**REPORT 1213-1: ** **On the Time-Domain Response
of Havriliak-Negami Dielectrics**

Matthew F.
Causley^{1} and Peter G. Petropoulos^{2}

^{1}Department
of Mathematics, Michigan State University, East Lansing, MI
48824

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

Abstract: We apply a combination of asymptotic and
numerical methods to study electromagnetic pulse propagation in the Havriliak-Negami permittivity model of

fractional
relaxation. This dielectric model contains the Cole-Cole and Cole-Davidson
models as special cases. We analytically determine the impulse response at short
and long distances behind the wavefront, and validate
our results with numerical methods for performing inverse Laplace transforms and
for directly solving the time-domain Maxwell equations in such dielectrics. We
find that the time-domain response of Havriliak-Negami
dielectrics is significantly different from that obtained for Debye dielectrics.
This makes possible using pulse propagation measurements in TDR setups in order
to determine the appropriate dielectric model, and its parameters, for the
actual dielectric whose properties are being measured.

**REPORT 1213-2: ** **On Equality in Distribution of Some Ratios
Involving the Sum of Components of a Random Vector**

Manish C. Bhattacharjee and Sunil K. Dhar

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

Abstract: Motivated by a classical result in the i.i.d. case for a pair random variables X; Y; we look for a simple sufficient condition, allowing for possible dependence between X and Y, under which the ratios of the components X; Y to their sum are equal in distribution. Our finding is easily extended to random vectors of higher dimensions to show that exchangability of a finite sequence X1,…, Xn is sufficient to guarantee the desired result. Any Archimedian copula can be used as a generator of such random vectors. Our main result is applicable in many Bayesian contexts, where the observations are conditionally i.i.d. given an environmental variable with a prior.

MSC 2010 subject classifications Primary, Secondary, 62A01, 62N05

Keywords and phrases : random vectors, equality in distribution, exchangable finite sequences, copulas, Bayesian framework.

**REPORT 1213-3: ** **Generalized Linear Model under the Extended
Negative Multinomial Model and Cancer Incidence**

S. Lahiri & Sunil K. Dhar

Department of Mathematical Sciences, CAMS, New Jersey Institute of Technology, Newark, NJ-07111, USA

E-mail: dhar@njit.edu

Abstract: The generalized linear model for a multi-way contingency table for several independent populations that follow the extended negative multinomial distributions is introduced. This model represents an extension of negative multinomial log-linear model. The parameters of the new model are estimated by the quasi-likelihood method and the corresponding score function which gives a close from estimate of the regression parameters. The goodness-of-fit test for the model is also discussed. An application of the log-linear model under the generalized inverse sampling scheme representing cancer incidence data is given as an example of this model. Keywords: generalized inverse sampling, extended negative multinomial distribution, quasi-likelihood, asymptotic distribution.

**REPORT 1213-4: ** **Persistence of Force Networks in Compressed
Granular Media**

M.
Kramar^{1}, A. Goullet^{2}, L. Kondic^{2}, K.
Mischaikow^{1}

^{1}
Department of Mathematics, Rutgers University, Piscataway, NJ
08854-8019

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

Abstract: We utilize the tools of persistent
homology to analyze features of force networks in dense granular matter, modeled
as a collection of circular, inelastic frictional particles. The proposed approach describes
these networks in a precise and tractable manner, allowing to
identify novel features which are difficult or impossible to characterize
by other means. In contrast to other techniques that consider each force
threshold level separately, persistent homology allows us to consider all
threshold levels at once to describe the force network in a complete and
insightful manner. We
consider continuously compressed system of particles characterized by varied
polydispersity and friction in two spatial dimensions.
We find significant differences between the force networks in these systems,
suggesting that their mechanical response may differ considerably as
well.

**REPORT 1213-5: ** **Stability of a Liquid Ring on a
Substrate**

A.
G. Gonzalez^{1}, J. A. Diez^{1} and L.
Kondic^{2}

^{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,
University Heights, Newark, NJ 07102

Abstract: We study the stability of a viscous
incompressible fluid ring on a partially wetting substrate within the framework
of long-wave theory. We discuss the
conditions under which a static equilibrium of the ring is possible in the
presence of contact angle hysteresis. A linear stability analysis (LSA) of this
equilibrium solution is carried out by using a slip model to account for the
contact line divergence. The LSA provides specific predictions regarding the
evolution of unstable modes. In order to describe the evolution of the ring for
longer times, a quasi-static approximation of Wentzel-Kramers-Brillouin (WKB) type is implemented. This approach assumes a
quasi-static evolution and takes into account the concomitant variation of the
instantaneous growth rates of the modes responsible for either collapse of the
ring into a single central drop or breakup into a number of droplets along the
ring periphery. We compare
the results of the LSA and WKB with those obtained from nonlinear numerical
simulations using a complementary disjoining pressure model. We find remarkably good agreement
between the predictions of the two models regarding the expected number of drops
forming during the breakup process.

**REPORT 1213-6: ** **Directed Assembly of One- and
Two-dimensional Nanoparticle Arrays from Pulsed Laser Induced Dewetting of Square Waveforms**

N.
A. Roberts (Department of Materials Science and Engineering, University of
Tennessee)

J.
D. Fowlkes (Center for Nanophase Materials Sciences, Oak Ridge National
Laboratory)

K.
Mahady, S. Afkhami, L. Kondic (Department of Mathematical Sciences, New Jersey
Institute of Technology)

P.
D. Rack (Department of Materials Science and Engineering, University of
Tennessee and Center for Nanophase Materials Sciences,
Oak Ridge National Laboratory)

Abstract:
The directed assembly of arrayed nanoparticles is demonstrated by dictating the
flow of a liquid phase filament on the nanosecond time scale. Results for the
assembly of Ni nanoparticles on SiO2 are presented. Previously, we have
implemented a sinusoidal perturbation on the edge of a solid phase Ni, thin film
strip to tailor nanoparticle assembly. Here, a nonlinear square waveform is
explored. This waveform made it
possible to expand the range of nanoparticle spacing-radius combinations
attainable, which is otherwise limited by the underlying Rayleigh-Plateau type
of instability. Simulations of full
Navier-Stokes equations based on volume of fluid
method were implemented to gain further insight regarding the nature of
instability mechanism leading to particle formation in
experiments.

**REPORT 1213-7: ** **On the Dewetting
of Liquefied Metal Nanostructures**

S.
Afkhami and L. Kondic (Department of Mathematical Sciences, New Jersey Institute
of Technology)

Abstract:
Direct numerical simulations of liquefied metal nanostructures dewetting a substrate are carried out. Full three-dimensional Navier-Stokes equations are solved and a volume-of-fluid
method is used for tracking and locating the interface. Substrate wettability is varied to study
the influence of the solid/liquid interaction. The effects of
initial geometry on the retraction dynamics is numerically investigated.
It is shown that the dewetting velocity increases with
increasing the contact angle, and that the retraction dynamics is governed by an
elaborate interplay of initial geometry, inertial and capillary forces, and the
dewetting phenomena. Numerical results are presented
of the dewetting of nanoscale Cu and Au liquefied structures on a
substrate.

**REPORT 1213-8: ** **O****ptimal Data Assimilation Control for Ocean
Gliders**

Richard Moore (Department of
Mathematical Sciences, New Jersey Institute of Technology)

Abstract: Data assimilation is a process in which
observational data is incorporated into a physical model to provide the optimal
estimate for the state and the uncertainty. Common applications of data
assimilation are seen in weather prediction, climate models, control and
economic models. The estimates that are obtained through a model are updated as
more data is collected. In our
project, we will explore a simple inference problem, motivated by the recent
experiment in Monterey Bay, California. The experiment was conducted using
gliders to measure various ocean parameters such as temperature, salinity and
pressure over a month-long period. One of the goals of this experiment was to
find out an efficient way of using autonomous ocean vehicles. The gliders used
in this experiment can communicate with a shore station when at surface. Such
gliders are directed using trajectories that target specific regions to reduce
the overall uncertainty. We employ an unknown scalar field and a known velocity
field to develop heuristics for controlling autonomous observers toward the
regions of high uncertainty. The goal of our project is to formulate a
well-posed optimal control problem for assimilation of a scalar field by
controlled observers in a stationary flow where we also develop simple
heuristics for guiding the observers towards regions of high uncertainty. We
then test some ideas for control in Matlab, using the
Kalman Filter as our data assimilation
technique.