected running time of the algorithm is O(n\alpha (n) log n).
Abstract: This paper presents a randomized incremental algorithm for computing a single face in an arrangement of n line segments in the plane that is fairly simple to implement. The expected running time of the algorithm is $O(n\alpha (n)\log n)$. The analysis of the algorithm uses a novel approach that generalizes and extends the Clarkson–Shor analysis technique [in Discrete Comput. Geom., 4 (1989), pp. 387–421]. A few extensions of the technique, obtaining efficient randomized incremental algorithms for constructing the entire arrangement of a collection of line segments and for computing a single face in an arrangement of Jordan arcs are also presented.
TL;DR: In this article, a large number of particles diffusing on a circle interacting through a drift resulting from the gradient of a pair potential whose support is of the order of the interparticle distance is derived.
Abstract: We consider a large number of particles diffusing on a circle interacting through a drift resulting from the gradient of a pair potential whose support is of the order of the interparticle distance. We derive a nonlinear bulk diffusion equation for the density of the particle distribution on the circle. The diffusion coefficient is determined as a function of density in terms of standard thermodynamical objects.
03 Jun 1991
TL;DR: An experimental and theoretical analysis of the noise model in the geometric hashing technique is presented and it is shown that its introduction significantly reduces the expected burden of the verification stage.
Abstract: An experimental and theoretical analysis of the noise model in the geometric hashing technique is presented. The efficacy of the technique is discussed in the cases of affine transformations, similarities, and grid motions. The efficacy of the voting procedure, in geometric hashing is discussed, and it is shown that its introduction significantly reduces the expected burden of the verification stage. The discussion is illustrated by results of simulations which have been performed to test the expected performance of the algorithm. >
TL;DR: In this paper, the authors compared the results of a von Neumann-Richtmyer calculation with the weak limit of calculations performed by Von Neumann's original method; although there are discernible differences, they are surprisingly small.
Abstract: We have compared the results of a von Neumann-Richtmyer calculation with the weak limit of calculations performed by von Neumann's original method; although there are discernible differences, they are surprisingly small. We analyze von Newmann's algorithm for solving the initial value problem for the Lagrangian equations of compressible flow
TL;DR: It is proved that for any setS ofn points in the plane and n3−α triangles spanned by the points inS there exists a point (not necessarily inS) contained in at leastn3−3α/(c log5n) of the triangles.
Abstract: We prove that for any setS ofn points in the plane andn3?? triangles spanned by the points inS there exists a point (not necessarily inS) contained in at leastn3?3?/(c log5n) of the triangles. This implies that any set ofn points in three-dimensional space defines at most $$\sqrt[3]{{(c/2)}}n^{8/3} \log ^{5/3} n$$ halving planes.
01 May 1991
TL;DR: An improvement of existing methods that leads to more effective storage rechunation in the absence of tags is described that relies on compiler-generated garbage collection routines specific to the program bekg executed and incurs no time overhead during execution other then the cost of the garbage collection process itself.
Abstract: With the emergence of a number of strongly typed kmguages with very dynamic storage allocation, efficient methods of storage reclamation have become especially important, Even though no type tags are required for type checking programs written in these languages, current implementations douse tags to support run time garbage collection, This often inflicts a high time and space overhead on program execution. Since the early days of LISP (and Algo168 later on), there have been schemes for performing tag-free garbage collection, In this paper, we describe an improvement of existing methods that leads to more effective storage rechunation in the absence of tags. Garbage collection has also traditionally been viewed as being independent of the particular program being executed. This means that results of compile-time analyses which could increase the effectiveness of garbage collection cannot be incorporated easily into the garbage collection process. This paper describes a method for performing garbage collection 1) in the absence of tagged data, and 2) using compile-time information. This method relies on compiler-generated garbage collection routines specific to the program bekg executed and incurs no time overhead during execution other then the cost of the garbage collection process itself. We describe tag-free garbage collection methods for monomorphically typed and polymorphically typed languages, and suggest how they might be extended to support parallel languages.
TL;DR: In this paper, the authors describe a method by which the fully two-and three-dimensional cubic Schrodinger equations can be accurately integrated numerically up to times very close to the formation of singularities.
TL;DR: Recurrence-time statistics provide a local measurement to make evident the existence of anomalous transport and in the anomalous case the distribution of recurrence times has a power tail.
Abstract: For a model stationary flow with hexagonal symmetry, we study the recurrence-time statistics. This model flow has been shown elsewhere to provide a sharp transition from normal to anomalous transport. We show here that this transition from normal to anomalous transport is accompanied by a corresponding change of the recurrence-time statistics from ``normal'' to ``anomalous.'' In the anomalous case the distribution of recurrence times has a power tail. Recurrence-time statistics provide a local measurement to make evident the existence of anomalous transport.