## Center for Applied Mathematics and Statistics

REPORT 0102-1: Low-pressure iris deformation, part 1

D. Stickler

The iris, assumed to have cylindrical symmetry, is a thin membrane tied down at the iris root muscle and at the pupil margin. The root radius will be  denoted by b and the pupil radius by a with b>a. These two circumferential muscles are assumed to lie in parallel planes. In addition the iris has a set of radial or azimuthal muscles which extend from the pupil root  to the iris root and are joined longitudinally  by connective tissue, which is  called collagen. The elastic properties of muscles and collagen are different. The iris therefore has different elastic properties in these two directions. When there in no outside light the iris pupil margin moves form a to a smaller radius. At this radius, in the absence of  any other forces the assumption will be made that the iris has zero stresses. When there is light present it moves to a which, of course, depends on the  amount of light. This induces a stress in the membrane. In addition, the ciliary body excretes a fluid in the chamber behind the iris which can exert an  outward pressure  on the iris. In this analysis the pressure difference across the  iris will be denoted by p. This pressure difference is not the same as the corneal pressure. However it may be possible to relate these two pressures. Four measures of the distortion of the iris by the pressure p are defined: the change in the surfacearea and the change in arc-length of an azimuthal line, the value of the  maximum displacement and the location of the maximum. A small pressure assumption leads to a regular perturbation problem. This expansion will be derived in general and numerical examples given for a Hookean  stress law. For a Hookean model the four distortion measures are plotted  as a function of the stress parameters for some typical values of iris radii.

REPORT 0102-2: The effect of discretization on travelling wave solutions of bistable partial differential equations

C. E. Elmer & E. S. Van Vleck

This article is concerned with effect of spatial and temporal discretizations on traveling wave solutions to parabolic PDEs possessing bistable nonlinearities, in particular for Nagumo type PDEs. Solution behavior is compared in terms of waveforms and in terms of the so called (a,c) relationship where a is a parameter controlling the bistable nonlinearity and c is the wave speed of the traveling wave. Uniform spatial discretizations and A(alpha) stable linear multistep methods in time are considered. Results obtained show the effect of spatial discretization at zero wave speed and the effect of temporal discretization for large wave speeds. An analysis of a complete discretization points out the potential for nonuniquess in the (a,c) relationship.

REPORT 0102-3: Burst synchrony patterns in hippocampal pyramidal cell model networks

V. Booth & A. Bose

Types of burst synchrony are investigated in networks composed of  two-compartment CA3 pyramidal cells and interneurons.  Mechanisms for and  stability of different synchrony patterns are discussed and analyzed. We show that the strength and timing of inhibitory and excitatory synaptic inputs contribute to the generation of either perfectly synchronized or nearly synchronized  network firing, across different burst or spiking modes. Our analysis shows that excitatory inputs tend to desynchronize cells, but that common, slowly decaying  inhibition can be used to synchronize them. We also introduce the concept of equivalent networks'' whereby networks with different architectures and synaptic connections display identical firing patterns.

REPORT 0102-4: LAn affine analoque of the Hartman-Nirenberg cylinder theorem

M. A. Akivis & V. V. Goldberg

Let X be a smooth, complete, connected submanifold of dimension n < N in a complex affine space A^N (C), and r is the rank of its Gauss map \gamma, \gamma (x) = T_x (X). The authors prove that if 2 \leq r \leq n - 1, N - n \geq 2, and in the pencil of the second fundamental forms of X, there are two forms defining a regular pencil all eigenvalues of which are distinct, then the submanifold X is a cylinder with (n-r)-dimensional plane generators erected over a smooth, complete, connected submanifold Y of rank r and dimension r. This result is an affine analogue of the Hartman-Nirenberg cylinder theorem proved for X \subset R^{n+1} and r = 1. For n \geq 4 and r = n - 1, there exist complete connected submanifolds X \subset A^N (C) that are not cylinders.

REPORT 0102-5: Existence of monotone traveling fronts for BDF discretizations of bistable reaction-diffusion equations

C. E. Elmer & E. S. Van Vleck

This article is concerned with the effect of temporal discretization on traveling wave solutions to parabolic PDEs possessing bistable nonlinearities. The focus is on the application of backward differentiation formulas to Nagumo type PDEs with two different bistable nonlinearities. Existence of monotone traveling fronts is shown and the efficacy of different methods of proof discussed.

REPORT 0102-6: Fully nonlinear gravity-capillary solitary waves in a two-fluid system of finite depth

L. Barannyk & D. T. Papageorgiou

We study large amplitude waves at the interface between two laminar immisible inviscid streams of different densities and velocities, bounded together in a straight infinite channel, when surface tension and gravity are both present. A long wave approximation is used to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across it. We study traveling waves of permanent form and show that solitary waves are possible for a range of physical parameters. All solitary waves can be expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions have been found whose amplitudes are equal to h/4 or h/9 where 2h is the channel thickness. In the absence of gravity solitary waves are not possible but periodic ones are. Numerically constructed solitary waves are given for representative physical parameters.

REPORT 0102-7: Transitions between different synchronous firing modes using synaptic depression

V. Booth & A. Bose

In this paper, we show how various synaptic mechanisms interact with intrinsic mechanisms of two-compartment pyramidal cells to produce multiple, stable synchronous firing patterns.  These solutions differ in their frequencies, in the type of firing pattern and in their degree of synchrony. We identify the network elements governing the frequency of each mode, and discuss ways to transition between the two modes.

REPORT 0102-8: An Analytical Study of the Discrete Perfectly Matched Layer for the Time-Domain Maxwell Equations in Cylindrical Coordinates

P. Petropoulos

We present an analysis of the discrete perfectly matched layer in cylindrical coordinates. The discretization is effected with a staggered second-order accurate finite difference time domain method. For fixed discretization parameters, layer width, and a quadratic loss function, we find the numerical reflection produced by the discrete layer is accurately predicted by the analytical reflection coefficient for $\sigma_{max}\in[0,\sigma_{max}^c]$, where $\sigma_{max}$ is the maximum value of the absorption parameter in the layer. We also find that the numerical reflection coefficient achieves its minimum value at a $\sigma^m_{max}>\sigma_{max}^c$. The analysis determines $\sigma^c_{max}$ and $\sigma_{max}^m$, given the discretization parameters and layer width. Numerical experiments validate the analysis.

REPORT 0102-9: Computation of Mixed Type Functional Differential Boundary Value Problems

C. E. Elmer, A. R. Humphries, and E. S. Van Vleck

We describe a general purpose code for boundary value differential-difference equations where the difference terms may contain both advances and delays. The code is an extension of the COLMOD, COLNEW, and COLSYS family. The method employed is piecewise polynomial collocation at Gauss points. Mesh selection and continuation ia as in COLMOD. Because of the nonlocal nature of the difference trems the linear algebra has been extensively modified, and both boundary conditions and boundary functions must be specified. Special attention is paid to computation of connecting orbits. Aproximation issues are discussed, the code and its uses are described, and several examples are shown to illustrate the utility of the code.

REPORT 0102-10: A Variant of Newton Method for the Computation of Traveling Waves of Bistable Differential Difference Equations

C. E. Elmer and E. S. Van Vleck

We consider a variant of Newton's method for solving nonlinear differential-difference equations arising from the traveling wave equations of a large class of nonlinear evolution equations. Building on the Fredholm theory recently developed by J. Mallet-Paret we prove convergence of the method. Several examples are considered and the utility of the method is demonstrated with a series of examples.

REPORT 0102-11: Short-Term Synaptic Dynamics Promote Phase Maintenance in Multi-Phasic Rhythms

Y. Manor, V. Booth, A. Bose, and F. Nadim

We show that in an inhibitory network synaptic depression promotes phase constancy. As oscillation period increases, the synapse recovers from depression and becomes more effective in delaying the postsynaptic cell's firing. As a result, the delay between the pre- and postsynaptic bursts increases as period increases. We discuss the dependence of the bursting phase of the postsynaptic cell on the strength and kinetics of the depressing synapse.

REPORT 0102-12: Smooth Lines on Projective Planes Over Two-Dimensional Algebras and Submanifolds with Degenerate Gauss Maps

M. A. Akivis and V. V. Goldberg

The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines forming point three-dimensional submanifolds X^3 with degenerate Gauss maps of rank r \leq 2. The authors study focal properties of these submanifolds and prove that they represent examples of different types of submanifolds X^3 with degenerate Gauss \maps. Namely, the submanifold X^3, corresponding in RP^5 to a smooth line \gamma of the projective plane CP^2, does not have real singular points, the submanifold X^3, corresponding in RP^5 to a smooth line \gamma of the projective plane C^1 P^2, bears two plane singular lines, and finally the submanifold X^3, corresponding in RP^5 to a smooth line \gamma of the projective plane C^0 P^2, bears one singular line. It is also proved that in all three cases, the rank of X^3 is equal to the rank of the curvature of the line \gamma.

REPORT 0102-13: Mechanical Properties of Diseased Hearts During Adaptation

R. Chaudhry, B. Bukiet, A. B. Ritter, and R. Arora

In this paper, a method is developed to determine the material constants of normal and diseased human hearts in vivo. With these material constants, one can compute regional stresses and strains through the left ventricular (LV) wall. This knowledge provides a better understanding of the heart adaptation process and may lead to earlier diagnosis of heart disease. It also may enable earlier treatment of heart disease, and evaluation of the efficacy of treatments for it. The heart is modeled as a thick cylindrical shell and large deformation theory, incorporating residual stresses is employed to compute the regional stresses and strains through the LV wall. These stresses and strains at the end diastolic state for the normal, hypertensive and congestive heart failure cases are presented. The average circumferential stress is also computed at the end systolic state for these cases.

REPORT 0102-14: Simulations of Coating Flows and Drops in Higher Dimensions

L. Kondic and J. A. Diez

We present a computational method for quasi 3D unsteady flows of thin liquid films on a solid substrate. This method includes surface tension as well as gravity forces in order to model realistically the spreading on an arbitrarily inclined substrate. The method uses a positivity preserving scheme to avoid possible negative values of the fluid thickness near the fronts. The `contact line paradox', i.e., the infinite stress at the contact line, is avoided by using the precursor film model which also allows for approaching problems that involve topological changes. After validating the numerical code on problems for which the analytical solutions are known, we present results of fully nonlinear time-dependent simulations of merging liquid drops using both uniform and nonuniform computational grids.

REPORT 0102-15: A Class of Four-Dimensional Warped Products

F. Defever, R. Deszcz, M. Glogowska, V. V. Goldberg, and L. Verstraelen

The authors investigate properties of 4-dimensional warped product manifolds satisfying a particular set of curvature conditions. As an application, they obtain a generalization of a pseudosymmetric property for Ricci-flat warped product spacetimes which was established previously in a number of special cases, including the Schwarzschild metric.

REPORT 0102-16: Equations of Interface Dynamics for Quasi-Stationary Stefan Problem

R. Andrushkiw, V. Gafiychuk, A. Shnyr, and R. Zabrodsky

The interface dynamics in a Laplacian growth model is investigated, using conformal mapping techniques. Starting from the governing equation of B.Shraiman and D.Bensimon, we derive integrodifferential evolution equations of interphase dynamics. It is shown that such representation based on conformal mapping techniques is convenient for computer simulation of quasistationary Stefan problems. Application of the technique to anisotropic growth of crystals is considered.

REPORT 0102-17: New Order Preserving Properties of Geometric Compounds

M.C. Bhattacharjee, S. Ravi, R. Vasudeva, and N.R. Mohan

Description to be provided

REPORT 0102-18: On the Unique Constructive Solvability of Hammerstein Equation

P. S. Milojevic

Description to be provided

REPORT 0102-19: Application of the Duality Principle for Construction of Varieties with Degenerate Gauss Maps

M. A. Akivis and V. V. Goldberg

The authors apply the duality principle to construct new examples of varieties with degenerate Gauss maps. In particular, they prove that the variety which is dual to the projection of the Veronese variety onto a four-dimensional subspace is a cubic hypersurface with a degenerate Gauss map of rank two.

REPORT 0102-20: Discrete Convex Ordered Lifetimes: Characterizations, Equivalence and Applications

M. C. Bhattacharjee

Description to be provided