Showing papers by "Courant Institute of Mathematical Sciences published in 1998"

On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations

Abstract: The authors consider the length, $l_N$, of the length of the longest increasing subsequence of a random permutation of $N$ numbers. The main result in this paper is a proof that the distribution function for $l_N$, suitably centered and scaled, converges to the Tracy-Widom distribution [TW1] of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest decent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 [DZ1] in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel [Ge] for the Poissonization of the distribution function of $l_N$.

A new proof of the Caffarelli‐Kohn‐Nirenberg theorem

Abstract: Here we give a self-contained new proof of the partial regularity theorems for solutions of incompressible Navier-Stokes equations in three spatial dimensions. These results were originally due to Scheffer and Caffarelli, Kohn, and Nirenberg. Our proof is much more direct and simpler. © 1998 John Wiley & Sons, Inc.

Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density

Abstract: We present a sufficient condition on the blowup of smooth solutions to the compressible Navier-Stokes equations in arbitrary space dimensions with initial density of compact support. As an immediate application, it is shown that any smooth solutions to the compressible Navier-Stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support, and an upper bound, which depends only on the initial data, on the blowup time follows from our elementary analysis immediately. Another implication is that there is no global small (decay in time) or even bounded (in the case that all the viscosity coefficients are positive) smooth solutions to the compressible Navier-Stokes equations for polytropic fluids, no matter how small the initial data are, as long as the initial density is of compact support. This is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum. The blowup of smooth solutions to the compressible Euler system with initial density and velocity of compact support is a simple consequence of our argument. © 1998 John Wiley & Sons, Inc.

Geometric wave equations

Abstract: The wave equation Conservation laws Function spaces The linear wave equation Well-posedness Semilinear wave equations Wave maps Wave maps with symmetry Notes Bibliography

Spectral Deferred Correction Methods for Ordinary Differential Equations

Abstract: We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples. For non-stiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with orders between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with stiff problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a state-of-the-art extrapolation code (at least, at moderate to high precision).

The role of dendrites in auditory coincidence detection

Abstract: Coincidence-detector neurons in the auditory brainstem of mammals and birds use interaural time differences to localize sounds Each neuron receives many narrow-band inputs from both ears and compares the time of arrival of the inputs with an accuracy of 10-100 micros Neurons that receive low-frequency auditory inputs (up to about 2 kHz) have bipolar dendrites, and each dendrite receives inputs from only one ear Using a simple model that mimics the essence of the known electrophysiology and geometry of these cells, we show here that dendrites improve the coincidence-detection properties of the cells The biophysical mechanism for this improvement is based on the nonlinear summation of excitatory inputs in each of the dendrites and the use of each dendrite as a current sink for inputs to the other dendrite This is a rare case in which the contribution of dendrites to the known computation of a neuron may be understood Our results show that, in these neurons, the cell morphology and the spatial distribution of the inputs enrich the computational power of these neurons beyond that expected from 'point neurons' (model neurons lacking dendrites)

Accelerating fast multipole methods for the Helmholtz equation at low frequencies

Abstract: The authors describe a diagonal form for translating far-field expansions to use in low frequency fast multipole methods. Their approach combines evanescent and propagating plane waves to reduce the computational cost of FMM implementation. More specifically, we present the analytic foundations for a new version of the fast multipole method for the scalar Helmholtz equation in the low frequency regime. The computational cost of existing FMM implementations, is dominated by the expense of translating far field partial wave expansions to local ones, requiring 189p/sup 4/ or 189p/sup 3/ operations per box, where harmonics up to order p/sup 2/ have been retained. By developing a new expansion in plane waves, we can diagonalize these translation operators. The new low frequency FMM (LF-FMM) requires 40p/sup 2/+6p/sup 2/ operations per box.

On the Number of Incidences Between Points and Curves

Abstract: We apply an idea of Szekely to prove a general upper bound on the number of incidences between a set of m points and a set of n ‘well-behaved’ curves in the plane.

Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers

Abstract: Recent numerical simulations reveal remarkably different behavior in rotating stably stratified fluids at low Froude numbers for finite Rossby numbers as compared with the behavior at both low Froude and Rossby numbers. Here the reduced low Froude number limiting dynamics in both of these situations is developed with complete mathematical rigor by applying the theory for fast wave averaging for geophysical flows developed recently by the authors. The reduced dynamical equations include all resonant triad interactions for the slow (vortical) modes, the effect of the slow (vortical) modes on the fast (inertial gravity) modes, and also the general resonant triad interactions among the fast (internal gravity) waves. The nature of the reduced dynamics in these two situations is compared and contrasted here. For example, the reduced slow dynamics for the vortical modes in the low Froude number limit at finite Rossby numbers includes vertically sheared horizontal motion while the reduced slow dynamics i...

Attractors for non-compact semigroups via energy equations

Abstract: The energy-equation approach used to prove the existence of the global attractor by establishing the so-called asymptotic compactness property of the semigroup is considered, and a general formulation that can handle a number of weakly damped hyperbolic equations and parabolic equations on either bounded or unbounded spatial domains is presented. As examples, three specific and physically relevant problems are considered, namely the flows of a second-grade fluid, the flows of a Newtonian fluid in an infinite channel past an obstacle, and a weakly damped, forced Korteweg-de Vries equation on the whole line.

An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg

Abstract: We present a direct analytic approach to the Guillemin-Sternberg conjecture [GS] that `geometric quantization commutes with symplectic reduction', which was proved recently by Meinrenken [M1], [M2] and Vergne [V1], [V2] et al. Besides providing a new proof of this conjecture, our methods also lead immediately to further extensions in various contexts.

Minimum-Relative-Entropy Calibration of Asset-Pricing Models

Abstract: We present an algorithm for calibrating asset-pricing models to the prices of benchmark securities. The algorithm computes the probability that minimizes the relative entropy with respect to a prior distribution and satisfies a finite number of moment constraints. These constraints arise from fitting the model to the prices of benchmark prices are studied in detail. We find that the sensitivities can be interpreted as regression coefficients of the payoffs of contingent claims on the set of payoffs of the benchmark instruments. We show that the algorithm has a unique solution which is stable, i.e. it depends smoothly on the input prices. The sensitivities of the values of contingent claims with respect to varriations in the benchmark instruments, in the risk-neutral measure. We also show that the minimum-relative-entropy algorithm is a special case of a general class of algorithms for calibrating models based on stochastic control and convex optimization. As an illustration, we use minimum-relative-entropy to construct a smooth curve of instantaneous forward rates from US LIBOR swap/FRA data and to study the corresponding sensitivities of fixed-income securities to variations in input prices.

Voronoi Diagrams in Higher Dimensions under Certain Polyhedral Distance Functions

Abstract: The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance functions. Specifically, if S is a set of n points in general position in R d , the maximum complexity of its Voronoi diagram under the L ∞ metric, and also under a simplicial distance function, are both shown to be $\Theta(n^{\lceil d/2 \rceil})$ . The upper bound for the case of the L ∞ metric follows from a new upper bound, also proved in this paper, on the maximum complexity of the union of n axis-parallel hypercubes in R d . This complexity is $\Theta(n^{\left\lceil d/2 \right\rceil})$ , for d ≥ 1 , and it improves to $\Theta(n^{\left\lfloor d/2 \right\rfloor})$ , for d ≥ 2 , if all the hypercubes have the same size. Under the L 1 metric, the maximum complexity of the Voronoi diagram of a set of n points in general position in R 3 is shown to be $\Theta(n^2)$ . We also show that the general position assumption is essential, and give examples where the complexity of the diagram increases significantly when the points are in degenerate configurations. (This increase does not occur with an appropriate modification of the diagram definition.) Finally, on-line algorithms are proposed for computing the Voronoi diagram of n points in R d under a simplicial or L ∞ distance function. Their expected randomized complexities are $O(n \log n + n ^{\left\lceil d/2 \right\rceil})$ for simplicial diagrams and $O(n ^{\left\lceil d/2 \right\rceil} \log ^{d-1} n)$ for L ∞ -diagrams.

The dirichlet problem for singularly perturbed elliptic equations

Abstract: There has been much work on various singularly perturbed partial differential equations or systems. Such equations or systems depend on some small parameters ε > 0, solutions denoted as uε. There are at least two types of questions being investigated. The first type is to study possible behavior of uε as ε tends to zero. The second is to actually construct, by various methods, such solutions. In this paper we mainly present some results of the second type. We will study some specific singularly perturbed partial differential equations. However, the methods we used are useful in studying other problems as well. Let Ω ⊂ Rn be a bounded domain with smooth boundary. We consider { −ε2∆ũ+ ũ = ũq , ũ > 0 , in Ω , ũ ∣∣ ∂Ω = 0 , (0.1)

Non-Newtonian Hele-Shaw Flow and the Saffman-Taylor Instability

Abstract: We explore the Saffman-Taylor instability of a gas bubble expanding into a shear thinning liquid in a radial Hele-Shaw cell. Using Darcy{close_quote}s law generalized for non-Newtonian fluids, we perform simulations of the full dynamical problem. The simulations show that shear thinning significantly influences the developing interfacial patterns. Shear thinning can suppress tip splitting, and produce fingers which oscillate during growth and shed side branches. Emergent length scales show reasonable agreement with a general linear stability analysis. {copyright} {ital 1998} {ital The American Physical Society}

Decay estimates for the critical semilinear wave equation

Abstract: In this paper we prove that finite energy solutions (with added regularity) to the critical wave equation □u + u5 = 0 on R3 decay to zero in time. The proof is based on a global space-time estimate and dilation identity.

On the numerical evaluation of elastostatic fields in locally isotropic two-dimensional composites

Abstract: We present a fast algorithm for the calculation of elastostatic fields in locally isotropic composites. The method uses an integral equation approach due to Sherman, combined with the fast multipole method and an adaptive quadrature technique. Accurate solutions can be obtained with inclusions of arbitrary shape at a cost roughly proportional to the number of points needed to resolve the interface. Large-scale problems, with hundreds of thousands of interface points can be solved using modest computational resources.

Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds

Abstract: Here we study the asymptotic behavior of solutions to the complex Ginzburg-Landau equations and their associated heat flows in arbitrary dimensions when the Ginzburg-Landau parameter tends to infinity. We prove that the energies of solutions in the flow are concentrated at vortices in two dimensions, filaments in three dimensions, and codimension-2 submanifolds in higher dimensions. Moreover, we show the dynamical laws for the motion of these vortices, filaments, and codimension-2 submanifolds. Away from the energy concentration sets, we use some measure-theoretic arguments to show the strong convergence of solutions in both static and heat flow cases. © 1998 John Wiley & Sons, Inc.

Averaging over Fast Gravity Waves for Geophysical Flows with Unbalanced Initial Data

Abstract: Various facets of recent mathematical theories for averaging over fast gravity waves on advective time scales for geophysical flows with unbalanced initial data are presented here including nonlinear Rossby adjustment and simplified reduced dynamics. This work is presented within the context of simplified geophysical models involving the rotating shallow-water equations and the rotating stably stratified Boussinesq equations. Novel mechanisms for enhanced gravity wave dissipation through the catalytic interaction with potential vortical modes are also developed here within the context of the rotating shallow-water equations.

Mechanisms of Spontaneous Activity in the Developing Spinal Cord and Their Relevance to Locomotion

Abstract: The isolated lumbosacral cord of the chick embryo generates spontaneous episodes of rhythmic activity. Muscle nerve recordings show that the discharge of sartorius (flexor) and femorotibialis (extensor) motoneurons alternates even though the motoneurons are depolarized simultaneously during each cycle. The alternation occurs because sartorius motoneuron firing is shunted or voltage-clamped by its synaptic drive at the time of peak femorotibialis discharge. Ablation experiments have identified a region dorsomedial to the lateral motor column that may be required for the alternation of sartorius and femorotibialis motoneurons. This region overlaps the location of interneurons activated by ventral root stimulation. Wholecell recordings from interneurons receiving short latency ventral root input indicate that they fire at an appropriate time to contribute to the cyclical pause in firing of sartorius motoneurons. Spontaneous activity was modeled by the interaction of three variables: network activity and two activity-dependent forms of network depression. A "slow" depression which regulates the occurrence of episodes and a "fast" depression that controls cycling during an episode. The model successfully predicts several aspects of spinal network behavior including spontaneous rhythmic activity and the recovery of network activity following blockade of excitatory synaptic transmission.