The City University of New York

Applied Mathematics Seminar

The City University of New York

Applied Mathematics Seminar

Department of Mathematics

Graduate Center of CUNY

365 Fifth Avenue

New York, NY 10016

Fridays 4:00PM-5:00PM

Room: 8405

Organizers: Faranak Pahlevani, Jesenko Vukadinovic, Hüseyin Yüce

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Spring 2012

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Feb 10: Organizational meeting

Feb 17: Xing Zhong (New Jersey Institute Technology, New Jersey)

Threshold Phenomena for Symmetric Decreasing Solutions of Reaction-Diffusion Equations

We study the Cauchy problem for nonlinear reaction-diffusion equation (u_t = u_xx + f(u), u(x,0) = \phi (x), x \in R, t > 0), with different nonlinearities. By using energy functional and exponentially weighted functional, for symmetric decreasing initial conditions, we prove one-to-one relation between long time behavior of solution and limit value of energy. Then we study the threshold phenomena. This is a joint work with Cyrill Muratov.

Feb 24: No meeting today

Mar 02: TBA

Mar 09: Keith Promislow (Michigan State University, East Lansing, MI)

Network Formation and Ion Conduction in Ionomer Membranes

Many important processes in the physical world can be described as a gradient (overdamped) flow of a variational energy. We present a broad formalism for the generation of new classes of higher-order variational energies with a physically motivated structure. In particular we reformulate the Cahn-Hilliard energy, which is well know to describe the surface area of mixtures, into a higher-order model of interfacial energy for mixtures of charged polymers (ionomers) with solvent. These materials are important as selectively conductive membrane separators in a wide variety of energy conversion devices, including polymer electrolyte membrane fuel cells, Lithium ion batteries, and dye sensitized solar cells.

Our reformulated energy, called the Functionalized Cahn-Hilliard (FCH) energy, captures elastrostatic interactions between the charged groups and the complex entropic effects generated by solvent-ion interactions, and allows us to unfold the bilayer and pore networks formed by the solvent phase imbibed into the polymer matrix. We discuss sharp interface reductions of the FCH energy, its gradient flows, and sharp interface reductions of the gradient flows that give rise to higher-order curvature driven flows. We also describe extensions to models that couple to ionic transport and as well as to multiphase models suitable to describe a wide range of membrane casting processes.

References

[1] N. Gavish, J. Jones, Z. Xu, A. Christlieb, K. Promislow, submitted to Polymers Special issue on Thin Membranes (2012).

[2] H. Zhang and K. Promislow, Critical points of Functionalized Lagrangians,

Discrete and Continuous Dynamical Systems, A to appear.

[3] N. Gavish, G. Hayrapetyan, Y. Li, K. Promislow, Physica D, 240: 675-693 (2011).

[4] K. Promislow and B. Wetton, PEM fuel cells: A Mathematical Overview, Invited Review Paper to SIAM

Applied Math. 70: 369-409 (2009)

Mar 16: No seminar today

Mar 23: Levent Kurt (The Graduate Center of CUNY)

The Higher-Order Short Pulse Equation

We derive an equation, the higher-order short pulse equation (HSPE), from the nonlinear wave equation

to capture the dynamics of ultra-short solitons in cubic nonlinear media using both multiple

scaling technique and re-normalization group. The multiple scaling derivation will be presented. The

numerical solution of the HSPE as the exact one- and two-soliton solutions of the short pulse equation

(SPE) being the initial conditions and its comparison to the numerical solutions of the SPE and

original equation will also be demonstrated.

Mar 30: Christina Mouser (William Paterson University, New Jersey)

The Control of Frequency of a Conditional Oscillator Simultaneously Subjected to Multiple Oscillatory Inputs

The gastric mill network of the crab Cancer borealis is an oscillatory network with frequency ~ 0.1 Hz. Oscillations in this network require neuromodulatory synaptic inputs as well as rhythmic inputs from the faster (~ 1 Hz) pyloric neural oscillator. We study how the frequency of the gastric mill network is determined when it receives rhythmic input from two different sources but where the timing of these inputs may differ. We find that over a certain range of the time difference one of the two rhythmic inputs plays no role what so ever in determining the network frequency while in another range, both inputs work together to determine the frequency. The existence and stability of periodic solutions to model sets of equations are obtained analytically using geometric singular perturbation theory. The results are validated through numerical simulations. Comparisons to experiments are also presented.

Apr 20: Peter Gordon (New Jersey Institute Technology, New Jersey)

Local kinetics and self-similar dynamics of morphogen gradients

Some aspects of pattern formation in developing embryos can be described by nonlinear reaction-diffusion equations. An important class of these models accounts for diffusion and degradation of a locally produced single chemical species and describe formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues. At long times, solutions of such models approach a steady state in which the concentration decays with distance from the source of production. I will present our recent results that characterize the dynamics of this process. These results provide an explicit connection between the parameters of

the problem and the time needed to reach a steady state value at a given position. I will also show that the long time behavior of such models, in certain cases, can be described in terms of very singular self-similar solutions. These solutions are associated with a limit of infinitely large signal production strength.

This is a joint work with: C. Muratov, S. Shvartsman, C. Sample and A.Berezhkovskii.

Apr 27: Ionut Florescu (Stevens Institute of Technology, New Jersey)

Solving systems of PIDE's coming from regime switching jump diffusion models

In this talk we consider an underlying model where constant parameters are switching according to a continuous time Markov process. The times of switch are modeled using a Cox process. In addition the model features jumps. We examine the option pricing problem when the stock process follows this process and we find that a tightly coupled system of partially integro-differential equations needs to be solved. We exemplify the solution on several case studies. We also analyze two types of jump distributions the log double exponential due to Kou and a new distribution which we call a log normal mixture which seems to be useful in precisely modeling the jumps and distinguishing them from sampled variability.

May 04: Kasia Pawelek (University of Oakland, MI)

Mathematical Modeling of Virus Infections and Immune Responses

The first part of the talk is about mathematical models for the HIV infection. Such mathematical models have made considerable contributions to our understanding of HIV dynamics. Introducing time delays to HIV models usually brings challenges to both mathematical analysis of the models and comparison of model predictions with patient data. We incorporate two delays, one the time needed for infected cells to produce virions after viral entry and the other the time needed for the adaptive immune response to emerge to control viral replication, into an HIV-1 model. We begin model analysis with proving the local stability of the infection-free and infected steady states. By developing different Lyapunov functionals, we obtain conditions ensuring global stability of the steady states. We also fit the model including two delays to viral load data from 10 patients during primary HIV-1 infection and estimate parameter values.

The second part of the talk deals with mathematical models for the Influenza infection. The mechanisms underlying viral control during an uncomplicated influenza virus infection are not fully understood. We developed a mathematical model including both innate and adaptive immune responses to study the within-host dynamics of equine influenza virus infection in horses. By comparing modeling predictions with both interferon and viral kinetic data, we examined the relative roles of target cell availability, and innate and adaptive immune responses in controlling the virus. This study provides a quantitative understanding of the biological factors that can explain the viral and interferon kinetics during a typical influenza virus infection.

May 11: Kia Dalili (Stevens Institute of Technology, New Jersey)

Modeling network evolution

Networks constructed out of real world data often exhibit a number of properties not normally seen in random graphs. Amongst them are a tendency to have a modular structure and a small average shortest path length. We will introduce a model of network evolution using benefit-maximizing independent agents as nodes, and use it to explain how modularity emerges in complex networks and how the environment within which the agents interact controls the degree of modularity.

May 18: No meeting today

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Fall 2011

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Sept 02: Organizational meeting

Sept 09: No meeting today

Sept 16: No meeting today

Sept 23: No meeting today

Sept 30: No meeting today

Oct 07: No meeting today

Oct 14: Philippe G. LeFloch (Université Paris VI and CNRS)

Undercompressible shocks and moving phase boundaries

Regularization-sensitive wave patterns often arise in continuum physics, especially in complex fluid flows, which may contain undercompressive shock waves and moving phase boundaries. I will review here the theory of solutions to nonlinear hyperbolic systems of conservation laws, in the regime when small-scale effects like viscosity and capillarity drive the selection and dynamics of (nonclassical) shocks. The concept of a kinetic relation was introduced and provides the proper tool in order to characterize admissible shocks. The kinetic relation depends on higher-order terms that take additional physics into account. A general theory of the kinetic relation has been developed by the author and his collaborators, which covers various issues such as the Riemann problem, the Cauchy problem, the front tracking schemes, and several numerical strategies adapted to handle nonclassical shocks. Relevant papers are available at the link: philippelefloch.org.

Oct 21: Robert Numrich (College of Staten Island-CUNY)

Computer Performance Analysis and the PI Theorem of Dimensional Analysis

This talk applies the Pi Theorem of dimensional analysis to a representative set of examples from computer performance analysis. It takes a different look at problems involving latency, bandwidth, cache-miss ratios, and the efficiency of parallel numerical algorithms. The Pi Theorem is the fundamental tool of dimensional analysis, and it applies to problems in computer performance analysis just as well as it does to problems in other sciences. Applying it requires the definition of a system of measurement appropriate for computer performance analysis with a consistent set of units and dimensions. Then a straightforward recipe for each specific problem reduces the number of independent variables to a smaller number of dimensionless parameters. Two machines with the same values of these parameters are self-similar and behave the same way. Self-similarity relationships emphasize how machines are the same rather than how they are different. The Pi Theorem is simple to state and simple to prove, using purely algebraic methods, but the results that follow from it are often surprising and not simple at all. The results are often unexpected but they almost always reveal something new about the problem at hand.

Oct 28: No meeting today

Nov 04: No meeting today

Nov 11: Joab Winkler (The University of Sheffield, UK)

The computation of multiple roots of polynomials whose coefficients are inexact

This lecture will show by example some of the problems that occur when the roots of a polynomial are computed using a standard polynomial root solver. In particular, polynomials of high degree with a large number of multiple roots will be considered, and it will be shown that even roundoff error due to floating point arithmetic, in the absence of data errors, is sufficient to cause totally incorrect results. Since data errors are usually larger than roundoff errors (and fundamentally different in character), the errors encountered with real world data are significant and emphasise the need for a computationally robust polynomial root solver.

The inability of commonly used polynomial root solvers to compute high degree multiple roots correctly requires investigation of the cause of this failure. This leads naturally to a discussion of a structured condition number of a root of a polynomial, where structure refers to the form of the perturbations that are applied to the coefficients. It will be shown that this structured condition number, where the perturbations are such that the multiplicities of the roots are preserved, differs significantly from the standard condition numbers, which refer to random (unstructured) perturbations of the coefficients. Several examples will be given and it will be shown that the condition number of a multiple root of a polynomial due to a random perturbation in the coefficients is large, but the structured condition number of the same root is small. This large difference is typically several orders of magnitude.

A method developed by Gauss for computing the roots of a polynomial will be discussed. This method has an elegant geometric interpretation in terms of pejorative manifolds, which were introduced by William Kahan (Berkeley). The method is rarely used now, but it will be considered because it differs significantly from all other methods (Newton-Raphson, Bairstow, Laguerre, etc.) and is non-iterative. The computational implementation of this method raises, however, some non-trivial issues – the determination of the rank of a matrix in a floating point environment and the quotient of two inexact polynomials – and they will be discussed because they are ill-posed operations. They must be implemented with care because simple methods will necessarily lead to incorrect results. I will finish the talk by giving several non-trivial examples (polynomials of high degree, with several multiple roots of high degree, whose coefficients are corrupted by noise), and the results will be compared with other methods for the computation of multiple roots of polynomials whose coefficients are corrupted by noise.

Nov 18: No meeting today

Nov 25: No meeting today

Dec 02: Pam Cook (University of Delaware)

Complex (wormlike micellar) fluids: Shear banding and inertial effects

Pam Cook (University of Delaware) {with Lin Zhou, New York City College of Technology and Gareth McKinley, Massachusetts Institute of Technology)

Concentrated surfactants in solution, depending on the concentration, salinity and temperature, self-assemble into highly entangled wormy cylindrical micelles. In solution these "worms" entangle, exhibiting visco-elastic properties like polymer solutions. In addition to reptative and Roussian relaxation/disentanglement the worms break and reform and are thus known as "living" polymers. When sheared, as the applied shear rate increases experiments show that the steady state velocity profile across the gap of the shear cell transitions from a single shear-rate to a two banded profile. The transition to the two banded state is accompanied by a strong viscous thinning. The VCM (Vasquez, Cook, McKinley) model is a rheological equation of state capable of describing these fluids which specifically incorporates the rate-dependent breakage and reforming of the worms as well as non-local effects arising from coupling between the macroscopic stress in the deformed elastic network and the microstructure. The constitutive equations describe the evolution of the number density and stresses of two micellar species (a long species ‘A’ which breaks to form two worms of a shorter species ‘B’ which can then reform). The resulting system of coupled nonlinear partial differential equations includes conservation of mass, momentum, and the constitutive relations. Tracking of the spatio-temporal evolution of flow shows that this "simple" two-species description does exhibit many of the key features observed in the deformation-dependent nonlinear rheology of these wormlike micellar solutions. The model has been studied in detail under several flow conditions including elongational flow, pressure-driven channel flow, and in Large Amplitude Oscillatory Shear (LAOS)). In those studies the flow was assumed to be inertialess, so that boundary information travels throughout the sample at infinite speed.

In this talk, the predictions of the VCM model incorporating the effect of fluid inertia are presented as the flow evolves to steady state in a Couette cell following a controlled ramp in the shear rate. The presence of fluid inertia results in short time transient propagation of elastic shear waves (which damp and diffuse over longer time scales) between the boundaries and, as a result of the interaction of these shear waves with the spatio-temporal development of the shear-bands, the model predicts multiple shear banded states in steady shearing deformation over a wide range of parameter space. The dependence of the region of multiple banding on model parameters (elasticity, diffusivity, shear rate) and on initial conditions (ramp speed) is analyzed. Both three-band and four-band solutions are observed.

Dec 09: No meeting today

Dec 16: No meeting today