Center for Applied Mathematics and Statistics


REPORT 1011-1:    Multiple Imputations and the Missing Censoring Indicator Model


Sundar Subramanian

New Jersey Institute of Technology


Abstract:  Semiparametric random censorship (SRC) models (Dikta, 1998) provide an attractive framework for estimating survival functions when censoring indicators are fully or partially available. When there are missing censoring indicators (MCIs), the SRC approach employs a model-based estimate of the conditional expectation of the censoring indicator given the observed time, where the model parameters are estimated using only the complete cases. The multiple imputations approach, on the other hand, utilizes this model-based estimate to impute the missing censoring indicators and form several completed data sets. The Kaplan-Meier and SRC estimators based on the several completed data sets are averaged to arrive at the multiple imputations Kaplan-Meier (MIKM) and the multiple imputations SRC (MISRC) estimators. While the MIKM estimator is asymptotically as or less efficient than the standard SRC-based estimator that involves no imputations, here we investigate the performance of the MISRC estimator and prove that it attains the benchmark variance set by the SRC-based estimator. We also present numerical results comparing the performances of the estimators under several misspecified models for the above mentioned conditional expectation.



REPORT 1011-2:  Some Adjusted Marked Empirical Processes for Model Checking with Missing Binary Data


Sundar Subramanian

New Jersey Institute of Technology


Abstract:  To test the adequacy of a parametric model for the conditional expectation of a binary response variable given an explanatory variable, Kolmogorov--Smirnov and Cramer-von Mises statistics, which are based on a marked empirical process introduced by Stute can be used. Dikta, Kvesic, and Schmidt proposed a model-based resampling scheme for the bootstrap and approximated the critical values of tests formed using these statistics. However, their approach is inapplicable when the binary response variable is not always observed. Some missingness adjusted marked empirical processes are proposed under the framework that the missing binary responses are missing at random. We employ Dikta and Winkler's modified model-based resampling scheme for the bootstrap in this set up and prove its asymptotic validity. Numerical comparison studies between the competing adjustments lead to interesting conclusions.


REPORT 1011-3:  Warranty Servicing with a Brown-Proschan 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:  Reducing warranty servicing costs are of great interest to product manufacturers or, sellers who are contractually bound to provide post-sales support, up to a specifed warranty period, usually in the form of some remedial action that restores a failed item to a functioning condition. Here, in the spirit of Jack, Iskandar and Murthy (2009) strategy based on partitioning the effective warranty period into three intervals, we consider and analyze the cost of a new two-dimensional warranty servicing strategy, that probabilistically exercises a choice between a replacement and a minimal repair to rectify the first failure if any, in the middle interval. A numerical illustration of our analysis with Weibull failure model is included.



REPORT 1011-4:  Asymptotic Structure of Diffusion Flames at High Pressure

D. Fong and J. K. Bechtold
Department of Mathematical Sciences
New Jersey Institute of Technology
Newark, NJ 07102

C. K. Law
Department of Mechanical and Aerospace Engineering
Princeton University
Princeton, NJ 08544

Abstract:  We analyze the structure of diffusion flames at high pressure in the limit of large activation energy for the particular configuration of a steady flame in counterflow. We consider the fluid to be near the thermodynamic critical point in which case the effective mass diffusivity is negligible, and Soret diffusion is the dominant mechanism for fuel mass transport. Temperature and species profiles, as well as flame temperature and location, are determined as a function of Damkohler number and Soret coefficient. Our analysis includes a description of extinction phenomenon.


REPORT 1011-5:  Inhibitory Feedback Promotes Stability in an Oscillatory Network

F. Nadim1,2, S. Zhao2, L. Zhou2 and A. Bose1,3

1 Dept. of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA

2 Federated Dept. of Biological Sciences, 195 University Avenue, Rutgers University, Newark, NJ 07102, USA

3 School of Physical Sciences, Jawaharlal Nehru University, New Delhi, 110067, India

Abstract:  Reliability and variability of neuronal activity are both thought to be important for the proper function of neuronal networks. The crustacean pyloric rhythm (~1 Hz) is driven by a group of pacemaker neurons (AB/PD) which inhibit, and burst out of phase with, all follower pyloric neurons. The only chemical synaptic feedback to the pacemakers is an inhibitory synapse from the follower LP neuron.  Although this synapse has been extensively studied, its role in the generation and coordination of the pyloric rhythm is unknown. We examine the hypothesis that this synapse acts to stabilize the oscillation by reducing the variability in cycle period on a cycle-by-cycle basis. Our experimental data show that removing the LP-to-PD synapse increases the pyloric period variability. Using a reduced mathematical model we demonstrate that the increase in pyloric rhythm stability in the presence of the LP-to-PD synapse can be explained by a decrease in the amplitude of the phase response curve of the PD neuron.

REPORT 1011-6:  Peptide Neuromodulation of Synaptic Dynamics in an Oscillatory Network

Shunbing Zhao1,  Amir Farzad Sheibanie2 , Myongkeun Oh3 ,  Pascale Rabbah4 , and Farzan Nadim5

1Department of Biological Sciences, Rutgers University, Newark, NJ 07102.

2Department of Neuroscience, University of Medicine and Dentistry of New Jersey, Newark, NJ 07102.

3Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102.

 4Department of Biological Sciences, Rutgers University, Newark, NJ 07102.

5Department of Mathematical Sciences, New Jersey Institute of Technology and Department of Biological Sciences, Rutgers University, Newark, NJ 07102.

Abstract:  Neuromodulation of synaptic strength and short-term dynamics can have important consequences for the output of an oscillatory network. Although such effects are documented, few studies have examined neuromodulation of synaptic output in the context of network activity. The crab pyloric bursting oscillations are generated by a pacemaker group that includes the pyloric dilator (PD) neurons. The sole chemical synaptic feedback to this pacemaker group is the inhibitory synapse from the lateral pyloric (LP) neuron, which is comprised of an action-potential-mediated and a graded component. We show that the neuropeptide proctolin unmasks a surprising heterogeneity in its dynamics of the graded component depending on the magnitude of the presynaptic input: it switches the direction of short-term dynamics of this component by changing depression to facilitation. Whether the graded component shows depression or facilitation, however, depends on the amplitude of the slow voltage waveform of the presynaptic LP neuron and is correlated with a putative presynaptic calcium current. The spike-mediated component is strengthened as the baseline membrane potential is increased in control conditions and is also enhanced by proctolin at all baseline potentials. In addition to direct modulation of the synaptic components, proctolin also affects the amplitude of the LP waveform and its action potential frequency, both of which influence synaptic release. Acting through these multiple pathways, proctolin greatly enhances the strength of this synapse under natural biological conditions as evidenced in the significant increase in the synaptic current measured during ongoing oscillations.

REPORT 1011-7:   Excitable Nodes on Random Graphs: Relating Dynamics to Network Structure

Thounaojam Umeshkanta Singh1, Kaustubh Manchanda1 , Ramakrishna Ramaswamy1 and Amitabha Bose1, 2

1 School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India

2 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 01702 USA

Abstract: Rhythmic activity in complex systems is generated and sustained through interactions among the constituent units. In this paper we study the interplay between topology and dynamics of excitable nodes on random networks. The nodal dynamics are discrete, each node being in three possible states, active, refractory or silent. Loading rules are defined whereby a subset of active nodes may be able to convert a silent node into active one at the next time step. In the case of simple loading (SL) a silent node becomes active if it receives input from any neighbor. In the majority rules (MR) loading, a silent node fires when the majority of its neighbors are active. We address the question of whether a particular network design pattern confers dynamical advantage for the generation and sustainment of rhythmic activity. We find that the intrinsic properties of a node and the rules for interaction between them determine which structural features of the graph permit sustained activity.With SL the level of activity in the graph increases monotonically with the probability of connections between nodes, while for MR, the level of activity may be either monotonic or non-monotonic depending on parameters.

REPORT 1011-8:  Self– versus Directed– assembly of Nanoparticles via Pulsed Laser Induced Dewetting of Patterned Metal Films

J. D. Fowlkes1, L. Kondic2, J. Diez3, Y. Wu4, and P. D. Rack1,4 

1 Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 37831–6493

2 Department of Mathematical Sciences, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, New Jersey
3 Universidad Nacional del Centro de la Provincia de Buenos Aires (UNCBRA), Pinto 399, Tandil, Argentina

4 Materials Science and Engineering Department, The University of Tennessee, Knoxville, Tennessee, 37996–2200

Abstract:  A nanoscale, synthetic perturbation was all that was required to nudge a natural, self–assembly process toward significantly higher order. Nanolithography was used to impose the perturbation which ultimately led to an organized nanoparticle array. Specifically, liquid–phase pulsed laser induced dewetting (PLiD) was used to transform metallic thin film strips into nanoparticle arrays. Initially, thin film strips retracted into fluid rivulets. Rivulet breakup followed, forming linear nanoparticle arrays via a process resembling the Rayleigh-Plateau (RP) instability; nanoparticle diameter and pitch were poorly controlled and disperse. We then demonstrated that the assembly accuracy and precision could be drastically improved by merely imposing a synthetic sinusoidal perturbation onto the lateral surfaces of the thin film strip. The synthetic perturbations in the strip translated into an unstable varicose oscillation on the rivulet during retraction – a precise nanoparticle diameter and pitch emerged thereby superseding the otherwise naturally evolving modes predicted by the modified Rayleigh-Plateau instability.


REPORT 1011-9:  Evolution of Droplets of Perfectly Wetting Liquid under the Influence of Thermocapillary Forces

Shomeek Mukhopadhyay1,4, Nebojsa Murisic2, Robert P. Behringer3, Lou Kondic4
1 Department of Physics and Center for Complex Systems, Duke University, Durham, NC 27706
2 Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095
3 Department of Mathematical Sciences, Center for Applied  Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102
4 Department of Chemistry, Columbia University, New York, NY 10027

Abstract:  We consider evolution of sessile droplets of a nonvolatile perfectly wetting liquid on differentially radially heated solid substrates.  The heating induces thermocapillary Marangoni forces which affect the contact line dynamics. Our experiments involving a particular heating pattern reveal that the Marangoni effect suppresses the spreading of a drop, typical for perfectly wetting liquids.  The result is a rather slow receding motion, and a distinctive thinning of the liquid layer in the region close to the contact line.  Our theoretical model, based on the lubrication approximation and incorporating the Marangoni effect, recovers the main features observed in the experiments and in addition predicts novel features which are still to be observed.


REPORT 1011-10:  Defect Modeling in Spreading Nematic Droplets

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

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

Abstract:  Experiments by Poulard & Cazabat [Langmuir 21, 6270 (2005)] on spreading droplets of nematic liquid crystal (NLC) reveal a surprisingly rich variety of behavior, including at least two different emerging lengthscales resulting from a contact line instability. In earlier work [Phys. Fluids 23, 043102 (2011)] we modified a lubrication model for nematic liquid crystals due to Ben Amar & Cummings [Phys. Fluids 13, 1160 (2001)], and showed that, in a qualitative sense, it can account for 2D versions of the observed behavior. In the present work we propose a new approach that allows us to explore the effect of anchoring variations on the substrate, again in a 2D geometry. This in turn gives a simple way to model the presence of defects, which are nearly always present in such flows.   The new model leads to additional terms in the governing equation. We explore the influence of these additional terms for some simple flow scenarios to gain a basic understanding of their influence.

REPORT 1011-11:  Modeling and Simulations of the Spreading and Destabilization of Nematic Droplets

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

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

Abstract:  A series of experiments [C. Poulard & A.M. Cazabat, ``Spontaneous spreading of nematic liquid crystals,'' Langmuir 21, 6270-6276 (2005)] on spreading droplets of nematic liquid crystal (NLC) reveal a surprisingly rich variety of behaviors. Small droplets can either be arrested in their spreading, spread stably, destabilize without spreading (corrugated surface), or spread with a fingering instability and corrugated free surface. In this work, we discuss the problem of NLC drops spreading in a simplified two-dimensional (2D) geometry. The model that we present is based on a long-wavelength approach for NLC's due to Ben Amar & Cummings [M. Ben Amar & L.J. Cummings, ``Fingering instabilities in driven thin nematic films,'' Phys. Fluids 13, 1160-1166 (2001), L.J. Cummings, ``Evolution of a thin film of nematic liquid crystal with anisotropic surface energy,'' Europ. J. Appl. Math. 15, 651-677 (2004)]. The improvements of the model here permit fully nonlinear time-dependent simulations. These simulations, for the appropriate choice of parameter values, exhibit 2D versions of most of the phenomena mentioned above.

REPORT 1011-12:  Quasi-Optimal Convergence of Non-Overlapping Domain Decomposition Algorithm for the Helmholtz Equation

 Y. Boubendir1, X. Antoiney2, C. Geuzaineets3

1Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, NJIT, Univ. Heights. 323 Dr. M. L. King Jr. Blvd, Newark, NJ 07102, USA

2Institut Elie Cartan Nancy (IECN), Nancy University, INRIA Corida Team, B.P. 239, F-54506 Vandoeuvre les-Nancy, Cedex, France

3University of Liege, Department of Electrical Engineering and Computer Science, Monte ore Institute B28, B-4000, Liege, Belgium

Abstract: This paper presents a new non-overlapping domain decomposition method for the Helmholtz equation, whose effective convergence is quasi-optimal. These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Dirichlet to Neumann operator. A convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions.

REPORT 1011-13:  Second Kind Integral Equations for the First Kind Dirichlet Problem of the Biharmonic Equation in Three Dimensions

Shidong Jiang1 , Bo Ren1, Paul Tsuji2, and Lexing Ying2
1Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey  07102

2Department of Mathematics and ICES, University of Texas at Austin, Austin, TX  78712

Abstract: A Fredholm second kind integral equation (SKIE) formulation is constructed for the Dirichlet problem of the biharmonic equation in three dimensions. A fast numerical algorithm is developed based on the constructed SKIE. Its performance is illustrated via several numerical examples.

REPORT 1011-14:  Caslav V. Stanojevic - a Renaissance Man

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

Abstract:  Caslav was my god-father and was a great friend of my family. He was a renaissance man, erudite, and had a profound knowledge in many fields. He was like a walking encyclopedia.  Caslav had a great sense of humor, sharp wit as well as a sharp tongue and was enjoyable to be in his company. He was a renowned mathematician in the field of Fourier Analysis. His mentors were M. Frechet, N. Saltikov and D. Markovic. Through them, he was a mathematical grandson of J. Hadamard, V.A. Steklov, A. M. Lyapunov and M. Petrovic. He moved to America in 1963 and spent most of his professional career at the University of Missouri in Rolla. There he found his well known school in Fourier Analysis. He revitalized Fourier Analysis in the L1 spaces and made many fundamental contributions in the theory of Fourier series in these spaces. He had published about 50 research papers. His style of writing was very concise and lucid and he published only well polished results. There is a Gauss motto "Paula seed natural", or "Few, but ripe".  Caslav's motto was perhaps "Pauca sed egregia", or "Few, but outstanding". His work was published in the most prestigious mathematical journals and was quoted extensively. He was a founder of the series International Workshops in Analysis and Its Applications which took place in Kupari each summer. These workshops attracted many renown mathematicians from all over the world. His knowledge of Serbian and world history was profound. He was also an accomplished astrologer and had even made the architectural project of his house in Rolla. Reading was one of his many passions. Among others, he was fascinated by the Serbian romance poets (J. Ducic, B. Radicevic and others) so much that he even knew many of their verses by heart. He even wrote a novel "The Time Catchers" in 1974.  Caslav was a deeply religious man and lived his life according with the Serbian religious rituals. He had enormous zest for life, enjoyed new wines and healthy food and had great culinary skills. He knew how and liked to prepare nice dishes for his friends.  Caslav was wrestling with questions about death and life after death. He was hoping to visit the Holy Mountain on the peninsula Atos, and in particular the Serbian monastery Hilandar, and find some answers to these questions. Moreover, he told me that when he would sense the end was near, he would want to go to some Serbian monastery to live his final days. Unfortunately, he never fulfilled these two wishes.  Caslav has left us, but his contributions to the science are long lasting.