## Center for Applied Mathematics and Statistics

REPORT 1112-1:     Semi-analytic Solutions for Elastic Capsules in 2D Flow

Michael Higley, Michael Siegel, and Michael R. Booty

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

Abstract:  Elastic capsules occur in nature in the form of cells and vesicles and are manufactured for biomedical applications. They are widely modeled but there are few analytical results. In this paper, complex variable techniques are used to derive semi-analytic solutions for the steady-state response and time-dependent evolution of elastic capsules in 2D Stokes flow. This provides a complete picture of the steady response of initially circular capsules in canonical linear strain and shear flows as a function of the capillary number Q. The analysis is complemented by spectrally accurate numerical simulations of the time-dependent evolution. An imposed nonlinear strain that models the far-field velocity in Taylor’s four roller mill is found to lead to cusped steady shapes at a critical capillary number Qc. Numerical simulation of the time-dependent evolution for Q > Qc shows the development of finite-time cusp singularities. The dynamics immediately prior to cusp formation are asymptotically self-similar, and the similarity exponents are predicted analytically and confirmed numerically. This appears to be the first example of finite-time singularity formation in fluid flow with elastic interfaces.

REPORT 1112-2:     A Teaching Experience in the Medical Department: From Concepts to Computation

Sunil K. Dhar

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

Abstract: This work is based on the experience of teaching a biostatistics course to a group of medical professionals, including doctors, lab technicians, and medical students, with a minimal or rusty statistics background. The main goal of the course was to teach how to communicate in the language of statistics so that scientists can carry out their measurements, collect data according to a designed experiment, choose appropriate statistical analytic tools, and become proficient in statistical computation and interpretation of results. The topics that were covered and the computational resources that were used are discussed, with examples. The research is used to explore effective teaching tools and topics that are best suited for teaching statistics to those involved with the medical professions.

REPORT 1112-3:     Modeling Photon Generation

Shuchi Agrawal, New Jersey Institute of Technology
David A. Edwards, University of Delaware
Joseph D. Fehribach, Worcester Polytechnic Institute
John Gounley, Old Dominion University
Isaac Harris, University of Delaware
Richard Moore, New Jersey Institute of Technology
Takeshi Takahashi, University of Massachusetts
Jacek Wrobel, New Jersey Institute of Technology

Abstract:  This report explores the relationship between the linear approximate form of the coupled-mode equations modeling frequency conversion in optical parametric devices, and the corresponding input-output equations. In particular, the spectral decomposition of the linear operator in the coupled-mode equations has implications for the singular value decomposition of the input-output equations. This relationship is explored in general to the extent that linear algebraic tools will allow; specific cases of physical interest are explored in more depth.

REPORT 1112-4:     Canard-like Explosion of Limit Cycles in Two-dimensional Piecewise-linear Models of FitzHugh-Nagumo Type

Horacio G. Rotstein
1, Stephen Coombes 2, Ana Maria Gheorghe 2

1 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
2 School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK

Abstract: We investigate the mechanism of abrupt transition between small and large amplitude oscillations in fast-slow piecewise-linear (PWL) models of
FitzHugh-Nagumo (FHN) type. In the context of neuroscience, these oscillatory regimes correspond to subthreshold oscillations and action potentials (spikes) respectively. The minimal model that shows such phenomenon has a cubic-like nullcline (for the fast equation) with two or more linear pieces in the middle branch and one piece in the left and right branches. Simpler models with only one linear piece in the middle branch or a discontinuity between the left and right branches (McKean model) show a single oscillatory mode. As the number of linear pieces increases, PWL models of FHN type approach smooth FHN-type models. For the minimal model, we investigate the bifurcation structure, we describe the mechanism that leads to the abrupt, canard-like transition between subthreshold oscillations and spikes, and we provide an analytical way of predicting the amplitude regime of a given limit cycle trajectory which includes the approximation of the canard critical control parameter. We extend our results to models with a larger number of linear pieces. Our results for PWL-FHN type models are consistent with similar results for smooth FHN type models. In addition, we develop tools that are amenable for the investigation of avariety of related, and more complex, problems including forced, stochastic and coupled oscillators.

REPORT 1112-5:     Simple Modeling of Bistable Nematic Liquid Crystal Display Devices: What is the optimal design?

L.J. Cummings, C. Cai and L. Kondic

Department of Mathematical Sciences, New Jersey Institute of Technology

Abstract:  Bistable Liquid Crystal Displays (LCDs) offer the potential for considerable power savings, compared with conventional (monostable) LCDs. The existence of two stable field-free states that are optically-distinct means that contrast can be maintained in a display without an externally-applied electric field. An applied field is required only to switch the device from one state to the other, as needed. In this paper we examine a theoretical model of a possible bistable device, originally proposed by Cummings \& Richardson \cite{CR}, and explore means by which it may be optimized, in terms of optical contrast, manufacturing considerations, switching field strength and switching times. The compromises inherent in these conflicting design criteria are discussed.

REPORT 1112-6:     Influence of Electric Field Gradient on a Stretched Nematic Sheet

L.J. Cummings

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark NJ 07102-1982

J. Low & T.G. Myers

Centre de Recerca Matematica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona, Spain

Abstract:  Systematic asymptotic methods are used to formulate a model for the extensional flow of a thin sheet of nematic liquid crystal. With no external body forces applied, the model is found to be equivalent to the so-called Trouton model for Newtonian sheets (and fibres), albeit with a modified Trouton ratio". However, with a symmetry-breaking electric field gradient applied, behavior deviates from the Newtonian case, and the sheet can undergo finite-time breakup if a suitable destabilizing field is applied. Some simple exact solutions are presented to illustrate the results in certain idealized limits, as well as sample numerical results to the full model equations.

REPORT 1112-7:    Model-based Likelihood Ratio Confidence Intervals for Survival Functions

Sundar Subramanian

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

Abstract: We introduce an adjusted likelihood ratio procedure for computing pointwise confidence intervals for survival functions from censored data. The test statistic, scaled by a ratio of two variance quantities, is shown to converge to a chi-squared distribution with one degree of freedom. The confidence intervals are seen to be a neighborhood of a semiparametric survival function estimator and are shown to have correct empirical coverage. Numerical studies also indicate that the proposed intervals have smaller estimated mean lengths in comparison to the ones that are produced as a neighborhood of the Kaplan--Meier estimator. We illustrate our method using a lung cancer data set.

REPORT 1112-8:    Potential Theory for Initial-boundary Value Problems of Unsteady Stokes Flow in Two Dimensions

Shidong Jiang

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102

Abstract: Integral equation methods have been developed for unsteady Stokes flow. Specifically, the Green's formula was derived; and integral equation formulations have been developed for the initial problem, nonhomogeneous problem, Dirichlet problem, and Neumann problem. The jump relations of the single layer and double layer potentials have also been derived. The analytical apparatus developed here will be useful for the development of fast numerical algorithms (especially those related with Fast Multipole Methods) for unsteady Stokes flow.

REPORT 1112-9:    Swing, Release, and Escape Mechanisms Contribute to the Generation of Phase-locked Cluster Patterns in a Globally Coupled FitzHugh-Nagumo Model

Horacio G. Rotstein and Hui Wu

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102

Abstract:  We investigate the mechanism of generation of phase-locked cluster patterns in a globally coupled FitzhHugh-Nagumo model where the fast variable (activator) receives global feedback from the slow variable (inhibitor). We identify three qualitatively different mechanisms (swing-and-release, hold-and-release, and escape-and-release) that contribute to the generation of these patterns. We describe these mechanisms and use this framework to explain under what circumstances two initially out-of phase oscillatory clusters reach steady phase-locked and in-phase synchronized solutions, and how the phase difference between these steady state cluster patterns depends on the clusters relative size, the global coupling intensity, and other model parameters.

REPORT 1112-10:    Dynamic Mechanisms of Generation of Oscillatory Cluster Patterns in a Globally Coupled Chemical System

Horacio G. Rotstein and Hui Wu

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102

Abstract: We use simulations and dynamical systems tools to investigate the mechanisms of generation of phase-locked and localized clusters in a globally coupled Oregonator where the activator receives global feedback from the inhibitor, mimicking experimental results observed in the photosensitive Belousov-Zhabotinsky reaction. A homogeneous two-cluster system (two clusters with equal cluster size) displays antiphase patterns. Heterogenous two-cluster systems (two clusters with different sizes) display both phase-locked and localized patterns depending on the parameter values. In a localized pattern the magnitude of the largest cluster is roughly an order of magnitude smaller than the oscillation amplitude of the smaller cluster, reflecting the effect of self-inhibition. The transition from phase-locked to localized patterns occurs as the intensity of global feedback. Three qualitatively different basic mechanisms, described previously for a globally coupled FitzHugh-Nagumo model, are involved in the generation of the observed patterns. The swing-and-release mechanism is related to the canard phenomenon (canard explosion of limit cycles) in relaxation oscillators. The hold-and-release and hold-and-escape mechanisms are related to the release and escape mechanisms in synaptically connected neural models.

REPORT 1112-11:    Microstructure Evolution during Impact on Granular Matter

L. Kondic1, X. Fang1, W. Losert2, C. S. O'Hern3, R.P. Behringer4

1 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ  07102

2 Department of Physics, IPST, and IREAP, University of Maryland, College Park, MD, 20742

3  Departments of Mechanical Engineering & Materials Science and Physics, Yale University, New Haven, CT 06520-8284

4 Department of Physics and Center for Nonlinear and Complex Systems, Duke University, Durham NC, 27708-0305

Abstract: We study the impact of an intruder on a dense granular material.  The process of impact and interaction between the intruder and the granular particles is modeled using discrete element simulations in two spatial dimensions.  In the first part of the paper, we discuss how the intruder's dynamics depends on 1) the intruder's properties, including its size, shape and composition, 2) the properties of the grains, including friction, polydispersity, structural order, and elasticity, and 3) the properties of the system, including its size and gravitational field.  It is found that polydispersity and related structural order, and frictional properties of the granular particles, play a crucial role in determining impact dynamics.  In the second part of the paper we consider the response of the granular system itself.  We discuss the force networks that develop, including their topological evolution.  The influence of friction and structural order on force propagation, including the transition from hyperbolic-like to elastic-like behavior is discussed, as well as the affine and non-affine components of the grain dynamics.  Several broad observations include the following: tangential forces between granular particles are found to play a crucial role in determining impact dynamics; both force networks and particle dynamics are correlated with the dynamics of the intruder itself.

REPORT 1112-12:    Instability of a Transverse Liquid Rivulet on an Inclined Plane

Javier A. Diez1, Alejandro G. Gonzalez1, Lou Kondic2

1 Instituto de Fisica Arroyo Seco, Universidad Nacional del Centro de la Provincia de Buenos Aires, Pinto 399, 7000, Tandil, Argentina

2 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102

Abstract: This work concentrates on the stability of a viscous liquid rivulet positioned across an inclined plane under partial wetting conditions. The study is performed within the framework of lubrication approximation by employing a slip model. Both normal and parallel components of gravity are considered.   We find the stability regions for given area of the cross section of the rivulet, $A$, plane inclination angle, $\alpha$, and static contact angle, $\theta_0$, characterizing the wettability of the substrate.  For $\alpha$'s smaller than some critical angle, $\alpha^{\ast}$,  a static solution exists.   This solution is characterized by rear/front contact angles given by $\theta_0 \pm \delta$.  The linear stability analysis of this solution is performed using an efficient pseudo-spectral Chebyshev method. We analyze the effects of $A$, $\theta_0$ and $\alpha$ on the predictions of the model, such as the dominant wavelength, the maximum growth rate and the behavior of most unstable perturbation mode. To verify the predictions, we also carry out experiments with silicone oils spreading on a coated glass substrate for a number of different fluid volumes and inclination angles.    We find very good agreement between the wavelength of maximum growth rate given by the theory and the average distance between the drops after rivulet breakup.  An analysis of finite size effects allows to show that inclusion of normal gravity effects leads to better agreement between theoretical and experimental results.

Instability of Gravity Driven Flow of Liquid Crystal Films

Authors: Sean P. Naughton, Namrata K. Patel, Ivana Seric

Advisors: L. Kondic, T-S. Lin, L. J. Cummings

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102

Abstract:  This paper discusses modeling of spreading nematic liquid crystal films.   We concentrate on gravity driven spreading and consider various instabilities which occur during the spreading.   We find that nematic character of the spreading film leads to stronger instabilities of the film fronts, and that it also leads to surface instabilities.   We also present results of physical experiments involving spreading nematic films and find good agreement with the theoretical and computational predictions.

Thin Films Flowing down Inverted Substrates: Three Dimensional Flow

T.-S. Lin1, L. Kondic1 and A. Filippov

1Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102

Abstract: We study contact line induced instabilities for a thin film of fluid under destabilizing gravitational force in three-dimensional setting. In the previous work [T.-S. Lin and L. Kondic, Phys. Fluids 22, 052105 (2010)], we considered two-dimensional flow, finding formation of surface waves whose properties within the implemented long-wave model depend on a single parameter, D = (3Ca)1/3cotα, where Ca is the capillary number and α is the inclination angle. In the present work we consider fully 3D setting and discuss the influence of the additional dimension on stability properties of the flow. In particular, we concentrate on the coupling between the surface instabilities and the transverse (fingering) instabilities of the film front. We furthermore consider these instabilities in the setting where fluid viscosity varies in the transverse direction. It is found that the flow pattern strongly depends on the inclination angle and the viscosity gradient.

Model Checks via Bootstrap when There are Missing Binary Data

Gerhard Dikta1,  Sundar Subramanian2, and Thorsten Winkler1

1 Department of Medizintechnik und Technomathematik, Fachhochschule Aachen, Germany

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

Abstract:  Dikta, Kvesic, and Schmidt proposed a model-based resampling scheme to approximate critical values of tests for model checking involving binary response data. Their approach is inapplicable when the binary response variable is not always observed, however. We propose a missingness adjusted marked empirical process under the framework that the missing binary responses are missing at random. We introduce a resampling scheme for the bootstrap and prove its asymptotic validity. We present some numerical comparisons and illustrate our methodology using a real data set.

Managing Warranty Costs with Variable Usage Rates

Sonia Bandha and Manish C. Bhattacharjee

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

Abstract: We investigate a Copula based approach to model expected warranty costs and their corresponding minimization under a two-dimensional warranty regime defined by a base warranty period and a maximum allowable use limit. Contrary to the standard paradigm adopted by most researchers on the subject, we assume that customers' individual usage rates  - although assumed constant, is unknown to the seller except through a distributional specification of usage rates among all customers. A family of warranty strategies that partition the warranty period into three intervals, which have been considered by several researchers (Jack et al.(2009), Yun et al.(2008), Banerjee and Bhattacharejee ((2012-a) & (2012-b)) is explored with minimal repairs and a replacement option, via a Copula approach and illustrated using an accelerated Weibull lifetime conditional on usage rate, with the latter being uniformly distributed. We provide a numerical example, which is identical to the example used by others - to contrast our results with theirs.

Dynamics of the Primary Cilium in Shear Flow

Yuan-Nan Young1, M. Downs2 , and C. R. Jacobs2

1 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey, 07102
2 Department of Biomechanical Engineering, Columbia University, New York, New York, 10027

Abstract:  In this work the equilibrium shape and dynamics of a primary cilium under flow are investigated using both theoretical modeling and experiment.  The cilium is modeled as an elastic beam that may undergo large deflection due to the hydrodynamic load. Equilibrium results show that the anchoring effects of the basal body on the cilium axoneme behave as a nonlinear rotational spring. Details of the rotational spring are elucidated by coupling the elastic beam with an elastic shell. We further study the dynamics of cilium under shear flow with the cilium base angle determined from the nonlinear rotational spring, and obtain good agreement in cilium bending and relaxing dynamics when comparing between modeling and experimental results. These results potentially shed light on the physics underlying the mechanosensitive ion channel transport through the ciliary membrane.  Key words: Primary Cilium; Stokes Flow; Modeling; Slender Body; Mechanosensing.

On the Dimension of Solutions of Nonlinear Equations

P. S. Milojevic

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey, 07102

Abstract:  We study the covering dimension of (positive) solutions to various classes of nonlinear equations based on the nontriviality of the fixed point index of a certain condensing map. Applications to semilinear equations and to nonlinear perturbations of the Wiener-Hopf integral equations are given.