Showing papers by "Courant Institute of Mathematical Sciences published in 1996"

A subexponential bound for linear programming

Abstract: We present a simple randomized algorithm which solves linear programs withn constraints andd variables in expected $$\min \{ O(d^2 2^d n),e^{2\sqrt {dIn({n \mathord{\left/ {\vphantom {n {\sqrt d }}} \right. \kern- ulldelimiterspace} {\sqrt d }})} + O(\sqrt d + Inn)} \}$$ time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorithm computes the lexicographically smallest nonnegative point satisfyingn given linear inequalities ind variables. The expectation is over the internal randomizations performed by the algorithm, and holds for any input. In conjunction with Clarkson's linear programming algorithm, this gives an expected bound of $$O(d^2 n + e^{O(\sqrt {dInd} )} ).$$ The algorithm is presented in an abstract framework, which facilitates its application to several other related problems like computing the smallest enclosing ball (smallest volume enclosing ellipsoid) ofn points ind-space, computing the distance of twon-vertex (orn-facet) polytopes ind-space, and others. The subexponential running time can also be established for some of these problems (this relies on some recent results due to Gartner).

Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics

Abstract: We study systems of conservation laws arising in two models of adhesion particle dynamics. The first is the system of free particles which stick under collision. The second is a system of gravitationally interacting particles which also stick under collision. In both cases, mass and momentum are conserved at the collisions, so the dynamics is described by 2×2 systems of conservations laws. We show that for these systems, global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the Lax-Oleinik variational principle for scalar conservation laws. However, this weak solution is not unique among weak solutions satisfying the standard entropy condition. We also study a modified gravitational model in which, instead of momentum, some other weighted velocity is conserved at collisions. For this model, we prove both existence and uniqueness of global weak solutions. We then study the qualitative behavior of the solutions with random initial data. We show that for continuous but nowhere differentiable random initial velocities, all masses immediately concentrate on points even though they were continuously distributed initially, and the set of shock locations is dense.

On viscosity solutions of fully nonlinear equations with measurable ingredients

Abstract: We study fully nonlinear, uniformly elliptic equations with measurable ingredients. Caffarelli's recent work on W2,p estimates for viscosity solutions has led to significant progress in this area. Here we present a unified treatment of this theory based on an appropriate notion of viscosity solution. For instance, it is shown that strong solutions are viscosity solutions, that viscosity solutions are twice differentiable a.e., and that the pointwise derivatives satisfy the equation a.e. An important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations. This extends results of this type due to Evans and Krylov. Our work is to some extent expository, the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients. © 1996 John Wiley & Sons, Inc.

The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions

Abstract: We consider the fourth-order degenerate diffusion equation, in one space dimension. This equation, derived from a lubrication approximation, models the surface-tension-dominated motion of thin viscous films and spreading droplets [15]. The equation with f(h) = |h| also models a thin neck of fluid in the Hele-Shaw cell [10], [11], [23]. In such problems h(x,t) is the local thickness of the the film or neck. This paper considers the properties of weak solutions that are more relevant to the droplet problem than to Hele-Shaw. For simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets. We consider two problems: The first has initial data h0 ≥ 0 and f(h) = |h|n, 0 < n < 3. We show that there exists a weak nonnegative solution for all time. Also, we show that this solution becomes a strong positive solution after some finite time T*, and asymptotically approaches its means as t ∞. The weak solution is in the classical sense of distributions for 3/8 < n < 3 and in a weaker sense introduced in [1] for the remaining 0 < n ≤ 3/8. Furthermore, the solutions have high enough regularity to just include the unique source-type solutions [2] with zero slope at the edge of the support. They do not include any of the less regular solutions with positive slope at the edge of the support. Second, we consider strictly positive initial data h0 ≥ m > 0 and f(h) = |h|n, 0 < n < ∞. For this problem we show that even if a finite-time singularity of the form h 0 does occur, there exists a weak nonnegative solution for all time t. This weak solution becomes strong and positive again after some critical time T*. As in the first problem, we show that the solution approaches its mean as t ∞. The main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity. We show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result. © 1996 John Wiley & Sons, Inc.

Surface order large deviations for Ising, Potts and percolation models

Abstract: We derive uniform surface order large deviation estimates for the block magnetization in finite volume Ising (or Potts) models with plus or free (or a combination of both) boundary conditions in the phase coexistence regime ford≧3. The results are valid up to a limit of slab-thresholds, conjectured to agree with the critical temperature. Our arguments are based on the renormalization of the random cluster model withq≧1 andd≧3, and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability. The results are new even for the case of independent percolation (q=1). As a byproduct of our methods, we obtain further results in the FK model concerning semicontinuity (inp andq) of the percolation probability, the second largest cluster in a box and the tail of the finite cluster size distribution.

Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes

Abstract: We present and analyze several ways of discretizing first-order Hamilton-Jacobi equations on unstructured meshes. We first discuss two Godunov-type Hamiltonians: the first one is an extension of a result by Bardi and Osher, where a particular decomposition of the initial condition is assumed, and we point out its practical limits; the other one arises from a particular decomposition of the Hamiltonian. Despite its complexity this decomposition enables us to construct a Lax-Friedrichs Hamiltonian. These schemes all share common properties: They are consistent, monotonic, and independent of the geometrical interpretation of the piecewise linear initial condition. Under these assumptions and classical ones on the mesh, we show these schemes are convergent. We describe their high-order extensions using the ENO technique and provide numerical illustrations. © 1996 John Wiley & Sons, Inc.

Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model

Abstract: We present an algorithm for hedging option portfolios and custom-tailored derivative securities, which uses options to manage volatility risk. The algorithm uses a volatility band to model heteroskedasticity and a non- linear partial differential equation to evaluate worst-case volatility scenarios for any given forward liability structure. This equation gives sub-additive portfolio prices and hence provides a natural ordering of prefer- ences in terms of hedging with options. The second element of the algorithm consists of a portfolio optim- ization taking into account the prices of options available in the market. Several examples are discussed, including possible applications to market-making in equity and foreign-exchange derivatives.

Coarsening and persistence in the voter model

Abstract: The voter model is a simple model for coarsening with a nonconserved scalar order parameter. We investigate coarsening and persistence in the voter model by introducing the quantity ${\mathit{P}}_{\mathit{n}}$(t), defined as the fraction of voters who changed their opinion n times up to time t. We show that ${\mathit{P}}_{\mathit{n}}$(t) exhibits scaling behavior that strongly depends on the dimension as well as on the initial opinion concentrations. Exact results are obtained for the average number of opinion changes, 〈n〉, and the autocorrelation function, A(t)\ensuremath{\equiv}\ensuremath{\sum}(-1${)}_{\mathit{n}}^{\mathit{nP}}$\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathit{d}/2}$ in arbitrary dimension d. These exact results are complemented by a mean-field theory, heuristic arguments, and numerical simulations. For dimensions d\ensuremath{\gtrsim}2, the system does not coarsen, and the opinion changes follow a nearly Poissonian distribution, in agreement with mean-field theory. For dimensions d\ensuremath{\le}2, the distribution is given by a different scaling form, which is characterized by nontrivial scaling exponents. For unequal opinion concentrations, an unusual situation occurs where different scaling functions correspond to the majority and the minority, as well as for even and odd n. \textcopyright{} 1996 The American Physical Society.

On the editing distance between undirected acyclic graphs

Abstract: We consider the problem of comparing CUAL graphs (Connected, Undirected, Acyclic graphs with nodes being Labeled). This problem is motivated by the study of information retrieval for bio-chemical and molecular databases. Suppose we define the distance between two CUAL graphs G1 and G2 to be the weighted number of edit operations (insert node, delete node and relabel node) to transform G1 to G2. By reduction from exact cover by 3-sets, one can show that finding the distance between two CUAL graphs is NP-complete. In view of the hardness of the problem, we propose a constrained distance metric, called the degree-2 distance, by requiring that any node to be inserted (deleted) have no more than 2 neighbors. With this metric, we present an efficient algorithm to solve the problem. The algorithm runs in time O(N1N2D2) for general weighting edit operations and in time for integral weighting edit operations, where Ni, i=1, 2, is the number of nodes in Gi, D=min{d1, d2} and di is the maximum degree of Gi.

Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations

Abstract: The Maxwell-Dirac system describes the interaction of an electron with its own electromagnetic field. We prove the existence of soliton-like solutions of Maxwell-Dirac in (3+1)-Minkowski space-time. The solutions obtained are regular, stationary in time, and localized in space. They are found by a variational method, as critical points of an energy functional. This functional is strongly indefinite and presents a lack of compactness. We also find soliton-like solutions for the Klein-Gordon-Dirac system, arising in the Yukawa model.

Large manifolds with positive Ricci curvature

Abstract: The main purpose of this paper is to show that an n-dimensional manifold with Ricci curvature greater or equal to (n−1) which is close (in the Gromov– Hausdor topology) to the unit n-sphere has volume close to that of the sphere. This shows the converse of the theorem in [C1]. Namely together with [C1] it shows that an n-manifold with Ricci curvature greater or equal to (n − 1) is close to the sphere if and only if the volume is close to that of the sphere. In particular, by [P], such a manifold is homeomorphic to a sphere. Further, as an application of this and the result of [C1], we prove a Radius Theorem saying that if an n-manifold with Ricci curvature greater or equal to (n − 1) has radius almost equal to ; then the volume is close to that of the sphere. In order to obtain these results we further develop and apply the estimates of [C1]. Whereas the main concern in [C1] were with the large scale geometry the main concern of this paper is with the small scale geometry. Let !n be the volume of the round n-sphere, Sn; with sectional curvature one.

Spatial organization in cyclic Lotka-Volterra systems

Abstract: We study the evolution of a system of {ital N} interacting species which mimics the dynamics of a cyclic food chain. On a one-dimensional lattice with {ital N}{lt}5 species, spatial inhomogeneities develop spontaneously in initially homogeneous systems. The arising spatial patterns form a mosaic of single-species domains with algebraically growing average size, {l_angle}l({ital t}){r_angle}{approximately}{ital t}{sup {alpha}}, where {alpha}=3/4 (1/2) and 1/3 for {ital N}=3 with sequential (parallel) dynamics and {ital N}=4, respectively. The domain distribution also exhibits a self-similar spatial structure which is characterized by an additional length scale, {l_angle}{ital L}({ital t}){r_angle}{approximately}{ital t}{sup {beta}}, with {beta}=1 and 2/3 for {ital N}=3 and 4, respectively. For {ital N}{ge}5, the system quickly reaches a frozen state with noninteracting neighboring species. We investigate the time distribution of the number of mutations of a site using scaling arguments as well as an exact solution for {ital N}=3. Some relevant extensions are also analyzed. {copyright} {ital 1996 The American Physical Society.}

Persistent homoclinic orbits for a perturbed nonlinear schrodinger equation

Abstract: The persistence of homoclinic orbits for certain perturbations of the integrable nonlinear Schrodinger equation under even periodic boundary conditions is established. More specifically, the existence of a symmetric pair of homoclinic orbits is established for the perturbed NLS equation through an argument that combines Melnikov analysis with a geometric singular perturbation theory for the PDE. © 1996 John Wiley & Sons, Inc.

Non-mean-field behavior of realistic spin glasses.

Abstract: We provide rigorous proofs which show that the main features of the Parisi solution of the Sherrington-Kirkpatrick spin glass, as applied to more realistic spin glass models, are not valid in any dimension and at any temperature. {copyright} {ital 1996 The American Physical Society.}

Efficient randomized algorithms for some geometric optimization problems

Abstract: In this paper we first prove the following combinatorial bound, concerning the complexity of the vertical decomposition of the minimization diagram of trivariate functions: Let $$\mathcal{F}$$ be a collection ofn totally or partially defined algebraic trivariate functions of constant maximum degree, with the additional property that, for a given pair of functionsf, fźź $$\mathcal{F}$$ , the surfacef(x, y, z)=fź(x, y, z) isxy-monotone (actually, we need a somewhat weaker property). We show that the vertical decomposition of the minimization diagram of $$\mathcal{F}$$ consists ofO(n3+ź) cells (each of constant description complexity), for any ź>0. In the second part of the paper, we present a general technique that yields faster randomized algorithms for solving a number of geometric optimization problems, including (i) computing the width of a point set in 3-space, (ii) computing the minimum-width annulus enclosing a set ofn points in the plane, and (iii) computing the "biggest stick" inside a simple polygon in the plane. Using the above result on vertical decompositions, we show that the expected running time of all three algorithms isO(n3/2+ź), for any ź>0. Our algorithm improves and simplifies previous solutions of all three problems.

Interface proliferation and the growth of labyrinths in a reaction-diffusion system

Abstract: In the bistable regime of the FitzHugh-Nagumo model of reaction-diffusion systems, spatially homogeneous patterns may be nonlinearly unstable to the formation of compact "localized states." The formation of space-filling patterns from instabilities of such structures is studied in the context of a nonlocal contour dynamics model for the evolution of boundaries between high and low concentrations of the activator. An earlier heuristic derivation [D. M. Petrich and R. E. Goldstein, Phys. Rev. Lett. 72, 1120 (1994)] is made more systematic by an asymptotic analysis appropriate to the limits of fast inhibition, sharp activator interfaces, and small asymmetry in the bistable minima. The resulting contour dynamics is temporally local, with the normal component of the velocity involving a local contribution linear in the interface curvature and a nonlocal component having the form of a screened Biot-Savart interaction. The amplitude of the nonlocal interaction is set by the activator-inhibitor coupling and controls the "lateral inhibition" responsible for the destabilization of localized structures such as spots and stripes, and the repulsion of nearby interfaces in the later stages of those instabilities. The phenomenology of pattern formation exhibited by the contour dynamics is consistent with that seen by Lee, McCormick, Ouyang, and Swinney [Science 261, 192 (1993)] in experiments on the iodide-ferrocyanide-sulfite reaction in a gel reactor. Extensive numerical studies of the underlying partial differential equations are presented and compared in detail with the contour dynamics. The similarity of these phenomena (and their mathematical description) with those observed in amphiphilic monolayers, type I superconductors in the intermediate state, and magnetic fluids in Hele-Shaw geometry is emphasized.

The toda rarefaction problem

Abstract: In the Toda shock problem (see [7], (1 11, [S], and also 131) one considers a driving particle moving with a fixed velocity 2a and impinging on a one-dimensional semi-infinite lattice of particles, initially equally spaced and at rest, and interacting with exponential forces. In this paper we consider the related Toda rarefaction problem in which the driving particle now moves away from the lattice at fixed speed, in analogy with a piston being withdrawn, as it were, from a container filled with gas. We make use of the Riemann-Hilbert factorization formulation of the related inverse scattering problem. In the case where the speed 21al of the driving particle is sufficiently large (la1 > l), we show that the particle escapes from the lattice, which then executes a free motion of the type studied, for example, in [5]. In other words, in analogy with a piston being withdrawn too rapidly from a container filled with gas, cavitation develops. 0 1996 John Wiley & Sons, Inc.

Kinetics of a diffusive capture process: lamb besieged by a pride of lions

Abstract: The survival probability, , of a diffusing prey (`lamb') in the proximity of N diffusing predators (a `pride of lions') in one dimension is investigated. When the lions are all to one side of the lamb, the survival probability decays as a non-universal power law, , with the decay exponent proportional to . The crossover behaviour as a function of the relative diffusivities of the lions and the lamb is also discussed. When , the lamb survival probability exhibits a log-normal decay, .

Spatial inhomogeneity and thermodynamic chaos.

Abstract: We present a coherent approach to the competition between thermodynamic states in spatially inhomogeneous systems, such as the Edwards-Anderson spin glass with a fixed coupling realization. This approach, modeled on chaotic dynamical systems, leads to a classification of the allowable structures for replicas and their overlaps.

Rectilinear and polygonal p-piercing and p-center problems

Abstract: We consider the p-pie~cing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or near-linear algorithms for small values of p in cases where the given regions are either axispazallel rectangles or convex c-oriented polygons in the plane (i.e., convex polygons with sides from a fixed finite set of directions). We also investigate the planar rectilinear (and polygonal) p-center problem, in which we are given a set S of n points in the plane, and wish to find p axis-parallel congruent squares (isothetic copies of some given convex polygon, respectively) of smallest possible size whose union covers S. We also study several generalizations of these problems. New results aze a linear-time solution for the rectilinear 3-center problem (by showing that this problem can be formulated w an LP-type problem and by exhibiting a relation to Helly numbers). We give O(n log n)-time solutions for 4-piercing of translates of a square, as well as for the rectilinear 4-center problem; this is worst-case optimal. We give O(n polylog n)-time solutions for 4and 5-piercing of axis-pazallel rectangles, for more general rectilinear 4center problems, and for rectilinear 5-center problems. 2pierceability of a set of n convex c-oriented polygons can be decided in time 0(c2n log n), and the 2-center problem for a convex c-gon can be solved in O(c5n log n) time. The first solution is worst-case optimal when c is fixed. *Both authors acknowledge support by G.I.F. — the German Israeli Foundation for Scientific Research and Development, and by a Max Planck Research Award. Work by Micha Sharir haa also been supported by National Science Foundation Grants CCR-9424398 and CCR-93-11 127, and by grants from the U.S.–Israeli Binational Science Foundation, and the Israel Science Fund administered by the Israeli Academy of Sciences. Emo Welzl has also been supported by a Leibniz Prize from the German Research Society, We 1265/5-1. Part of the work on this paper was done during the participation of Emo Welzl in the Special Semester on Combinatorial and Computational Geometry, organized by the Mathematical Research Institute of Tel Aviv University. tschool of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA. $Institut fur Informatik, Freie Universitat Berlin, and Department Informatik, ETH Zurich, CH-8092 Zurich, Switzerland. Permission to make dlgkal/hard copies of all or part of tlds material for personal or clasarenm use is granted without fee provided that the copies are net made or distributed for profit or commercial advantage, the cepyright notice, the title of the publication and its date appear, and notice is given that copyright is by permission of the ACM, fnc. To copy otherwise, to ~publish, to peat on servers or to nxiistributa to lists, tequires specific permission and/or fee. Computational Geometry’96, Philadelphia PA, USA Q 1996 ACM 0-89791-804-5/96/05. .$3.50 Emo Welzl$

Superdiffusivity in first-passage percolation

Abstract: In standard first-passage percolation on \({\Bbb Z}^d\) (with \(d\geq 2\)), the time-minimizing paths from a point to a plane at distance \(L\) are expected to have transverse fluctuations of order \(L^\xi\). It has been conjectured that \(\xi(d)\geq 1/2\) with the inequality strict (superdiffusivity) at least for low \(d\) and with \(\xi(2)=2/3\). We prove (versions of) \(\xi(d)\geq 1/2\) for all \(d\) and \(\xi(2)\geq 3/5\).