Showing papers by "Courant Institute of Mathematical Sciences published in 1988"

Geometric Hashing: A General And Efficient Model-based Recognition Scheme

Abstract: A general method for model-based object recognition in occluded scenes is presented. It is based on geometric hashing. The method stands out for its efficiency. We describe the general framework of the method and illustrate its applications for various recogni- tion problems both in 3-D and 2-D. Special attention is given to the recognition of 3-D objects in occluded scenes from 2-D gray scale images. New experimental results are included for this important case.

A Fast Adaptive Multipole Algorithm for Particle Simulations

Abstract: This paper describes an algorithm for the rapid evaluation of the potential and force fields in systems involving large numbers of particles whose interactions are described by Coulomb's law. Unlike previously published schemes, the algorithm of this paper has an asymptotic CPU time estimate of $O(N)$, where N is the number of particles in the simulation, and does not depend on the statistics of the distribution for its efficient performance. The numerical examples we present indicate that it should be an algorithm of choice in many situations of practical interest.

An instanton-invariant for 3-manifolds

Abstract: To an oriented closed 3-dimensional manifoldM withH1(M, ℤ)=0, we assign a ℤ8-graded homology groupI*(M) whose Euler characteristic is twice Casson's invariant. The definition uses a construction on the space of instantons onM×ℝ.

The unregularized gradient flow of the symplectic action

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.

Weak solutions and development of singularities of the SU(2) σ‐model

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

Parallel computational geometry

Abstract: We present efficient parallel algorithms for several basic problems in computational geometry: convex hulls, Voronoi diagrams, detecting line segment intersections, triangulating simple polygons, minimizing a circumscribing triangle, and recursive data-structures for three-dimensional queries.

Variational bounds on the effective moduli of anisotropic composites

Abstract: The vritional inequalities of Hashin and Shtrikman are transformed to a simple and concise form. They are used to bound the effective conductivity tensor σ∗ of an anisotropic composite made from an arbitrary number of possibly anisotropic phases, and to bound the effective elasticity tensor C ∗ of an anisotropic mixture of two well-ordered isotropic materials. The bounds depend on the conductivities and elastic moduli of the components and their respective volume fractions. When the components are isotropic the conductivity bounds, which constrain the eigenvalues of σ∗, include those previously obtained by Hashin and Shtrikman, Murat and Tartar, and Lurie and Cherkaev. Our approach can also be used in the context of linear elasticity to derive bounds on C ∗ for composites comprised of an arbitrary number of anisotropic phases. For two-component composites our bounds are tighter than those obtained by Kantor and Bergman and by Francfort and Murat, and are attained by sequentially layered laminate materials.

An n -dimensional Borg-Levinson theorem

Abstract: We show that the potentialq is uniquely determined by the spectrum, and boundary values of the normal derivatives of the eigenfunctions of the Schrodinger operator −Δ+q with Dirichlet boundary conditions on a bounded domain Ω in ℝ n . This and related results can be viewed as a direct generalization of the theorem in the title, which states that the spectrum and the norming constants determine the potential in the one dimensional case.

Inverse boundary value problems at the boundary—continuous dependence

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.

A relative morse index for the symplectic action

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.

Small sets supporting fary embeddings of planar graphs

Abstract: Answering a question of Rosenstiehl and Tarjan, we show that every plane graph with n vertices has a Fary embedding (i.e., straight-line embedding) on the 2n - 4 by n - 2 grid and provide an O(n) space, O(n log n) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any set F, which can support a Fary embedding of every planar graph of size n, has cardinality at least n + (1 - o(1)) √n which settles a problem of Mohar.

On uniqueness of recovery of a discontinuous conductivity coefficient

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

Stability of binary exponential backoff

Abstract: Binary exponential backoff is a randomized protocol for regulating transmissions on a multiple-access broadcast channel. Ethernet, a local-area network, is built upon this protocol. The fundamental theoretical issue is stability: Does the backlog of packets awaiting transmission remain bounded in time, provided the rates of new packet arrivals are small enough? It is assumed n ≥ 2 stations share the channel, each having an infinite buffer where packets accumulate while the station attempts to transmit the first from the buffer. Here, it is established that binary exponential backoff is stable if the sum of the arrival rates is sufficiently small. Detailed results are obtained on which rates lead to stability when n = 2 stations share the channel. In passing, several other results are derived bearing on the efficiency of the conflict resolution process. Simulation results are reported that, in particular, indicate alternative retransmission protocols can significantly improve performance.

A rescaling algorithm for the numerical calculation of blowing-up solutions

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

Three‐dimensional centrifugal‐type instabilities in inviscid two‐dimensional flows

Abstract: In this paper the classical Rayleigh centrifugal instability theory is extended to general inviscid two‐dimensional flows. Sufficient conditions for centrifugal instability are that the streamlines be convex closed curves in some region of the flow, with the magnitude of the circulation decreasing outward. If these conditions are satisfied, a class of three‐dimensional short‐wave instabilities can be constructed, which are localized near the streamline on which the exponent of a certain matrix Floquet problem is maximized.

Nonlinear Second-Order Elliptic Equations V. The Dirichlet Problem for Weingarten Hypersurfaces

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

Parallel construction of a suffix tree with applications

Abstract: Many string manipulations can be performed efficiently on suffix trees. In this paper a CRCW parallel RAM algorithm is presented that constructs the suffix tree associated with a string ofn symbols inO(logn) time withn processors. The algorithm requires ź(n2) space. However, the space needed can be reduced toO(n1+ź) for any 0< ź ≤1, with a corresponding slow-down proportional to 1/ź. Efficient parallel procedures are also given for some string problems that can be solved with suffix trees.

Linearized instability for nonlinear schrödinger and klein‐gordon equations

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.

Planar realizations of nonlinear Davenport-Schinzel sequences by segments

Abstract: LetG={l1,...,ln} be a collection ofn segments in the plane, none of which is vertical. Viewing them as the graphs of partially defined linear functions ofx, letYG be their lower envelope (i.e., pointwise minimum).YG is a piecewise linear function, whose graph consists of subsegments of the segmentsli. Hart and Sharir [7] have shown thatYG consists of at mostO(n?(n)) segments (where?(n) is the extremely slowly growing inverse Ackermann's function). We present here a construction of a setG ofn segments for whichYG consists ofΩ(n?(n)) subsegments, proving that the Hart-Sharir bound is tight in the worst case. Another interpretation of our result is in terms of Davenport-Schinzel sequences: the sequenceEG of indices of segments inG in the order in which they appear alongYG is a Davenport-Schinzel sequence of order 3, i.e., no two adjacent elements ofEG are equal andEG contains no subsequence of the forma ...b ...a ...b ...a. Hart and Sharir have shown that the maximal length of such a sequence composed ofn symbols is ?(n?(n)). Our result shows that the lower bound construction of Hart and Sharir can be realized by the lower envelope ofn straight segments, thus settling one of the main open problems in this area.

The evaluation of program-based software test data adequacy criteria

Abstract: In earlier work, a preliminary set of axioms for software test data adequacy was introduced in order to formalize properties which should be satisfied by any good program-based adequacy criterion. Here, we extend this work by augmenting the set with additional axioms which substantially strengthen the set. In doing so, we rule out several types of unsuitable notions of adequacy.

Solving jigsaw puzzles by computer

Abstract: An algorithm to assemble large jigsaw puzzles using curve matching and combinatorial optimization techniques is presented. The pieces are photographed one by one and then the assembly algorithm, which uses only the puzzle piece shape information, is applied. The algorithm was experimented successfully in the assembly of 104-piece puzzles with many almost similar pieces. It was also extended to solve an intermixed puzzle assembly problem and has successfully solved a 208-piece puzzle consisting of two intermixed 104-piece puzzles. Previous results solved puzzles with about 10 pieces, which were substantially different in shape.

Variational calculations for solid and liquid 4He with a "shadow" wave function.

Abstract: A new class of trial wave functions is introduced to compute variationally the ground-state energy of solid /sup 4/He. This wave function is symmetric under particle exchange, translationally invariant, and does not require the a priori introduction of a crystal lattice. It gives a lower energy than and has properties comparable with those given by previous calculations in which atoms are explicitly localized. The same functional form of the wave function is used to investigate the liquid phase, where a lower energy than those given by a wave function of the Jastrow form is obtained as well.

Stable defects of minimizers of constrained variational principles

Abstract: We prove energy and density bounds for minimizers of certain constrained variational problems, and we deduce limitations for their topological degree.