Title: Kinematics Data Fusion and Weighting process on Covariance
Ellipses
Title: Global Minimization of Hopf Surfaces, with Application to Nematic
Electroconvection.
Abstract: A cell exclusion algorithm is used to numerically find the
global minimum of multiparametric surfaces arising in Hopf bifurcation
problems. The method is then applied to the oscillatory neutral
stability surface resulting from the linear stability analysis of the
weak electrolyte model for the electroconvection of nematic liquid
crystals (NLC).
Title: A Deterministic Approach to the Spread of Rumors.
Abstract: Ideas and messages spread in ways that resemble the
transmission dynamics of viruses. We begin with the same framework
as Daley Kendall, which classifies individuals as susceptibles,
"spreaders", and "stiflers", and models rumor spreading as an
epidemic. We look at the spread of a rumor, an aspect not
considered in the Daley Kendall model. Finally, the dynamics of
rumor spreading in chat rooms that are accessible to a large number
of groups are explored under the assumption of simple, local
(neighborhood) dynamics. Characterization of the dynamics is
carried out through a combination of analytical and numerical
results. Efforts to determine the most effective ways to stop or
accelerate the spread of rumors are also discussed.
Title: Rumors on Complex Attractors
Abstract: We consider rumor "invasion" into populations
with strong fluctuations in density. Prior
to the rumor arrival, the dynamics of the target
population is assumed to be at a demographic
"steady-state". In fact, it will be modeled by
a preselected attractor. We assume that while the
rumor circulates it divides the target population into
two classes; spreaders, and nonspreaders. The
transitions between classes are functions of the contact rates
and the proportion of spreaders. Will the
spreader population survive? and if it does;
Will it settle on a different attractor?; How does the
dynamics of the rumor compare to the dynamics of
its analogue epidemic process?
Title: Waiting-time statistics in scheduling with applications to
QoS assurance in computer systems
Abstract: Scheduling a set of jobs on a single machine to minimize the waiting time
variance (WTV) has acquired importance theoretically and practically.
We find a very interesting phenomenon: the plot of variance over mean of
the waiting times of the jobs always follows an eye
shape. The optimal point which has the minimal WTV is located at the
bottom of the eye shape. Several questions come to us: why WTV over mean
always scatters like an eye? Can we derive the optimal sequence from this
phenomenon, e.g., given the function for the bottom edge of the eye curve,
solve for the minimum point and use the mean and variance values of the
minimum point to derive the job sequence for it? We will try to answer
these questions in this paper
Title:Effects of education, vaccination and treatment on HIV transmission in
homosexuals with genetic heterogeneity
Abstract: Genetic studies report the existence of a mutant allele 32 of CCR5 chemokine
receptor gene at high allele frequencies (~10%) in Caucasian populations.
The presence of this allele is believed to provide partial or full
resistance to HIV. In this study, we look at the impact of education,
temporarily effective vaccines and therapies on the dynamics of HIV in
homosexually active populations. In our model, it is assumed that some
individuals possess one or two mutant alleles (like 32 of CCR5) that prevent
the successful invasion or replication of HIV. Our model therefore
differentiates by genetic and epidemiological status and naturally ignores
the reproduction process. Furthermore, HIV infected individuals are
classified as rapid, normal or slow progressors. In this complex setting,
the basic reproductive number is derived in various situations. The
separate or combined effects of therapies, education, vaccines, and genetic
resistance are analyzed. Our results support the conclusions of Hsu Schmitz
that some integrated intervention strategies are far superior to those based
on a single approach. However, treatment programs may have effects which
counteract each other, as may genetic resistance.
Title: Radial Basis Function Model Order Determination using
Statistical Hypothesis Testing.
Abstract: Several factors contribute to make the on-line, non-linear and
non-stationary function approximation a challenging and
interesting problem. These include: change of the statistical
characteristics of arrival stream of data points,
non-repeatability and variation of the received signal over time,
competing noise mixed with the original signal, and lack of any
{\it a priori} knowledge about the properties of the signal.
Among the growing networks, growing radial basis functions have
special features, which make them very suitable for this task. In
the current methods of growing RBF there are many ad-hoc
parameters, which require accurate tuning for a specific data set.
As a result the current methods are very sensitive and their
performance depends on the choice of parameters.
Our intention is to provide a new prospective of the on-line
growing RBFs by releasing the limiting and ad-hoc assumption in
the literature. The primary goal of this research is to study a
model validation test based on linear autocorrelations as an
objective to determine the optimal number of units in the hidden
layer of a radial basis function in the real time. The data to be
fit is assumed to contain a desire signal with additive,
independent identically distributed (iid) or white noise. The
resulted automated system works based on the statistics of the
residuals rather than on ad-hoc parameters. This network is shown
to capture an accurate model and neither over fit nor under fit
the data. Specific canonical data sets were chosen to demonstrate
the performance and the robustness of the method.
The wide range of application of this method justifies the
usefulness of this method in the practical fields of mathematics
and industry. Applications in fault detection and control systems,
times series analysis, mathematics of large data sets, signal
processing and communications.
Title: Heteroclinic Cycles in an Eight Dimensional Vector Field Derived
From
Nematic Electroconvection.
Abstract:The symmetry-breaking Hopf bifurcation from a spatially uniform
steady-state in spatially extended, anisotropic systems is described by an
eight dimensional equivariant vector field. This bifurcation occurs at the
onset of electroconvection in the weak electrolyte model for nematic liquid
crystals, thus the vector field can be used to predict electroconvective
wave
patterns above onset. The dynamics of the vector field are investigated,
with
a focus on the existence and asymptotic stability of structurally stable
heteroclinic cycles.
Title: Beneficial role of noise
in promoting species diversity
Abstract: Species diversity in nature is accomplished by
coexistence. In a spatially-extended environment, inferior but rapidly
movable species can coexist with superior bur relatively stationary species.
Recent work showed that chaotic dynamics can provide the spatiotemporal
variation in the fitness required for coexistence, via the dynnamical
mechanism of synchronization and intermittency. tilizing a realistic model
that consist of two interacting species in a two-patch environment, we
address the role of small noise in coexistence. We focus on the effect of
noise on the intermittent patchy dynamics and obtain quantative result
that noise can effectively facilitate species coexistence by enhancing the
intermittency.
Title: Change in Host Behavior and its Impact on the Co-evolution of Dengue
Abstract:The joint evolutionary dynamics of dengue strains are poorly understood
despite its high prevalence around the world. Two dengue strains are put in
competition in a population where behavioral changes can affect the
probability of infection. The destabilizing dynamic effect of even "minor"
behavioral changes is discussed and their role in dengue control is
explained.
Title: Immune-response based approach to multiple influenza strain dynamics.
Abstract: We develop a mathematical model that incorporates partial
cross-immunity to next-to-kin strains.
The immunity status of the host is the antibody levels corresponding to
all the strains that each host has immunity to.
Antibody immune response of the host population
is captured by an index-set notation where the index specifies the
immune-competence level against a particular strain.
In contrast to previous modeling frameworks, the population here is
structured into non-intersecting subclasses. Since multiple infection with
influenza strains is unlikely to occur, we do not imbed superinfection
with the same or different strains as part of our model.
This framework allows for a biologically interpretable range of
cross-immunity levels as observed during the yearly influenza
epidemics (antigenic drift).
Using the proposed framework of cross-immunity, we provide specific
cases to illustrate the impact of the assumptions on the strain state
space. Furthermore, we utilize the proposed notation to provide
threshold conditions for the invasion of a new strain and show the
existence of an endemic multi-strain equilibrium in a particular case.
Title: Wave Patterns in Anisotropic Systems, with Application to Nematic
Electroconvection
Abstract: Oscillatory instabilities in 2d extended, anisotropic systems are
analyzed for the case in which the neutral stability surface has four
minima. In this case the linearized system admits solutions in the form
of two pairs of counter-propagating traveling waves in oblique
directions. In a weakly nonlinear analysis, the post--threshold dynamics
is described by a system of four globally coupled complex Ginzburg
Landau equations for the waves' envelopes. If spatial variations are
ignored, these equations reduce to the normal form for a Hopf
bifurcation with O(2) x O(2) symmetry.
The dynamics of the normal form is rich and shows six different basic
wave patterns, quasiperiodic waves, and heteroclinic cycles. The goal of
our future work is to compare the normal form dynamics to the
spatio--temporal dynamics of the Ginzburg Landau system. This system is
simulated using a pseudo--spectral method.
As application for our approach to study wave instabilities in
anisotropic systems we consider the weak electrolyte model for
electroconvection in nematic liquid crystals. The complicated numerical
task of finding the minimum of the neutral stability surface is solved
by means of a cell exclusion algorithm combined with a Nelder Meid
method and an augmented system for detecting Hopf bifurcations.
Title: Theoretical Aspects of Blind Source Separation Using Second Order Statistics
Abstract: Blind identification of source signals is studied from the theoretical point of view using the
information drawn from second order statistics.
SOBI algorithm is a source separation technique exploiting the time coherence of the
source signals, based on joint diagonalization of a set of covariance matrices. This paper presents
the geometric representation of this algorithm
and a characterization of the optimization problem that leads to the joint diagonalization algorithm.
Title: Weighted complex networks and characterization Abstract: To account for possible distinct functional roles played by different nodes and links in complex networks, we introduce and analyze a class of weighted scale-free networks. The weight of a node is assigned as a random number, based on which the weights of links are defined. We utilize the concept of betweenness to characterize the weighted networks and obtain the scaling laws governing the betweenness as functions of the weight and of the degree as well. The scaling results may be useful for identifying influential nodes in terms of physical functions in complex networks.
Title: Models for Dengue Transmission and Control Abstract: A model for the transmission dynamics of a single strain of dengue, a mosquito-transmitted disease, that includes some stages of the life-history of the vector is introduced. Vector life-histories that support multiple steady states, generated by biological vector control measures, are used to highlight the dependence of host-dengue levels on the vectors' density. The implications of our results on dengue control are discussed.
Title: The Effects of Student-Teacher Ratio on Student and Faculty Performance in High School Scenarios Abstract: We develop a model that incorporates the impact of student-teacher ratio on the performance dynamics of both teachers and students.The model assumes that the members of both populations may be found in three dynamic states: positive, discouraged and reluctant. The role of complex nonlinear interactions between students and teachers, as well as the role of recruitment and intervention, are studied via analytic and numerical studies. Using center manifold theory we find conditions for the existence of a backward bifurcation that support endemic stationary states below the critical threshold value R0 < 1, when normally only a positive environment would be supported. Our simulations show that in order to maintain a positive environment for students and teachers, R0 must be reduced significantly.
Title: A Pseudospectral Method for the Globally Coupled Ginzburg Landau Equations in 2D
Abstract: In spatially extended systems a basic, homogeneous state becomes unstable if a
distinguished control parameter passes through the minimum of a neutral stability surface in parameter-wave number
space. If the critical eigenvalue at this minimum is imaginary, the system encounters a Hopf bifurcation
and to each minimum with nonzero critical wave vector there corresponds a traveling wave solution of
the linearized system. In the case of two-dimensional anisotropic systems the neutral stability
surface typically has either two minima, giving rise to a pair of counterpropagating waves in
the direction of a reflection axis, or four minima, in which case the linearized system has two
counterpropagating pairs of traveling wave solutions in two oblique directions. A paradigm system
of PDEs in which both of these types of Hopf instabilities can occur is the weak electrolyte model
for electroconvection in nematic liquid crystals. In the dynamically more interesting case of
four minima, the spatio-temporal evolution of this system slightly above threshold is described
by a system of four globally coupled Ginzburg Landau equations for the envelopes of the four
traveling waves. We numerically simulate this system using a pseudo-spectral method to efficiently compute
the solution to these equations, and provide a visualization tool to observe the bifurcations in the system.
Title: Avalanches in Complex Networks
Abstract: Avalanche failure is a severe problem in complex networks, such as
the breakdown of power grid and telecommunication networks, the
spreading of disease in social networks and spreading of forest fires,
etc. We propose a model to simulate avalanches in complex networks,
which accounts for not only the spreading process, but also the effect
of accumulative damage to each node in the network. Our model exhibits
a phase transition from the state without avalanche to the one with
avalanche.