TECHNICAL REPORTS of the

Center for Applied Mathematics and Statistics

REPORT 1415-1: An M^X/G/1 Queueing System with Disasters and Repairs under a Multiple Adapted Vacation Policy

George C. Mytalas - Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102

Michael A. Zazanis - Department of Statistics, Athens University of Economics and Business, Athens 10434, Greece

Abstract: We consider a queueing system with batch Poisson arrivals subject to disasters which occur only when the server is busy and clear the system. Following a disaster the server initiates a repair period during which arriving customers accumulate without receiving service. The server operates under a Multiple Adapted Vacation policy. We analyze this system using the supplementary variables technique and obtain the probability generating function of the number of customers in the system in stationarity the fraction of customers who complete service, and the Laplace transform of the system time of a typical customer in stationarity. Finally, we examine a variation of the model in which the system is subject to disasters even when the server is taking a vacation or is under repair.

REPORT 1415-2: Dynamics of Vortex Dipoles in Anisotropic Bose-Einstein Condensates

Roy H. Goodman - Department of Mathematical Sciences, New Jersey Institute of Technology

Panayotis G. Kevrekidis - Department of Mathematics and Statistics, University of Massachusetts and Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory

R. Carretero-Gonzalez - Nonlinear Dynamical System Group, Computational Science Research Center, and Department of Mathematics and Statistics, San Diego State University

Abstract: We study the motion of a vortex dipole in a Bose-Einstein condensate confined to an anisotropic trap. We focus on a system of ordinary differential equations describing the vortices' motion, which is in turn a reduced model of the Gross-Pitaevskii equation describing the condensate's motion. Using a sequence of canonical changes of variables, we reduce the dimension and simplify the equations of motion. We uncover two interesting regimes. Near a family of periodic orbits known as guiding centers, we find that the dynamics is essentially that of a pendulum coupled to a linear oscillator, leading to stochastic reversals in the overall direction of rotation of the dipole. Near the separatrix orbit in the isotropic system, we find other families of periodic, quasi-periodic, and chaotic trajectories. In a neighborhood of the guiding center orbits, we derive an explicit iterated map that simplifies the problem further. Numerical calculations are used to illustrate the phenomena discovered through the analysis. Using the results from the reduced system we are able to construct complex periodic orbits in the original, partial differential equation, mean-field model for Bose-Einstein condensates, which corroborates the phenomenology observed in the reduced dynamical equations.

REPORT 1415-3: Convergence of a Boundary Integral Method for 3D Interfacial Darcy Flow with Surface Tension

Michael Siegel, Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102

David M. Ambrose and Yang Liu, Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104

Abstract: We study convergence of a boundary integral method for 3D interfacial flow with surface tension when the fluid velocity is given by Darcy’s Law. The method is closely related to a previous method developed and implemented by Ambrose, Siegel, and Tlupova, in which one of the main ideas is the use of an isothermal parameterization of the free surface. We prove convergence by proving consistency and stability, and the main challenge is to demonstrate energy estimates for the growth of errors. These estimates follow the general lines of estimates for continuous problems made by Ambrose and Masmoudi, in which there are good estimates available for the curvature of the free surface. To use this framework, we consider the curvature and the position of the free surface each to be evolving, rather than attempting to determine one of these from the other. We introduce a novel substitution which allows the needed estimates to close.

REPORT 1415-4: A Hybrid Numerical Method for Interfacial Flow with Soluble Surfactant and its Application to an Experiment in Microfluidic Tipstreaming

Michael R. Booty and Michael Siegel, Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102

Shelley L. Anna, Center for Complex Fluids Engineering and Departments of Chemical Engineering and Mechanical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

Abstract: This review article describes a hybrid numerical method for solving problems of two-phase flow with soluble surfactant in the limit of large bulk Peclet number. It summarizes recent work on development of the method and aspects of its application to canonical examples of the deformation of an isolated drop by an imposed flow. Experiments with a microfluidic flow focusing device that is designed to produce a monodisperse stream of micron-size droplets by surfactant-mediated tipstreaming are also described. Preliminary predictions of the numerical method on conditions for which tipstreaming can occur are compared with results from the experiments.

REPORT 1415-5: Semiparametric Simultaneous Confidence Bands for the Difference of Survival Functions Using Empirical Likelihood

Nubyra Ahmed - Center for Applied Mathematics and Statistics, Department of Mathematical Sciences, New Jersey Institute of Technology, USA

Sundar Subramanian - Center for Applied Mathematics and Statistics, Department of Mathematical Sciences, New Jersey Institute of Technology, USA

Abstract: In the analysis of censored survival data, simultaneous confidence bands are useful devices to help determine the efficacy of a treatment over a control. Semiparametric confidence bands are developed for the difference of two survival curves using empirical likelihood and compared with the nonparametric counterpart. Simulation studies are presented to show that the proposed semiparametric approach is superior, with the new confidence bands giving empirical coverage closer to the nominal level. Further comparisons reveal that the semiparametric confidence bands are tighter and, hence, more informative. For censoring rates between 10% and 40%, the semiparametric confidence bands provide a relative reduction in enclosed area amounting to between 2% and 7% over their nonparametric bands, with increased reduction attained for higher censoring rates. The methods are illustrated using an University of Massachusetts AIDS data set.

REPORT 1415-6: Single-Index Model for Inhomogeneous Spatial Point Processes

Yixin Fang - New York University

Ji Meng Loh - Center for Applied Mathematics and Statistics, Department of Mathematical Sciences, New Jersey Institute of Technology, USA

Abstract: We introduce a single index model for the intensity of an inhomogeneous spatial point process, relating the intensity function to an unknown function rho of a linear combination of measurements of a p-dimensional spatial covariate process. Such a model extends and generalizes a commonly used model where rho is known. We derive an estimating procedure for rho and the coefficient parameters beta and show consistency and asymptotic normality of estimates of beta under some regularity assumptions. We present results of some simulation studies showing the effectiveness of the procedure. Finally, we apply the procedure to a dataset of fast food restaurant locations in New York City.

REPORT 1415-7: Bandwidth Selection for Estimating the Two-Point Correlation Function of a Spatial Point Pattern Using AMSE

Ji Meng Loh - Center for Applied Mathematics and Statistics, Department of Mathematical Sciences, New Jersey Institute of Technology, USA

Abstract: We introduce an asymptotic mean squared error (AMSE) approach to obtain closed-form expressions of the adaptive optimal bandwidths for estimating the two-point correlation function of a homogeneous spatial point pattern. The two-point correlation function is one of several second-order measures of clustering of a point pattern and is commonly used in astronomy where cosmologists have found a relationship between the clustering of galaxies and the evolution of the universe. The AMSE approach is adapted from an AMISE approach that is well-known in density estimation and our approach provides a simple and quick method for optimal bandwidth selection for estimating the two-point correlation function. Using optimal bandwidths for estimation will allow more information about clustering to be extracted from the data. Results from numerical studies suggest that the mean squared error of estimates obtained using AMSE optimal bandwidths is competitive with those obtained using more computationally intensive methods and is close to the empirical optimal bandwidths. We illustrate its use in an application to a galaxy cluster catalog from the Sloan Digital Sky Survey.

REPORT 1415-8: Fast Food and Liquor Store Density, Co-Tenancy, and Turnover: Vice Store Operations in Chicago, 1995-2008

Ji Meng Loh - Center for Applied Mathematics and Statistics, Department of Mathematical Sciences, New Jersey Institute of Technology, USA

Abstract: Fast food restaurants and liquor stores—vice stores—have been shown to be more prevalent in predominantly Black and low income U.S. neighborhoods, and are associated with a number of health risks and social ills. The purpose of this study was to investigate the association between vice store density and spatial distribution as a function of racial, socioeconomic, and other population characteristics; to examine spatial clustering among these outlets; and to study how store turnover follows population change over a 13-year period in Chicago. We found heterogeneous associations between stores and population characteristics, with the most consistent finding being a positive association between percent Black and liquor store exposure. A high degree of spatial clustering was evident, and liquor stores were more likely to stay in business over time than fast food restaurants. However, when liquor stores closed, they were more likely to be replaced by non-vice businesses. Results suggest that vice stores are associated with lower positions in racial and socioeconomic hierarchies, and this patterning is often durable over time.

REPORT 1415-9: Mathematical Model of Cardiovascular and Metabolic Responses to Umbilical Cord Occlusions in Fetal Sheep

Qiming Wang†, Nathan Gold†, Martin G. Frasch‡,
Huaxiong Huang†, Marc Thiriet*, and Steven Wang†

Affiliations:
†Department of Mathematics and Statistics, York University, Ontario, Canada, M3J 1P3
‡Department of Obstetrics and Gynecology and Department of Neurosciences, Faculty of Medicine at CHU Sainte-Justine Research Center and Centre de Recherche en Reproduction Animale (CRRA), Faculty of Veterinary Medicine, Universit ´e de Montr ´eal, Quebec, Canada, H3T 1C5
*Sorbonne University, UPMC, Laboratoire Jacques-Louis Lions,CNRS, UMR 7598, INRIA, EPI REO, 75252, Paris, France

Abstract: Fetal acidemia during labour is associated with an increased risk for brain injury and lasting neurological deficits. This is in part due to the repetitive occlusions of the umbilical cord (UCO) induced by the uterine contractions. While fetal heart rate (FHR) monitoring is widely used clinically, it fails to detect fetal acidemia. Hence, new approaches are needed for early detection of fetal acidemia during labour. We built a mathematical model of the UCO effects on FHR, mean arterial blood pressure (MABP), oxygenation and the metabolism. Mimicking fetal experiments, our in silico model reproduces salient features of experimentally observed fetal cardiovascular and metabolic behavior including FHR overshoot, gradual MABP decrease and mixed metabolic and respiratory acidemia during UCO. Combined with statistical analysis, our model provides valuable insight of the labour-like fetal distress and guidance for refining FHR monitoring algorithms to improve detection of fetal acidemia and cardiovascular decompensation.

REPORT 1415-10: Debye Potentials for the Time Dependent Maxwell Equations

Leslie Greengard Courant Institute of Mathematical Sciences, New York University, New York, NY 10012.

Thomas Hagstrom Department of Mathematics, Southern Methodist University, PO Box 750156, Dallas, TX 75275.

Abstract: The explicit solution to the scattering problem of time dependent Maxwell equations on a sphere is derived. The derivation of the explicit solution is based on a generalization of Debye potentials for the time harmonic case and reduces the problem to two scalar wave problems - one with the Dirichlet condition and the other with the Robin condition. A high-order and stable numerical scheme is constructed to evaluate the solution at an arbitrary point r outside the unit sphere at any time t>0, without marching in the whole space-time domain. Several numerical examples are presented to illustrate the performance of the algorithm. The solution can be served as a reference for checking the accuracy of other numerical methods. More importantly, it will provide some insight towards better integral equation formulations for the wave equation and time-dependent Maxwell equations in a general domain.

REPORT 1415-11: Computing the Ground State and Dynamics of the Nonlinear Schrodinger Equation with Nonlocal Interactions via the Nonuniform FFT

Weizhu Bao
Department of Mathematics, National University of Singapore, Singapore 119076, Singapore

Shidong Jiang
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey, 07102, USA

Qinglin Tang
Universite de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-les-Nancy, F-54506, France Inria Nancy Grand-Est/IECL-CORIDA, France

Yong Zhang
Wolfgang Pauli Institute c/o Fak. Mathematik, University Wien, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

Abstract: We present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear Schrodinger equation (NLSE) with nonlocal interactions based on a fast and accurate evaluation of the long-range interactions via the nonuniform fast Fourier transform (NUFFT). We begin with a review of the fast and accurate method, proposed recently by two of the authors, which evaluates the Coulomb interaction in three (3D) and two (2D) dimensions via the NUFFT. Then we extend the method to compute other nonlocal interactions which do not decay at far field, such as the interaction whose kernel is the Green's function of the Laplace operator in 2D and 1D. We compare numerically the performance of this method and other existing methods with particular attention on the effect of the size of the truncated bounded computational domain. For computing the ground state and dynamics, we propose efficient and accurate numerical methods based on the normalized gradient flow with backward Euler Fourier pseudospectral discretization and time-splitting Fourier pseudospectral method, respectively, together with the fast and accurate NUFFT method for evaluating the nonlocal interactions. Extensive numerical comparisons are carried out between these methods and other existing methods for computing the ground state and dynamics of the NLSE with nonlocal interactions. Numerical results show that the methods via the NUFFT perform much better than those existing methods in terms of accuracy and/or efficiency for computing the ground state and dynamics as well as for evaluating the nonlocal interactions, especially when the bounded computational domain is chosen smaller.

REPORT 1415-12: On the Integral Equation Derived from the Linearized BGK Equation for the Steady Couette Flow

Shidong Jiang
Department of Mathematics Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA

Abstract: The integral equation for flow velocity u(x,k) derived from the linearized Bhatnagar-Gross-Krook kinetic equation for the steady Couette flow is studied in detail both theoretically and numerically in a wide range of the Knudsen number k. First, it is shown that the integral equation is a Fredholm equation of the second kind in which the norm of the compact integral operator is less than 1 and thus there exists a unique solution to the integral equation via the Neumann series. Second, it is shown that the solution is logarithmically singular at the end points. And third, a high-order adaptive numerical algorithm is designed to compute the solution numerically to high precision. The solutions for the flow velocity, the stress, and the half-channel total mass flow rate are obtained in a wide range of the Knudsen number 0.003 <= k <= 10.0; and these solutions are accurate for at least twelve significant digits or better, thus they can be used as benchmark solutions.

Aminur Rahman, Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102
Denis Blackmore, Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102

Abstract: Bouncing droplets on a vibrating fluid bath can exhibit wave-particle behavior, such as being propelled by the waves they generate. These droplets seem to walk across the bath, and thus are dubbed walkers. These walkers can exhibit exotic dynamical behavior which give strong indications of chaos, but many of the interesting dynamical properties have yet to be proven. In recent years discrete dynamical models have been derived and studied numerically. We prove the existence of a Neimark--Sacker bifurcation for a variety of eigenmode shapes of the Faraday wave field from one such model. Then we reproduce numerical simulations and produce new numerical simulations and apply our theorem to the test functions used for that model in addition to new test functions. Evidence of chaos is shown by numerically studying a global bifurcation.

Abstract: Logical RS flip-flop circuits are investigated once again in the context of discrete planar dynamical systems, but this time starting with simple bilinear (minimal) component models based on fundamental principles. The dynamics of the minimal model is described in detail, and shown to exhibit some of the expected properties, but not the chaotic regimes typically found in simulations of physical realizations of chaotic RS flip-flop circuits. Any physical realization of a chaotic logical circuit must necessarily involve small perturbations - usually with quite large or even nonexisting derivatives - and possibly some symmetry-breaking. Therefore, perturbed forms of the minimal model are also analyzed in considerable detail. It is proved that perturbed minimal models can exhibit chaotic regimes, sometimes associated with chaotic strange attractors, as well as some of the bifurcation features present in several more elaborate and less fundamentally grounded dynamical models that have been investigated in the recent literature. Validation of the approach developed is provided by some comparisons with (mainly simulated) dynamical results obtained from more traditional investigations.

REPORT 1415-15: On the Dimension of the Set of (Positive) Solutions of Nonlinear Equations with Applications

P. S. Milojevic, Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102

Abstract: We study the covering dimension of the set of (positive ) so- lutions to varoius classes of nonlinear equations involving condensing and A- proper maps. It is based on the nontriviality of the fixed point index of a certain condensing map or on oddness of a nonlinear map. Applications to non- linear singular integral equations and to semilinear ordinary and elliptic partial differential equations are given with finite or infinite dimensional null space of the linear part.

REPORT 1415-16: Bootstrap Likelihood Ratio Confidence Bands for Survival Functions Under Random Censorship and its Semiparametric Extension

Sundar Subramanian - Center for Applied Mathematics and Statistics, Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA

Abstract: Simultaneous confidence bands (SCBs) for survival functions, from randomly right censored data, can be computed by inverting likelihood ratio functions based on appropriate thresholds. Sometimes, however, the requisite asymptotic distributions are intractable, or thresholds based on Brownian bridge approximations are not easy to obtain when SCBs over only sub-regions are possible or desired. We obtain the thresholds by bootstrapping (i) a nonparametric likelihood ratio function via censored data bootstrap and (ii) a semiparametric adjusted likelihood ratio function via a two-stage bootstrap that utilizes a model for the second stage. These two scenarios are grounded respectively in standard random censorship and its semiparametric extension introduced by Dikta. The two bootstraps, which are different in the way resampling is done, are shown to have asymptotic validity. The respective SCBs are neighborhoods of the well-known Kaplan--Meier estimator and the more recently developed Dikta's semiparametric counterpart. As evidenced by a validation study, both types of SCBs provide approximately correct coverage. The model-based SCBs, however, are tighter than the nonparametric ones. Two sensitivity studies reveal that the model-based method performs well when standard binary regression models are fitted, indicating its robustness to misspecification as well as its practical applicability. An illustration is given using real data.