**REPORT 0203-1: ** 4-webs in the plane and their linearizability.

Vladislav V. Goldberg

We investigate the linearizability problem for different classes of 4-webs in the plane. In particular, we prove that a 4-web MW with equal curvature forms of its 3-subwebs and a covariantly nonconstant basic invariant is always linearizable, and such a 4-web with a constant basic invariant is linearizable if and only if it is parallelizable. We also consider four classes of the so-called almost parallelizable 4-webs APW_a, a = 1, 2, 3, 4 (for them the curvature K = 0 and the basic invariant is covariantly constant on the web foliation X_a), and prove that a 4-web APW_a is linearizable if and only if it coincides with a 4-web MW_a of the corresponding special class of 4-webs MW. The existence theorems are proved for all the classes of 4-webs considered in the paper.

**REPORT 0203-2:** Maximum rank webs are not necessarily almost
Grassmannizable.

Vladislav V. Goldberg

We construct an example of a 6-web of codimension two and of maximum rank on a six-dimensional manifold which is not almost Grassmannizable.

**REPORT 0203-3:**
Anisotropy, Propagation Failure, and Wave Speedup in Traveling Waves
of Discretizations of a Nagumo PDE.

Christopher E. Elmer and Erik S. Van Vleck

This article is concerned with effect of spatial and temporal discretizations on traveling wave solutions to parabolic PDEs (Nagumo type) possessing piecewise linear bistable nonlinearities. 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 by varying the potential energy difference of the two phases and $c$ is the wave speed of the traveling wave. Uniform spatial discretizations and $A(\alpha)$ stable linear multistep methods in time are considered.

**REPORT 0203-4:** Cancer
diagnostic method based on pattern recognition of DNA changes in
buccal epithelium in the pathology of the thyroid and mammary glands.

Dmitri A. Klyushin, Yuri I. Petunin, Roman I. Andrushkiw, Natalya V. Boroday, and Karelia P. Ganina

The object of this investigation is to study, from the point of view of statistical and geometrical theory of pattern recognition, the peculiarities of the distribution of optical density of DNA in the interphase nuclei of buccal epithelium present in the pathology of the thyroid and mammary glands. Two new indices characterizing this distribution (ratio of modal class volumes and relief index) are proposed. It is shown that in malignant neoplasms of the thyroid and mammary glands the changes in the nuclei of buccal epithelium are characterized by an increase in the optical density of DNA in a range from 0.15 to 0.30 in conventional units of measure, as compared with its values in benign pathological processes. The sensitivity of the proposed criterion in the case of the diseases of the thyroid gland is equal to 76.2 % and the specificity is equal to 85.8 %. In the case of diseases of the mammary gland (excluding IDLC) we have discovered that the sensitivity of the method is equal to 94.29 % and its specificity equal to 90.91 %. In the case of diseases of the mammary gland (including IDLC) we have discovered that the sensitivity of the method is equal to 71.42% and its specificity is equal to 90.91%.

**REPORT 0203-5:** On the
use of infinite elements for the determination of optimal closure
patterns based on stress analysis.

D. A. Lott and H. R. Chaudhry

Solid infinite elements are used in conjunction with finite elements to compute the stress and displacement distribution resulting from the suturing of wounds of symmetric and nonsymmetric shapes in orthotropic, abdominal human skin. The optimal pattern of suturing of wounds are investigated from a stress perspective. Highly accurate, quantitative and qualitative improvements over the use of finite elements to approximate distant boundaries are obtained. Numerical results quantitatively agree with analytic results computed using complex analysis techniques. The technique used and the results obtained will aid surgeons in closing nonsymmetrical wounds on regions of the body that exhibit orthotropy.

**REPORT 0203-6:** The
effect of electric fields on the rupture of thin viscous films by van
der Waals forces.

K. Savettaseranee, D. T. Papageorgiou, P. G. Petropoulos, and B. S. Tilley

We examine the stability of a thin two-dimensional incompressible liquid film when an electric field is applied in a direction parallel to the initially flat bounding fluid interfaces, and study the competition between surface tension, van der Waals, viscous and electrically induced forces. The film is assumed to be sufficiently thin, and the surface tension and electrically induced forces are large enough that gravity can be ignored to the leading order. We analyze the nonlinear stability of the flow by deriving and numerically solving a set of nonlinear evolution equations for the local film thickness and the horizontal velocity. We find that the electric field forces enhance the stability of the flow and can remove rupture. If rupture occurs then the form of the singularity, to leading order, is that found in the absence of an electric field.

**REPORT 0203-7:** On the
long-time behavior of unsplit Perfectly Matched Layers.

E. Becache, P. G. Petropoulos, S. D. Gedney

We show how to eliminate an undesirable long-time linear growth of the electromagnetic field in a class of unsplit Perfectly Matched Layers (PML) typically used as Absorbing Boundary Conditions in Computational Electromagnetics codes. For the new PML equations we give energy arguments that show the fields in the layer are bounded by a time-independent constant hence they are long-time stable. Numerical experiments confirm the elimination of the linear growth, and the long-time boundedness of the fields.

**REPORT 0203-8:**
Instabilities in gravity driven flow of thin fluid films.

L. Kondic

This paper presents theoretical, computational and experimental aspects of the instability development in the flow of thin fluid films. The theoretical part involves basic fluid mechanics and presents derivation of the thin film equation using lubrication approximation. Simplified version of this equation is then analyzed analytically using linear stability analysis, and also numerically. The results are then compared directly to experiments. The experimental part outlines the setup, as well as data acquisition and analysis. This immediate comparison to experiments is very useful for gaining better insight into interpretation of various theoretical and computational results.

**REPORT 0203-9:**
Computing three dimensional thin film flows including contact lines.

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 0203-10:**
On the Characterization of a Bivariate Geometric Distribution

Sunil K. Dhar and Srinivasan Balaji

The discrete analogue of the bivariate Freund's models is generalized by way of characterization. This characterization is achieved by extending a key lemma in this area and is in terms of constant total failure rate and mixture geometric marginals. The generalized bivariate Freund's model includes a bivariate geometric distribution previously characterized.

**REPORT 0203-11:**
High-frequency, depressing inhibition facilitates synchronization in
globally inhibitory networks

Steve Kunec and Amitabha Bose

Motivated by the study of Sharp Wave-Associated Ripples, a high-frequency extracellular field oscillations observed in the CA1 region of the rat hippocampus during slow-wave sleep and periods of behavioral immobility, we consider a single inhibitory, neuron synapsing onto a network of uncoupled, excitatory neurons. The inhibitory synapse is depressing and has a small synaptic delay. Each excitatory cell provides instantaneous, positive feedback to the inhibitory cell. We show that the interneuron can rapidly synchronize the action potentials of the pyramidal cells if the frequency of inhibitory input is increased in a ramp like manner as occurs during the ripple. We show that the basin of attraction of the synchronous solution is larger when the inhibtion frequency is gradually increased as opposed to if it remained constant. This mechanism for synchrony is robust in the presence of heterogeneities.

**REPORT 0203-12:**
An analytical study of the inductive pattern formation
mechanism in Drosophila egg development

C. B. Muratov and S. Y. Shvartsman

The complexity and nonlinear dynamics of patterning networks in development make modeling an important approach for the evaluation of the experimentally-derived pattern formation mechanisms. As a rule, models of patterning networks have large numbers of uncertain parameters and model analysis requires extensive computational searches of the parameter space. Analytical techniques can circumvent these difficulties and offer important insights into the networks' functional capabilities. Here, we present an asymptotic analysis of the multiple steady states and transitions between them in a mechanistic model of patterning events specifying the formation of a pair organ in {\em Drosophila} oogenesis. The model describes the interaction between the spatially nonuniform inductive signal and a network of feedback loops within the layer of epithelial cells. Our approach dramatically reduces the complexity of the problem and provides an explicit analytical method for the construction and parametric analysis of the patterned states responsible for signaling. The analysis reveals a skeleton structure for the patterning capability of the considered regulatory module and demonstrates how a single regulatory network can be used to generate a variety of developmental patterns.

**REPORT 0203-13:**
A shallow water ocean acoustics inverse problem

David Stickler

A shallow water ocean acoustics experiment is described from which it is possible to recover the scattering data necessary to recover the sound speed in the ocean bottom.

**REPORT 0203-14:**
The contribution of synaptic depression to phase maintenance in a model
rhythmic network

Yair Manor, Amitabha Bose, Victoria Booth, and Farzan Nadim

In many rhythmic neuronal networks that operate in a wide range of frequencies, the time of neuronal firing relative to the cycle period (the phase) is invariant. This invariance suggests that, when frequency changes, firing time is precisely adjusted, either by intrinsic or synaptic mechanisms. We study the maintenance of phase in a computational model in which an oscillator neuron (O) inhibits a follower neuron (F) by comparing the dependency of phase on cycle period in two cases: when the inhibitory synapse is depressing and when it is non-depressing. Of the numerous ways of changing the cycle period, we focus on three cases where either the duration of the active state, the inactive state, or the duty cycle of O remains constant. In each case, we measure the phase at which F fires with respect to the onset of firing in O. With a non-depressing synapse this phase is generally a monotonic function of cycle period, except in a small parameter range in the case of the constant inactive duration. In contrast, with a depressing synapse, there is always a parameter regime in which phase is a cubic function of cycle period: it decreases at short cycle periods, increases in an intermediate range and decreases at long cycle periods. This complex shape for the phase-period relationship arises because of the interaction between synaptic dynamics and intrinsic properties of the postsynaptic neuron. By choosing appropriate parameters, the cubic shape of the phase-period curve results in a small variation in phase for a large interval of periods. Consequently, we find that although a depressing synapse does not produce perfect phase maintenance, it is generally superior to a non-depressing synapse in promoting a constant phase difference.

**REPORT 0203-15:**
Short-term dynamics of a mixed chemical and electrical synapse in a rhythmic
network

Akira Mamiya, Yair Manor, and Farzan Nadim

In the rhythmically active pyloric circuit of the spiny lobster, the synapse between the lateral pyloric (LP) neuron and pyloric constrictor (PY) neurons has an inhibitory depressing chemical and an electrical component. We characterized the dynamics of the LP to PY synapse after blocking the rhythmic activity and correlate these dynamics with the relative timing of activity of these two neurons before the rhythm was blocked. When a train of voltage pulses was applied to the voltage-clamped LP neuron, the inhibitory chemical component of the postsynaptic potential (PSP) in the PY neuron rapidly depressed. Thus, after the first few pulses, the PSP was either hyperpolarizing or depolarizing depending on the interpulse duration, with higher interpulse durations producing depolarizing PSPs. To characterize the synaptic response during rhythmic activity, we played back pre-recorded realistic waveforms in the voltage-clamped LP neuron. After an initial transient, the resulting PSP in PY was always depolarizing, suggesting that in an ongoing rhythm the electrical component of the synapse is dominant. However, using pharmacological tools we unmasked a significant contribution of the chemical component of the synapse to the fine-tuning of the PY PSP. Our results indicate that the chemical component of the mixed synapse acts to delay the peak time of the PSP and to reduce its amplitude, and that these effects become more important at slower cycle periods. We then compared the properties of the synapse as measured in the quiescent preparation with the relative activity phase of the LP and PY neurons in the ongoing pyloric rhythm. Interestingly, the phase between the LP waveform and the PY PSP correlated well with the phase between the LP and PY bursts in the ongoing rhythm. This suggests that mixed synapses may dynamically affect the phase between two neurons if the rhythm frequency is altered.

**REPORT 0203-16:**
Discrete models of autocrine cell communication in epithelial layers

Michal Pribyl, Cyrill Muratov and Stanislav Shvartsman

Pattern formation in epithelial layers heavily relies on cell communication by secreted ligands. While the experimentally observed signaling patterns can be visualized at single cell resolution, a biophysical framework for their interpretation is currently lacking. To this end, we develop a family of discrete models of cell communication in epithelial layers. The models are based on the introduction of cell-cell coupling coefficients that characterize the spatial range of intercellular signaling by diffusing ligands. We derive the coupling coefficients as functions of geometric, cellular, and molecular parameters of the ligand transport problem. Using these coupling coefficients, we analyze a nonlinear model of positive feedback between ligand release and binding. In particular, we study criteria of existence of the patterns consisting of clusters of a few signaling cells, as well as the onset of signal propagation. We use our model to interpret recent experimental studies of EGFR/Rhomboid/Spitz module in Drosophila development.

**REPORT 0203-17:**
Atomistic Theory of Elasticity for Thin Epitaxial Films

Cameron Connell and Russel E. Caflisch

Strain has significance for both the growth characteristics and material properties of thin epitaxial films. Current methods for modelling and simulating strain in thin films are either continuum or atomistic. Continuum theories of strain are not applicable to very thin films. Purely atomistic theories do not really lend themselves to analysis and are computationally slow. In this work, a uniform approach that combines atomistic and continuum aspects of elasticity is developed and applied to epitaxial thin films. In its present form, the theory is developed for linear elasticity and for a material system in which the only defects are step edges and material interfaces. This theory is outlined for a general lattice, but an explicit version of the theory is described for a 2D cubic lattice for which the energy has 2D cubic symmetry. Force balance equations are derived for a material system with material/vacuum and material/material interfaces. Intrinsic surface stress in a material/vacuum interface is included in the theory by treating atoms at the surface as belonging to a different material. The Keating model is found to be a special case of the present theory.

**REPORT 0203-18:**
Theory of Strain Relaxation in Epitaxial Growth

A. C. Schindler, M. F. Gyure, G. D. Simms, D. D. Vvedensky, R. E. Caflisch, C. Connell, Erding Luo

We introduce a general approach to calculating the morphological consequences of coherent strain relaxation in heteroepitaxial thin films based on lattice statics using linear elasticity. The substrate and film are described by a simple cubic lattice of atoms with localized interactions. The boundary conditions at concave and convex corners that appear as a result of this construction, those along straight interfacial segments, and the governing equations are obtained from a variational calculation applied to a discretized form of the total elastic energy. The continuum limit of the equations and the boundary conditions along straight boundaries reproduces standard results in elasticity theory, but the boundary conditions at corners have no such analogue. Our method enables us to calculate quantities such as the local strain energy density for any surface morphology once the lattice misfit and the elastic constants of the constituent materials are specified. The methodology is illustrated by examining the strain, displacement and energies of one-dimensional strained vicinal surfaces. We discuss the effects of epilayer thickness on the energy of various step configurations and suggest that coupling between surface and substrate steps should affect the equilibiration of the surface toward a bunched state.

**REPORT 0203-19:**
A Discrete Elastic Model for Epitaxial Systems and the Force Field at a Step

Cameron R. Connell, Russel E. Caflisch, Erding Luo, and Geoff Simms

Strain has significance for both the growth characteristics and material properties of thin epitaxial films. In this work, the method of lattice statics is applied to an epitaxial system with cubic symmetry, using harmonic potentials. The energy density and force balance equations are written using a finite difference formalism that clearly shows their consistency with continuum elasticity. For simplicity, the atomic interactions are assumed to be maximally localized. For a layered material system with a material/vacuum interface and with surface steps, force balance equations are derived, and intrinsic surface stress at the material/vacuum interface is included by treating the atoms at the surface as having different elastic properties. By defining the strain relative to an appropriately chosen nonequilibrium lattice, as in the method of eigenstrains, analytic formulas in terms of microscopic parameters are found for the local force field near a step and for the macroscopic monopole and dipole moment forces due to a step. These results provide an atomistic validation of the Marchenko-Parshin formula for the dipole moment in terms of the elastic surface stress.

**REPORT 0203-20:**
How Random is "Random"?

Sunil K. Dhar and Varun Oberoi

Three random number generators are analyzed for data sets of 500 numbers. Their performances are evaluated through several tests. All generators create uniform numbers. The various results are then interpreted to find out which random number generator creates numbers that are in close agreement with the random theory. Using these results and interpretations some ideas are explored to improve on the randomness of numbers. A new test is proposed to check for randomness.

**REPORT 0203-21:**
Episodic Bouts of Activity Accompany Recovery of Rhythmic Output by a
Neuromodulator and Activity Deprived Adult Neural Network

Jason A. Luther, Alice A. Robie, John Yarotsky, Christopher Reina, Eve Marder, and Jorge Golowasch

The pyloric rhythm of the stomatogastric ganglion of the crab, Cancer borealis, slows or stops when descending modulatory inputs are acutely removed. However, the rhythm spontaneously resumes after one or more days in the absence of neuromodulatory input. We recorded continuously for days to characterize quantitatively this recovery process. Activity bouts lasting 40 to 900 seconds began several hours after removal of neuromodulatory input and were followed by stable rhythm recovery after 1-2 days. Bout duration was not related to the intervals (0.3 to 800 minutes) between bouts. During an individual bout the frequency rapidly increased and then decreased more slowly. Photoablation of back-filled neuromodulatory terminals in the STG neuropil had no effect on bouting or recovery, suggesting that these processes are intrinsic to the STG neuronal network. After removal of neuromodulatory input the phase relationships of the components of the triphasic pyloric rhythm were altered, and then over time the phase relationships moved towards their control values. Although at low pyloric rhythm frequency the phase relationships among pyloric network neurons depended on frequency, nges in frequency during recovery could not account for the change in phase seen after rhythm recovery. We suggest that activity bouts represent underlying mechanisms controlling the restructuring of the pyloric network to allow resumption of an appropriate output following removal of neuromodulatory input.

**REPORT 0203-22:**
The Influences of I_h on Temporal Summation in Hippocampal CA1 Pyramidal
Neurons: A Modeling Study

Adrien E. Desjardins , Yue-Xian Li, Stefan Reinker, Robert M. Miura, and Richard S. Neuman

Recent experimental and theoretical studies have found that active dendritic ionic currents can compensate for the effects of electrotonic attenuation. In particular, temporal summation, the percentage increase in peak somatic voltage responses invoked by a synaptic input train, is independent of location of the synaptic input in hippocampal CA1 pyramidal neurons under normal conditions. This independence, known as normalization of temporal summation, is destroyed when the hyperpolarization-activated current, I_h, is blocked [J.C. Magee (1999), Nature Neurosci. 2:508-514]. Using a compartmental model derived from morphological recordings of hippocampal CA1 pyramidal neurons, we examined the hypothesis that Ih was primarily responsible for normalization of temporal summation. We concluded that this hypothesis was incomplete. With a model that included Ih, the persistent Na+ current (I_{NaP}), and the transient A-type K+ current (I_A), however, we observed normalization of temporal summation across a wide range of synaptic input frequencies, in keeping with experimental observations.

**REPORT 0203-23:**
Membrane Resonance and Stochastic Resonance Modulate Firing Patterns of
Thalamocortical Neurons

Stefan Reinker, Ernest Puil, and Robert M. Miura

We examined the interactions of subthreshold membrane resonance and stochastic resonance using whole-cell patch clamp recordings in thalamocortical neurons of rat brain slices, as well as with a Hodgkin-Huxley-type mathematical model of thalamocortical neurons. The neurons exhibited subthreshold resonance when stimulated with small amplitude sine wave currents of varying frequency, and stochastic resonance when noise was added to sine wave inputs. Stochastic resonance was manifest as a maximum in signal-to-noise ratio of output response to subthreshold periodic input combined with noise. Stochastic resonance in conjunction with subthreshold resonance resulted in action potential patterns that showed frequency selectivity for periodic inputs. Stochastic resonance was maximal near subthreshold resonance frequency and a high noise level was required for detection of high frequency signals. We speculate that combined membrane and stochastic resonances have physiological utility in coupling synaptic activity to preferred firing frequency and in network synchronization under noise.

**REPORT 0203-24:**
A Construction Technique for Heteroclinic Solutions to Continuous and
Differential-Difference Damped Wave Equations

Marianito Rodrigo, Christopher Elmer, and Robert M. Miura

In this paper, we give a systematic method for generating continuous and diffrential-difference damped wave equations for which explicit travelling wave solutions can be obtained. We demonstrate the procedure with several examples. In some specific cases, we recover the well-known solutions of the continuous Nagumo and sine-Gordon equations.

**REPORT 0203-25:**
Clustering in Small Networks of Excitatory Neurons with Heterogeneous
Coupling Strengths

Yue-Xian Li, Yuqing Wang, and Robert M. Miura

Excitatory coupling with a slow rise time destabilizes synchrony between coupled neurons. Thus, the fully synchronous state is usually unstable in networks of excitatory neurons. Phase-clustered states, in which neurons are divided into multiple synchronized clusters, have also been found unstable in numerical studies of excitatory networks in the presence of noise. The question arises as to whether synchrony is possible in networks of neurons coupled through slow, excitatory synapses. In this paper, we show that robust, synchronous clustered states can occur in such networks. The effects of non-uniform distributions of coupling strengths are explored. Conditions for the existence and stability of clustered states are derived analytically. The analysis shows that a multi-cluster state can be stable in excitatory networks if the overall interactions between neurons in different clusters are stabilizing and strong enough to counter-act the destabilizing interactions between neurons within each cluster. When heterogeneity in the coupling strengths strengthens the stabilizing inter-cluster interactions and/or weakens the destabilizing in-cluster interactions, robust clustered states can occur in excitatory networks of all known model neurons. Numerical simulations were carried out to support the analytical results.

**REPORT 0203-26:**
Heat-Induced Stretching of a Glass Tube Under Tension: Applications to
Glass Microelectrodes

Huaxiong Huang, Robert M. Miura, William P. Ireland, and Ernest Puil

Deformation of glass using heat occurs in many industrial and artistic applications, including the manufacturing of laboratory glass products, drawing of fiber optics, and hand-blown artistic creations. The formation of glass objects is an art, but the trial-and-error procedures can be reduced by development of a systematic theory, especially when the objects are formed using mechanical means. Glass microelectrodes are ubiquitous in experimental studies of the electrophysiology of biological cells and their membranes, and the ``pulling" of these electrodes is based on trial-and-error. To make this process more systematic, we derive a model for glass microelectrode formation using a coil heater with a gravity-forced electrode puller assuming the glass tube is an incompressible, viscous fluid. The model is one dimensional, and the effects of thermal radiation from the coil heater are essential in the formation process. A breaking stress criterion is imposed to fracture the glass tube, forming the electrode tip. The difficulty with the moving free end is avoided by introducing a quasi-Lagrangian coordinate system. The model equations are solved using an adaptive moving grid to account for the local stretching of the glass. A number of examples using a double-pull paradigm have been computed to illustrate the dependence of the electrode shape and tip diameter on the heater temperature and the ratio between the inner and outer radii.

**REPORT 0203-27:**
Spermine modulates neuronal excitability and NMDA receptors in juvenile
gerbil auditory thalamus

Israeli Ran, Robert M. Miura, and Ernest Puil

Medial geniculate body (MGB)neurons process synaptic inputs from auditory cortex. Corticothalamic stimulation evokes glutamatergic excitatory postsynaptic potentials (EPSPs) that vary markedly in amplitude and duration during development. The EPSP decay phase is prolonged during second postnatal week but then shortens, significantly, until adulthood. The EPSP prolongation depends on spermine interactions with a polyamine-sensitive site on receptors for N-methyl-D-aspartate (NMDA). We examined effects of spermine application on EPSPs, firing modes, and membrane properties in gerbil MGB neurons during the P14 period of highest polyamine sensitivity. Spermine slowed EPSP decay and promoted firing on EPSPs, without changing passive membrane properties. Spermine increased membrane rectification on depolarization, which is mediated by tetrodotoxin (TTX)-sensitive, persistent Na conductance. As a result, spermine lowered threshold and increased tonic firing evoked with current injection by up to V. These effects were concentration-dependent (ED_{50} =100 \mu M), reversible, and eliminated by NMDA receptor antagonist, 2-amino-5-phosphonovalerate (APV). In contrast, spermine increased dV/dt of the low threshold Ca^{2+} spike (LTS) and burst firing, evoked from hyperpolarized potentials. LTS enhancement was greater at 3 mV than at hyperpolarized potentials and did not result from persistent Na conductance or glutamate receptor mechanisms. In summary, spermine increased excitability by modulating NMDA receptors in juvenile gerbil neurons.

**REPORT 0203-28:**
Implementation of the Barrier Method to the Variational Inequality (VI) Parking
Spatial Price Equilibrium Problem

David Bernstein, Kyriacos C. Mouskos, and John Tavantzis

The game played between parking facility owners and travelers who want to park in a specific geographic area such as the Central Business District (CBD) is formulated as a Variational Inequality (VI) spatial price equilibrium problem. The parking facilities owners place a price for parking (supply price) and the users set their own price that they are willing to pay (demand price). After a certain period of trade-offs, usually the system reaches equilibrium, where the supply price for each parking facility equals the demand price of the user groups that have decided to accept the price. This means that there is a flow from each of these user groups to the specific parking facilities, otherwise there is no flow. The user groups whose demand prices are lower than the supply prices will not park in this area and either use another parking facility further away, park on the street, visit the area through another transportation means (e.g. ride share, transit, bicycle, etc.), or avoid the area and go somewhere else to conduct their business and personal functions. This VI mathematical formulation is solved using the barrier method that forces the solution to stay in the feasible region. Two small problems are solved that demonstrate the implementation of the algorithms to this application.

**REPORT 0203-29:**
A note on surface water waves for finite depth in the presence of an ice-cover

A. Chakrabarti, D.S. Ahluwalia, and S.R. Manam

A class of boundary value problems involving propagation of two-dimensional surface water waves, associated with water of uniform finite depth, against a plane vertical wave maker is investigated under the assumption that the surface is covered by a thin sheet of ice. It is assumed that the ice-cover behaves like a thin isotropic elastic plate. Then the problems under consideration lead to those of solving the two-dimensional Laplace equation in a semi-infinite strip, under Neumann boundary condition on the vertical boundary as well as on one of the horizontal boundaries, representing the bottom of the fluid region, and a condition involving upto fifth order derivatives of the unknown function on the top horizontal ice-covered boundary, along with two appropriate edge-conditions, at the ice-covered corner, ensuring the uniqueness of the solutions. The mixed boundary value problems are solved completely, by exploiting the regularity property of the Fourier cosine transform.

**REPORT 0203-30:**
Role of weakly singular integral equations in surface water wave scattering.

A. Chakrabarti, D.S. Ahluwalia and S.R. Manam

A class of boundary value problems arising in the study of scattering of surface water waves by barriers, under the assumption of the linearized theory, is reduced to singular integral equations of the first kind, involving only " weakly singular " kernels. The unified treatment presented here is observed to be most suitable to handle the class of scattering problems under consideration and it is shown that the reduction of these scattering problems to either the Cauchy type " strongly singular" integral equation or the so called " hyper singular" integral equations can be avoided altogether.