Text Box: TECHNICAL REPORTS of the Center for Applied Mathematics and Statistics   REPORT 0910-1:  Inference for Comparing Two Treatments Using Kernel Density Estimation   Sibabrata Banerjee Schering-Plough Research Institute, Kenilworth, NJ 07033   Sunil Dhar Department of Mathematical Sciences, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102   Farid Kianifard Biometrics, U.S. Clinical Development, Novartis Pharmaceuticals, East Hanover, NJ 07936   Abstract:   In randomized clinical trials, nonparametric approaches are considered when assumptions of a parametric approach are not reasonable. Nonparametric approaches have typically concentrated on hypothesis testing and, unlike parametric approaches, have not been amenable to providing measures of treatment efficacy. If X and Y denote the random variables representing the responses on two treatments A and B, respectively, P(Y>X) is an intuitive measure of efficacy. We consider point and interval estimation of P(Y>X) using kernel density estimation and bootstrapping. We illustrate this methodology on a data set, where comparison is made with point and interval estimates obtained by inverting the nonparametric Wilcoxon-Mann-Whitney test.    REPORT 0910-2:  Development of a Recursive Finite Difference Pharmacokinetic Model from an Exponential Model: Application to a Propofol Infusion   Glen Atlas Department of Anesthesiology, University of Medicine and Dentistry of New Jersey, Newark, NJ and Department of Chemistry, Chemical Biology and Biomedical Engineering, Stevens Institute of Technology, Hoboken, NJ   Sunil Dhar Department of Mathematical Sciences, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102   Abstract:   Pharmacokinetic models, using recursive finite difference equations (RFDEs), can be derived directly from traditional exponential models. This method has been successfully applied to propofol infusion data. Furthermore, this technique yields identical accuracy, on a subject-specific basis, as the exponential model from which each RFDE model was derived. Specifically, these infusion models are based upon an inhomogenous RFDE: P(k+3) = A·P(k+2) + B·P(k+1) + C·P(k) + R, where A, B, C, and R are non-zero constants and P represents plasma propofol levels for each kth unit of time. When applied to propofol infusions, RFDE modeling has advantages, over traditional exponential models, in that fewer coefficients are needed and patient-to-patient variation of these coefficients is reduced. However, initial conditions for RFDEs have to be specified. These characteristics, of RFDE modeling of propofol infusions, are similar to those for RFDE modeling of propofol boluses. Based on these findings, as well as those of our prior study, RFDE pharmacokinetic modeling can be applied to both infusion and bolus data of propofol. Further research, on the applications of RFDEs in pharmacokinetics, appears warranted.    REPORT 0910-3:  Homeomorphisms and Fredholm Theory for Perturbations of Nonlinear Fredholm Maps of Index Zero with Applications   P.S. Milojevic Department of Mathematical Sciences and CAMS, New Jersey Institute of Technology, Newark, NJ 07102   Abstract:   We develop a nonlinear Fredholm alternative theory involving k-ball and k-set perturbations of general homeomorphisms as well as of homeomorphisms that are nonlinear Fredholm maps of index zero. Various generalized first Fredholm theorems are given and finite solvability of general (odd) Fredholm maps of index zero is also studied. We apply these results to the unique and finite solvability of potential and semilinear problems with strongly nonlinear boundary conditions and to quasilinear elliptic equations. The basic tools used are the Nussbaum degree and the recent degree theories for nonlinear Fredholm maps of index zero and their perturbations.   REPORT 0910-4:  An Efficient Algorithm for the Evaluation of Certain Convolution Integrals with Singular Kernels   Shidong Jiang Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102   Abstract:   An efficient algorithm is developed for the evaluation of a broad class of convolution integrals with singular kernels.  The algorithm is then applied to study the Havriliak-Negami model for dielectric medias.    REPORT 0910-5:  A Hybrid Numerical Method for Interfacial Fluid Flow with Soluble Surfactant   M.R. Booty and M. Siegel Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA   Abstract:   We address a significant difficulty in the numerical computation of fluid interfaces with soluble surfactant that occurs in the physically representative limit of large bulk Peclet number Pe. At the high values of Pe in typical fluid-surfactant systems, there is a transition layer near the interface in which the surfactant concentration varies rapidly, and large gradients at the interface must be resolved accurately to evaluate the exchange of surfactant between the interface and bulk flow. We use the slenderness of the layer to develop a fast and accurate `hybrid' numerical method that incorporates a separate, singular perturbation analysis of the dynamics in the transition layer into a full numerical solution of the interfacial free boundary problem. The accuracy and efficiency of the method is assessed by comparison with a more `traditional' numerical approach that uses finite differences on a curvilinear coordinate system exterior to the bubble, without the separate transition layer reduction. The traditional method implemented here features a novel fast calculation of fluid velocity off the interface.    REPORT 0910-6:  Stability of Fronts and Transient Behavior in KPP Systems   Anna Ghazaryan Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045, USA; e-mail: aghazaryan@math.ku.edu   Peter Gordon Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA; e-mail: peterg@njit.edu; corresponding author   Alexander Virodov Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA; e-mail: av89@njit.edu   Abstract:   We consider a system of two reaction diffusion equations with the KPP type nonlinearity which describes propagation of pressure driven flames. It is known that the system admits a family of traveling wave solutions parameterized by their velocity. In this paper we show that these traveling fronts are stable under the assumption that perturbations belong to an appropriate weighted L2 space. We also discuss an interesting meta-stable pattern the system exhibits in certain cases. Key words: KPP systems, traveling fronts, stability, meta-stable regimes. AMS Subject Classifications: 35K57, 35B35, 35B40    REPORT 0910-7:  A Non-Stiff Boundary Integral Method for 3D Porous Media Flow with Surface Tension   D. M. Ambrose Department of Mathematics, Drexel University, Philadelphia, PA 19104   M. Siegel Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102   Abstract:   We present an effcient, non-stiff boundary integral method for 3D porous media flow with surface tension. Surface tension introduces high order (i.e., high derivative) terms into the evolution equations, and this leads to severe stability constraints for explicit time-integration methods. Furthermore, the high order terms appear in nonlocal operators, making the application of implicit methods difficult. Our method uses the fundamental coefficients of the surface as dynamical variables, and employs a special isothermal parameterization of the interface which enables efficient application of implicit or linear propagator time-integration methods via a small-scale decomposition.  The method is tested by computing the relaxation of an interface to a flat surface under the action of surface tension. These calculations employ an approximate interface velocity to test the stiffness reduction of the method. The approximate velocity has the same mathematical form as the exact velocity, but avoids the numerically intensive computation of the full Birkhoff-Rott integral. The algorithm is found to be effective at eliminating the severe time-step constraint that plagues explicit time-integration methods.     REPORT 0910-8:  Effects of Neuronal Morphology on the Passive Properties of Neurons   Krutanjali Shah, Amir Farzad Sheibanie, and Farzan Nadim New Jersey Institute of Technology, University Heights, Newark, NJ 07102   Abstract:   The passive properties of a neuron are defined as the electrical (resistive and capacitive) aspects of cell membrane of neuron when it is in passive state. The cell is in passive state when all the other channels but the leak channels are blocked or disabled. The neurons consist of very complex anatomical structure. The morphology of neurons (the level of complexity in anatomical structure) varies in every neuron. The variation in morphology produces essential changes in the passive properties of neurons as well. We examined the passive properties of Pyloric Dilator (PD) neurons in Stomatogastric Nervous System (STNS) of the crab Cancer borealis. We analyzed the change in passive properties by injecting the hyperpolarizing current into the soma of PD neurons. We characterized the input resistance of neurons and the time constant (τ) by studying the obtained voltage response of the neurons when the current (Iapp) was injected. In order to get the specifications about the morphology, we dye-filled the same neuron and imaged it using a confocal microscope. The boundaries of the neuron in the image were traced by utilizing the software Neurolucida and the specifications about the morphology of that specific neuron were attained using Neuroexplorer. We transported the morphology of indentified PD neuron in the NEURON software and obtained the voltage traces in response to  hyperpolarizing currents in order to compare the results with our experimental data. We built a simplified model neuron and compared its output with our experimental data as well. In this study, we came up with a conclusion that the morphology greatly affects the passive properties of neuron. If we do not take the fine details of the morphology into account, the response of the model is very different than the experimental response; however, this error can be reduced by adding additional surface area that would account for the ignored fine details of the neuron.    REPORT 0910-9:  The Effect of Morphology on the Passive Properties of Neurons   Yamin Noor, Amir Farzad Sheibanie, and Farzan Nadim New Jersey Institute of Technology, University Heights, Newark, NJ 07102   Abstract:   Passive properties of neuron refer to the inherent capacitive and resistive characteristics of the cell membrane and the cytoplasm of neuron. As the morphology of neuron varies drastically from one to another, the passive properties also vary. We studied the passive properties of Pyloric Dilator (PD) neurons of Cancer borealis. We injected negative current pulses (Iapp) into the soma of identified PD neurons and recorded the resulting voltage response. We then filled the same neuron with a fluorescent dye and imaged its anatomical structure using a confocal microscope. Using the software Neurolucida, we traced the 3D image of the neuron and then analyzed the tracing and obtained a detailed morphology with the software Neuroexplorer.  Based on this detailed morphology, a detailed model neuron was developed.  We determined the passive membrane properties of the PD neuron by comparing the experimental with the model voltage response for three different negative currents injected at the soma of the neuron in the Neuron simulation envrionment. We then used the cable equation to perform a spatial analysis of the compartments to determine the voltage along the length of each compartment at the steady state. We found that if an extension was added to the model, its response to current pulses better fit the experimental measurements. Such an extension may account for the fine morphological details that were neglected during tracing. We also reduce the complicated morphological model into a simplified model neuron based on the passive membrane properties.    REPORT 0910-10:  A PRC Description of How Inhibitory Feedback Promotes Oscillation Stability   Farzan Nadim Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102   Shunbing Zhao Federated Department of Biological Sciences, Rutgers University and New Jersey Institute of Technology, Newark, NJ 07102   Amitabha Bose Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102   Abstract:   Using methods of geometric dynamical systems modeling, we demonstrate the mechanism through which inhibitory feedback synapses to oscillatory neurons stabilize the oscillation, resulting in a flattened phase-resetting curve. In particular, we use the concept of a synaptic phase-resetting curve to demonstrate that periodic inhibitory feedback to an oscillatory neuron locks at a stable phase where it has no impact on cycle period and yet it acts to counter the effects of extrinsic perturbations. These results are supported by data from the stable bursting oscillations in the crustacean pyloric central pattern generator.    REPORT 0910-11:  Modeling with Bivariate Geometric Distributions   Jing Li and Sunil Dhar Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey, USA   Abstract:   We study systems with several components which are subject to different types of failures. Examples of such systems include twin engines of an airplane or the paired organs in a human body. We find that such a system, using conditional arguments, can be characterized as multivariate geometric distributions. We prove that the characterizations of the geometric model can be achieved using conditional probabilities, conditional failure rates, or probability generating function. These new models are fitted to real-life data using the Method of Moments Estimators, Maximum Likelihood Estimators, and Bayes Estimators. The last two estimators are obtained by solving score equations. We also compare two Methods of Moments Estimators in each of the several bivariate geometric models to evaluate their performance using the bias vector and variance-covariance matrix. This comparison is done through a Monte-Carlo simulation for increasing sample sizes. The Chi-square goodness-of-fit tests are used to evaluate model performance.    REPORT 0910-12:  Network Frequency can be Predicted from the Preferred Frequency of Pacemaker Neurons in Response to Input Waveforms   Hua-an Tseng Federated Department of Biological Sciences, Rutgers University and New Jersey Institute of Technology, Newark, NJ 07102   Farzan Nadim Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102   Abstract:   The frequency of an oscillating network is often crucial for its function. Many oscillatory networks involve neurons with some intrinsic rhythmicity, but such neurons typically possess a large variety of voltage-gated currents which interact in a complex fashion making it difficult to determine which factors control frequency. Yet, these neurons often exhibit preferred (resonance) frequencies that can be close to the network frequency. We examine the hypothesis that parameters that shift the preferred frequency of constituent neurons affect network frequency in the same way. The frequency of the crustacean pyloric network is correlated with the preferred frequency of its bursting pacemaker neurons AB and PD. We measure the preferred frequency of the PD neuron under voltage-clamp conditions, which allows us to have control of the oscillation voltage range and waveforms and show that 1) the preferred frequency increases when the upper or lower limit of the oscillating voltage waveform is increased;  2) the slope of the waveform near its peak has a strongly negative correlation with the preferred frequency; and 3) correlations between waveform parameters and preferred frequency can be used to predict shifts in the network frequency when the PD waveform changes. Changing the upper or lower limits of ongoing PD neuron oscillations using dynamic clamp shifts the network frequency in a manner consistent with the predicted changes in the PD neuron preferred frequency. These results demonstrate that an oscillatory neuron’s burst waveform, which can be recorded readily, can be used to predict shifts in the network frequency.    REPORT 0910-13:   Self-Organized Density Relaxation by Tapping   Anthony D. Rosato, Oleksandr Dybenko, Vishagan Ratnaswamy Granular Science Laboratory, Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102   David J. Horntrop and Lou Kondic Department of Mathematical Sciences and the Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102   Abstract:   The density relaxation phenomenon is modeled using both Monte Carlo and discrete element simulations to investigate the effects of regular taps applied to a vessel having a planar floor filled with monodisperse spheres.  Results suggest the existence of a critical tap intensity which produces a maximum bulk solids fraction. We find that the mechanism responsible for the relaxation phenomenon is evolving quasi-ordered packing structure propagating upwards from the plane floor.   REPORT 0910-14:   Thin films Flowing Down Inverted Substrates:  Two Dimensional Flow   Te-Sheng Lin and Lou Kondic Department of Mathematical Sciences and the Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102   Abstract:   We consider free surface instabilities of films flowing on inverted substrates within the framework of lubrication approximation.  We allow for the presence of fronts and related contact lines, and explore the role which they play in instability development. It is found that a contact line, modeled by a commonly used precursor film model, leads to free surface instabilities without any additional natural or excited perturbations.  A single parameter  is identified as a governing parameter in the problem.  This parameter may be interpreted to reflect the combined effect of inclination angle, film thickness, Reynolds number and the fluid flux.  Variation of this parameter leads to change of the wave-like properties of the instabilities, allowing us to observe traveling wave behavior, mixed waves, and the waves resembling solitary ones.    REPORT 0910-15:  On the Breakup of Patterned Nanoscale Copper Rings into Nanoparticles: Competing Instability and Transport Mechanisms   Yueying Wu, Jason D. Fowlkes, Philip D. Rack Department of Materials Science and Engineering, The University of Tennessee, Knoxville, TN 37996 The Oak Ridge National Laboratory, Oak Ridge, TN 37831   Javier A. Diez Instituto de Fisica Arroyo Seco, Universidad Nacional del Centro de la Provincia de Buenos Aires, Pinto 399, 7000, Tandil, Argentina   Lou Kondic Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102   Abstract:   Nanolithographically patterned copper rings were synthesized and the self assembly of the rings into ordered nanoparticle arrays was accomplished via nanosecond pulsed laser heating above the melt threshold. The resultant length scale was correlated to the transport and instability growth that occurs during the liquid lifetime of the melted copper rings. For 13 nm thick rings, a change in the nanoparticle spacing with the ring width is attributed to a transition from a Raleigh-Plateau instability to a spinodal thin film instability due to competition between the cumulative transport and instability time scales. To further explore this competition between instability mechanisms, we carried out experiments with 7 nm thick rings.   In agreement with the theoretical predictions, these rings break up in both azimuthal and radial direction, confirming that a simple hydrodynamic model captures the main features of the processes responsible for instabilities.   REPORT 0910-16:  Homeomorphisms and Fredholm Theory for Perturbations of Nonlinear Fredholm Maps of Index Zero and of A-Proper Maps with Applications   P. S. Milojević Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102   Abstract:   1. Summary. In Part I, we develop a nonlinear Fredholm alternative theory involving k-ball and k-set perturbations of general homeomorphisms as well as of homeomorphisms that are nonlinear Fredholm maps of index zero. Various generalized first Fredholm theorems are given and finite solvability of general (odd) Fredholm maps of index zero is also studied. We apply these results to the unique and finite solvability of potential and semilinear problems with strongly nonlinear boundary conditions and to quasilinear elliptic equations. The results of Section 2 are based on the Browder and Banach-Mazur homeomorphism theorems. The basic tools used in section 3 are the recent degree theories for nonlinear C1 Fredholm maps of index zero and their perturbations as defined by Fitzpatrick, Pejsachowicz-Rabier ([13],[22]), Benevieri-Furi [2,3], Rabier-Salter [24] and Benevieri-Calamai-Furi [4]. The results discussed in this part are based on Milojević [31,32].    REPORT 0910-17:  Optimal Costs of a Two-dimensional Warranty Servicing Strategy with an Imperfect Repair Option   Rudrani Banerjee and Manish C. Bhattacharjee Department of Mathematical Sciences, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA.   Abstract:   A warranty policy for a product should balance the interests of both producer and consumer. Consumer protection is typically provided by a guarantee of replacement or some form of repair of the product failing  within a promised warranty period, while an approach to provide a corresponding protection for the manufacturer is to limit  the maximum usage allowed under warranty.  Such warranty policies are two-dimensional, and the warranty expires at the end of the promised warranty period or upon reaching the maximum usage allowed, whichever occurs sooner.  From a manufacturer's point of view, reducing warranty costs is an issue of great interest. In this paper, we look at two different servicing strategies for a two-dimensional warranty scheme involving minimal and imperfect repairs. Our work demonstrates the modeling and analysis of costs under these servicing strategies and compare their performance to other strategies that have been investigated in the literature.   REPORT 0910-18:   On the Complex Dynamics of a Red Blood Cell in Simple Shear Flow   Petia M. Vlahovska (Thayer School of Engineering, Dartmouth College, 8000 Cummings Hall, Hanover, NH 03755) Yuan-Nan Young (Dept of Mathematical Sciences, NJIT, Newark, NJ, 07102) Gerrit Danker and Chaoqi Misbah (Laboratoire de Spectrometrie Physique, UMR, 140 avenue de la physique, Universite Joseph Fourier and CNRS, 38402 Saint Martin d'Heres, France) Abstract: Motivated by the reported peculiar dynamics of a red blood cell in shear flow, we develop an analytical theory for the motion of a nearly-spherical fluid particle enclosed by a visco-elastic incompressible interface in linear flows.  The analysis explains the effect of particle deformability on the transition from tumbling to swinging as shear rate increases.  Near the transition, intermittent behavior is predicted only if the particle has a fixed shape; the intermittency disappears for a deformable particle. Comparison with available phenomenological models based on the fixed shape assumption highlights their physical foundations and limitations.   REPORT 0910-19:   Dynamics of a Semi-flexible Polar Filament in Stokes Flow   Yuan-Nan Young (Dept. of Mathematical Sciences, NJIT, Newark, NJ, 07102)     Abstract: In this work the dynamics and transport of a polarly driven filament is examined using a continuum slender-body model.  Immersed in a viscous fluid, the filament gains polar propulsion from the motor proteins (anchored on the motility assay) while experiencing a viscous drag from the bottom wall.  Results from the linear analysis on a straight polar filament illustrate the necessity of spatial inhomogeneity in the polar forcing for the buckling instability.  The ensuing buckling leads to filament deformation, undulation, and change of its direction of motion in the numerical simulations.  Repeated filament buckling in two types of motor protein concentration landscape results in diffusive transport of a polar filament on scales much larger than the mean-free-path and the average duration between filament buckling events.   REPORT 0910-20:     Objective Method for Determining the Most Valuable Player in Major League Baseball Kevin Fritz Hillsborough, NJ 08844 Bruce Bukiet Department of Mathematical Sciences Center for Applied Mathematics and Statistics New Jersey Institute of Technology Newark, NJ 07012   Abstract:   The sportswriters who select the Most Valuable Player (MVP) and Cy Young award winners for Major League Baseball do not use any mandated criteria to make their selections and as a result, many observers feel they do not select the most appropriate players.  In this paper, we introduce an objective criterion for selecting the MVP and Cy Young award winners.  We extended the Markov Chain model developed by Bukiet et al. (1997) to include more realistic aspects of baseball, improving its authenticity.  We used the model to analyze the various candidates for the two awards over a period of 20 years to determine the players whose performance would have added the greatest number of expected wins to an average team and call these players the “objective winners” of the awards.  We found that the sportswriters’ selections matched the objective criterion just under half the time and that the sportswriters selected one of the top three performers nearly 70% of the time.