Center for Applied Mathematics and Statistics

REPORT 9899-1: The Geometry of Lightlike Hypersurfaces of the de Sitter Space

M. A. Akivis & V. V. Goldberg.

We study lightlike hypersurfaces in the de Sitter space and prove that their geometry is closely connected with the geometry of hypersurfaces of the conformal space. This connection is applied for a construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces as well as for studying singularities of lightlike hypersurfaces.

REPORT 9899-2: A Geometric Approach to Singularly Perturbed Non-local Reaction Diffusion Equations

A. Bose

In the context of a microwave heating problem, a geometric method to construct a spatially localized, 1-pulse steady state solution of a singularly perturbed, non-local reaction diffusion equation is introduced. The 1-pulse is shown to lie in the transverse intersection of relevant invariant manifolds. The transverse intersection encodes a consistency condition that all solutions of non-local equations must satisfy. A detailed stability analysis of the 1-pulse is conducted by locating the spectrum of a non-local linear operator. A general Oscillation Theorem for the non-local eigenfunctions of the linear operator is established. The theorem is used to prove that the linear operator associated with the 1-pulse solution possesses an exponentially small principal eigenvalue. The existence and instability of $n$-pulse solutions is also proved. A second application of the existence and stability theory to the Gierer-Meinhardt equations is provided.

REPORT 9899-3: Cusp Formation for Evolving Bubbles in 2-D Stokes Flow: The Effect of Variable Surface Tension

M. Siegel

Analytical and numerical methods are applied to investigate the effect of variable surface tension, induced by the presence of surfactant, on a bubble evolving in 2-D Stokes flow. Of particular interest is the possible spontaneous occurance of a cusp singularity on the bubble surface. In the case of straining flows, it is found that the bubble achieves an unsteady cusped formation in finite time. The numerical computations are greatly simplified by exploiting the analytic structure of the governing equations and interfacial shape. This enables the evolution to be followed nearly up to cusp formation. Comparison with experiment reveals some interesting similarities, and a possible connection between the observed behavior and the phenomenon of tip streaming is discussed.

REPORT 9899-4: Reflectionless Sponge Layers as Absorbing Boundary Conditions for the Numerical Solution of Maxwell's Equations in Rectangular, Cylindrical and Spherical Coordinates

P. G. Petropoulos

A scaling argument is used to derive reflectionless sponge layers to absorb outgoing time-harmonic waves in numerical solutions of the three-dimensional elliptic Maxwell equations in rectangular, cylindrical and spherical coordinates. We also develop our reflectionless sponge layers to absorb outgoing transient waves in numerical solutions of the time-domain Maxwell equations and prove that these absorbing layers are described by causal, strongly well-posed hyperbolic systems. Representative numerical results for transient and time-harmonic waves scattering from compact obstacles demonstrate the many orders of magnitude improvement offered by our approach over standard techniques used to truncate a computational domain in which Maxwell's equations are solved.

REPORT 9899-5: Plane-Wave Analysis and Comparison of the Split-Field, Biaxial and Uniaxial PML ABC Methods for Pseudospectral Electromagnetic Wave Simulations in Curvilinear Coordinates

P. G. Petropoulos

In this paper, we discuss and compare the split-field, biaxial and uniaxial perfectly matched layer (PML) methods for absorbing outgoing vector waves in cylindrical and spherical coordinates. We first extend Berenger's split-field formulation into spherical and cylindrical coordinates in such a way that it maintains all the desirable properties it exhibits in rectangular coordinates. Then we discuss the biaxial and the uniaxial medium PML methods in Cartesian coordinates and extend them to spherical and cylindrical coordinates. Properties of plane-wave solutions of the PML methods are analyzed. In particular, the decay and boundness properties of the solutions are considered in order to provide further insight into the different formulations presented herein. Moreover, we propose a set of symmetric hyperbolic equations for both the biaxial and the uniaxial PML methods in the time-domain, which is very suitable for time-domain problems. All three types of spherical and cylindrical PML methods are applied in simulations of plane wave scattering as well as radiating dipole problems. We use a multidomain pseudospectral (Chebyshev) numerical scheme, and the effectiveness of the PML methods is demonstrated through the accurate numerical results obtained. The order of outer-boundary reflection is as low as $0.1\%$ of the exact solution.

REPORT 9899-6: On the Behavior of Some Estimators for the Index of Stability

N. Crato & L. Dowling-DaCosta

Heavy-tailed distributions have been used to model phenomena in which extreme events occur with high probability. In these type of occurrences, it is likely that extreme events are not observable after a certain threshold. Appropriate estimators are needed to deal with this type of truncated data. By means of simulation, it is shown that the well-known Hill-Hall estimator yields highly biased estimates in the presence of truncated data. An unbiased modified maximum likelihood estimator and the tail regression estimator are studied. The expected value and variance of the estimators is assessed in the cases of stable- and Pareto-distributed data.

REPORT 9899-7: Numerical Dispersion and Absorbing Boundary Conditions

P. G. Petropoulos

Predictions of performance of both exact and approximate Absorbing Boundary Conditions (ABCs) do not take into account the fact that in an actual simulation it is numerical rather than analytical waves that are incident on the computational domain boundary. Via a model problem in rectangular coordinates we identify and frame this issue. Then, we study the reflection produced by discrete local ABCs in cylindrical coordinates using as a model the Bayliss-Turkel operator. We find that the analytical reflection coefficient of the ABC significantly underestimates the actual reflection on the grid, and that the additional error decays slowly with increasing resolution.

REPORT 9899-8: Nonresident Semilinear Equations and Applications to Boundary Value Problems

P. S. Milojevic

Description not available.

REPORT 9899-9: A Mechanism for Temporal Control of the Phase Precession of Hippocampal Place Cells

A. Bose, V. Booth & M. Recce

The phase relationship between the activity of hippocampal place cells and the hippocampal theta rhythm systematically precesses as the animal runs through the region in an environment called the place field of the cell. We present a minimal biophysical model of the phase precession of place cells in region CA3 of the hippocampus. The model describes the dynamics of two coupled point neurons, namely a pyramidal cell, and an interneuron the latter of which is driven by a pacemaker input. Outside of the place field, the network displays a stable, background firing pattern which is locked to the theta rhythm. The pacemaker input drives the interneuron, which in turn activates the pyramidal cell. A single stimulus to the pyramidal cell from the dentate gyrus, simulating entrance into the place field, reorganizes the functional roles of the cells in the network for a number of cycles of the theta rhythm. In the reorganized network, the pyramidal cell drives the interneuron at a higher frequency than the theta frequency, thus causing a systematic precession relative to the theta input. The frequency of the pyramidal cell can vary to account for changes in the animal's running speed. The transient dynamics end after up to $ 360 ^ o $ of phase precession when the pacemaker input to the interneuron occurs at a phase to return the network to the stable background firing pattern, thus signaling the end of the place field. Our model, in contrast to others, reports that phase precession is a temporally,and not spatially, controlled process. We also predict that like pyramidal cells, some interneurons phase precess. Our model provides a mechanism for shutting off place cell firing after the animal has crossed the place field and it explains the observed nearly $ 360 ^o $ of phase precession. We also describe how this model is consistent with a proposed auto-associative memory role of the CA3 region.

REPORT 9899-10: Slope Effects in Sidescan Bathymetry

D. Alexandrou, Z.-H. Michalopoulou & D. Pantzartis

This report investigates the effect of a sloping seafloor on the quality of bathymetric estimates. A sidescan sonar is employed for bathymetric purposes, estimating the sea bottom depth by calculating the phase difference between signals received at two spatially separated receivers. The report shows that a sloping seafloor leads to loss of spatial cross-correlation of bottom reverberation, resulting in degradation in the phase difference, and, subsequently, depth, computations.

REPORT 9899-11: Measurement of Modal Power in Optical Fibers

J. Abbot, A. Bose, J. Haus, T. Witelski & S. Zabrenske

Increasing demands on communications networks to achieve higher carrying capacities, faster transmission rates and better signal reliability require the development of improved testing procedures and the establishment of new communication standards. Over the coming decade, local area networks (LAN's) will be upgraded to operate at bit rates one hundred times faster than current standards. The gigabit ethernet (GBE) will achieve transmission rates of one gigabit per second over multimode optical fibers with high speed laser signal sources. To optimize the signal characteristics of laser pulses in the optical fibers, it is important to understand how dispersion, radiation, tunneling and other effects work to distort the signal. We address questions concerning what properties of the input laser pulse can be recovered from measurements of the output signal intensity. In particular, we examine a method to calculate the power sent through each of the propagating linear modes in a multimode optical fiber from a measurement of the near-field intensity $I(r)$. The goal of our work is to give a robust estimate of the modal power distribution which is not overly sensitive to measurement noise, is internally self-consistent, and provides an improvement over existing techniques.

REPORT 9899-12: Construction of 3D Solutions for the Maxwell Equations

A. Yefet & E. Turkel

We consider numerical solutions for the three dimensional time dependent Maxwell equations. We construct a fourth order accurate implicit scheme and compare it to the Yee scheme for free space in a box.

REPORT 9899-13: Adaptation of Passive Rat Left Ventricle in Diastolic Dysfunction

H. R. Chaudhry, B. Bukiet, M. Siegel, T. Findley, A. B. Ritter & N. Guzelsu

This article deals with providing a theoretical explanation for the clinically observed phenomenon of wall thickening of the rat ventricular wall during adaptation of the passive left ventricle in diastolic dysfunction. Large deformation theory is applied. Transmural stress and strain distribution in an assumed homogeneous, incompressible, transversely isotropic, nonlinear elastic left ventricular wall are analyzed. Relevant parameters are determined for normotensive, hypertensive and adaptive left ventricle. The analysis predicts that during adaptation, wall thickness and wall mass of the ventricle increase, in agreement with experimental findings. These are the indications of initiation of congestive heart failure.

REPORT 9899-14: Effects of Stoichiometry on Stretched Premixed Flames

J. K. Bechtold & M. Matalon

We derive a generalized equation for the burning rate of stretched premixed flames under near-stoichiometric conditions. Our theory is a slowly-varying-flame theory in which Lewis numbers are bounded away from unity, and the derived equation exhibits a nonlinear dependence on Lewis numbers, equivalence ratio and stretch. We demonstrate that the burning rate is significantly influenced by both of the species involved in the reaction. Furthermore, for a flame in a strained flow, the different diffusivities of the two species can lead to one or the other being locally deficient, depending on the magnitude of the strain rate. This has important implications on the extinction characteristics of these flames. We also calculate burning velocities with parameter values typical of hydrocarbon-air mixtures, and our results are in good agreement with experiments.

REPORT 9899-15: A Genetic Algorithm Study on the Influence of Dendritic Plateau Potentials on Bistable Spiking in Motoneurons

V. Booth

In this report, a genetic algorithm (GA) is applied to a compartmental model of a vertebrate motoneuron. The GA is designed to search the model parameter space for values that lead to bistable behaviors in the firing patterns of the model neuron. The goal of parameter search is to investigate the necessity of dendritic ionic conductances for the generation of bistable spiking patterns. The preliminary results presented in this report indicate that bistable spiking is robustly obtained in the model when spike-generating conductances are compartmentally segregated from plateau-generating conductances.

REPORT 9899-16: Stability of Numerical Boundary Conditions Imposed on the Ty(2.4) Scheme

A. Yefet

We consider the one dimensional Maxwell equations. We approximate the spatial derivatives using the Ty(2,4) scheme. We use the same mesh stencil as used in the standard Yee scheme and study the stability of the numerical boundary conditions imposed by the Ty(2,4) scheme.

REPORT 9899-17: Numerical Implementation of Delta Functions

A. Yefet

We propose an approximation for the delta function. Many use different approximations, but one can not compare an approximated solution with an exact solution when when one deals with sources or pulses. Here we propose a way to approximate the delta function and consider its implementation in a computer code.

REPORT 9899-18: Synaptic Depression Creates a Switch That Controls the Frequency of an Oscillatory Circuit

F. Nadim, Y. Manor, N. Kopell & E. Marder

Synaptic depression is a property of many synapses, but the functional contributions of synaptic depression to circuit dynamics are only now beginning to be understood. When depressing synapses are found in rhythmically active networks, the relationship between the kinetics of depression and its recovery and the period of the network may result in a steady-state condition in which the synapse is significantly depressed. We demonstrate that synaptic depression in a recurrent inhibitory network that includes an intrinsic oscillator can give rise to a switch between two distinct modes of network operation. When the depressing synapse is weak, the oscillation period is determined by the oscillator component. Increasing the strength of the depressing synapse beyond a threshold value activates a positive feedback mechanism. In this regime the oscillation period is determined by the depressing synapse.

REPORT 9899-19: Differential Geometry of Webs

A. Akivis & V. Goldberg

The authors present a very detailed description of numerous works in the field of webs and local differentiable quasigroups. The paper is organized in such a way that it can be used both as an introductory text and an encyclopaedia on this subject. The authors give an introduction to the theory of webs with the history of research in this subject, indicate the most interesting directions of this theory and the results obtained and formulate some open problems. In their presentation of the differential geometry of webs, the authors describe those topics of the general theory that in their opinion are the most important. In addition, certain new notions and thoughts are introduced.

REPORT 9899-20: Strict Stability of High-Order Compact Implicit Finite-Difference Schemes: The Role of Boundary Conditions for Hyperbolic PDEs

S. Abarbanel, A. Chertock & A. Yefet

In this paper we shall continue the discussion about numerical methods for hyperbolic initial boundary value problems(IBVPs). This paper is devoted to solving one- and two-dimensional hyperbolic systems. Numerical examples show that the forth and the sixth-order schemes presented here are effective and provide time stability even when a theoretical foundation is lacking. As in the scalar case, the fourth- and the sixth-order schemes are used to solve model problems. The formal accuracy of each scheme is determined by doing a grid refinement study. The numerical results show that the convergence rate of the schemes used here agrees well with the theory. As an application where high-order accurate approximations are needed we consider the two-dimensional Maxwell's equation in free space. The problem is solved using both the forth- and the sixth-order schemes. Numerical results are compared with the forth order compact implicit Ty(2,4) scheme.

REPORT 9899-21: Matched-Impulse-Response Processing for Shallow Water Localization and Geoacoustic Inversion

Z.-H. Michalopoulou

In this work, we extract the impulse response of the ocean by processing signals received from linear frequency modulated transmissions. The impulse response estimation is performed through a cross-correlation procedure facilitated by the impulse-like form of the autocorrelation of the lfm pulses. Once the ocean impulse response is estimated, it is matched to replica impulse responses computed using normal modes and Fourier synthesis. The matching process is a time-domain correlation procedure. The search space, including source location, water column depth, and sediment properies, is searched with a hierarchical scheme, that layers parameters according to their impact on the acoustic field. The proposed inversion method gives excellent results with the SWellEX-96 data. Its performance is superior to that of the conventional, linear, incoherent matched-field processor.

REPORT 9899-22: Existence and the Number of Solutions of Nonresonant Semilinear Equations and Application to Boundary value Problems

P. S. Milojevic

Description not available.

REPORT 9899-23: Stability of Some Green's-Function-Based Methods Applied to Nondispersive Linear Wave Equations

J. H. C. Luke

The \epd ~method, a finite difference method for highly dispersive linear wave equations, is introduced and analyzed. Motivated by the problem of simulating the propagation of microwave pulses through water, the method attempts to relieve the computational burden of resolving fast processes, such as dipole relaxation or oscillation, occurring in a material with dynamic structure. This method, based on a novel differencing scheme for the time step, is considered primarily for problems in one spatial dimension with constant coefficients. It is defined in terms of the solution of an initial value problem for a system of ordinary differential equations that, in an implementation of the method, need be solved only once in a preprocessing step. For certain wave equations of interest (nondispersive systems, the telegrapher's equation, and the Debye model for dielectric media) explicit formulas for the method are presented. The dispersion relation of the method exhibits a high degree of low-wavenumber asymptotic agreement with the dispersion relation of the model to which it is applied. Comparisons with a finite-difference time-domain (FDTD) approach and an approach based on Strang splitting demonstrate the potential of the method to substantially reduce the cost of simulating linear waves in dispersive materials. A generalization of the \epd ~method for problems with variable coefficients appears to retain many of the advantages seen for constant coefficients.

REPORT 9899-24: Almost-Synchronous Solutions for Networks of Neurons Coupled by Excitation

A. Bose, N. Kopell & D. Terman

We consider synchronization in a pair of neurons described by voltage-gated conductance equations and coupled by mutual excitation. Our model neurons have three time scales: the very fast transition between active and inactive states, an intermediate scale during the active portion of a cell's trajectory, and the slowest during the inter-burst interval. We show that the interplay of time scales can lead to stable ``almost-synchronous" solutions in which the jumps between active and inactive states of the two cells happen with a time difference that is a small fraction of the total period of the coupled system. Furthermore, modulation of parameters not affecting time scales can change the stable solution from almost-synchronous to synchronous. We use a geometric analysis that enables us to identify the parts of the trajectories over which the interactions move the coupled trajectory away from synchrony, the parameters responsible for this phenomenon and how the distance from synchrony depends on the time scales and can be modulated.

REPORT 9899-25: Pattern Formation in Microwave Heated Ceramics: Cylinders and Slabs

G. A. Kriegsmann

Analyses of microwave heating of a thin ceramic cylinder and a thin ceramic slab in a single mode, highly resonant cavity are presented.  Realistic assumptions regarding the effective electrical conductivity, thermal parameters, and physical dimensions are adhered to throughout.  Consequently, the models developed herein incorporate most of the features of actual experiments.  They incorporate both the effects of cavity detuning and a local electric field perturbation on the heating process. The models presented take the form of one and two dimensional reaction diffusion equation which contain a functional and an inhomogeneous source
term for the cylinder and slab, respectively.  The development of these equations is the product of a systematic modeling process that involves S-matrix theory, a small Biot number asymptotic analysis, an a matched asymptotic analysis of a non-standard electromagnetic scattering problem. The one dimensional equation for the cylinder reveals both the mathematical
structure and physical mechanism for the formation of hot spots.  The two dimensional equation supports a hot stripe pattern, due to preferential electromagnetic heating, which becomes unstable and evolves into a oval-like spot.  Accurate numerical methods which approximate the solutions of these equations and their stability are presented and these agree qualitatively with experiments and predict observed trends.

REPORT 9899-26: A Classification and Examples of Four-Dimensional Isoclinic Three-Webs of Codimension Two

V. V. Goldberg

A classification and examples of four-dimensional isoclinic three-webs of codimension two are given. The examples considered prove the existence theorem for many classes of webs for which the general existence theorems are not proved yet.

REPORT 9899-27: Lightlike Hypersurfaces on Manifolds Endowed with a Conformal Structure of Lorentzian Signature

M. A. Akivis & V. V. Goldberg

The authors study the geometry of lightlike hypersurfaces on manifolds $(M, c)$ endowed with a pseudoconformal structure $c = CO (n - 1, 1)$ of Lorentzian signature. Such hypersurfaces are of interest in general relativity since they can be models of different types of physical horizons. On a lightlike hypersurface, the authors consider the fibration of isotropic geodesics and investigate their singular points and singular submanifolds. They construct a conformally invariant normalization of a lightlike hypersurface intrinsically connected with its geometry and investigate affine connections induced by this normalization. The authors also consider a special class of lightlike hypersurfaces for which the elements of the constructed normalization are integrable.

REPORT 9899-28: Continuity of Generations: Calapso's Family of Geometers (Pasquale, Renato, and Maria Teresa)

V. V. Goldberg

The author describes the lives of three generations of geometers Calapso: Pasquale, Renato, and Maria Teresa, establishes connections between works of these three geometers and shows how Renato developed and enriched ideas of his father Pasquale and how Maria Teresa developed and enriched the ideas of both her grandfather and her father. He also shows a great impact of Pasquale Calapso's work on the development of contemporary mathematics.

REPORT 9899-29: Lightlike Hypersurfaces on a Four-Dimensional Manifold Endowed with a Pseudoconformal Structure of Signature (2,2)

M. A. Akivis & V. V. Goldberg

The authors study the geometry of lightlike hypersurfaces on a four-dimensional manifold $(M, c)$ endowed with a pseudoconformal structure $c = CO (2, 2)$. They prove that a lightlike hypersurface $V \subset (M, c)$ bears a foliation formed by conformally invariant isotropic geodesics and two isotropic distributions tangent to these geodesics, and that these two distributions are integrable if and only if $V$ is totally umbilical. The authors also indicate how, using singular points and singular submanifolds of a lightlike hypersurface $V \subset (M, c)$, to construct an invariant normalization of $V$ intrinsically connected with $V$.

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