## Center for Applied Mathematics and Statistics

REPORT 9900-1: A Non-Dissipative Staggered Fourth-Order Accurate Explicit Finite Difference Scheme for the Time-Domain Maxwell's Equations

A. Yefet & P. G. Petropoulos

We consider an explicit fourth-order accurate staggered finite difference scheme for the hyperbolic Maxwell's equations. Appropriate fourth-order accurate extrapolation and one-sided difference operators are derived in order to complete the scheme near metal boundaries and dielectric interfaces. A stability analysis of the overall scheme shows it is long-time stable. Numerical results verify the stability analysis, and the scheme's fourth-order convergence rate over complex domains including dielectric interfaces and perfectly conducting surfaces. It is determined that, for a fixed error level, this particular fourth-order scheme is computationally cheaper in comparison to the Yee scheme by more than an order of magnitude.

REPORT 9900-2: Optimal Patterns for Suturing Wounds of Arbitrary Configuration: Finite Element technique

D. A. Lott-Crumpler & H. R. Chaudhry

A numerical model for computing the principal stresses in sutured abdominal human skin wounds of complex geometries is presented. The finite element method is utilized to compute the principal stresses and displacements resulting from suturing circular and triangular wounds in a finite skin sheet, in order to determine optimal suturing patterns. The model uses the basic equations of classical linear elasticity theory for orthotropic materials and elastic constants valid for the {\em in vivo} configuration. The results show that the average stress indices for a circular wound sutured towards the diameter are smaller than those of a circular wound sutured in O to T closures. In addition, it is observed that the average stress indices of a triangular wound sutured in an A to T closure are smaller than those of a triangular wound sutured vertically or horizontally. The A to T closure of a triangle showed a much smaller average stress index than the O to T closure of a circle of the same height. Since regions of high stresses in surgical closures produce adverse affects on healing and scar production, it is necessary to predict areas of high stresses to allow for comparison of stress distributions to help predict regions of slow healing in wounds of complex geometries.

REPORT 9900-3: On Four-Dimensional Three-Webs with Integrable Transversal Distributions

M. A. Akivis & V. V. Goldberg

For a four-dimensional (nonisoclinicly geodesic) three-web $W (3, 2, 2)$, a transversal distribution $\Delta$ is defined by the torsion tensor of the web. In general, this distribution is not integrable. The authors find necessary and sufficient conditions of its integrability and prove the existence theorem for webs $W (3, 2, 2)$ with integrable distributions $\Delta$. They prove that for a web $W (3, 2, 2)$ with the integrable distribution $\Delta$, the integral surfaces $V^2$ of $\Delta$ are totally geodesic in an affine connection of a certain bundle of affine connections. They also consider a class of webs $W (3, 2, 2)$ for which the integral surfaces $V^2$ of $\Delta$ are geodesicly parallel with respect to the same affine connections and a class of webs for which two-dimensional webs $W (3, 2, 1)$ cut by the foliations of $W (3, 2, 2)$ on $V^2$ are hexagonal. They prove the existence theorems for webs of the latter class as well as for webs of the subclass which is the intersection of two classes mentioned above. The authors also establish relations between three-webs considered in the paper.

REPORT 9900-4: Decay of Velocity Fluctuations in a Stably Stratified Suspension

J. H. C. Luke

Sedimentation theory for a well-stirred suspension predicts that the variance in the particle velocities is proportional to the diameter of the container holding the suspension. Numerical simulations support this result, but physical experiments generally do not. In this paper, a possible explanation for this discrepancy is considered. In a stably stratified suspension, we show that sedimentation dynamics modify the particle distribution in such a way as to eliminate the dependence of velocity fluctuations on the size of the container.

REPORT 9900-5: Phase Precession and Phase Locking of Hippocampal Pyramidal Cells

A. Bose & M. Recce

We propose that the activity patterns of CA3 hippocampal pyramidal cells in freely running rats can be described as a speed-modulated, temporal phenomenon. With this hypothesis, we explain why pyramidal cells fire in specific spatial locations, and how place cells phase precess with respect to the EEG theta rhythm for rats running on linear tracks. We are also able to explain why wheel cells phase lock with respect to the theta rhythm for rats running in a wheel. Using biophysically minimal models of neurons, we show how the same network of neurons displays these activity patterns. The different rhythms are the result of inhibition being used in different ways by the system. The inhibition is produced by anatomically and physiologically diverse types of interneurons, whose role in controlling the firing patterns of hippocampal cells we analyze. Each firing pattern is characterized by a different set of functional relationships between the network elements. Our analysis suggests a way to understand these functional relationships and transitions between them.

REPORT 9900-6: Hippocampal Place Cells and the Generation of a Temporal Code

M. Recce, A. Bose & V. Booth

Pyramidal cells in the hippocampus of freely moving rats have a spatially specific activity pattern which provides information to downstream in the phase of cell activity. We present a minimal biophysical model for the generation of the phase information from a combination of two inputs: a short duration spatial trigger and the animal's running speed. This single input is shown to determine the start and end of spatial firing, and a transient phase code for location. Three different simple networks are shown to produce this behavior, without changes in synaptic conductance or connectivity.

REPORT 9900-7: On th estimation of the ocean impulse response

Z.-H. Michalopoulou

This paper discusses different approaches for the direct estimation of the impulse response of an oceanic waveguide from received data. The sources considered are wideband. The performance of the employed techniques is found to be dependent on the exact source signature and how close this is to an impulse. Impulse response estimation is demonstrated with both linear and hyperbolic frequency modulated source sequences; both synthetic and real data results are obtained.

REPORT 9900-8: Numerical methods for the quasilinear wave equation: Antiplane shearing of nonlinearly elastic bodies

D. A. Lott-Crumpler, S. S. Antman & W. G. Szymczak

We formulate an efficient numerical algorithm based on finite-difference approximations and inspired by algorithms from gas dynamics to treat the quasilinear wave equation governing antiplane motions of nonlinearly elastic bodies in two-dimensions. We employ body-fitted meshes to handle computation in domains with irregular geometries and show how to avoid serious numerical errors caused by singularities in the transformations. Finally, we develop robust and effective numerical methods to capture the shocks that arise. We validate our method by comparing with the axisymmetric version of this equation in polar coordinates.

REPORT 9900-9:Acoustic scattering by baffled flexible surfaces: The discrete optical theorem

G. A. Kriegsmann

The optical theorem for acoustic scattering by baffled membranes and plates relates the total cross-section of the scattered field, the directivity factor in the specular direction, and the energy dissipated by the structure. It is basically a statement of conservation of power. In this paper it is demonstrated that the discrete formulation of these problems, obtained by a Galerkin approximation, exactly satisfies the optical theorem.  This discrete relationship holds regardless of the choice of basis functions and the size of the truncated system $N$.  Thus, the adherence of the approximate numerical results to the power conservation law does not necessarily imply its accuracy.

REPORT 9900-10: Spike autosolitons in the Grey-Scott model

C. B. Muratov & V. V. Osipov

We performed a comprehensive study of the spike autosolitons: self-sustained solitary inhomogeneous states, in the classical reaction-diffusion system --- the Gray-Scott model. We developed singular perturbation techniques based on the strong separation of the length scales to construct asymptotically the solutions in the form of a one-dimensional static autosolitons, higher-dimensional radially-symmetric static autosolitons, and two types of traveling autosolitons. We studied the stability of the static autosolitons in one and three dimensions and analyzed the properties of the static and the traveling autosolitons.

REPORT 9900-11: On some methods of construction of invariant normalizations of lightlike hypersurfaces

M. A. Akivis & V. V. Goldberg

The authors study the geometry of lightlike hypersurfaces on pseudo-Riemannian manifolds (M, g) of Lorentzian signature. Such hypersurfaces are of interest in general relativity since they can be models of different types of physical horizons. For a lightlike hypersurface V \subset (M, g) of general type and for some special lightlike hypersurfaces (namely, for totally geodesic, umbilical, and belonging to a manifold (M, g) of constant curvature), in a third-order neighborhood of a point x \in V, the authors construct invariant normalizations intrinsically connected with the geometry of V and investigate affine connections induced by these normalizations. For this construction, they used relative and absolute invariants defined by the first and second fundamental forms of V. The authors show that if \dim M = 4, their methods allow to construct three invariant normalizations and affine connections intrinsically connected with the geometry of V. Such a construction is given in the present paper for the first time. The authors also consider the fibration of isotropic geodesics of V and investigate their singular points and singular submanifolds.

REPORT 9900-12: Algebraic aspects of web geometry

M. A. Akivis & V. V. Goldberg

Algebraic aspects of web geometry, namely its connections with the quasigroup and loop theory, the theory of local differential quasigroups and loops, and the theory of local algebras are discussed.

REPORT 9900-13: Large Amplitude Solutions of Spatially Non-homogeneous Non-local Reaction Diffusion Equations

A. Bose & G. A. Kriegsmann

Existence and stability of a pulse solution of a spatially non-homogeneous non-local reaction diffusion equation are considered. Geometric singular perturbation theory is used to construct a large amplitude solution which lies in the transverse intersection of relevant invariant manifolds. The transverse intersection encodes a consistency condition of the non-local equation which determines the height of the pulse solution. An oscillation theorem for non-local eigenfunctions is used to prove the stability of the pulse.  The results show that the spatial non-homogeneity in the non-linear and non-local terms is essential for stability of the pulse. Two different applications are considered, both of which are related to the microwave heating of ceramic materials.

REPORT 9900-14: On the well-posedness of the equations for the smoothed phase space distribution function and irreversibility in classical statistical mechanics

C. B. Muratov

We obtain an exact equation for a smoothed phase space distribution function for a system of $N$ non-relativistic particles of unit mass obeying Hamiltonian dynamics. We show that this equation is well-posed only in one time direction in the sense of the continuous dependence of the solution on the initial data. We interpret the ill-posedness of this equation in the backward time direction as the manifestation of irreversibility of the observable statistical quantities for systems obeying time-reversible dynamical laws.

REPORT 9900-15: Cusp formation for time evolving bubbles in two-dimensional Stokes flow

M. Siegel

Analytical and numerical methods are applied to investigate the transient evolution of an inviscid bubble in two dimensional Stokes flow. The evolution is driven by extensional flow in a four roller mill. Of particular interest is the possible spontaneous occurrence of a cusp singularity on the bubble surface. The role of constant as well as variable surface tension, induced by the presence of surfactant, is considered. For constant surface tension, a conjecture concerning the existence of a critical capillary number above which all symmetric steady bubble solutions are linearly unstable is found to be false. Steady bubbles for large capillary number Q are found to be susceptible to finite amplitude instability, with the dynamics often leading to cusp or topological singularities. The evolution of an initially circular bubble at zero surface tension is found to culminate in unsteady cusp formation. In contrast to the clean flow problem, for variable surface tension there exists an upper bound Q_c for which steady bubble solutions exist. Theoretical considerations as well as numerical calculations for Q>Q_c verify that the bubble achieves an unsteady cusped formation in finite time. The numerical computations are greatly simplified by exploiting a special analytic structure of the governing equations and interfacial shape. The role of a nonlinear equation of state and the influence of surface diffusion of surfactant are both considered. A possible connection between the observed behavior and the phenomenon of tip streaming is discussed.

REPORT 9900-16: Distributed-parameter model for formaldehyde uptake and distribution in rat nasal lining

A. Georgieva, J. Kimbell & P. Schlosser

DNA-protein crosslinks (DPX) are used as a dosimeter for inhaled formaldehyde and are associated with the presence of tumors induced in rat nasal passages in chronic studies. We present a model that links the airflow-driven uptake of formaldehyde with the DPX formation in two different regions  of the rat nose; one corresponds to high tumor incidence and the other corresponds to low tumor incidence. Three major components are put together by this study: 1) three-dimensional, anatomically accurate computational fluid dynamics model of rat nasal airflow and inhaled gas uptake 2) a distributed-parameter mathematical model that incorporates tissue thickness and DNA-distribution in the nasal mucosa to predict the formation of DPX and to compare them with 3) the experimentally measured DPX in the two regions.

REPORT 9900-17: Dynamics of a two species oscillating  particle system

D. Blackmore, R. Samulyak, R. Dave & A. Rosato

A binary mixture of inelastic particles in a vibrating bed is analyzed. Qualitative properties of the dynamics such as bifurcations and chaotic motions are shown to exist.

REPORT 9900-18: Singularity theory approach to swept volumes

D. Blackmore, R. Samulyak & M. Leu

Singularity theory is applied to study the geometry of objects moving smoothly through space. Computer programs are developed to implement the analytical methods.

REPORT 9900-19: KAM theory analysis of the dynamics of three coaxial vortex rings

D. Blackmore & O. Knio

KAM theory is used to prove the existence of certain configurations of three coaxial vortex ring in an ideal fluid that lead to periodic, quasiperiodic, and chaotic motion.

REPORT 9900-20: Approximate inertial manifolds in finite differences for granular flow dynamical systems

D. Blackmore, R. Samulyak & R. Dave

A numerical algorithm, based on the theory of inertial manifolds, is developed for dissipative evolutionary equations in a finite-difference framework.

REPORT 9900-21: The Lax solution to a Hamilton-Jacobi equation and its generalizations. Part 2.

Y. Mykytiuk, A. Prykarpatsky & D. Blackmore

Continuing our work in Part 1, we prove the existence almost everywhere of Lax type generalized solutions to a class of Cauchy problems for certain types of Hamilton-Jacobi equations.

REPORT 9900-22: Closure properties of uniform convergence of empirical means and PAC learnability under a family of probability measures

M. Vidyasagar, S. Balaji & B. Hammer

Open Problem No:12.2 of [8] asks:" Are the properties of uniform convergence of empirical means, and learnability preserved when the family of probability measures is replaced by its closure?". In this note, the question is answered in the affirmative. Further, it is shown that these properties are not preserved in general if the family of probability measures id replaced by its convex closure. An open question is posed to whether it is possible to replace the family of probability measures by its convex closure in case the family is compact.

REPORT 9900-23: Existence and the number of solutions of semilinear equations and applications to boundary value problems

P. S. Milojevic

Description not available.

REPORT 9900-24: Continuation theory for a-proper mappings and their uniform limits and nonlinear perturbations of fredholm operators

P. S. Milojevic

Description not available.

REPORT 9900-25: Global models for moving contact lines

J. A. Diez, L. Kondic & A. L. Bertozzi

We consider thin films flows driven by surface tension and gravity. Within the framework of the lubrication approximation, we study moving contact line motion using global models for both precursor films and contact lines with slip.  In the latter case, we show  that completely wetting films can be simulated with a slip model that does not require direct tracking of the contact line interface. We perform a comparative study of standard and positivity preserving numerical methods for these problems in one space dimension, with the ultimate goal of choosing the best method for two dimensional computations.  We find a considerable computational advantage to the precursor model vs. global slipping models.

REPORT 9900-26: Unsteady Stokes flow near an oscillating, heated contact line

B. S. Tilley, S. H. Davis & S. G. Bankoff

A contact line on a heated oscillating plate is investigated. The interface is a nondeformable plane and the contact angle is $\pi/2$.  The amplitude of the oscillation and the temperature deviation of the plate from the ambient temperature of the fluid are assumed to be much smaller than the viscous velocity scale.  This flow is then governed by the unsteady Stokes equations coupled to the heat equation in a frame of reference moving with the contact line.  Evaporation is assumed to be neglible, but the effects of heat transfer across the interface and unsteadiness are assumed to be significant.  For a stationary heated plate, there are three distinct regions of flow that is induced by Marangoni stresses.  Two counter-rotating vortical regions are found near the contact line within ten thermal boundary-layer lengths.  Beyond this region, a stagnation-point type flow is found.  For an oscillatory, isothermal plate, vortices are generated at the plate during plate reversal and are propagated along the interface. Dissipation of these vortices occurs on the Stokes's layer scale.  The order-Peclet number correction in the thermal field is also found, and the presence of the flow field leads to a localized cooled region in the steady case.  For the unsteady case, an analogous region that propagates into the bulk with a trajectory determined by the relative scale of the thermal diffusive scale and the rate of heat transfer across the interface.

REPORT 9900-27: A quantitative approximation scheme for the travelling-wave solutions in the Hodgkin-Huxley model

C. B. Muratov

We introduce an approximation scheme for the Hodgkin-Huxley model of nerve conductance which allows to calculate both the speed of the traveling pulses and their shape in quantitative agreement with the solutions of the model. We demonstrate that the reduced problem for the front of the traveling pulse admits a unique solution. We obtain an explicit analytical expression for the speed of the pulses which is valid with good accuracy in a wide range of the parameters.

REPORT 9900-28: The role of short-term synaptic dynamics in motor control