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SCIENTIFIC PROGRAMS AND ACTIVITIES |
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| November 22, 2024 |
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ABSTRACTSPatrizia Daniele, University of Catania,
IT: We outline the importance of the evolutionary variational inequalities
theory from the beginning in 1967 to the numerous applications in the
field of operational research, engineering, economy, finance. The study
of the evolutionary variational inequalities has been given a substantial
boost in the last decade when it was proved that dynamic economic, financial
and transportation equilibrium problems on networks can be expressed
by means of evolutionary variational inequalities. So we have the possibility
to follow the evolution in time of economic phenomena, since the theory
of evolutionary variational inequalities allows for the study of existence,
uniqueness and sensitivity analysis of the equilibrium trajectory. Also
the problem of the calculation of the solution in a time interval can
be solved by means of appropriate computational techniques and we present
two computational methods based on the discretization procedure. The
first one requires no regularity conditions on the solution and makes
use of the mean value operators. The second one requires the continuity
of the solutions and exploit the relation between the solutions to an
evolutionary variational inequality and the stationary points of a projected
dynamical system. Some numerical examples of traffic networks and spatial
price models will be presented too. It is well known (to those that know it well) that nonlinear dynamical
systems acting on points have corresponding to them linear operators
acting on functions. Systems of differential equations have corresponding
to them partial differential operators, and when these systems are stochastic,
the resulting partial differential operators lead to famous financial
math results such as the Black -Scholes equation. The topic of this
talk will be discrete dynamical systems, to which correspond the so-called
Perron-Frobenius Operator. After giving a quick review of this theory
I will use it to provide some results on weakly coupled logistic map
systems. We will demonstrate how the paradigm of complex networks can be used
to model some aspects of the process of acquisition of the second language.
When learning a new language, knowledge of 3000-4000 of the most frequent
words appears to be a significant threshold, necessary to transfer reading
skills from the first to the second language. This threshold corresponds
to the transition from Zipf's law to a non-Zipfian regime in the rank-frequency
plot of words of the English language. Using a large dictionary, one
can construct a graph G representing this dictionary, and study topological
properties of subgraphs of G generated by the $k$ most frequent words
of the language. Since the vocabulary grows with time, one can also
think of this as a dynamical process in which $k$ increases as a function
of time. The clustering coefficient of subgraphs of G reaches a minimum
in the same place as the crossover point in the rank-frequency plot.
We conjecture that the coincidence of all these thresholds may indicate
a change in the language structure, which occurs when the vocabulary
size reaches about 3000-4000 words. Lattice differential equations consist of an infinite number of coupled
ODEs parametrized by a planar lattice (square and hexagonal being the
most popular) such as would be obtained by discretizing a planar system
of PDEs. We assume the system is homogeneous; that is, the ODEs at each
site are identical. Herb Kunze, University of Guelph A hyperbolic iterated function system (IFS) has a unique fixed point
that we refer to as its set attractor. To each point on the set attractor
we can associate a code that tells us the order in which we should apply
the IFS maps in order to reach or approach the point. By traveling through
code space and using the notion of fractal tops (a very recent idea
introduced by M. Barnsley), we connect the dynamics of two different
IFSs: we paint the points of (an approximation of) the attractor of
the first IFS by stealing colours from a digital image via the dynamics
of the second hyperbolic IFS. Besides developing assorted theory, I
will explore various examples and applications, producing some stunning
images along the way. Greg Lewis, University of Ontario Institute
of Technology: Mathematical models of fluid systems that isolate the effects of differential
heating and rotation are useful tools for studying the behaviour of
large-scale geophysical fluids, such as the Earth's atmosphere. In this
talk, I discuss the bifurcations of steady axisymmetric solutions that
occur in one such model that considers a fluid contained in a rotating
spherical shell. A differential heating of the fluid is imposed between
the pole and equator of the shell, and gravity is assumed follow axisymmetric
solutions through parameter space, and numerical approximations of normal
form coefficients are computed for a cusp bifurcation that acts as an
organizing centre for the observed dynamics. Pietro Lio, University of Cambridge, UK: During the HIV infection several quasispecies of the virus arise, which
are able to use different coreceptors, in particular the CCR5 and CXCR4
coreceptors (R5 and X4 phenotypes, respectively). The switch in coreceptor
usage has been correlated with a faster progression of the disease to
the AIDS phase. As several pharmaceutical companies are starting large
phase III trials for R5 and X4 drugs, models are needed to predict the
co-evolutionary and competitive dynamics of virus strains. We present
a model of HIV early infection which describes the dynamics of R5 quasispecies
and a model of HIV late infection which describes the R5 to X4 switch.
We report the following findings: after superinfection or coinfection,
quasispecies dynamics has time scales of several months and becomes
even slower at low number of CD4+ T cells. The curve of CD4+ T cells
decreases, during AIDS late stage, and can be described taking into
account the X4 related Tumor Necrosis Factor dynamics. Phylogenetic
inference of chemokine receptors suggests that virus mutational pathway
may generate R5 variants able to interact with chemokine receptors different
from CXCR4. This may explain the massive signalling disruptions in the
immune system observed during AIDS late stages and may have relevance
for vaccination and therapy. Xinzhi Liu, University of Waterloo With the rapid development of personal communications and the Internet,
information security has become an increasingly important issue of the
telecommunication industry. Recently, there has been tremendous worldwide
interest in exploiting chaos for secure communications. The idea is
to use chaotic systems as transmitters and receivers, where the message
signal is added to a chaotic carrier generated by the transmitter system
and it is recovered at the receiver through synchronization. This talk
will discuss the method of impulsive synchronization for chaos-based
secure communication systems in the presence of transmission delay and
sampling delay and the related theory of impulsive dynamical systems
with time delay, which provides the main framework for modeling the
error dynamics between the driving and response systems employed in
such communication systems. Marcus Pivato, Trent University: Let L:= Z^D be the D-dimensional lattice, and let A^L be the Cantor
space of L-indexed configurations in some finite alphabet A, with the
natural L-action by shifts. A `cellular automaton' is a continuous,
shift-commuting self-map F of A^L, and an `F-invariant subshift' is
a closed, F-invariant and shift-invariant subset X of A^L. Suppose x
is a configuration in A^L that is X-admissible everywhere except for
some small region we call a `defect'. It has been empirically observed
that such defects persist under iteration of F, and often propagate
like `particles' which coalesce or annihilate on contact. We construct
algebraic invariants for these defects, which explain their persistence
under F, and partly explain the outcomes of their collisions. Some invariants
are based on the cocycles of multidimensional subshifts; others arise
from the higher-dimensional (co)homology/homotopy groups for subshifts,
obtained by generalizing the Conway-Lagarias tiling groups and the Geller-Propp
fundamental group. Mary Pugh, Univerity of Toronto If ten genes affect an individual's height and one has a population
with average height 6'5" that is forced to live in low-ceilinged
caves, how might this selective pressure act at the genetic level of
individuals? What happens if the different genes have different magnitudes
of effect? A nonlocal diffusion model is constructed and studied for
the joint distribution of absolute gene effect sizes and allele frequencies
for genes contributing to an quantitative trait in a haploid population
where there is a selection pressure on the quantitative trait. We present a simple yet efficient dynamic system for real-time collision-free
robot path planning applicable to situations where targets and barriers
are permitted to move. The algorithm requires no prior knowledge of
target or barrier movements. In the static situation, where both targets
and barriers do not move, our algorithm is a dynamic programming solution
to the shortest path problem, but restricted by lack of global knowledge.
In this case the dynamic system converges in a small number of iterations
to a state where the minimal distance to a target is recorded at each
grid point, and our robot path-planning algorithm can be made to always
choose an optimal path. In the case that barriers are stationary but
targets can move, the algorithm always results in the robot catching
the target provided it moves at greater speed than the target, and the
dynamic system update frequency is sufficiently large. We also look
at how the algorithm can be modified to choose paths that not only reach
the target via the shortest possible route, but also shun obstacles.
The effectiveness of the algorithm is demonstrated through a number
of simulations. Gail Wolkowicz, McMaster University: After pointing out the problems with the classical delayed logistic
equation as a model of population growth, an alternative expression
for a delayed logistic equation is derived. It is assumed that the rate
of change of the population depends on three components: growth, death,
and intraspecific competition, with the delay in the growth component.
In our formulation, we incorporate the delay in the growth term in a
manner consistent with the rate of instantaneous decline in the population
given by the model. After a complete global analysis of the model, the
dynamics are compared with the dynamics of the classical logistic delay
differential equation model, the classical logistic ordinary differential
equations growth model, and various other more recent formulations.
Implications of our analysis for including delay in such models is also
discussed. This is joint work with Julien Arino and Lin Wang
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