0$ and $a,b$ are continuous
and bounded coefficients. We assume that $0 \le a(x) \le \lim_{|x|
\to \infty} a(x)$ and $b(x) \ge \lim_{|x| \to \infty} b(x) >0$. In
particular, $a$ may vanish somewhere in $\mathbb{R}^N$; in such a
case, we impose additional conditions on the location of zeros of
$a$ and the local summability of some negative power of $a$. We
study the effect of the properties of the coefficients $a$ and $b$
on the existence and multiplicity of solutions to (1).
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\newpage
\begin{center}
{\Large \bf Special Sessions}
\end{center}
\begin{center}
{\Large \bf Mathematical Aspects of Wave Propagation}\\
Organizer: Boris Belinskiy, University of Tennessee at Chattanooga
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Exact control of a string under an axial stretching tension}\\ \\
%author
{\bf Sergei A. Avdonin}\\
%affiliation
University of Alaska at Fairbanks, USA \\
%e-mail
email:ffsaa@uaf.edu \\
%coauthor
{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We study the exact controllability problem for a string
under an axial stretching tension. We are looking for an exterior
force $g(x)f(t)$ that drives the state solution to the rest. The
tension is a sum of two terms. The first term is a positive
constant and the second one is small and slowly varies in time.
The function $f(t)$ is considered as a control. The problem of
exact controllability is reduced to a moment problem for $f(t)$.
If the tension is a constant, the moment problem for a system of
non-harmonic exponentials appears (see the classical papers by
D.L. Russell). In our case, the tension varies in time, and the
moment problem is more complex. Its solution requires the proof of
the basis property for a system of special solutions of a second
order differential equation with variable coefficients.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Stability of mechanical system with multiplicative white noise}\\ \\
%author{
{\bf Peter Caithamer}\\
%affiliation
US Military Academy, West Point, USA\\
%e-mail
email:ap6939@exmail.usma.army.mil \\
%coauthor
{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The asymptotic behavior as $t\to \infty$ of some linear mechanical
systems described by stochastic PDE with multiplicative white
noise is considered. Ito and Stratonovich interpretations are
used. They lead to different conditions of stability with respect
to the expected energy. If the initial data are random
(independent on white noise) but the parameters are not, the
systems are stable. Otherwise the expected energy may be infinite,
approach zero, remain bounded, or increase with no bound. The
necessary and sufficient conditions for stability in terms of the
structure of roots of an auxiliary equation are formulated.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Analyticity of semigroup for Mead-Markus sandwich plate }\\ \\
%author
{\bf Scott Hansen}\\
%affiliation
Iowa State University, USA\\
%e-mail
email:shansen@iastate.edu
%coauthor
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The classical sandwich plate model of Mead and Markus has been
very useful in explaining vibrational properties of composite
structures. In particular this model explains why damping due to
shear in the middle layer (i.e., constrained layer damping) leads
to much higher dissipation of energy at low frequencies than other
forms of damping. In this paper we formulate a general version
(for $n$ layers) of the sandwich plate model and show that shear
damping in alternate layers leads to analyticity of the associated
semigroup.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent{\bf The resolvents of operators in several classes of integrodifferential equations }\\ \\
%author{
{\bf Min He }\\
%affiliation
Kent State University Trumbull Campus, USA \\
%e-mail
email:mhe@kent.edu
%coauthor
%{\bf P. Puri}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The resolvent of an operator plays a crucial role in determining
properties of the semigroup generated by the operator. The
interest of this paper is to develop an effective way of
determining properties of semigroups (thus leading to the
properties of solutions) with respect to parameters through
obtaining the properties of resolvents in abstract
integrodifferential equations. A specific discussion focuses on a
problem arising in viscoelasticity .
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent{\bf Some recent findings concerning unsteady dipolar fluid flows}\\ \\
%author{
{\bf Pedro Jordan}\\
%affiliation
Naval Research Laboratory, USA \\
%e-mail
email:pjordan@nrlssc.navy.mil\\
%coauthor
{\bf P. Puri}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Stokes' first and second problems, as well as that of plane
Couette flow, are considered for incompressible dipolar fluids,
dipolar fluids being the simplest example of a class of
non-Newtonian fluids known as multipolar fluids. Laplace transform
methods are used to determine exact solutions, for arbitrary
values of the dipolar constants $d(\geq 0)$ and $l(>0)$, to these
unsteady flow problems. In considering special/limiting cases of
the dipolar constants, exact solutions are also determined for
Rivlin-Ericksen, couple stresses, and viscous Newtonian fluids. In
particular, steady-state development time, displacement
thickness/mass flux, and boundary layer thickness are determined
for the various fluid types and results for all fluids are then
compared. The influence of $d$, $l$, and the other physical
parameters on the velocity field is illustrated, as well as the
effects of start/stop plate motion in the Couette flow case. Most
significantly, we show that the velocity field suffers a jump
discontinuity at start-up when $d>0$, a backflow condition is
possible when $l>d$, a phase velocity term is negative when $l>d$,
and for special values of the physical parameters the flow
instantly attains steady-state. Lastly, bounds are placed on the
value of the dipolar traction at the plate(s).
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf An averaging method for the Helmholtz equation}\\ \\
%author{
{\bf Sikimeti Ma'u}\\
%affiliation
University of Auckland, New Zealand \\
%e-mail
email:sikimeti@math.auckland.ac.nz\\
%coauthor
{\bf Boris Pavlov}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The well-known J. Schauder result on the existence of
Lip$_\alpha(\Omega)$ solutions of the Dirichlet problem for
bounded domains with smooth boundaries is obviously true for the
Helmholtz equation $-\triangle u = \lambda u$, with spectral
parameter different from eigenvalues of the corresponding
homogeneous problem. We suggest a new method of construction of
the solution, based on an averaging procedure and mean-value
theorem. We show that, for $0 < \alpha < 1$, and $\lambda \le
\lambda_0$, where $\lambda_0$ is a positive parameter depending on
the "effective width" of the domain, the sequence of iterated
averages converges in Lip$_\alpha(\Omega)$ as a geometric
progression.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Optimal design of turbines}\\ \\
%author
{\bf C. Maeve McCarthy}\\
%affiliation
Murray State University, USA\\
%e-mail
email:maeve.mccarthy@murraystate.edu \\
%coauthor
{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We minimize, with respect to shape, the lowest frequency of the
torsional oscillations of a turbine, subject to fixed moment of
inertia. The problem is modelled by a Sturm-Liouville problem with
eigenparameter dependence in the boundary conditions.
Rearrangements with respect to a weight lead to the existence of
an optimal design. Optimality conditions are also established.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Some aspects of output determination for distributed parameter}\\ \\
%author{
{\bf David Russell}\\
%affiliation
Virginia Tech, USA \\
%e-mail
email:russell@calvin.math.vt.edu
%coauthor
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider a linear input-output system in the abstract form
$dx/dt=Ax+bu$, $x$,$b$ in $H$, $y=_H$, $c$ in $H$, $u$ scalar,
where H is a Hilbert space. The goal is to be able to determine
the output function $y(t)$ on an interval $[t_0,\infty)$, $t_0
>0$, through appropriate choice of $u(t)$ on $[0,\infty)$. The values
of $y(t)$ on $[0,t_0)$ are ignored. Clearly we need $U(s)=Y(s)
/T(s)$, where $T(s) = <(sI-A)^{-1}b,c>$ is the transfer function
and $Y(s)$ and $U(s)$ are the Laplace transforms of $y(t)$ and
$u(t)$, respectively. Since $T(s)$ will, in most cases of
interest, have an infinite collection of right half plane zeros;
the requirement $u$ in $L^2(0,\infty)$ implies that $Y(s)$ must
have zeros at those same points. This is arranged, if it can be
arranged, through the choice of $y(t)$ on $[0,t_0)$; the choice of
$t_0$ depends on the system in question. Additionally $T(s)$ may
tend to zero at various rates depending on how $s$ approaches
infinity; this leads to further requirements on $y(t)$, not
necessarily restricted to $[0,t_0)$; e.g. differentiability and/or
related requirements on the whole output $y(t)$ may thereby be
indicated. A number of specific cases related to applications will
be considered.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Realization theory of Herglotz-Nevanlinna matrix-valued functions}\\ \\
%author{
{\bf Eduard Tsekanovskii}\\
%affiliation
Niagara University, USA \\
%e-mail
email:tsekanov@niagara.edu
%coauthor
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
A survey of recent results in realization theory of
Herglotz-Nevanlinna matrix- valued functions, interpolation
problems and explicit system solutions is presented. An extension
of Brodski\u{\i}-Liv\v{s}ic systems (operator colligations),
involving an unbounded main operator and an additional orthogonal
projection, is studied. It leads to new types of representation
and realization results for certain classes of Herglotz-Nevanlinna
functions and for the associated transfer functions. The
realization criterion of general Herglotz-Nevanlinna matrix-valued
function in terms of linear fractional transformation of transfer
function of the systems involving triplets of Hilbert spaces
(rigged operator colligations) is established. We consider also a
new type of solutions of Nevanlinna--Pick interpolation problem,
so--called explicit system solutions generated by Brodskii--Livsic
time-invariant scattering systems(colligations), and find
conditions on interpolation data of their existence, uniqueness
and restoration.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Nonlinear Boundary Value Problems
} \\
Organizer: Anna Capietto, University of Torino, Italy
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Critical and subcritical nonlinear Schr\"odinger
equation with magnetic field
}\\ \\
%author
{\bf Gianni Arioli }\\
%affiliation
Universit\'a del Piemonte Orientale, Italy\\
%e-mail
email:gianni@unipmn.it
%coauthor
%{\bf Boris Belinskiy}
%abstract
\\
We consider the nonlinear stationary Schr\"odinger equation in a
magnetic field $\hat H\psi=f(\psi)$, where $\hat
H=\frac{1}{2m}\left(-i \hbar \nabla-\frac{e}{c}A\right)^2+eV$. $V$
is the scalar potential and $A$ is the vector potential. We study
the existence and nonexistence of solutions for the equation under
different assumptions on the potentials and on the nonlinear part
$f$. We consider both the critical case $f(x)=|x|^{2^*-2}x$ and
the subcritical case $|f(x)|<|x|^p$, $p<2^*-1$. The results are
obtained by variational methods; in particular the solutions are
obtained either as constrained minima or as mountain pass points.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Multiple positive solutions for quasilinear
elliptic boundary value problems
}\\ \\
%author
{\bf Maya Chhetri }\\
%affiliation
UNC Greensboro, USA\\
%e-mail
email:maya@uncg.edu \\
%coauthor
{\bf R. Shivaji}
%abstract
\\
We will discuss multiplicity of positive solutions for a class of
quasilinear elliptic boundary value problem. We prove our result
by using the method of sub and super solutions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On the existence of solutions and their long time
behavior of a nonlocal thermistor inequality
}\\ \\
%author
{\bf Shuqing Ma }\\
%affiliation
University of Alberta, Canada\\
%e-mail
email:shuqing@ualberta.ca
%coauthor
%{\bf Shivaji, R.}
%abstract
\\
We study an obstacle problem which models the behavior of certain
micromachined microsensor devices: \begin{eqnarray}{r}
\frac{du}{dt}-\Delta u + \eta \int_{\Omega}G(x,y)u(y,t)dy + \gamma
u^4 \nonumber\\ \geq \nabla[\sigma(u) \phi \nabla \phi], \nonumber\\
-\nabla[\sigma(u)\nabla \phi] = 0. \nonumber \end{eqnarray} Here
the unknown functions $u$ and $\phi$ denotes the distributions of
the temperature and the electrical potential in the electrical
devices respectively. Let $\Omega$ be a domain in $R^3$. Its
boundary $\partial \Omega$ is divided into three parts, i.e.,
$\partial \Omega = \Gamma_0\cup \Gamma_1\cup\Gamma_N$. On the
boundary, the temperature $u$ satisfies a homogenous Dirichlet
boundary condition. While the potential $\phi$ satisfies
$\phi|_{\Gamma_0} = \phi_0(x,t)$, $\frac{\partial \phi}{\partial
n}|_{\Gamma_N} =0$ and $\phi|_{\Gamma_1} = \xi(t)$. Here $\xi(t)$
is an unknown constant for each $t$. But the total current $I(t)$
through $\Gamma_1$ is given for each time $t$. Thus another
nonlocal boundary condition is given by \begin{eqnarray} && I(t) =
\int_{\Gamma_1}\sigma(u)\frac{d \phi}{dn} ds. \nonumber
\end{eqnarray} We first consider the initial value case, i.e.,
$u(x,0) = u_0(x)$ with $u_0(x)$ a known function. The existence of
a unique solution is established by a penalized method. Next, if
$\phi_0(x,t)$ and $I(t)$ are time periodic functions with period
$T$, by setting up a Poicar\'{e} map and applying Schauder's fixed
point theorem we show that there exists a time periodic solution
such that $u(x,t+T) = u(x,t)$. Finally the long time behavior of
the solutions is studied. Actually it is characterized by a
uniform attractor. The upper bound of the dimensions of the
attractors are also investigated.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Some multi-point boundary value problems containing
the operator $-(\phi(u'))'$
}\\ \\
%author
{\bf Raul Man\'asevich }\\
%affiliation
Universidad de Chile, Chile \\
%e-mail
email:manasevi@dim.uchile.cl
%coauthor
%{\bf Shivaji, R.}
%abstract
\\
In this talk we will consider some result for problems of the form
$$(\phi(u'))'=f(t,u,u'),\ \ t\in(a,b), $$ under the multi-point
boundary conditions $$u(a)=0, \quad u'(\eta)=u'(b),$$ and
$$u'(a)=0, \quad u(\eta)=u(b),$$ where $\eta \in (a,b)$ is given.
We will consider the case when problem $(P)$ is at resonance.
Three-point boundary value problems at resonance have been studied
in several papers, we present some new result as well as
generalizations to the general operator of other results valid for
particular forms of the operator $-(\phi(u'))'$.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Boundary value problems on sequence spaces}\\ \\
%author
{\bf Jesus Rodriguez }\\
%affiliation
North Carolina State University, USA \\
%e-mail
email:rodrigu@math.ncsu.edu
%coauthor
%{\bf Shivaji, R.}
%abstract
\\
In this paper we study the solvability and qualitative properties
of solutions of nonlinear operator equations of the form $$
Lx=F(u,x) $$ subject to $$ H(u,x)=0 $$ where $x$ belongs to a
sequence space,$u$ is a parameter, $L$ is a linear map from the
sequence space into itself, $F$ and $H$ are smooth nonlinear maps,
the range of $F$ is contained in the sequence space and the range
of $H$ is contained in some $n$-dimensional Euclidean space.
Connections with differential equations will be established.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Nonlinear boundary value problems of the Calculus
of Variations
}\\ \\
%author
{\bf Felix Sadyrbaev }\\
%affiliation
University of Latvia, Latvia \\
%e-mail
email:felix@cclu.lv
%coauthor
%{\bf Shivaji, R.}
%abstract
\\
We consider various nonlinear boundary value problems arising in
the extremal theory for the functional $$ I(x) = \int_a^b L
(t,x,x') \, dt, $$ where $L$ is sufficiently smooth. The free end
point problem, the basic problem of the calculus of variations,
the Bolza problem and others are considered. We are interested
mainly in the existence of extremals (solutions of the Euler
equation $L_x = \frac{d}{dt}L_{x'}$ together with related boundary
conditions) and properties of extremals. First, we discuss the
method of upper and lower solutions (functions) and its
variational meaning. Second, we derive several the Bernstein -
Nagumo type conditions, which are formulated directly in terms of
the Lagrangian $L.$ We show the role of those conditions in the
theory of coercive, non-coercive (slow-growth) variational
problems and discuss also the so called problem of regularity of
solutions in the calculus of variations. Third, we consider
properties of extremals, which are expressed in terms of the
linearized equation (the Jacobi equation) and discuss several
problems arising in the theory of the second variation for the
functional $I(x).$
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Existence and uniqueness for a class of quasilinear
elliptic boundary value problems
}\\ \\
%author
{\bf Ratnasingham Shivaji }\\
%affiliation
Mississippi State University, USA \\
%e-mail
email:shivaji@Math.MsState.Edu
%coauthor
%{\bf Shivaji, R.}
%abstract
\\
We prove existence and uniqueness of positive solutions for the
boundary value problem $(r^{N-1}\phi (u^{\prime }))^{\prime
}=-\lambda r^{N-1}f(u),\;u^{\prime }(0)=u(1)=0$, where $\phi
(x)=|x|^{p-2}x$, $\frac{f(x)}{x^{p-1}}$ may not be decreasing on
$(0,\infty ),$ and $\lambda $ is a large parameter.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Some multiplicity results for polyharmonic elliptic
problems with broken of symmetry
}\\ \\
%author
{\bf Marco Squassina }\\
%affiliation
Universit\'a Cattolica del S.C. - Brescia, Italy \\
%e-mail
email:squassin@dmf.bs.unicatt.it
%coauthor
%{\bf Shivaji, R.}
%abstract
\\
By means of a perturbation argument devised by P. Bolle we
investigate the existence of infinitely many solutions for the
problem $(-\Delta)^Ku=|u|^{\sigma-2} u+\varphi \quad\,\, \mbox{in
$\Omega$} $ with nonhomogeneous Dirichlet boundary conditions
$\left.\left(\frac{\partial}{\partial \nu}\right)^j u\right
|_{\partial\Omega}=\phi_j \quad\quad j=0,\ldots,K-1\,, $ provided
that suitable growth restrictions on $\sigma$ are assumed. \par
Moreover, when $\sigma$ reaches the critical growth and
$\phi_j=0$, we show the existence of multiple solutions when the
domain and the nonhomogenous term are invariant with respect to
some group of symmetries.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Periodic solutions to some N body problems
}\\ \\
%author
{\bf Susanna Terracini }\\
%affiliation
Universit\'a di Milano-Bicocca, Italy \\
%e-mail
email:suster@matapp.unimib.it
%coauthor
%{\bf Shivaji, R.}
%abstract
\\
A remarkable new solution of the 3-body problem was recently
discovered by Chenciner and Mongomery, using some symmtery
argument. We shall show how many non collision periodic solutions
can be constructed exploiting symmetries of the systems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On the existence of periodic solutions to second
order ordinary differential equations
}\\ \\
%author
{\bf James Ward }\\
%affiliation
University of Alabama at Birmingham, USA \\
%e-mail
email:ward@vorteb.math.uab.edu
%coauthor
%{\bf Shivaji, R.}
%abstract
\\
We consider second order differential equations and systems of the
form $$ u''+g(u)=p(t), $$ \noindent where $g$ is a continuous
function and $p(t+T)=p(t)$. In the scalar case we assume $g(s)s
\ge 0$ for $|s|$ sufficiently large. We provide sufficient
conditions for the existence of $T$-periodic solutions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Positive solutions for second order nonlinear
equations with singularities
}\\ \\
%author
{\bf Fabio Zanolin}\\
%affiliation
Universit\'a di Udine, Italy \\
email:zanolin@dimi.uniud.it
%coauthor
%{\bf Shivaji, R.}
%abstract
\\
We consider the two-point (Dirichlet) boundary value problem for
the second order scalar equation
$$u'' + f(t,u) = 0,\quad u(0) = u(1) = 0$$
where the presence of singularities both in $t$ and in $u$ is
allowed.
In particular, we present some recent results (jointly with
Gaudenzi and Habets) on the existence of positive solutions in the
case when $f(t,u) = a(t) g(u)$ and $a:[0,1]\to {I\!\!R}$ is not
necessarily of constant sign and may have singularities at $t=0$
and $t=1.$
Proofs are based on a development of the method of lower and upper
solutions to the singular case. The situation in which
$\alpha\not\leq \beta$ (with $\alpha(t)$ and $\beta(t)$ a lower
and an upper solution, respectively), is considered as well.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Stabilization and Optimal Control of Dynamical Systems}\vskip
-0.15in \
$$\begin{array}{rl}
\mbox{Organizer:}& \mbox{Yacine Chitour, University of Paris Sud} \\
& \mbox{Ugo Boscain, University of Bourgogne} \end{array} $$
\end{center}
\vskip .2in
\begin{multicols}{2}
% ----------------------------------------------------------------
%\title
\noindent {\bf Optimal control and generalized rigid body dynamics }\\ \\
%author
{\bf Anthony Bloch }\\
%affiliation
University of Michigan, USA \\
%e-mail
email:abloch@math.lsa.umich.edu
%coauthor
%{\bf P. Perera Texas Tech University }
%abstract
%\begin{center}{\bf Abstract} \end{center}
\\
In this talk I will discuss the geometry of certain optimal
control problems and relate them to geodesic flows on symmetric
spaces, smooth and discrete generalized rigid body problems and
inviscid fluid flow. In particular I will show how to obtain a
discrete symmetric form of the rigid body equations and relate
this form to the discrete rigid body equations of Moser and
Veselov. In the fluid case I will relate the problem to the
so-called impulse equations. This is joint work with P. Crouch and
also with D. Holm, J. Marsden and T. Ratiu.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Lower and upper bounds for the number of switchings for time optimal trajectories of the Dubin's problem on SO(3) }\\ \\
%author
{\bf Ugo Boscain }\\
%affiliation
University of Bourgogne , France \\
%e-mail
email:uboscain@u-bourgogne.fr \\
%coauthor
{\bf Yacine Chitour}
%abstract
%\begin{center} \bf Abstract} \end{center}
\\
In this paper, we investigate the structure of time-optimal
trajectories for the Dubin's control system on $SO(3)$: $\dot
x=x(f_1+u f_2)$, $|u|\leq 1$, where $f_1,f_2\in SO(3)$ define two
linearly independent left-invariant vector fields on $SO(3)$. In
particular we are interested to find a lower and an upper bound on
the number of switchings for bang-bang trajectories as function of
the parameters of the problem. The lower bound is obtained by
studying the projection of this control system on the two
dimensional sphere by an appropriate Hopf fibration. For the
projected problem we compute the optimal synthesis. This is a
joint work with Yacine Chitour.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On the stabilizability of controllable switching systems }\\ \\
%author
{\bf Wijesuriya Dayawansa }\\
%affiliation
Texas Tech University, USA\\
%e-mail
email:daya@math.ttu.edu \\
%coauthor
{\bf P. Perera }
%abstract
%\begin{center}\bf Abstract} \end{center}
\\
We consider a linear switching system of the form $\dot{x} = Ax +
Bu$, where $A \in \{A_i, i=1, \cdots, k\}$, and $B$ is fixed. It
has been shown recently that the controllability of such systems
is equivalent to the Lie ideal consisting of constant vector
fields contained in the Lie algebra generated by $\{A_ix,~i=1,
\cdots, k\}$ and $B$, span the state space. An important
unresolved question is whether controllability implies
stabilizability in an appropriate sense, and if so, what is the
analog of the pole placement theorem for fixed linear control
systems. In this note we take an indirect approach to studying
this problem. Our first aim is to ask whether it is possible to
find positive feedback functions $\alpha_i(x), ~i=1, \cdots, k$
such that $\dot{x} = \sum_{i=1}^k \alpha_i(x) A_i(x) + Bu$ is
controllable. In this note it will be shown that such feedback
functions always exist, and in addition, they can be found in such
a way that the closed loop system becomes a weighted homogeneous
control system. It will be shown that the trajectories of the
closed loop system can be approximated via switched trajectories
of the open loop system. This will be used to derive a
stabilization theorem.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Verification theorems for Hamilton-Jacobi-Bellman equations }\\ \\
%author
{\bf Mauro Garavello }\\
%affiliation
SISSA, Italy \\
%e-mail
email:mgarav@sissa.it
%coauthor
%{\bf Elisha FALBEL University of Paris 6 }
%abstract
%\begin{center}{\bf Abstract} \end{center}
\\
We study an optimal control problem in Bolza form and we consider
the value function associated to this problem. We prove two
verification theorems which ensure that, if a function $W$
satisfies some suitable weak continuity assumptions and a
Hamilton-Jacobi-Bellman inequality outside a rectifiable set of
codimension one, then it coincides with the value function.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On noncoercive quadratic forms }\\ \\
%author
{\bf Manuel Guerra }\\
%affiliation
ISEG/UTL, Portugal\\
%e-mail
email:mguerra@iseg.utl.pt
%coauthor
%{\bf Yacine Chitour}
%abstract
%\begin{center}{\bf Abstract} \end{center}
\\
Nonautonomous linear-quadratic (LQ) problems are an important
class of optimal control problems that arise naturally in the
study of second-order optimality conditions for nonlinear
problems. The properties of LQ problems that satisfy a
strengthened Legendre condition are, in most aspects, well
understood. In particular, the cost functional is coercive for
time intervals up to the first conjugate point. LQ problems that
don't satisfy this strengthened Legendre condition are far less
understood. In this cases it can be difficult to ascertain whether
the second variation of the cost is nonnegative. The loss of
coercivity also implies that, even if the second variation is
nonnegative, the existence of a minimum is not guaranteed. We
present some conditions that are necessary for the second
variation of a problem of this type to be nonnegative. These
conditions are extensions of the well known Goh and generalized
Legendre-Clebsch conditions. For LQ problems that satisfy these
conditions, we show how to provide the space of controls with a
new topology, weaker then the usual topology of $L_{2}$, such that
the problem can be extended by continuity onto the topological
completion of $L_{2}$. If the second variation is positive
definite, then the cost functional is coercive with respect to
this new topology and hence standard theorems can be used to prove
existence and uniqueness of generalized optimal solutions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Measures of transverse paths in sub-Riemannian geometry }\\ \\
%author
{\bf Frederic Jean }\\
%affiliation
ENSTA, France \\
%e-mail
email:fjean@ensta.fr \\
%coauthor
{\bf Elisha Falbel }
%abstract
%\begin{center}{\bf Abstract} \end{center}
\\
We define a class of lengths of paths in a sub-Riemannian
manifold. It includes the length of horizontal paths but it also
measures the length of transverse paths. It is obtained by
integrating an infinitesimal measure which generalizes the norm on
the tangent space. This requires to define and study the metric
tangent space (in Gromov sense). As an example we compute
geometrically those measures in the case of contact sub-Riemannian
manifolds.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On flatness in the smallest nontrivial dimensions
}\\ \\
%author
{\bf Jean-Baptiste Pomet}\\
%affiliation
INRIA, France \\
%e-mail
email:pomet@sophia.inria.fr
\\
%coauthor
{\bf D. Avanessoff
}
%abstract
%\begin{center}{\bf Abstract} \end{center}
\\
This talk will be a review of results on flatness of control
systems with two inputs and three states. According to a
previously known necessary condition for flatness, systems of
these dimensions which are flat may be transformed into a
\emph{control affine} system with four states and two inputs. A
characterization of $(x,u)$-flatness for these systems have been
given by the second author (1997). The results are presented here
in more geometrical manner, and with much simpler proofs.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Control of quantum mechanical processes: models, techniques and practice }\\ \\
%author
{\bf Viswanath Ramakrishna }\\
%affiliation
University of Dallas, USA \\
%e-mail
email:vish@utdallas.edu
%coauthor
%{\bf Elisha Falbel University of Paris 6 }
%abstract
%\begin{center}{\bf Abstract} \end{center}
\\
Several applications motivate the control, observation and
optimization of quantum mechanical systems. Included among them
are ultrafast photochemistry and spectroscopy in the visible,
infrared, ultrviolet and radiofrequency regimes; quantum
information processing and of course, quantum computation. The
talk will first address the issue of models for the control of
quantum systems in such applications. It will be argued that for
many problems (though by no means all) finite-dimensional models
have good predictive power. The talk will, besides the usual
unitary models, include other finite-dimensional models which
arise when certain physical objectives can be reformulated as the
control over certain observables, as opposed to the wave function
itself. A brief synopsis of the kinds of approximations (at least
within spectroscopy) that go into unitary models will be given,
with emphasis on the issue that control of such systems has to
naturally respect the regimes for these approximations. Next, the
talk will dwell on two techniques for the control of such systems.
The first is centered around the notion of factorization of unitary
matrices and leads to new techniques even for classical systems.
The second, dubbed learning control is an immensely popular method
which has found immediate applications but is yet to be
thoroughly, mathematically investigated. Finally, time permitting
some discussion of pulse shaping laboratory techniques will be
provided.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip 0.1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Optimal control of a spatial shuttle }\\ \\
%author
{\bf Emmanuel Trelat }\\
%affiliation
University Paris XI, France \\
%e-mail
email:emmanuel.trelat@math.u-psud.fr
%coauthor
%{\bf Yacine Chitour}
%abstract
%\begin{center}{\bf Abstract} \end{center}
\\
The aim of this talk is to make some geometric remarks and some
numerical simulations in order to construct the optimal
atmospheric arc of a spatial shuttle (problem of reentry on Earth
or Mars Sample Return project). The system describing the
trajectories is in dimension 6, the control is the bank angle and
the cost is the total thermal flux. Moreover there are state
constraints (thermal flux, normal acceleration and dynamic
pressure). Our study is mainly geometric and is founded on the
evaluation of the accessibility set taking into account the state
constraints. We make an analysis of the extremals of the Minimum
Principle in the non-constrained case, and give a version of the
Minimum Principle adapted to deal with the state constraints.
Numerical simulations are done using shooting methods.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Recent Developments in Mechanical Systems and Geometric
Control Theory
}\\
Organizer: Manuel de Leon and Alberto Ibort,Spain
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Advances on numerical integrators for nonholonomic systems }\\ \\
%author
{\bf Jorge Cort\'es }\\
%affiliation
University of Twente, The Netherlands \\
%e-mail
email:j.cortesmonforte@math.utwente.nl \\
%coauthor
{\bf S. Martínez Consejo}
%abstract
\\
In the last few years, variational integrators derived from
Veselov's discretization technique has grown out to be a very
large and active area of research. These are numerical integrators
that preserve the symplectic structure and the momentum conserved
quantities for a given mechanical system. In this talk, we will
focus our attention on one of the many interesting avenues of
exploration that variational integrators open: namely, that which
deals with mechanical systems subject to nonholonomic constraints.
We will present a discretization of the Lagrange-d'Alembert
principle, which will allow us to derive the discrete nonholonomic
equations of motion. We will also present a study of the geometric
invariance properties of this discrete flow, which provide an
explanation for the good performance of the proposed method.
Several examples will illustrate the results.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Cartan’s approach applied to nonholonomic geometry }\\ \\
%author
{\bf Kurt Ehlers }\\
%affiliation
TMCC, Reno Nevada, USA \\
%e-mail
email:kehlers@scs.unr.edu
%coauthor
%{\bf S. Martínez Consejo Superior de Investigaciones Científicas}\\
%abstract
\\
I shall discuss the application of Cartan’s method equivalence to
nonholonomic geometry. The method of equivalence is an algorithmic
procedure for uncovering differential invariants associated to
geometric structures. As an example, the method applied to
Riemannian manifolds yields the Levi-Civita connection and
Riemannian curvature apparatus. The problem can be described as
follows. Let $(M, D, g_{1}) $ and $(N, C, g_{2})$ be Riemannian
manifolds with completely non-integrable distributions $D$ and
$C$. We take as metrics on $D$ and $C$ the restrictions of $g_{1}$
and $g_{2}$ to the distributions. The basic problem is then to
find invariants characterizing the existence of local
diffeomorphisms from $M$ to $N$ that preserve the nonholonomic
geometry. As particular examples, I will discuss the examples of
nonholonomic geometry on contact and Engel manifolds. Robert
Bryant and his students have successfully applied the method to
the study of sub-Riemannian geometries on several types of
distributions. In the nonholonomic case, one begins the algorithm
with a smaller structure group and this leads to additional
differential invariants.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On the topology and geometry of singular optimal control problems }\\ \\
%author
{\bf Alberto Ibort }\\
%affiliation
Universidad Carlos III de Madrid, Spain \\
%e-mail
email:albertoi@math.uc3m.es
%coauthor
%{\bf S. Martínez Consejo Superior de Investigaciones Científicas}\\
%abstract
\\
We will discuss some of the geometrical and topological properties
of singular optimal control problems. We will analyze a
geometrical recursive algorithm, inspired both in the theory of
implicit differential equations and Dirac's constraints algorithm,
whose result will be a reduced Pontriaguine maximum principle for
singular systems. A global singular perturbation theory will be
discussed that will allow us to solve explicitly such reduced
systems. Finally, some examples and applications will be
presented.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf An axially symmetric rolling sphere hitting a rough wall }\\ \\
%author
{\bf Ernesto Lacomba }\\
%affiliation
University Autonoma Metropolitana, Mexico \\
%e-mail
email:lace@xanum.uam.mx
%coauthor
%{\bf S. Martínez Consejo Superior de Investigaciones Científicas}\\
%abstract
\\
Non-Holonomic systems with symmetry are described by a reduced
Lagrange-d'Alembert variational principle. The case of rolling
constraints has a long history and it has been the purpose of many
works in recent times, in part because of its applications to
robotics. In this paper we study the case of a symmetric sphere,
that is, a sphere where two of its three moments of inertia are
equal, rolling on a plane, using an abelian group of symmetry. The
presence of some impulsive constraints upon hitting a rough wall
is also studied. This is joint work with H. Cendra and W. Reartes.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Some geometric aspects of control theory }\\ \\
%author
{\bf Bavo Langerock }\\
%affiliation
Department of mathematical physics and astronomy, Belgium \\
%e-mail
email:bavo.langerock@rug.ac.be
%coauthor
%{\bf S. Martínez Consejo Superior de Investigaciones Científicas}\\
%abstract
\\
We introduce a quasi-order relation associated with an everywhere
defined family of vector fields appearing in a geometric
formulation of a control problem. A notion of variations to a
concatenation of integral curves of vector fields is considered
and is used to develop necessary conditions for an optimal control
problem with fixed time.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Control theory for affine connection control systems }\\ \\
%author
{\bf Andrew Lewis }\\
%affiliation
Queen's University, Canada \\
%e-mail
email:andrew@mast.queensu.ca
%coauthor
%{\bf S. Martínez Consejo Superior de Investigaciones Científicas}\\
%abstract
\\
It is well-known that affine differential geometry is useful in
classical mechanics as a means for organizing the description of
certain systems, e.g., the geodesics of the Levi-Civita connection
are the solutions of the Euler-Lagrange equations for kinetic
energy Lagrangians. In this talk an overview will be given of how
affine differential geometry is useful in the control theory for
mechanical systems. As control systems, the problems considered
are challenging as they are not amenable to well-established
methods in control. The emphasis will be on optimal control,
controllability, and motion planning for these systems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Reduction of symmetries }\\ \\
%author
{\bf Jedrzej Sniatycki }\\
%affiliation
University of Calgary, Canada \\
%e-mail
email:sniat@math.ucalgary.ca
%coauthor
%{\bf S. Martínez Consejo Superior de Investigaciones Científicas}\\
%abstract
\\
I shall describe the role played by manifolds of a given symmetry
type and manifolds of a given orbit type in singular reduction of
symmetries of differential equations. Reduction techniques for
ODEs will be illustrated by the symplectic reduction of
Hamiltonian systems, and the non-holonomic reduction of
distributional Hamiltonian systems. PDEs will be represented by
the Dedonder equation (multisymplectic reduction).
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Analysis and Computation of Nonlinear Elliptic PDEs} \vskip
-.15in\
$$\begin{array}{rl}
\mbox{Organizer}:&\mbox{Zhonghai Ding, University of Nevada} \\
&\mbox{Hossein Tehrani, University of Nevada} \end{array}$$
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf A Bolza--type problem in a Riemannian manifold }\\ \\
%author
{\bf Anna Maria Candela }\\
%affiliation
Universit\'a di Bari, Italy \\
%e-mail
email:candela@dm.uniba.it \\
%coauthor
%{\bf P. Girao }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Consider the nonlinear equation
\[ D_s\dot x + \lambda \nabla_x V(x,s) = 0 \]
in a complete Riemannian manifold $M$. If $V$ has a
quadratic growth with respect to $x$, then a ``best constant''
$\bar\lambda > 0$ exists such that if $\lambda < \bar\lambda$ any
couple of points in $M$ can be joined by a curve which is solution
of the given equation.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
%\title
\noindent {\bf On a class of critical perturbations of a Neumann elliptic problem with limiting Sobolev exponent }\\ \\
%author
{\bf David Costa }\\
%affiliation
Univ of Nevada at Las Vegas,USA \\
%e-mail
email:costa@nevada.edu\\
%coauthor
{\bf P. Girao }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider the question of existence of least-energy solutions
for the Neumann problem \[\left\{\begin{array}{ll} -\Delta u + a u
= u^{2^{\ast}-1} - \alpha u^{q-1} &\mbox{ in }\Omega,\\ u>0
&\mbox{ in }\Omega,\\ \frac{\partial u}{\partial \nu}=0 &\mbox{ on
}\partial\Omega, \end{array}\right.\] where $\Omega$ is a bounded
smooth domain in $\Re^N$, $N\geq 5$, $a>0$, $\alpha\geq 0$ and
$\displaystyle 2^{\ast}=\frac{2N}{N-2}$ is the limiting Sobolev
exponent for the embedding $H^{1}(\Omega)\subset L^{r}(\Omega)$.
We show that the exponent $\displaystyle q=\frac{2(N-1)}{N-2}$
plays a critical role on existence of such ground-state solutions
for these problems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Multiple nonlinear periodic oscillations in a suspension bridge system }\\ \\
%author
{\bf Zhonghai Ding }\\
%affiliation
University of Nevada, Las Vegas,USA \\
%e-mail
email:dingz@unlv.edu
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this talk, we present a recent result on multiple nonlinear
periodic oscillations in a suspension bridge system governed by
the coupled nonlinear wave and beam equations describing
oscillations in the supporting cable and roadbed under periodic
external forces. By applying a variational reduction method, it is
proved that the suspension bridge system has at least three
periodic oscillations.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Schr\"odinger type equations with asymptotically linear nonlinearities }\\ \\
%author
{\bf Francois Heerden }\\
%affiliation
Utah State University, USA\\
%e-mail
email:hfvan@cc.usu.edu \\
%coauthor
{\bf Z.Q Wang.}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider existence of solutions for the nonlinear Schr\"odinger
type equation \begin{equation} -\Delta u~+~(\lambda
g(x)~+~1)u~=~f(u), \ \ \ x\in{\bf R}^N \label{eq:nls}
\end{equation} which satisfy $u(x) \to 0$ as $|x|\to \infty$. The
potential $\lambda g(x)+1$ is strictly positive, $\lambda\geq 0$
is a real parameter and there exists a nonempty potential well
$\Omega:=\mbox{int $g^{-1}(0)$}$. No limit for $g(x)$ is assumed
as $|x|\to\infty$. The nonlinearity $f$ is assumed to be
asymptotically linear, i.e. there exists an $\alpha\in(0,\infty)$
such that $\lim_{|s|\to\infty}f(s)s^{-1}=\alpha$. The constant
$\alpha$ will be further restricted in terms of the spectrum of
the linear operator $-\Delta+1$ under Dirichlet boundary
conditions in $\Omega$. I will show that for $\lambda$
sufficiently large, a positive solution to (\ref{eq:nls}) is
obtained with minimal assumptions on $f$. I will also illustrate
how, under the assumption that $f$ is odd, the number of
eigenvalues of the operator $-\Delta+1$ beneath $\alpha$ effects
the number of solutions for (\ref{eq:nls}). The limiting behavior
of solutions as $\lambda\to\infty$ will also be considered.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Solutions with internal jump for an autonomous elliptic system of bistable type }\\ \\
%author
{\bf Carolus Reinecke }\\
%affiliation
University of Potchefstroom, South Africa \\
%e-mail
email:wskcjr@puknet.ac.za
%coauthor
%{\bf Z.-Q Wang.}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider the following system of semilinear elliptic equations
\begin{equation}
\left\{\begin{array}{rlll}
-\varepsilon^2\Delta u & = & f(u)-v & \textrm{in}\quad\Omega;\\
\gamma v-\Delta v & = & \delta u & \textrm{in}\quad\Omega;\\
u &=& v = 0& \textrm{on}\quad\partial \Omega.\\
\end{array}\right.
\end{equation}
We assume $\Omega$ to be a smooth bounded domain in $\Re^N$, with $N\ge 1$, while $\gamma$
is larger than the first eigenvalue of $-\Delta$ on $\Omega$ subjected to homogeneous
Dirichlet boundary conditions. We take $\varepsilon>0$ and $\delta \ge 0$ as parameters. The
nonlinearity we assume for simplicity to be $f(u)=u(u-1)(a-u)$ with $00$. Since the fishing quota is
fixed but may vary geographically, the harvesting rate of the fish
is assumed to be density independent and spatially nonhomogeneous.
We discuss the existence and uniqueness of stable steady state
solution and bifurcation diagram with bifurcation parameter $c$.
Related dynamics will also be discussed.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On a class of Schr\"{o}dinger equations with indefinite nonlinearities }\\ \\
%author
{\bf Hossein Tehrani }\\
%affiliation
University of Nevada, Las Vegas,USA \\
%e-mail
email:tehranih@unlv.edu
%coauthor
%{\bf Z.-Q Wang.}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We will consider Schr\"{o}dinger equations like the model
\[-\Delta u +V(x)u=a(x)|u|^{p-2}u \hspace{0.8in} x\in R^N \] where
the function $a(x)$ changes sign in $\Re^N$ ( hence the indefinite
nature of the nonlinearity) with $\lim_{|x|\rightarrow \infty}a(x)
=a_{\infty}<0$. We will investigate existence and multiplicity
results for the solutions under the assumption that
$\sigma(-\Delta +V(x))\cap (-\infty, 0]$ is a finite set.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A minimax method for computing multiple critical points in Banach space }\\ \\
%author
{\bf Xudong Yao }\\
%affiliation
Texas A\&M University, USA\\
%e-mail
email:xdyao@math.tamu.edu
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
A minimax method in Hilbert space setting has been developed by
Li-Zhou, and successfully applied to find multiple numerical
solutions to many nonlinear problems. Motivated by a semilinear
p-Laplacian equation, whose solutions coincide with critical
points of a variational functional in Lp space, this method is
generalized to fit Banach space setting. Implementation of the
method in Lp space setting will be discussed.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Invariant properties in computing multiple critical points }\\ \\
%author
{\bf Jianxin Zhou }\\
%affiliation
Texas A\&M University, USA \\
%e-mail
email:jzhou@math.tamu.edu
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Invariant sets of negative gradient flow have been introduced to
prove the existence of multiple solutions to a class of semilinear
elliptic PDE by Z. Liu and J. Sun, and are generalized and
intensively studied by S. Li and Z. Wang. In this talk, invariant
sets of negative gradient-type numerical algorithms will be
studied. It will be shown that how an invariant set may help or
may collapse in computing multiple critical points.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\newpage
\begin{center}
{\Large \bf Hyperbolic Systems of Conservation Laws and
Related Problems}\\
\hspace{-0.3in} Organizer: Haitao Fan, Georgetown University
\\
\hspace{0.75in} Tong Yang, City University of Hong Kong
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Ring formation in flows with liquid/vapor phase transitions }\\ \\
%author
{\bf Haitao Fan }\\
%affiliation
Georgetown University, USA \\
%e-mail
email:fan@archimedes.math.georgetown.edu
%coauthor
%{\bf Athanasios E. Tzavaras }
%abstract
\\
In this talk, we shall explain the peculiar multi-dimensional
phenomenon in flows with phase transitions: ring formation. The
flow is described by a reactive flow type systems. The key to the
explanation is the existence and properties of various travelling
waves of the model, which we shall discuss.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
%\title
\noindent {\bf $L^1$ stability estimate of semilinear hyperbolic systems with quadratic nonlinear source terms }\\ \\
%author
{\bf Seung-Yeal Ha }\\
%affiliation
University of Wisconsin-Madison, USA \\
%e-mail
email:ha@math.wisc.edu \\
%coauthor
{\bf Athanasios E. Tzavaras }
%abstract
\\
In this talk, we consider semilinear hyperbolic systems with
quadratic nonlinear source terms: $\partial_t f_i + \partial_x
(v_i(x,t)f_i) = \sum_{j,k} S_i^{jk} f_j f_k $. This system is
assumed to be strictly hyperbolic in the sense that all
characteristic velocities are different and this system contains
the general one-dimensional discrete Boltzmann models as an
example. Under various assumptions on the coefficients $S_i^{jk}$
and initial data, we construct nonlinear functional which is
equivalent to the $L^1$ distance and non-increasing in time $t$.
Using this nonlinear functional, we prove the $L^1$ stability of
mild solutions: $|| f(\cdot,t) - \bar f(\cdot,t)||_{L^1(\mathbb
R)} \leq G ||f_0(\cdot) - \bar f_0(\cdot)||_{L^1(\mathbb R)}$,
where $f$ and $\bar f$ are mild solutions corresponding to two
initial data $f_0$ and $\bar f_0$ and C is a constant independent
of time $t$.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Stability of the Dafermos regularization of
conservation laws
}\\ \\
%author
{\bf Xiao-Biao Lin }\\
%affiliation
North Carolina State University, USA \\
%e-mail
email: xblin@math.ncsu.edu \\
%coauthor
{\bf Stephen Schecter }
%abstract
\\
In contrast to a viscous regularization of conservation laws, a
Dafermos regularization admits many self-similar solutions. We
refer to these smooth solutions as Riemann-Dafermos solutions.
After a change of coordinates, Riemann-Dafermos solutions become
stationary, and their stability can be studied by linearization.
We study the stability of Riemann-Dafermos solutions near Riemann
solutions consisting of n Lax shock waves. We show, by studying
the essential spectrum of the linearized system in a weighted
function space, that stability is determined by eigenvalues only.
We then use asymptotic methods to study the eigenvalues and
eigenfunctions. We find there are fast eigenvalues of order
1/epsilon and slow eigenvalues of order one. The fast eigenvalues
correspond to fast convergence of initial data to travelling wave
solutions in singular layers, while the slow eigenvalues
correspond to convection in regular layers connected by travelling
waves in singular layers. For an example from gas dynamics, we
show that all the slow eigenvalues are stable.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Global structure and asymptotic behavior to the solutions of flood wave equations}\\ \\
%author
{\bf Tao Luo }\\
%affiliation
University of Michigan, USA \\
%e-mail
email:taoluo@math.lsa.umich.edu \\
%coauthor
{\bf Tong Yang }
%abstract
\\
In this talk, the global structure and large time asymptotic
behavior of solutions to the Riemann problem of the flood wave
equations will be discussed by solving the free boundary problems.
The zero relaxation asymptotic behavior of the Cauchy problem with
a class of initial data will be presented by using a modified
Glimm's scheme.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Atomic-scale localization of high-energy solitary waves on lattices }\\ \\
%author
{\bf Karsten Matthies }\\
%affiliation
University of Warwick, United Kingdom \\
%e-mail
email:matthies@maths.warwick.ac.uk \\
%coauthor
{\bf Gero Friesecke }
%abstract
\\
One-dimensional monatomic lattices with Hamiltonian $H=\sum_{n\in
Z}(\frac{1}{2}p_n^2+V(q_{n+1}-q_n))$ are known to carry localized
travelling wave solutions, for generic nonlinear potentials $V$.
In this paper we derive the asymptotic profile of these waves in
the high-energy limit $H\to\infty$, for Lennard-Jones type
interactions. The limit profile is proved to be a universal,
highly discrete, piecewise linear wave concentrated on a single
atomic spacing. The limiting equation for the profile is a
spatially discrete analogue of a system of hyperbolic conservation
laws. This shows that dispersionless energy transport in these
systems is not confined to the long-wave regime on which the
theoretical literature has hitherto focused, but also occurs at
atomic-scale localization.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Dispersive effects in a modified Kuramoto-Sivashinsky equation }\\ \\
%author
{\bf Judith Miller }\\
%affiliation
Georgetown University, USA \\
%e-mail
email:miller@math.georgetown.edu \\
%coauthor
{\bf A. Iosevich}
%abstract
\\
We study the limiting behavior of the
Kuramoto-Sivashinsky/Korteweg-de Vries (KS/KdV) equation
$$u_t=-\beta_1 u_{xx}-\beta_2 u_{xxxx}-\delta u_{xxx}-uu_x.$$ We
show that in the appropriate sense, the solutions of KS/KdV tend
to the solutions of the standard Korteweg-de Vries equation $$
v_t=-\delta v_{xxx}-vv_x, $$ as $\delta \to \infty$. The proof
relies, to a large extent, on precise estimates for oscillatory
integrals that yield pointwise bounds on Green's functions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Remarks on the Chapman Enskog expansion }\\ \\
%author
{\bf Marshall Slemrod }\\
%affiliation
UW Madison, USA \\
%e-mail
email:slemrod@math.wisc.edu
%coauthor
%{\bf A. Iosevich) }\\
%abstract
\\
This talk gives a survey on the use of the Chapman Enskog
expansion to derive reduced orders models of the Boltzmann
equation. Numerical results using this will also be given.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Asymptotic analysis and regularity of solutions to models of compressible flows }\\ \\
%author
{\bf Konstantina Trivisa }\\
%affiliation
University of Maryland, USA \\
%e-mail
email:trivisa@math.umd.edu
%coauthor
%{\bf A. Iosevich) }\\
%abstract
\\
Systems of conservation laws result from the balance laws of
continuum physics and govern a broad spectrum of physical
phenomena in compressible fluid dynamics, nonlinear materials
science, etc. Such equations admit solutions that may exhibit
various kinds of shocks and other nonlinear waves, which play an
significant role in multiple areas of physics; astrophysics,
dynamics of (solid-solid) material interfaces, multiphase
(liquid-vapor) flows, combustion theory, etc. In this talk we
present results on the well-posedness and qualitative behavior of
solutions to various systems of conservation laws, including
models for compressible flows with large, discontinuous initial
data.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Large-time behavior of real compressible reacting flows in combustion }\\ \\
%author
{\bf Dehua Wang }\\
%affiliation
University of Pittsburgh, USA \\
%e-mail
email:dwang@math.pitt.edu
%coauthor
%{\bf A. Iosevich) }\\
%abstract
\\
The equations for viscous compressible, heat-conductive, real
reactive flows in combustion are considered, where the equations
of state are nonlinear in temperature. The initial-boundary value
problem with Dirichlet-Neumann mixed boundaries in a finite
one-dimensional domain is studied. The existence, uniqueness, and
regularity of global solutions are established with general large
initial data in $H^1$. It has been proven that, although the solutions
have large oscillations, there is no shock wave, turbulence,
vacuum, mass or heat concentration developed in a finite time.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Boundary layer solutions to Boltzmann equation }\\ \\
%author
{\bf Tong Yang }\\
%affiliation
City University of Hong Kong, Hong Kong \\
%e-mail
email:matyang@cityu.edu.hk \\
%coauthor
{\bf Seiji Ukai Yokohama } and {\bf Shih-Hsien Yu }
%abstract
\\
We study the half-plane problem of the nonlinear Boltzmann
equation, assigning the Dirichlet data for outgoing particles at
the boundary and a Maxwellian as the far field. We will show that
the solvability condition of the problem changes with the Mach
number ${\mathbb M}^\infty$ of the far field Maxwellian. If
${\mathbb M}^\infty<-1$, there exists a unique smooth solution
connecting the Dirichlet data and the far field Maxwellian for any
Dirichlet data sufficiently close to the far field Maxwellian.
Otherwise, such a solution exists only for the Dirichlet data
satisfying certain admissible conditions. The set of admissible
Dirichlet data forms a smooth manifold of co-dimension 1 for the
case $-1<{\mathbb M}^\infty<0$, 4 for $0<{\mathbb M}^\infty<1$ and
5 for ${\mathbb M}^\infty>1$, respectively. We also show that the
same is true for the linearized problem at the far field
Maxwellian, and the manifold is, then, a hyperplane. The proof is
essentially based on the macro-micro or hydrodynamic-kinetic
decomposition of solutions combined with an artificial damping
term and a spatially exponential decay weight.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Nonlinear Evolution Equations and Related
Topics}\vspace{-0.2in} \\
$$\begin{array}{rl}
\mbox{Organizer:}&\mbox{Alain Haraux, Universit Pierre et Marie Curie}\\
&\mbox{Mitsuharu Otani, Waseda University} \end{array}$$
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Long-time convergence of solutions to a phase-field system }\\ \\
%author
{\bf Sergiu Aizicovici }\\
%affiliation
Ohio University, USA \\
%e-mail
email:aizicovi@math.ohiou.edu \\
%coauthor
{\bf E. Feireisl}
%abstract
\\
We discuss the long-time stabilization of bounded solutions to a
phase-field model with memory. The approach relies on the use of
analyticity, in the spirit of L. Simon [Ann. of Math. 118 (1983),
525-571.]
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Evolution equations and subdifferentials in Banach spaces }\\ \\
%author
{\bf Goro Akagi }\\
%affiliation
Waseda University, Japan \\
%e-mail
email:akagi@otani.phys.waseda.ac.jp \\
%coauthor
{\bf M. \^Otani}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this talk, we introduce a treatment of evolution equations
governed by subdifferentials in reflexive Banach spaces. Since the
pioneering work of H. Br\'ezis, many results are obtained within
the Hilbert space setting. However it seems that the study in
reflexive Banach space setting is not fully pursued yet. On the
other hand, J. L. Lions established some methods to solve
evolution equations in reflexive Banach spaces by using
Faedo-Galerkin's method. In our framework, we can obtain more
detailed information on the regularity of solutions. Our approach
relies on an approximation procedure in the Hilbert space in which
the reflexive Banach space is embedded.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Qualitative properties of a wave equation }\\ \\
%author
{\bf Jorge Esquivel-Avila }\\
%affiliation
UAM-Azcapotzalco, Mexico \\
%e-mail
email:jaea@correo.azc.uam.mx
%coauthor
%{\bf M. \^Otani}\\ Waseda University
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We present a qualitative analysis of the weak solutions of a
nonlinear dissipative wave equation. We give a characterization of
blow-up that improves previous versions. Also we characterize all
the solutions which are global and unbounded, and those which are
global and bounded. We use the concepts of stable (potential well)
and unstable sets. We give interesting characterizations of these
sets.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Blow-up rates of solutions of a semilinear parabolic equation with nonlocal nonlinearity }\\ \\
%author
{\bf Isamu Fukuda }\\
%affiliation
Kokushikan University, Japan \\
%e-mail
email:ifukuda@kokushikan.ac.jp
%coauthor
%{\bf M. \^Otani}\\ Waseda University
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We treat total vs. single point blow-up of solutions of a
semilinear parabolic equation with nonlocal nonlinearity. Moreover,
we obtain blow-up rates of solutions depending on space dimensions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations }\\ \\
%author
{\bf Takahiro Hashimoto }\\
%affiliation
Ehime University, Japan \\
%e-mail
email:taka@math.sci.ehime-u.ac.jp
%coauthor
%{\bf M. \^Otani}\\ Waseda University
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this talk, we are concerned with the existence and nonexistence
of nontrivial solutions for nonlinear elliptic equations involving
a biharmonic operator. Concerning the second order equations, a
complementary result was obtained for the problem of interior,
exterior and whole space. The main purpose of this talk is to
discuss whether the complementary result mentioned above is still
valid for the nonlinear fourth order equations. We introduce
``Kelvin type transformation'' for a biharmonic operator to
convert an exterior problem to an interior problem. The existence
results in case of super-critical exterior problem are shown by
introducing a weighted version of Sobolev-Poincar\'e type
inequality, and the nonexistence results are shown by giving a
Pohozaev-type identity for fourth order equations.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Analysis of nonlinear polarization and magnetization models }\\ \\
%author
{\bf Frank Jochmann }\\
%affiliation
Institut f\"ur angewandte Mathematik Humboldt Universit\"at Berlin, Germany \\
%e-mail
email:jochmann@mathematik.hu-berlin.de
%coauthor
%{\bf M. \^Otani}\\ Waseda University
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
This talk is concerned with the anharmonic oscillator model and
the Landau-Lifschitz equation coupled with Maxwell's equations
describing the electromagnetic field in generally nonlinear
polarizable or magnetized media. The main subject are existence
and the asymptotic behavior of the solutions to these models.
References F. Jochmann, Long time asymptotics of solutions to the
anharmonic oscillator model from nonlinear optics, SIAM J. Math.
Anal., 32, 4, (2000), 887-915 . F. Jochmann, Asymptotic behaviour
of solutions to a class of semilinear hyperbolic systems in
arbitrary domains J. Diff. Equations, 160, (2000), 439-466. G.
Carbou and P. Fabrie and F. Jochmann, A remark on the weak
$\omega$-limit set in a micromagnetism model, to appear in Appl.
Math. Lett., Vol. 15, no. 1, (2001). F. Jochmann, Convergence to
stationary states in the Maxwell Bloch system from nonlinear
optics, to appear in Quart. Appl. Math. F. Jochmann, Existence of
solutions and a quasistationary limit for a hyperbolic system
describing ferromagnetism, in preparation.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Principle of symmetric criticality and evolution equations}\\ \\
%author
{\bf Jun Kobayashi }\\
%affiliation
Waseda University, Japan \\
%e-mail
email:jun@otani.phys.waseda.ac.jp\\
%coauthor
{\bf Mitsuharu \^Otani}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Let X be a Banach space where a symmetry group G linearly acts and
let J be a G-invariant functional defined on X. In 1979, R. Palais
gave some sufficient conditions to guarantee the so-called
"Principle of Symmetric Criticality": every critical point of J
restricted on the subspace of symmetric points becomes also a
critical point of J on the whole space X. In this talk this
principle is extended to the case where J is non-smooth and the
problem does not have full variational structure. This "extended"
principle is applied to a parabolic problem associated with
p-Laplacian in unbounded domains.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .3in
% ----------------------------------------------------------------
%\title
\noindent {\bf Asymptotic behavior of solutions to semilinear Euler-Poisson-Darboux type equations }\\ \\
%author
{\bf Akisato Kubo}\\
%affiliation
Fujita Health University, Japan \\
%e-mail
email:akikubo@fujita-hu.ac.jp
%coauthor
%{\bf M. \^Otani}\\ Waseda University
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We investigate the asymptotic property of the solution to the
following boundary value problem $(BV)$ for a semilinear
Euler-Poisson-Darboux type of equation as $t \rightarrow \infty $.
$$ P[u(t,x)]=g(t,x,\partial _tu,D_xu,u) \hskip 1pt \mbox{in} \quad
(t,x) \in [0, \infty ) \times \Omega \hskip 2.5pt \eqno (0.1) $$
$(BV)$ $$ u(t,x)=0 \hskip 6pt \mbox{on} \hskip 4.5pt [0, \infty )
\times \partial \Omega \hskip 4pt \eqno (0.2) $$ where $\Omega $
is assumed to be a bounded domain in $\mbox{\boldmath $R$}^n$ with
a smooth boundary $\partial \Omega ,$ $$ P[\cdot ]=\partial
_t^2\cdot -\sum_{i,j=1}^{n} \partial _i(a_{ij}(x) \partial _j\cdot
)+\frac{2\beta }{t+T}\partial _t\cdot \mbox{,} \hskip 5.5pt \eqno
(0.3) $$ $T$ and $\beta >0,\ \partial _t= \frac{\partial
}{\partial t} ,\ \partial _i= \frac{\partial }{\partial x_i} ,\
i=1,\cdot \cdot \cdot ,n, \ D_xu=(\partial _1u,\cdot \cdot \cdot
,\partial _nu).$ Based on this result, we discuss the optimality
of decay estimates and lower bounds of solutions to the mixed
problem corresponding to $(BV)$.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Strong solutions of magneto-micropolar fluid equation }\\ \\
%author
{\bf Kei Matsuura }\\
%affiliation
Waseda University, Japan \\
%e-mail
email:kino@otani.phys.waseda.ac.jp \\
%coauthor
{\bf Mitsuharu \^Otani} and {\bf Hiroshi Inoue }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We show the existence and the uniqueness of strong solutions for
the initial-boundary value problems of magneto-micropolar fluid
equations. Our result is similar to Fujita-Kato's result for the
classical Navier-Stokes equations. The method of proof relies on
the abstract results in the nonmonotone perturbation theory
developed by M. Otani.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Global attractor for one-dimensional Fremond model of shape memory alloys }\\ \\
%author
{\bf Ken Shirakawa }\\
%affiliation
Tokyo Denki University, Japan \\
%e-mail
email:Kenboich@aol.com \\
%coauthor
{\bf Pierluigi Colli}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this talk, we consider a one-dimensional Fremond model of shape
memory alloys. Here let us imagine a bar (with the length 1) of a
shape memory alloy whose left hand side is fixed, and assume that
the external stress on the right hand side vanishes as time goes
to infinity. Under the above assumptions, we shall discuss the
asymptotic stability for the dynamical system from the viewpoint
of the global attractor. More precisely, we first show the
existence of the global attractor for the limiting autonomous
dynamical system (the case of zero external stress), and secondly,
we shall characterize the asymptotic stability for nonautonomous
case by the limiting global attractor.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Global solutions for quasilinear parabolic systems with cross-diffusion }\\ \\
%author
{\bf Yoshio Yamada }\\
%affiliation
Waseda University, Japan \\
%e-mail
email:yamada@mn.waseda.ac.jp
%coauthor
%{\bf M. \^Otani}\\ Waseda University
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
My talk is concerned with the following Lotka-Volterra competition
system with cross-diffusion effects: $u_t = d_1\Delta[(1+\alpha
v)u] + u(a_1-b_1 u-c_1 v)$ in $\Omega\times (0,\infty)$, $v_t =
d_2\Delta v + v(a_2-b_2 u-c_1 v)$ in $\Omega\times (0,\infty)$,
where $a_i,b_i, c_i, d_i (i=1,2)$ and $\alpha$ are positive
constants and $\Omega$ is a bounded domain in $R^N$ with smooth
boundary. This system is supplemented with Neumann or Dirichlet
boundary conditions and nonnegative initial functions. Note that
the global existence of solutions has been established only in
case $N$ is less than or equal to 3. In this talk we can prove
that the system admits a unique global solution without any
restrictions on $N$ and norms of initial functions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Global attractors for non-autonomous multivalued dynamical systems generated by subdifferentials }\\ \\
%author
{\bf Noriaki Yamazaki}\\
%affiliation
Muroran Institute of Technology, Japan \\
%e-mail
email:noriaki@mmm.muroran-it.ac.jp
%coauthor
%{\bf M. \^Otani}\\ Waseda University
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In a real separable Hilbert space we treat non-autonomous
evolution equations including time-dependent subdifferentials and
their non-monotone multivalued perturbations. In this talk we
consider the non-autonomous multivalued dynamical systems
associated with time-dependent subdifferentials, in which the
solution is not unique for a given initial state. In particular we
discuss the asymptotic behavior of our multivalued semiflows from
the view-point of attractors. In fact, assuming that the
time-dependent subdifferential converges asymptotically to a
time-independent one (in a sense) as time goes to infinity, we
shall construct global attractors for non-autonomous multivalued
dynamical systems and its limiting autonomous system.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\newpage
\begin{center}
{\Large \bf Invariant Manifolds and Their Applications
} \\
Organizer: Kresimir Josic, Boston University
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Noise induced chaos }\\ \\
%author
{\bf Lora Billings }\\
%affiliation
Montclair State University, USA \\
%e-mail
email:lora@nls1.nrl.navy.mil \\
%coauthor
{\bf Ira Schwartz} and {\bf Erik Bollt}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We identify a global mechanism to induce chaos by stochastic
perturbations. Two systems in which we find this are the class B
laser and the SEIR population dynamics model. The bifurcation to
chaos requires two co-existing saddle periodic orbits in a
multistable system, which we call a bi-instability. The noise
induces a heteroclinic connection between the invariant manifolds
of the saddle periodic orbits, therefore inducing a chaotic
attractor. To refine the possibility of control, we have also
analyzed the stochastic transport between basins. This is joint
work with Ira Schwartz and Erik Bollt.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Approximating the dynamics of thin elastic media
}\\ \\
%author
{\bf R. E. Lee DeVille}\\
%affiliation
Rensellaer Polytechnic Institute, USA \\
%e-mail
email:devilr@rpi.edu\\
%coauthor
{\bf C. Eugene Wayne}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this talk, we describe a method for deriving and justifying a
hierarchy of "reduced equations" for the dynamical motion of thin
elastic media, i.e., starting with a PDE defined on a
three-dimensional domain, we will show that its solutions can be
approximated by the solutions of equations defined on a
two-dimensional domain, and, furthermore, there is a sequence of
approximating equations, each of which affords a successively
better approximation. The approach is based on ideas from
Hamiltonian mechanics, and is related to Nekhoroshev theory.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Bubbling bifurcations
}\\ \\
%author
{\bf Brian Hunt }\\
%affiliation
University of Maryland, USA \\
%e-mail
email:bhunt@ipst.umd.edu
%coauthor
%{\bf T. Gallay.}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
For a one-parameter family of dynamical systems with a persistent
invariant submanifold, I will characterize how a chaotic attractor
in the invariant manifold loses asymptotic stability transverse to
the manifold as the parameter is varied. After this bifurcation,
the attractor generically remains weakly stable, having a basin of
attraction that is "riddled" -- it has positive Lebesgue measure
but is not open. Small perturbations of the system can then lead
to intermittent behavior called "bubbling" -- trajectories spend
most of their time near the (formerly) invariant manifold but
occasionally burst far away. I will describe different types of
bifurcations that can lead to bubbling and the resulting size and
frequency of bursts near the bifurcation. The results are
relevant to the synchronization of coupled chaotic systems, where
bursting represents temporary loss of synchronization.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Limits to the detection of nonlinear synchrony
}\\ \\
%author
{\bf Kresimir Josic}\\
%affiliation
Boston University, USA \\
%e-mail
email:josic@math.bu.edu
%coauthor
%{\bf C. Eugene Wayne}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
It has long been recognized that the phenomenon of synchronization
of chaotic systems can be naturally described in terms of smooth
invariant manifolds. Recent evidence suggests that systems
exhibiting complex behavior may be synchronized in a weaker sense.
A number of such examples and the analytical methods needed to
study them will be discussed. I will address the effect of
nonsmoothnes of the synchronization manifold on the detectability
of the synchronized state. Moreover, in the case the driving
system is not invertible the synchronization set is no longer even
a manifold but a far more complicated set. I will discuss how the
usual graph transform methods can be extended to this case to gain
information about the structure of this set. In conclusion I will
discuss how these examples provide clues about the dynamical
nature of weak synchrony and discuss practical methods for the
detection of such coherent states.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Oscillation properties of the complex
Ginzburg-Landau equation
}\\ \\
%author
{\bf Igor Kukavica}\\
%affiliation
The University of South Dakota, USA \\
%e-mail
email: kukavica@math.usd.edu
%coauthor
%{\bf C. Eugene Wayne}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We present estimates on complexity of solutions of the 1D Complex
Ginzburg-Landau equation. We will discuss optimality of bounds and
discuss extensions to the 2D case. The methods are based on
analyticity and unique continuation properties of solutions to the
equation.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On the viscous shock profiles and viscous wave fan
profiles of Riemann solutions
}\\ \\
%author
{\bf Weishi Liu}\\
%affiliation
University of Kansas, USA \\
%e-mail
email:wliu@math.ukans.edu\\
%coauthor
{\bf Marchesin, Plohr and Schecter}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this talk, I will first describe briefly the Exchange Lemmas
for singularly perturbed systems with a class of turning points.
As an application, we consider a system of conservation laws in
one space dimension and study the structural stability of Riemann
solutions. We show that, in particular, there are Riemann
solutions which are generically structurally unstable in terms of
viscous shock profile criterion but are generically structurally
stable in terms of viscous wave fan profile criterion.
This research is closely related and motivated by some of the
works of Marchesin, Plohr, and Schecter.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Invariant manifolds and the Navier-Stokes equation
}\\ \\
%author
{\bf C. Eugene Wayne }\\
%affiliation
Boston University, USA \\
%e-mail
email:cew@math-pc10.bu.edu \\
%coauthor
{\bf T. Gallay}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We will explain some recent work on the construction of finite
dimensional invariant manifolds in the phase space of the
Navier-Stokes equation on ${\bf R}^n$. These manifolds control the
long-term behavior of small solutions, give geometric insight into
the host of existing results on the asymptotics of such solutions,
and allow one to extend those results in a number of ways. Our
results also allow us to prove the stability of certain vortex
solutions of the Navier-Stokes equation, even at very large
Reynolds's number.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Approximate normally hyperbolic invariant manifolds
}\\ \\
%author
{\bf Chongchun Zeng}\\
%affiliation
University of Virginia, USA \\
%e-mail
email:cz3u@weyl.math.virginia.edu
%coauthor
%{\bf C. Eugene Wayne}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this talk, we consider a semiflow in a Banach space where a
$C^1$ submanifold is approximately invariant and normally
hyperbolic. Assuming the semiflow is inflowing (overflowing) along
the boundary, we prove there exists a unique positively invariant
stable (unstable) manifold which has an invariant stable
(unstable) foliation.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Smooth Dynamical Systems}\\
Organizer: Vadim Y. Kaloshin, MIT
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Critical saddle-node bifurcations and Morse-Smale maps }\\ \\
%author
{\bf Brian Hunt }\\
%affiliation
University of Maryland at College Park, USA \\
%e-mail
email:bhunt@ipst.umd.edu
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We study the dynamics of a family of diffeomorphisms just beyond a
saddle-node bifurcation that destroys a stable periodic orbit,
assuming that at the bifurcation parameter the orbit has a
homoclinic tangency. We show that if the tangency is near critical
(cubic), the family generically includes diffeomorphisms that are
locally Morse-Smale for a set of parameters with positive Lebesgue
density at the bifurcation parameter, while if the tangency is
sufficiently far from critical, there are no Morse-Smale
diffeomorphisms in the family. These results rely heavily on
projecting the dynamics to circle endomorphisms. We conclude with
some numerical results that indicate how common the Morse-Smale
property is for near-critical circle endomorphisms.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On the derivative formula of SRB measures }\\ \\
%author
{\bf Miaohua Jiang }\\
%affiliation
Wake Forest University, USA \\
%e-mail
email:jiangm@wfu.edu
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We give the derivative formula of the Sinai-Ruelle-Bowen (SRB)
measure with respect to the hyperbolic dynamical system when the
foliation of the unstable manifold is smooth. Ruelle's simple
formula is valid when the potential function of the SRB measure is
a constant. As a consequence of this formula, we obtain the
derivative formula of the entropy of the SRB measure for
hyperbolic attractors.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Dynamics of an oil spill
}\\ \\
%author
{\bf Vadim Kaloshin}\\
%affiliation
MIT, USA \\
%e-mail
email: kaloshin@Math.Princeton.EDU\\
%coauthor
{\bf D. Dolgopyat} and {\bf L.Koralov}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider the evolution of a passive scalar, e.g. an oil
spill, on surface of ocean (the plane), where the motion is modelled
by a periodic incompressible stochastic flow. We show that for almost
every realization of the random flow at time $t$ most of the particles
are at a distance of order $\sqrt{t}$ away from the origin and there is
a measure zero and full Hausdorff dimension set of points, which escape
to infinity at the linear rate. We study the set of points visited by the
original set by time $t$, and show that such a set, when scaled down by
the factor of $t$, has a limiting non random shape.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Random perturbations of 2-dimensional Hamiltonian flows }\\ \\
%author
{\bf Leonid Koralov }\\
%affiliation
Princeton University, USA\\
%e-mail
email:koralov@Math.Princeton.edu \\
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We find the asymptotics of the effective diffusivity (which is the
main term of the homogenized flow) of 2-dimensional incompressible
flows perturbed by small diffusion. In the case when the stream
function of the flow has some symmetry properties, the asymptotics
of the effective diffusivity was obtained by Fannjiang and
Papanicolaou using variational methods. We consider the general
case using the probabilistic approach.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Smooth dynamical systems }\\ \\
%author
{\bf Waclaw Marzantowicz }\\
%affiliation
A. Mickiewicz University of Pozna\'n, Poland\\
%e-mail
email:marzan@main.amu.edu.pl
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In the early seventies Michael Shub posed the following conjecture
about the topological entropy: If the universal covering space of
a manifold $M$ is homeomorphic to the Euclidean space then the
topological entropy $h(f)$ of any continuous self-map $f: M\to M$
is estimated from below by the logarithm of the spectral radius
$\rho(f)$ of the map $H^*(f)$ induced by $f$ on real cohomology.
This conjecture was confirmed for a torus map by Misiurewicz and
Przytycki in 1977. In this work we prove the conjecture in a
weaker form, in the case when $M$ is a compact nilmanifold. An
assumption we need requires that the Lefschetz number $L(f)\neq 0
\,,$ i.e. $f $ is not homotopic to a fixed point free map. The
presented method uses a notion of linearization of a self-map of
nilmanifold to compare the spectral radius $\rho(f)$ with the
asymptotic Nielsen number $N^\infty(f)\,,$ and apply the Ivanov
theorem. In a second theorem we drop out the assumption $L(f)\neq
0\,, $ but we can prove then only that $h(f) > 0$ if $ \rho(f) >
0\,$.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Dynamics of Dominated Splitting}\\ \\
%author
{\bf Martin Sambarino }\\
%affiliation
University of Maryland at College Park,USA\\
%e-mail
email:samba@math.umd.edu
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We will show that surface diffeomorphisms with dominated splitting
admit a spectral decomposition and we present some consequences of
it. Also, we will discuss the robustness of sets having dominated
splitting and the integrability of these subbundless in dimensions greater
than 2.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\newpage
\begin{center}
{\Large \bf Chaos in Classic and Quantum Systems
}\\
Organizer: Helmut Kroger, Laval University
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Open quantum dots as a probe of quantum chaos:
going beyond convenient mathematical approaches
}\\ \\
%author
{\bf J. Bird }\\
%affiliation
Arizona State University, USA \\
%e-mail
email:bird@asu.edu\\
%coauthor
{\bf R. Akis}, {\bf D. K. Ferry}\\
{\bf A. P. de Moura } and {\bf Y.-C.Lai}
%abstract
\\
Semiconductor quantum dots are sub-micron sized structures that
consist of a mesoscopic scattering region, fabricated on length
scales much smaller than the electron mean-free path. The relevant
current-flow process through such structures is therefore one in
which electrons are injected into the cavity, and undergo multiple
scattering from the walls of the dot before finally escaping to
the external reservoirs. For this reason, these ballistic
structures may be viewed as the solid-state analog of classical
scattering billiards, and their electrical properties have
attracted much interest as an experimental probe of quantum chaos
(for a recent review, see Ref. [1]).In the development of a
theoretical description of these structures, there has
unfortunately been a tendency to introduce a number of limiting
assumptions, which provide for mathematical simplicity at the
expense of physical reality. In this presentation, we will
therefore discuss the results of experimental and theoretical
studies of open quantum dots, which go beyond these artificial
approaches to develop a better understanding of electron dynamics
in these structures. From quantum-mechanical simulations of our
experimental results, we find strong evidence for the role of
scarred wavefunction states in transport through the dots [2].
This behavior is in turn shown to be related to an effect
analogous to resonance trapping, in which transport through the
open system can be understood in terms of selected eigenstates of
the closed structure. We will also discuss the results of a more
resent semiclassical analysis, which has pointed to a consistent
picture in which the scars are found to be associated with
classically-inaccessible orbits, which end up playing an important
role in transport due to the process of dynamical (or phase-
space) tunnelling [3].
[1] J. P. Bird, "Recent experimental studies of electron transport in open
quantum dots", J. Phys.: Condens. Matter 11, R413 (1999).
[2] J. P. Bird, R. Akis, D. K. Ferry, D. Vasileska, J. Cooper, Y. Aoyagi,
and T. Sugano, "Lead orientation dependent wavefunction scarring in open
quantum dots", Phys. Rev. Lett. 82, 4691 (1999).
[3] A. P. S. de Moura, Y.-C. Lai, R. Akis, J. P. Bird, and D. K. Ferry,
"Tunneling and nonhyperbolicity in quantum dots", submitted for
publication.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Chaotic ionization of hydrogen in parallel fields
}\\ \\
%author
{\bf Kevin Mitchell }\\
%affiliation
College of William and Mary, USA \\
%e-mail
email:kevinm@physics.wm.edu\\
%coauthor
{\bf John Delos }\\
%abstract
\\
We examine the classical ionization of photoexcited states of
hydrogen in parallel electric and magnetic fields. This system
reduces to an area-preserving chaotic map of the plane and is a
useful and experimentally accessible model of chaotic decay and
scattering. Decay is studied by examining segments of a line of
initial conditions that escape at various iterates of the map.
These segments exhibit what we call "epistrophic self-similarity":
the segments organize themselves into self-similar geometric
sequences, but the beginnings of these sequences are only
partially predictable. Points that remain from a Cantor set which
we call an "epistrophic fractal". These studies should be
important for the analysis of classical decay rates, which
numerical studies have shown typically exhibit algebraic rather
than exponential decay. They should also be important for the
analysis of semiclassical decay and scattering, including the
construction of the semi-classical propagator and S-matrix.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Semiclassical and anticlassical limits of the
Schr\"odinger equation
}\\ \\
%author
{\bf H. Friedrich }\\
%affiliation
Tech. Univ. Munich, Germany \\
%e-mail
email:harald\_friedrich@physik.tu-muenchen.de
%coauthor
%{\bf Boris Belinskiy}
%abstract
\\
The natural place to study how the chaotic nature of a classical
system manifests itself in the corresponding quantum system is in
or near the semiclassical limit of the quantum system. For
billiard systems this is, quite generally, the high-energy limit,
but for more realistic one- or many-body systems described by the
appropriate Schr\"odinger equation,the semiclassical limit may not
be so easy to identify. In this talk I identify the semiclassical
limit of the Schr\"odinger equation - and the anticlassical or
extreme quantum limit - for several examples including
one-electron atoms in external fields and two- or more-electron
atoms. As a curiosity I mention a class of one-dimensional bound
systems where the quantum number grows to infinity without
approaching the semiclassical limit.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Long period orbits from symbolic dynamics
}\\ \\
%author
{\bf Fritz Haake }\\
%affiliation
Fachbereich Physik, Universitaet Essen, Germany \\
%e-mail
email:fritz.haake@uni-essen.de \\
%coauthor
{\bf Petr Braun} and {\bf Stefan Heusler}
%abstract
\\
Working with a sequence of local sections of the symbol sequence
one can construct periodic orbits of extremely long periods. The
accuracy attained grows exponentially with the length of the local
sections. By properly including correlations between symbols one
obtains orbits behaving ergodically.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf One-dimensional quantum chaos: solved
}\\ \\
%author
{\bf Roderick Jensen }\\
%affiliation
Weslyan University, USA \\
%e-mail
email:rjensen@wesleyan.edu\\
%coauthor
{\bf Yuri Dabaghian}
%abstract
\\
We have identified a class of quantum graphs (corresponding to
particles in one-dimensional potential wells) with unique and
precisely defined spectral properties that we call "regular
quantum graphs". Although these physical systems are chaotic in
the classical limit (with positive topological entropy), the
regular quantum graphs are explicitly solvable. Exact, convergent
periodic orbit expansions can not only be developed for the
quantum density of states (a la Gutzwiller) but explicit periodic
orbit formulas can be developed for individual quantum energy
levels.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Quantum chaos at finite temperature in the Paul
trap
}\\ \\
%author
{\bf Helmut Kr\"oger }\\
%affiliation
Laval University, Canada \\
%e-mail
email:hkroger@phy.ulaval.ca
%coauthor
%{\bf Petr Braun}\\{\bf Stefan Heusler}\\
%abstract
\\
In quantum chaos, we have no "local" information of the degree of
chaoticity, being available in classical chaos via Lyapunov
exponents and Poincar\'e sections from phase space. Also little is
known about the role of temperature in quantum chaos. For example,
an analysis of level densities is insensitive to temperature. Here
we present a new method aiming to overcome those
problems.Recently, my co-workers and I have proposed the concept
of a quantum action. This action has a mathematical structure like
the classical action, but takes into account quantum effects
(quantum fluctuations) via renormalized action parameters. This
bridges the gap between classical physics and quantum physics.
Lyapunov exponents, Poincar\'{e} sections etc. can be computed to
study quantum chaos via the quantum action. The quantum action has
been applied and proven useful to precisely define and
quantitatively compute quantum instantons.Here we use the quantum
action to study quantum chaos at finite temperature. We present a
numerical study of 2-D Hamiltonian systems which are classically
chaotic. First we consider harmonic oscillators with weak
anharmonic coupling. We construct the quantum action
non-perturbatively and find temperature dependent quantum
corrections in the action parameters. We compare Poincar\'{e}
sections of the quantum action at finite temperature with those of
the classical action. We observe chaotic behavior for both.
Secondly, we consider the Hamiltonian of the Paul trap, which is
quite important for atomic clocks, in Bose-Einstein condensation,
etc. We present Poincar\'{e} sections as function of temperature
and compare its chaotic behavior with its classical counterpart.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Classical and quantum chaos in fundamental field
theories
}\\ \\
%author
{\bf Harald Markum}\\
%affiliation
Institut f"ur Kernphysik Technische Universit"at Wien, Austria\\
%e-mail
email:markum@kph.tuwien.ac.at
%coauthor
%{\bf Petr Braun}\\{\bf Stefan Heusler}\\
%abstract
\\
The role of chaotic field dynamics for the confinement of quarks
is a longstanding question. Concerning classical chaos, we analyze
the leading Lyapunov exponents of Yang-Mills field configurations
on the lattice. Concerning the quantum case, we investigate the
eigenvalue spectrum of the staggered Dirac operator in QCD at
nonzero temperature. The quasi-zero modes and their role for
chiral symmetry breaking and the deconfinement transition are
examined. The bulk of the spectrum and its relation to quantum
chaos is considered. Our results demonstrate that chaos is present
when particles are confined, but it persists also into the
quark-gluon-plasma phase. Further, we decompose U(1) gauge fields
into a monopole and photon part across the phase transition from
the confinement to the Coulomb phase. We analyze the leading
Lyapunov exponents of such gauge field configurations on the
lattice which are initialized by quantum Monte Carlo simulations.
It turns out that there is a strong relation between the sizes of
the monopole density and the Lyapunov exponent.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Photon induced chaotic scattering
}\\ \\
%author
{\bf Linda Reichl }\\
%affiliation
University of Texas, USA\\
%e-mail
email:reichl@mail.utexas.edu
%coauthor
%{\bf Petr Braun}\\{\bf Stefan Heusler}\\
%abstract
\\
Strong electromagnetic fields, interacting with atoms, can induce
chaotic structures in open space. These chaotic structures may
support new quasi-bound electronic states and delay ionization of
electrons. Scattering processes in the presence of electromagnetic
fields can be analyzed using Floquet theory. Signatures of
chaos-induced quasi-bound states appear in scattering phase shifts
and delay times.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Quantum chaos in fractal repellers and disordered
systems: experiments using microwaves
}\\ \\
%author
{\bf S. Sridhar}\\
%affiliation
Northeastern University, USA \\
%e-mail
email:srinivas@neu.edu
%coauthor
%{\bf Petr Braun}\\{\bf Stefan Heusler}\\
%abstract
\\
I discuss some recent themes from our microwave experiments
designed to explore issues in Quantum Chaos. In n-disk geometries,
we have seen the quantum fingerprints of classical
Ruelle-Pollicott resonances. The wave experiments thus are
laboratory realizations of a fractal repeller, a prototypical
Axiom A dynamical system. In disordered geometries, which are
realizations of electrons in disordered solids, we have studied
correlations of eigenfunctions which manifest localization. The
results are shown to be well described by recent theories based
upon non-linear sigma models of supersymmetry. Earlier microwave
experiments have led to observation of scars in quantum
eigenfunctions of Sinai and other chaotic billiards, precision
tests of random matrix theory in microwave spectra and
eigenfunctions, and studies of experimental mathematics of
isospectral domains. (1) Quantum fingerprints of classical
Ruelle-Pollicot resonances, K.Pance, W.T.Lu and S.Sridhar, Phys.
Rev. Lett., 85, 2737 (2000) (2) Correlations due to localization
in quantum eigenfunctions of disordered microwave cavities",
Prabhakar Pradhan and S. Sridhar, Phys.Rev. Lett., 85, 2360 (2000)
(3) Quantum Correlations and Classical Resonances in an open
chaotic system", W.T.Lu, K.Pance, P.Pradhan and S.Sridhar, Physica
Scripta :Special Issue : Nobel Symposium on "Quantum Chaos Y2K",
T90, 238 (2001).
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A simple scheme for extracting internal motions
from spectroscopic Hamiltonians
}\\ \\
%author
{\bf Howard Taylor}\\
%affiliation
University of Southern California, USA \\
%e-mail
email:taylor@usc.edu
%coauthor
%{\bf Petr Braun}\\{\bf Stefan Heusler}\\
%abstract
\\
The problem addressed is to determine the internal vibrational
motions that when quantized yield the vibrational bound and
resonant states of a molecule. In particular we consider systems
with two or more vibrational resonances due to frequencies in
rational ratio, where motions at variance with simple normal or
local modes exist. We restrict ourselves to problems where a
spectroscopic normal form Hamiltonians (Heff) can be obtained
either from fits to experimental lines, or with the use of a PES,
to calculated eigenvalues. The application of canonical
perturbation theory to a system with a known PES can also supply
such an Heff. Presently we also require that the number of degrees
of freedom minus the number of constants of motion, one being the
polyad quantum number (P) that will exist when resonances are
present, is no greater than three. We treat the non-trivial cases
where more than one resonance exists and hence chaos can occur and
where the reduced phase space, obtained by canonical transform
using constants of motion, corresponds to two or three degrees of
freedom. We further limit ourselves to the challenging cases where
the dynamics is not susceptible to any simple adiabatic or other
separation scheme and where wave functions and trajectories in the
full dimension are too complex to be interpreted by any graphical
representations. In such cases it will be demonstrated that most
if not all of the dynamics can be uncovered and that dynamic
quantum numbers representing quasiconserved quantities can be
assigned given only the already existing eigenfunction-basis
transformation matrix used in fitting the Heff to the
experimentally or theoretically generated spectrum. The method in
conception depends on the ability previously gained(1), in
studying problems where nonlinear dynamics was used to find the
motions underlying the simple patterns seen in plots of the
density and the phases of semiclassical eigenfunctions created
from the information in the transformation matrix calculated in
the above "fit". These eigenfunctions are parametric in the
constants of the motion (the polyad number P in particular) and
lie for DCO on a 2D toroidal configuration space described by two
angle variables. Since the features of these 2D wave functions are
generally simple to recognize once one is comfortable working in
this unusual space; just viewing the patterns allows the sorting
of the interleaved states of different dynamics into suits (like a
deck of cards) each based on different dynamics. Then nodal counts
and/or phase advances (since we are in a space for angles) allows
sequential quantum number assignment. The contours of the 2D angle
space wave function density and phases can be used for each suite
to infer the type of classical internal motion that the atoms are
undergoing in normal mode, local mode or displacement coordinate
space. These motions are those that when quantized gives rise to
the levels in the suit. A discussion, as to why wave functions
represented in compact angle (of action-angle) spaces as opposed
to the usual open coordinate spaces are so much simpler to
interpret is given. The assignments and dynamics of DCO are
presented(4). Because of a strong 1:1:2 resonance assignment and
interpretation alluded previous workers who computed
eigenfunctions from both a high quality potential surface(2) and
from a spectroscopic Hamiltonian(3). Again we stress no serious
computation was needed to extract dynamics and to assign once the
Heff was available. Graphical representations of the phase and
densities of eigenfunctions in reduced configuration (angle)
space, the principles of nonlinear dynamic and semiclassical ideas
on how wave functions accumulate about phase space organizing
structure are the keys to the analysis. 1. (a) M.P. Jacobson, C.
Jung, H.S. Taylor and R.W. Field, J. Chem. Phys., 111, 600 (1999).
(b) C. Jung, H.S. Taylor and M. Jacobson, J. Phys. Chem. A., 105,
681 (2001). 2. H.-M. Keller, H. Floethmann, A. J. Dobbyn, R.
Schinke, H.-J. Werner, C. Bauer and P. Rosmus, J.Chem. Phys. 105,
4983 (1996). 3. A. Troellsch and F. Temps, Zeitschreft for
Physikalische Chemie, 215, 207 (2000). 4. E.Atligan, C. Jung and
H.S. Taylor, J. Phys. Chem. (2002) in press.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Mathematical Fluid Dynamics}\\
Organizer:Igor Kukavica, University of Southern California\\
\hspace {6 pt}James Robinson, University of Warwick \\
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Boundary layer for the Navier-Stokes-alpha model of fluid turbulence }\\ \\
%author
{\bf Alexey Cheskidov }\\
%affiliation
Indiana University, USA \\
%e-mail
email:acheskid@indiana.edu
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
We study a boundary layer problem for the Navier-Stokes-alpha
model obtaining a generalization of the Prandtl equations which we
conjecture to represent the averaged flow in a turbulent boundary
layer. We study the equations for the semi-infinite plate, both
theoretically and numerically. Solutions agree with some
experimental data in a part of the turbulent boundary layer.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Inertial manifolds and Gevrey regularity for the Moore--Greitzer model of turbo--machine engine }\\ \\
%author
{\bf Yeojin Chung }\\
%affiliation
University of California, Irvine, USA \\
%e-mail
email:ychung@math.uci.edu \\
%coauthor
{\bf Edriss S. Titi }
%abstract
\\
We study the regularity and long-time behavior of the solutions to
the Moore--Greitzer model of turbo-machine engine. In particular,
we prove that this dissipative system of evolution equations
possesses a global invariant inertial manifold, and therefore its
underlying long-time dynamics reduces to that of an ordinary
differential system. Furthermore, we show that the solutions of
this model belong to a Gevrey class of regularity (real analytic
in the spatial variables). As a result, one can show the
exponentially fast convergence of the Galerkin approximation
method to the exact solution, an evidence of the reliability of
the Galerkin method as a computational scheme in this case. The
rigorous results presented here justify the readily available low
dimensional numerical experiments and control designs for
stabilizing certain states and travelling wave solutions for this
model.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Remarks on rotating fluids }\\ \\
%author
{\bf Peter Constantin }\\
%affiliation
University of Chicago, USA \\
%e-mail
email:const@math.uchicago.edu
%coauthor
%{\bf Edriss S. Titi }
%abstract
\\
We will discuss bounds for transport and spectra in rotating
Navier-Stokes equations forced at the boundary.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On interpolation between algebraic and geometric
conditions for smoothness of the vorticity in the 3D NSE }\\ \\
%author
{\bf Zoran Grujic }\\
%affiliation
University of Virginia, USA \\
%e-mail
email:zg7c@virginia.edu
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
We formulate some sufficient conditions for smoothness of the
vorticity consisting of space-time integrability mixed with the
regularity of the vorticity directions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Recurrent estimates for the Navier-Stokes equations }\\ \\
%author
{\bf Michael Jolly }\\
%affiliation
Indiana University, USA \\
%e-mail
email:msjolly@indiana.edu \\
%coauthor
{\bf C. Foias} and {\bf O. P. Manley}
%abstract
\\
In the Kraichnan theory of two-dimensional turbulence it is
assumed that the enstrophy, rather than the energy that is the
most relevant quantity. We briefly recall recent rigorous results
which show that indeed the enstrophy cascade is more pronounced
than that of energy, in the 2-D case. These results pertain to
averages of the enstrophy transfer, or flux, through a specified
wavenumber, and thus do not provide any information on how these
quantities can fluctuate in time. Toward answering this question,
we also present two estimates relating the enstrophy beyond a
given wavenumber (the so-called high modes) and the enstrophy flux
through that wavenumber. They are recurrent in that they hold at
least once within certain bounded time intervals. The first
effectively provides a bound on how long the enstrophy flux can
remain negative, the second a bound on the enstrophy of the high
modes, valid for wavenumbers on the order of the square root of
the Grashof number. Some numerical results are presented to
illustrate these estimates. This work is joint with C. Foias and
O.P. Manley, in that much of it was completed before O.P.M. passed
away.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf The number of determining modes in 2D turbulence: A computational study }\\ \\
%author
{\bf Eric Olson }\\
%affiliation
University of Nevada, Reno, USA\\University of California, Irvine, USA \\
%e-mail
email:ejolson@unr.edu \\
%coauthor
{\bf Edriss Titi }
%abstract
\\
The method of continuous data assimilation from weather
forecasting is used to study the number of determining modes for
the two-dimensional incompressible Navier--Stokes equations. Our
focus is on how the body forcing affects the rate of continuous
data assimilation and the number of determining modes. These
quantities are shown to depend strongly on the length scales
present in the forcing.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Finite-dimensional dynamics on global attractors }\\ \\
%author
{\bf James Robinson }\\
%affiliation
University of Warwick, United Kingdom \\
%e-mail
email:jcr@maths.warwick.ac.uk \\
%coauthor
{\bf C. Foias }
%abstract
\\
The talk will discuss approaches to reproducing the dynamics on a
finite-dimensional attractor using a finite-dimensional dynamical
system. In particular, the aim is to obtain something akin to an
``inertial form'' for the 2d Navier-Stokes equations even though
the existence of an inertial manifold is still an open question.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Nonlinear wave equations and the Melnikov problem }\\ \\
%author
{\bf Alain Schenkel }\\
%affiliation
University of Helsinki, Finland \\
%e-mail
email:Alain.Schenkel@Helsinki.FI \\
%coauthor
{\bf Bricmont} and {\bf A. Kupiainen}
%abstract
\\
I will describe a new proof of the Melnikov problem in infinite
dimensional systems, namely, persistence of quasi-periodic, low
dimensional elliptic tori. Our result covers situations in which
the so-called normal frequencies are multiple. In particular, it
provides a new proof of the existence of small amplitude,
quasi-periodic solutions of nonlinear wave equations with periodic
boundary conditions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Sharp interface limits and global existence for the phase field Navier-Stokes equations }\\ \\
%author
{\bf Steve Shkoller }\\
%affiliation
University of California, Davis, USA\\
%e-mail
email: shkoller@math.ucdavis.edu
%coauthor
%{\bf Benjamin Luce LANL }\\
%abstract
\\
We will introduce the phase-field Navier-Stokes equations, and
prove that they possess Leray global weak solutions. We will then
show that for smooth initial data, solutions of the phase-field
model converge weakly to solutions of the sharp-interface
Navier-Stokes equations. In the convergence proof, an auxiliary
PDE is introduced which couples the Navier-Stokes equations with
the classical geometric problem of motion by mean curvature.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Attracting fixed points for Kuramoto-Sivashinsky equation - a computer assisted proof' }\\ \\
%author
{\bf Piotr Zgliczynski }\\
%affiliation
Jagiellonian University, Poland \\
%e-mail
email: zgliczyn@im.uj.edu.pl
%coauthor
%{\bf Benjamin Luce LANL }\\
%abstract
\\
We present a computer assisted proof of an existence of several
attracting fixed points for the Kuramoto-Sivashinsky equation
\[ u_t=(u^2)_x - u_{xx} -\nu u_{xxxx}, \]
\[ u(x,t)=u(x+2\pi,t), \quad u(x,t)=-u(-x,t),\]
where $\nu >0$. The approach based on the concept of
self-consistent a priori bounds introduced in \cite{ZM}. The
method is general and can be applied to other dissipative PDEs,
for example Navier-Stokes or Ginzburg-Landau equations, not only
to obtain fixed points, but also more complex dynamical objects
like periodic orbits and hopefully topological horseshoes. The
partial results concerning a rigorous steady-state bifurcation
diagram for Kuramoto-Sivashinsky equation will be also mentioned.
\hspace {20pt}\begin{thebibliography}{ZM} \bibitem[Z]{Z} P.\
Zgliczynski, Attracting fixed points for Kuramoto-Sivashinsky
equation - a computer assisted proof, submitted, {\em
http://www.im.uj.edu.pl/\~{}zgliczyn} \bibitem[ZM]{ZM} P.\
Zgliczynski and K. \ Mischaikow, Rigorous Numerics for Partial
Differential Equations: the Kuramoto-Sivashinsky equation, {\em
Foundations of Computational Mathematics,} (2001) 1:255-288
\end{thebibliography}
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Topological Methods for Boundary Value Problems
}\vspace{-0.2in}\\
$$\begin{array}{rl}
\mbox{Organizer:}&\mbox{J. R. L. Webb, University of Glasgow, UK} \\
&\mbox{K. Q. Lan, Ryerson University, CA} \end{array}$$
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Singular nonlinear boundary value problems with multiple positive solutions }\\ \\
%author
{\bf John Baxley }\\
%affiliation
Wake Forest University, USA \\
%e-mail
email:baxley@mthcsc.wfu.edu \\
%coauthor
{\bf Philip T. Carroll }
%abstract
\\
We extend recent results of Henderson and Thompson, Baxley and
Haywood, and Graef, Qian, and Yang, which provided conditions on
the nonlinear function $f(y)$ in order that the boundary value
problem $(-1)^n y^{(2n)} = f(y)$, $y^{(2k)} (0) = 0$, $y^{(2k)}
(1) = 0$, for $k=0,\cdots ,n-1$ have multiple symmetric positive
solutions. Since such solutions must satisfy $y^{(2k+1)} (1/2) =
0$, for $k = 0, \cdots , n-1$, we consider the more general
problem $L(y) = f(y)$, where $L$ is the $n$th iterate of the
Sturm-Liouville operator $\mbox{\\dis}$ $-\frac{1}{w} (p y')'$,
with boundary conditions $y^{(2k)} (0)= 0$, $y^{(2k+1)} (b) = 0$,
$b>0$, for $k = 0,\cdots,n-1$. The conditions we obtain allow
singular behavior in the operator $L$ at $x=0$. In the case $w
\equiv p \equiv 1$, $b=1/2$, our conclusions reduce to the earlier
results mentioned above. Previous work using the Leggett-Williams
fixed point theorem or a fixed point theorem of Krasnosel'ski\u{i}
has used properties of relevant Green's functions. Here we use a
refined version of the same theorem of Krasnosel'ski\u{i}, but
need no use of Green's functions to obtain the necessary
estimates.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Existence theorems for weakly inward semilinear operators }\\ \\
%author
{\bf Casey Cremins }\\
%affiliation
University of Maryland, USA \\
%e-mail
email:ctc@math.umd.edu
%coauthor
%{\bf Julian Lopez-Gomez }\\
%abstract
\\
We obtain existence theorems for semilinear equations of the form
Lx = Nx, where the operators L and N satisfy a weakly inward
condition and are such that L - N is A proper. In particular,
results involving positive and multiple solutions are proved.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Kelvin-Helmholtz instability waves and upstream propagating acoustic waves in supersonic multiple jets}\\ \\
%author
{\bf Joshua Du }\\
%affiliation
Kennesaw State University, USA\\
%e-mail
email:jdu@kennesaw.edu
%coauthor
%{\bf Julian Lopez-Gomez }\\
%abstract
\\
Jet aircraft were introduced right after that the Second World
War. Shortly after that, jet noise prediction and reduction became
an important research topic. Because of the need for large thrust,
many high performance military aircraft are propelled by two or
more jet engines housed close to each other. Three Physics Laws,
Conservation of Mass, Momentum and Energy, in differentiation form
are employed to formulate the Kelvin-Helmholtz Instability problem
of supersonic triple jets. The general solution of the system
about pressure of the jets and the dispersion relation for
instability waves about jet noise are derived.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Spectrum of positively homogeneous operators and applications }\\ \\
%author
{\bf Wenying Feng }\\
%affiliation
Trent University, Canada \\
%e-mail
email:wfeng@trentu.ca
%coauthor
%{\bf Julian Lopez-Gomez }\\
%abstract
\\
The spectral theory for nonlinear operators has been extensively
studied by many authors. After the theory of Furi, Martelli and
Vignoli, a new definition was introduced by the author. Later, the
work was generated to semilinear operators. In this paper, some
results on the relationship between the eigenvalues and the
spectrum of a positively homogeneous operators were obtained.
Applying the results, we prove a theorem that gives a condition
for a compact, positive operator to have a positive eigenvalue and
eigenvector. The theorem can be used in the study of a second
order differential equations with a three point boundary value
conditions that has been studied recently by some authors.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Positive solutions of differential equations with nonlinear boundary conditions }\\ \\
%author
{\bf Gennaro Infante }\\
%affiliation
Universita' della Calabria, Italy \\
%e-mail
email:infanteg@unical.it
%coauthor
%{\bf Julian Lopez-Gomez }\\
%abstract
\\
Using the theory of fixed point index, we establish new results
for some differential equations subject to nonlinear boundary
conditions. We obtain existence of at least one or of multiple
positive solutions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On the solvability of implicit complementarity problem and implicit variational inequalities ----- A unified approach and implicit projected dynamical system}\\ \\
%author
{\bf Gheorghe Isac }\\
%affiliation
Royal Military College of Canada, Canada \\
%e-mail
email:gisac@juno.com
%coauthor
%{\bf Julian Lopez-Gomez }\\
%abstract
\\
In this first part of this paper we will present a unified
approach of the study of Implicit Complementary Problems and
Implicit Variational Inequalities. This study is based on the
concept of ''Exceptional Family of Elements'' for a function. This
concept is obtained in this case using a kind of implicit
Leray-Schauder alternative. In the second part of this paper we
will present a study of solutions of Implicit Complementary
Problems and Implicit Variational Inequalities, from the dynamical
point of view. This study is obtained using an implicit global
projected dynamical system. This paper will be finished by
comments and open problems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Multiple positive solutions of conjugate boundary value problems with singularities }\\ \\
%author
{\bf K. Lan }\\
%affiliation
Ryerson University, Canada \\
%e-mail
email:klan@ryerson.ca
%coauthor
%{\bf Julian Lopez-Gomez }\\
%abstract
\\
We consider the existence of one or several nonzero positive
solutions for a higher order nonlinear ordinary differential
equation with conjugate boundary conditions. The conjugate
boundary value problems can be changed into a Hammerstein integral
equation with a suitable kernel. We shall show that the kernel has
upper and lower bounds. This enables us not only to exhibit a new
property of positive solutions for the conjugate boundary value
problems but also to derive new results on the conjugate boundary
value problems from the well-known results on the existence of one
or several positive solutions of Hammerstein integral equations
with singularities obtained by the author recently. Our results
generalize some known results where stronger conditions were
imposed and the theory of fixed point index for compact maps
defined on cones was used directly.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Riccati equations }\\ \\
%author
{\bf Allan Peterson }\\
%affiliation
University of Nebraska-Lincoln, USA \\
%e-mail
email:apeterso@math.unl.edu \\
%coauthor
{\bf Lynn Erbe }
%abstract
\\
We will use the Riccati equation to prove oscillation theorems for
self-adjoint vector differential equations on time scales.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Existence and uniqueness for a class of quasilinear elliptic boundary value problems }\\ \\
%author
{\bf Ratnasingham Shivaji }\\
%affiliation
Mississippi State University, USA \\
%e-mail
email:shivaji@Math.MsState.Edu
%coauthor
%{\bf Erbe }\\
%abstract
\\
We prove existence and uniqueness of positive solutions for the
boundary value problem
$$ (r^{N-1}\phi (u^{\prime
}))^{\prime }=-\lambda r^{N-1}f(u),\;u^{\prime }(0)=u(1)=0, $$
where $\phi (x)=|x|^{p-2}x$, $\frac{f(x)}{x^{p-1}}$ may not be
decreasing on $(0,\infty ),$ and $\lambda $ is a large parameter.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Remarks on positive solutions of some 3-point boundary value problem }\\ \\
%author
{\bf Jeff Webb }\\
%affiliation
University of Glasgow, United Kingdom \\
%e-mail
email:jrlw@maths.gla.ac.uk
%coauthor
%{\bf Erbe }\\
%abstract
\\
Some recent work on existence of one or of multiple solutions of a
nonlinear second order differential equation with nonlocal
boundary conditions will be discussed by the method of fixed point
index. An optimal value will be given for a constant that appears
in the definition of the cone being used and in some of the other
hypotheses.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On the existence of explosive solutions for semilinear elliptic problems }\\ \\
%author
{\bf Zhijun Zhang }\\
%affiliation
Yantai University, China \\
%e-mail
email:zhangzj@ytu.edu.cn
%coauthor
%{\bf Erbe }\\
%abstract
\\
TBA
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\newpage
\begin{center}
{\Large \bf Optimal Control and Control Systems } \\
Organizer: Urszula Ledzewicz,Southern Illinois University \\
\hspace{0.25in} Heinz Schaettler,Washington University
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf On optimality of bang-bang extremal controls for bilinear systems }\\ \\
%author
{\bf Andrey Sarychev }\\
%affiliation
University of Aveiro, Portugal \\
%e-mail
email:ansar@mat.ua.pt
%coauthor
%\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Application of Pontryagin Maximum Principle - a necessary
optimality condition for optimal control problems with constrained
controls - provides information on extremal controls, among which
optimal controls have to be searched. In some cases, like the one
of chained control systems with two inputs, one can provide
complete description of the structure of Pontryagin extremals for
time-optimal problems (see A.Sarychev, H.Nijmeijer, J. Dynam.
Control Systems, v.2, 1996, pp. 503-527). It is known however that
the bang-bang Pontryagin extremals may cease to be optimal. In our
talk we are going to present some results on optimality of
bang-bang extremals for bilinear control systems with constrained
controls and in particular on time optimality for chained systems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Local controllability of linear systems on Lie groups }\\ \\
%author
{\bf Fabiana Cardetti }\\
%affiliation
Louisiana State University, USA \\
%e-mail
email:fcardet@lsu.edu
%coauthor
%{\bf Heinz Schaettler}\\Washington University
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In a recent paper by Ayala and Tirao, the notion of linear control
systems on Lie groups was introduced. On a real connected
finite-dimensional Lie group $G$ with Lie algebra $g$, a linear
control system $\Sigma$ on $G$ is given by
$$ \label{linearsystem} \dot{x}=\vec{X}(x) + \sum_{j=1}^{k}u_j\vec{Y_j}(x), \eqno (1) $$
where $\vec{X}$ is an infinitesimal automorphism, $u_j$ are
piecewise constant functions, and the control vectors
$\vec{Y_j}\in g$ are left-invariant vector fields. Under
conditions similar to the Kalman condition that is needed for
controllability of linear control systems on $R^n$, Ayala and
Tirao showed local controllability of the system $\Sigma$ at the
group identity $e$. Another proof of this result is obtained using
Lie Theory of Semigroups. More importantly, using a Lie-wedge
approach, the result is extended to local controllability on $H$,
the subgroup of $G$ defined by $H=\langle
\exp(\mathcal{H})\rangle$. Here $\mathcal{H}\subseteq g$ denotes
the Lie subalgebra generated by the control vector fields, that
is, $\mathcal{H}=\langle Y_1,\dots,Y_k \rangle$. The fundamental
idea is to work in a related group using the Lie theory of
semigroups. The fact that $X$ is an infinitesimal automorphism is
used to construct a related group $\hat{G}$. In fact, $G$ is
diffeomorphic to a homogeneous manifold of the form $\hat{G}/K$.
The system $\Sigma$ on ${G}$ is then lifted into a system
$\hat{\Sigma}$ on $\hat{G}$. The latter has the advantage of being
a right-invariant system. Many results can be obtained for
$\hat{\Sigma}$. The conclusions on $\Sigma$ are drawn from these
results.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Relaxation of elliptic control systems }\\ \\
%author
{\bf Nikolas Papageorgiou }\\
%affiliation
National Tech. University, Greece \\
%e-mail
email:npapg@math.ntua.gr
%coauthor
%{\bf Heinz Schaettler}\\Washington University
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider relaxation methods for nonlinear optimal control
problems.We present four such methods, showing that they are
admissible and compare them.We also prove an general existence
theorem which motivates one of the relaxation methods.Also we
present a relaxation method based on lower semicontinuity
regularization techniques.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Optimal controller for third degree polynomial systems }\\ \\
%author
{\bf Michael Basin }\\
%affiliation
Autonomous University of Nuevo Leon, Mexico \\
%e-mail
email:mbasin@fcfm.uanl.mx \\
%coauthor
{\bf Maria Aracelia }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
This paper presents the optimal controller for a stochastic system
state given by a third degree polynomial equation with a linear
control input in the presence of linear observations. The optimal
controller equations are based on the separation principle applied
to the optimal filter for a third degree polynomial state and
linear observations and the optimal regulator for a third degree
polynomial system with a linear control input and a quadratic
criterion. The obtained results are applied to the problem of
controlling an automotive system with unobservable states and
compared with the best linear controller available for a
linearized model. Simulation graphs are given.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Minimization of the base transit time in semiconductor devices using optimal control }\\ \\
%author
{\bf Heinz Schaettler }\\
%affiliation
Washington University in St. Louis, USA \\
%e-mail
email:hms@cec.wustl.edu \\
%coauthor
{\bf Paolo Rinaldi }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this paper we consider the problem of determining the optimal
profile of doping concentration that minimizes the base transit
time in homojunction bipolar transistors. This is a well-studied
problem in the electronics literature , but attempts at giving
analytical solutions so far only have produced incorrect results
which were easily improved upon by numerical optimization. The
reason for this lies in the fact that the problem becomes an
optimal control problem with state-space constraints to which
standard results in the calculus of variations which have been
used in earlier attempts to find analytical solutions cannot
directly be applied. In this paper we give an explicit analytic
solution to the problem using the Pontryagin Maximum Principle
with state-space constraints and prove its optimality using
synthesis type arguments.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Optimal control for a general class of model in chemotherapy of cancer and HIV }\\ \\
%author
{\bf Urszula Ledzewicz}\\
%affiliation
Southern Illinois University, USA \\
%e-mail
email:uledzew@siue.edu \\
%coauthor
{\bf Heinz Schaettler}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In the past years there has been a strong interest in the analysis
of mathematical models arising in the problems of cancer
chemotherapy and treatment of HIV infections. Even for a specific
disease, like for example AIDS, in the literature an abundance of
models is considered which arises by considering different
specific aspects and neglecting others. Similarly, many different
models exist in the chemotherapy of cancer depending on the detail
of modelling like whether drug resistance is taken into account or
not. Still, all these models are normally based on one underlying
biological structure and many of them, although different in their
specifics, exhibit a significant common structure as mathematical
models. In particular this holds if optimal control problems are
considered. It therefore is natural to combine these different
approaches and analyze a general model which would encompasse
this common structure. In this paper we present such a general
optimal control problem and point out the main properties of
optimal controls which all these models have in common.\textsf{\
}We simultaneously consider optimal control problems with
objectives which are linear respectively quadratic in the control,
so-called $L_{1}$- and $L_{2}$-type objectives.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Numerical Solutions of Evolution Equations
}\vspace{-0.2in} \\
$$\begin{array}{rl}
\mbox{Organizer:}& \mbox{Yanping Lin, University of Alberta}\\
&\mbox{Tong Sun, Bowling Green State University} \end{array} $$
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Adaptive moving mesh methods for problems with blow-up solutions }\\ \\
%author
{\bf Weizhang Huang }\\
%affiliation
The University of Kansas, USA \\
%e-mail
email:huang@math.ukans.edu
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
We consider the numerical solution of partial differential
equations with blow-up solutions for which scaling invariance
plays a natural role in describing the underlying solution
structures. It is a challenging numerical problem to capture the
qualitative behavior in the blow-up region, and the use of
nonuniform meshes is essential. We consider moving mesh methods
for which the mesh is determined using so-called moving mesh
partial differential equations (MMPDEs). Based on error analysis
and motivated by the desire for the MMPDE to preserve the scaling
invariance of the underlying problem, we study the effect of
different choices of MMPDEs and monitor functions. Numerical
results are also presented to highlight features of moving mesh
methods.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Numerical Schemes for a highly coupled
elliptic-parabolic problem }\\ \\
%author
{\bf Shuqing Ma }\\
%affiliation
University of Alberta, Canada \\
%e-mail
email:shuqing@ualberta.ca
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
In this talk we study a finite element method for the following
strongly coupled nonlinear parabolic variational inequality:
\begin{eqnarray*}
\frac{\partial u}{\partial t}-\nabla[k(u)\nabla u] + \eta \int_{\Omega}G(x,y)u(y,t)dy +
\gamma u^4 \geq \\
\nabla[\sigma(u) \phi \nabla \phi].
\nonumber\\
-\nabla[\sigma(u)\nabla \phi] = 0. \nonumber
\end{eqnarray*}
This is a mathematical system which models the behavior of certain
micromachined thermistor sensors where heat losses to the
surrounding gas and radiation effects play a significant role. We
discuss the case where mixed boundary conditions are satisfied for
both $u$ and $\phi$. The difficulty arising from this case is the
low regularity of the exact solutions and hence the usual analysis
tools can not be applied here directly. We intend to apply a
nonstandard method and obtain suitable error estimate for the
scheme.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Numerical simulation of capillary formation during the onset of tumor angiogenesis }\\ \\
%author
{\bf Michael Smiley }\\
%affiliation
Iowa State University, USA \\
%e-mail
email:mwsmiley@iastate.edu
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
A mathematical model of the process of angiogenesis, the sprouting
of new capillary vessels from pre-existing capillaries, as it
relates to tumor vascularization will be briefly described. To
fully understand the implications of the model, which is a coupled
system of nonlinear ordinary and partial differential equations
describing biochemical and cellular processes in the extracellular
matrix between a tumor and an existing capillary, numerical
simulations are needed. Due to the nature of the modelling
equations both smooth and front-like regimes must be numerically
resolved. A discussion of these and other numerical issues will be
given and results of numerical simulations will be presented. This
is a report of work done in collaboration with H.A. Levine and M.
Nilsen-Hamilton, two of the proposers of the model.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Long-time error estimation for semi-linear parabolic PDEs }\\ \\
%author
{\bf Tong Sun }\\
%affiliation
Bowling Green State University, USA \\
%e-mail
email:tsun@math.bgsu.edu
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
The new long-time error estimation approach is applied here for
the error analysis and estimation of linear and semi-linear
parabolic partial differential equations. The new concepts and
techniques include the stability-smoothing indicator, the
smoothing assumption, the moving attractor, the exact error
propagation and the two-level error propagation analysis.
Moreover, an inverse elliptic projection is employed here as a key
technique in dealing with the spatial discretization error.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On numerical solutions of a dynamic contact problem }\\ \\
%author
{\bf Janos Turi }\\
%affiliation
University of Texas, Dallas, USA\\
%e-mail
email:turi@utdallas.edu
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
We consider the dynamics of a cantilever beam which is imperfectly
connected to a rigid wall. The contact condition consists in the
fact that the loose part of the beam cannot penetrate the wall. We
discuss a discretization procedure which reduces the originally
three dimensional problem to a two-point boundary value problem
with contact boundary conditions and establish the well-posedness
of the discretized problem.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On American put options on zero-coupon bonds }\\ \\
%author
{\bf HongTao Yang }\\
%affiliation
University of North Carolina, Charlotte, USA \\
%e-mail
email:hyang@uncc.edu
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
In this paper I shall study the existence and uniqueness of weak
solutions to parabolic free boundary problems for American put
options on zero-coupon bonds. Numerical methods and results are
also presented.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\newpage
\begin{center}
{\Large \bf Traveling Waves and Shock Waves
}\\
Organizer: Xiao-Biao Lin , North Carolina State University \\
\hspace{0.87in}Stephen Schecter , North Carolina State University
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Waves for bistable equations with non-local Mexican Hat interaction }\\ \\
%author
{\bf Peter Bates }\\
%affiliation
Michigan State University, USA\\
%e-mail
email:bates@math.msu.edu \\
%coauthor
{\bf Xinfu Chen } and {\bf Adam Chamj }
%abstract
\\
We establish the existence of stationary or travelling waves for a
nonlocal dissipative equation with bistable kinetics. The nonlocal
self-interaction is indefinite and so no maximum principle holds.
The equation arises in biological and material sciences.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Shock waves in a hemodynamics problem}\\ \\
%author
{\bf Suncica Canic }\\
%affiliation
University of Houston, USA \\
%e-mail
email: canic@math.uh.edu \\
%coauthor
%{\bf Xinfu Chen }\\{\bf Adam Chamj }\\
%abstract
\\
I will discuss formation of shock waves in a hyperbolic model of
blood flow through compliant axisymmetric large vessels. Although
the model and its closely related versions have been extensively
used by many authors, a mathematical analysis of shock formation
with pulsatile flow boundary data has not been performed. The aim
is to show that, in healthy humans, the model does not produce
shock waves. In this talk I will present estimates on the initial
and boundary data that imply strict hyperbolicity in the region of
smooth flow, and prove a general theorem which gives conditions
under which an initial-boundary value problem for a quasilinear
hyperbolic system admits a smooth solution. Using this result I
will show that pulsatile flow boundary data always gives rise to
shock formation (high gradients in the velocity and inner vessel
radius) but the time and the location of the first shock formation
is well outside the physiologically interesting region (2.8 meters
downstream from the inlet boundary). In the end I will present a
study of the influence of vessel tapering on shock formation and
show movies of the dynamics of the vessel wall and shock formation
in the pulsatile blood flow regime.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Nongeneric heteroclinic loop bifurcation }\\ \\
%author
{\bf Bo Deng }\\
%affiliation
University of Nebraska-Lincoln, USA \\
%e-mail
email: bdeng@math.unl.edu
%coauthor
%{\bf Xinfu Chen }\\{\bf Adam Chamj }\\
%abstract
\\
Homoclinic and heteroclinic bifurcations from a heteroclinic loop
are considered. The system in consideration has three parameters
two of which are not suitable for generic unfoldings. Analytical
criteria in terms of derivatives to Melnikov's functions are given
for nongeneric parameters. Four qualitatively distinct bifurcation
diagrams are obtained. The result is used to give an explanation
to a numerical finding on the generation of travelling pulsing
waves in a two-phase flow problem.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Exponential dichotomies and semidiscrete profiles of shock waves }\\ \\
%author
{\bf J\"org H\"arterich }\\
%affiliation
Free University Berlin, Germany \\
%e-mail
email: haerter@math.fu-berlin.de
%coauthor
%{\bf Xinfu Chen }\\{\bf Adam Chamj }\\
%abstract
\\
Looking for travelling waves in conservation laws which are
discretized in space leads to the study of forward--backward
functional differential equations. Even in the linear
constant-coefficient case the Cauchy problem for these equations
is not well posed. The main result of the talk states that for a
linear non-autonomous forward-backward functional differential
equation the state space can be split into a direct sum of two
subspaces: One of them contains all initial conditions that allow
for a solution in forward time, the other consists of those
initial conditions for which a solution exists for all negative
times. Moreover, solutions in both subspaces decay exponentially
fast. It is shown how this result can be used to prove the
existence of a center manifold containing the weak shock profiles
near a constant state. Such a center manifold has only recently
been constructed by Benzoni-Gavage and Huot using a different
method. It is also indicated how the exponential dichotomies may
be used to identify geometric objects which determine the linear
stability of semidiscrete shock profiles. The talk is based on
joint work with B.Sandstede and A.Scheel.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Local tracking and stability for degenerate viscous shock waves }\\ \\
%author
{\bf Peter Howard }\\
%affiliation
Texas A \& M University, USA \\
%e-mail
email: phoward@math.tamu.edu
%coauthor
%{\bf Xinfu Chen }\\{\bf Adam Chamj }\\
%abstract
\\
It is well known that the stability of certain distinguished waves
arising in evolutionary PDE can be determined by the spectrum of
the linear operator found by linearizing the PDE about the wave.
Indeed, work over the last fifteen years has shown that {\it
spectral stability} implies nonlinear stability in a broad range
of cases, including asymptotically constant travelling waves in
both reaction--diffusion equations and viscous conservation laws.
A critical step toward analyzing the spectrum of such operators
was taken in the late eighties by Alexander, Gardner, and Jones,
whose {\it Evans function} (generalizing earlier work of John W.
Evans) serves as a characteristic function for the above-
mentioned operators. Thus far, results obtained through working
with the Evans function have made critical use of the function's
analyticity at the origin (or its analyticity over an appropriate
Riemann surface). In the case of degenerate (or sonic) viscous
shock waves, however, the Evans function is certainly not analytic
in a neighborhood of the origin, and does not appear to admit
analytic extension to a Riemann manifold. I will show how this
obstacle can be surmounted by dividing the Evans function (plus
related objects) into two pieces: one analytic in a neighborhood
of the origin, and one sufficiently small.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Edge bifurcations for near integrable systems }\\ \\
%author
{\bf Todd Kapitula }\\
%affiliation
University of New Mexico, USA \\
%e-mail
email: kapitula@math.unm.edu \\
%coauthor
{\bf Bjorn Sandstede }
%abstract
\\
When studying the linear stability of waves for near integrable
systems, a fundamental problem is the location of the point
spectrum of the linearized operator. Internal modes may be created
upon the perturbation, i.e., eigenvalues may bifurcate out of the
continuous spectrum, even if the corresponding eigenfunction is
initially not localized. This phenomenon is also known as an edge
bifurcation. It has recently been shown that the Evans function is
a powerful tool when one wishes to detect an edge bifurcation and
track the resulting eigenvalues. It has been an open question as
to the role played by the solutions to the Lax pair, associated
with the integrable problem, in the construction of the Evans
function and the detection of edge bifurcations. Using the
Zakharov--Shabat eigenvalue problem as an illustration, we show
the connection between the inverse scattering formalism and the
linear stability analysis of waves. In particular, we show a
direct connection between the scattering coefficients and the
Evans function.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Travelling waves in thermal multiphase flow in porous media }\\ \\
%author
{\bf Dan Marchesin }\\
%affiliation
Instituto Nacional de Matem\'atica Pura e Aplicada- IMPA, Brazil \\
%e-mail
email: marchesi@impa.br
%coauthor
%{\bf Bjorn Sandstede }\\
%abstract
\\
Combustion in-situ and steam injection are methods used to improve
oil recovery, especially when it is very viscous. The main reason
for their effectiveness is that they reduce oil viscosity by
heating. We discuss the main nonstandard wave appearing in such
flows, a shock where the chemical reaction or the condensation
occurs. The internal structure of this shock is crucial to the
nature of the overall flow, so it has to be studied carefully as a
travelling wave. We show that such a wave is described as an orbit
connecting two equilibria of an associated system of ODE's.
Typically, at least one of the equilibria is not hyperbolic.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Shock structure in two- and three-phase flow with permeability hysteresis }\\ \\
%author
{\bf Bradley Plohr }\\
%affiliation
University at Stony Brook, USA \\
%e-mail
email: plohr@ams.sunysb.edu \\
%coauthor
{\bf Dan Marchesin }
%abstract
\\
Two-phase flow in a porous medium can be modelled, using Darcy's
law, in terms of the relative permeability functions of the two
fluids (say, oil and water). The relative permeabilities generally
depend not only on the fluid saturations but also on the direction
in which the saturations are changing. During water injection, for
example, the relative oil permeability falls gradually until a
threshold is reached, at which stage the begins to decrease
sharply. This stage is termed imbibition. If oil is subsequently
injected, then the relative oil permeability does not recover
along the imbibition path, but rather increases only gradually
until another threshold is reached, whereupon it rises sharply.
This second stage is called drainage, and the type of flow that
occurs between the imbibition and drainage stages is called
scanning flow. Changes in permeability during scanning flow are
approximately reversible, whereas changes during drainage an d
imbibition are irreversible. In our lecture, we describe a model
of permeability hysteresis based on relaxation. The distinctive
features of our model are that it allows the scanning flow to
extend beyond the drainage and imbibition curves and it treats
these two curves as attractors of states outside the scanning
region. Through a rigorous stud y of travelling waves, we
determine the shock waves that have diffusive profiles, and by
means of a formal Chapman-Enskog expansion, we make a connection
between our model and a standard one in the limit of vanishing
relaxation time. Numerical experiments confirm our analysis.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Traveling waves in networks of synaptically coupled spiking neurons }\\ \\
%author
{\bf Jonathan Rubin }\\
%affiliation
University of Pittsburgh, USA\\
%e-mail
email:rubin@math.pitt.edu \\
%coauthor
{\bf Bard Ermentrout }\\{\bf Remus Osan}
%abstract
\\
Experiments have suggested the existence of propagating waves of
activity in neuronal media. Such waves result from the intrinsic
behavior of the neuronal circuitry, as well as synaptic
interactions between neurons, which are nonlocal and
time-dependent. We analyze travelling waves in one-dimensional
models for such media, which take the form of integrodifferential
equations. Specifically, we use shooting methods to prove the
existence of smoothly propagating waves in the theta model for
neurons near a saddle-node bifurcation, and we consider stability
of these waves. Further, we study single-spike, multi-spike, and
periodic waves in an integrate-and-fire model.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Period-doubling bifurcations of spiral waves }\\ \\
%author
{\bf Bjorn Sandstede }\\
%affiliation
Ohio State University, USA \\
%e-mail
email:sandstede.1@osu.edu
%coauthor
%{\bf Bard Ermentrout }\\{\bf Remus Osan}
%abstract
\\
Period-doubling bifurcations of spiral waves have been observed in
light-sensitive BZ-type experiments. These observations are
puzzling since period-doubling bifurcations of spirals appear to
be impossible from a mathematical viewpoint. We report on
preliminary and partial explanations of these phenomena that
involve the computation of spectra, and of the associated
eigenfunctions, of spirals on bounded and unbounded domains.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Active and passive defects }\\ \\
%author
{\bf Arnd Scheel }\\
%affiliation
University of Minnesota, USA \\
%e-mail
email:scheel@math.umn.edu
%coauthor
%{\bf Bard Ermentrout }\\{\bf Remus Osan}
%abstract
\\
Motivated by the observation of one-dimensional spirals in the
CIMA reaction and travelling defects in convection experiments, we
study interaction between a large, standing pulse and
small-amplitude plain waves. We approximately describe the
dynamics of the plain waves by Burgers' equation and investigate
the role of the pulse as a point defect.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Thin film flow near a dynamic contact line }\\ \\
%author
{\bf Michael Shearer }\\
%affiliation
North Carolina State University, USA \\
%e-mail
email:shearer@math.ncsu.edu
%coauthor
%{\bf Bard Ermentrout }\\{\bf Remus Osan}
%abstract
\\
A fourth order nonlinear PDE describes the flow of a thin liquid
film up an inclined surface, under the opposing forces of gravity
and shear stress. The Navier slip condition allows a small amount
of slip between the liquid and the solid surface near the contact
line; it is designed to remove the stress singularity that occurs
under the usual no-slip condition. I present analytic and
numerical results for the ODE satisfied by travelling wave
solutions representing the contact line dynamics. Some of these
solutions can be identified with compressive shocks, and others
with undercompressive shocks, depending on whether upstream
characteristics approach the contact line, or are slower than the
contact line, respectively.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Travelling waves in random media }\\ \\
%author
{\bf Wenxian Shen }\\
%affiliation
Auburn University, USA \\
%e-mail
email:ws@math.auburn.edu
%coauthor
%{\bf Bard Ermentrout }\\{\bf Remus Osan}
%abstract
\\
The current talk deals with travelling waves in random diffusive
media, including time and/or space recurrent, almost periodic,
quasi-periodic, periodic ones as special cases. It first
introduces a notion of travelling waves in general random media. A
theory of existence of travelling waves is then established.
Applications of the general theory to bistable and KPP type media
are discussed. The results obtained generalize many existing ones
on travelling waves.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Spectral stability of small amplitude shock waves}\\ \\
%author
{\bf Peter Szmolyan}\\
%affiliation
Technical University of Vienna,Austria \\
%e-mail
email:szmolyan@tuwien.ac.at \\
%coauthor
%{\bf Bard Ermentrout }\\{\bf Remus Osan}
%abstract
\\
TBA
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Deformations of homoclinic and periodic solutions under singular perturbations}\\ \\
%author
{\bf Alexander Tovbis }\\
%affiliation
University of Central Florida, USA \\
%e-mail
email:tovbis@math.duke.edu \\
%coauthor
%{\bf Bard Ermentrout }\\{\bf Remus Osan}
%abstract
\\
We will consider a family of singular perturbations for certain
model second order ODE (quadratic nonlinearity). Deformatons of
both periodic and homoclinic solutions under these perturbations
will be discussed.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Stability of viscous shock and relaxation profiles }\\ \\
%author
{\bf Kevin Zumbrun }\\
%affiliation
Indiana University, USA \\
%e-mail
email:kzumbrun@indiana.edu \\
%coauthor
{\bf Corrado Mascia }
%abstract
\\
Under the weak assumption of spectral stability, or stable point
spectrum of the linearized operator about the wave, we establish
sharp pointwise Green's function bounds and consequent linear and
nonlinear stability for shock profiles of relaxation and real
viscosity systems satisfying the dissipativity condition of
Zeng/Kawashima. These include in particular compressible
Navier--Stokes and MHD equations, and essentially all standard
relaxation models: in particular, the discrete kinetic models of
Broadwell, Jin-Xin, Natalini, Bouchut, Platkowski--Illner, and the
moment closure models of Grad, Levermore, M\"uller--Ruggeri. A
consequence is stability of small-amplitude profiles of Broadwell
and Jin-Xin models and of general real viscosity systems, for each
of which spectral stability has been verified in other works.
These are the first complete stability results for profiles of a
real viscosity system, and the first for relaxation models with
nonscalar equilibrium equations. Our results apply also in
principle large-amplitude shocks, an important direction for
future investigation.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
\end{multicols}
\begin{center}
{\Large \bf Recent Progress in the Theory of Exponential Attractors}\\
Organizer: A. Miranville, University of Poitiers
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Chaotic attractors in discrete population models }\\ \\
%author
{\bf Jerry Chen }\\
%affiliation
University of Delaware, USA \\
%e-mail
email:jchen@math.udel.edu \\
%coauthor
{\bf Denis Blackmore }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We investigate and compare the dynamics of the discrete
Exponentially Self-Regulating (ESR), Lotka-Volterra (LV) and
Pioneer-Climax (PC) models for competing populations along with
some variants of these models. Special attention is paid to the
existence of chaotic dynamics for these models in certain ranges
of the critical parameters. It is shown, in particular, that
several of these models have strange attractors of the twisted
horseshoe with bending tail variety recently proven to exist for
special ESR models.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Exponential attractors for a class of
cross-diffusion reaction
systems on any dimensional domains }\\ \\
%author
{\bf Le Dung }\\
%affiliation
University of Texas at San Antonio, USA \\
%e-mail
email:dle@sphere.math.utsa.edu
%coauthor
%{\bf Denis Blackmore } \\New Jersey Institute of Technology
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
It will be proven that a class of triangular cross diffusion
systems possess exponential attractors in $W^{1,p}(\Omega)$. Here,
$\Omega$ is smooth bounded domain in $R^n$ with $n$ can be
arbitrary and $p>n$. Thus, the functional spaces considered here
are Banach spaces.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Gevrey regularity for the attractor of a damped wave equation }\\ \\
%author
{\bf Cedric Galusinski }\\
%affiliation
MAB, Universite Bordeaux I, France \\
%e-mail
email:galusins@math.u-bordeaux.fr \\
%coauthor
{\bf Sergei Zelik IITP }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\setcounter{equation}{0}
\\
The goal of this paper is to obtain a Gevrey regularity for the
attractor of the following singularly perturbed damped wave
equation in a cube domain $\Omega=[0,2\pi]^3$\\
\begin {equation}
\label {sys}
\begin {array} l
\eps \partial _t^2 u^\eps +\gamma \partial_t u^\eps -\lap u^\eps+f(u^\eps)=g,\\
u^\eps_{|t=0}=u_0,~\partial_tu^\eps_{|t=0}=u_1,
\end {array}
\end {equation}
with periodic boundary conditions. We assume that $\eps >0$ and
$\gamma>0$. The nonlinear function $f$ is required to be real
analytic,
\begin {equation}
f(u)=\sum_{j=0}^{\infty} a_j u^j,\mbox{ where }
\sum_{j=0}^{\infty} |a_j| s^j<+\infty ~\forall s\in \RR. \label
{hyp0}
\end {equation}
We assume furthermore that the nonlinearity $f$ satisfies
\begin {equation}
\begin {array} l
f'(u)\geq -K,\\
f(u)\cdot u \geq 0 \mbox{ if } |u|\geq L\\
|f''(u)|\leq C(1+|u|),
\end {array}
\label{hyp}
\end {equation}
where $C$, $K$, and $L$ are fixed positive constants. The
assumptions (\ref{hyp0}) and (\ref{hyp}) are fulfilled for cubic
nonlinearity $f(u)=u^3-\a u,~\a \in \RR$.
{\bf Remark} We can replace the assumption (\ref{hyp}) by an other
one, if we are able to obtain uniform (with respect to $\eps$)
absorbing sets in $L^\infty(\Omega)$. For example $f(u)=\sin u$.
We assume that
\begin {equation}
\label {hypg} g\mbox{ is periodic and analytic.}
\end {equation}
we already obtained, for this problem, the existence of
exponential attractors with a rate of attraction, a diameter and a
fractal dimension uniform with $\eps$, in the variable $u$ as well
as $u_t$. For such a result, the appropriated spaces for solutions
of (\ref{sys}) are ${\mathcal E}^k(\eps)=H^{k+1}_{per}(\Omega)
\times H^{k}_{per}(\Omega)$ equipped with the following norms
$$
||(u^\eps(t),u_t^\eps(t))||_{{\mathcal E}^k(\eps)}^2= $$ $$\eps
||u_t^\eps(t)||^2_{H^{k}_{per}}+||u_t^\eps(t)||^2_{H^{k-1}_{per}}+||u^\eps(t)||^2_{H^{k+1}_{per}},
$$
The aim here is to establish
{\bf Theorem} Let $k>\frac 5 2$, let $(u_0,u_1)$ be in ${\mathcal
B}\subset {\mathcal E}^k$, under assumptions (\ref{hyp0}),
(\ref{hyp}), (\ref{hypg}), there exists $(v,v_t)$ uniformly
bounded with respect to $\eps$ in $C^\infty(\RR^+,{\mathcal
F}_\sigma^k (\eps))$ such that
$$
\begin {array} l
||(u^\eps(t),u^\eps_t(t))-(v(t),v_t(t))||_{{\mathcal
E}^0(\eps)}\leq
d(t),~~~\forall t\geq 0,\vspace{0.1in}\\
\mbox{with }\displaystyle \lim_{t\to \infty} d(t)=0,
\end {array}
$$
where the function $d(t)$ is independent of $\eps$ but depends on
${\mathcal B}$, $g$, $f$. The Gevrey space ${\mathcal F}_\sigma^k
(\eps)$ is equipped with the norm
$$
||(u^\eps(t),u_t^\eps(t))||_{{\mathcal F}_\sigma^k(\eps)}^2=$$ $$
\eps ||u_t^\eps(t)||^2_{G^{\frac k
2}_{\sigma}}+||u_t^\eps(t)||^2_{G^{\frac {k-1}
2}_{\sigma}}+||u^\eps(t)||^2_{G^{\frac {k+1} 2}_{\sigma}}.
$$
{\bf Corollary} Under the same assumptions, the points of the
attractor of (\ref{sys}) are uniformly bounded for the
${\mathcal F}_\sigma^k (\eps)$-norm.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Necessary and sufficient conditions for the
existence of global attractors of semigroups and
applications }\\ \\
%author
{\bf Qingfeng Ma }\\
%affiliation
Indiana University, USA \\
%e-mail
email:qima@indiana.edu \\
%coauthor
{\bf Checkups Zhong } and {\bf Shouhong Wang }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
First we established some necessary and sufficient conditions for
the existence of the global attractor of an infinite dimensional
dynamical system, using the measure of noncompactness. Then we
gave a new method/recipe for proving the existence of the global
attractor. The main advantage of this new method/recipe is that
one needs only to verify a necessary compactness condition with
the same type of energy estimates as those for establishing the
absorbing set. In other words, one doesn't need to obtain
estimates in function spaces of higher regularity. In particular,
this property is useful when higher regularity is not available,
as demonstrated in the example on the Navier-Stokes equations on
nonsmooth domains.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A construction of exponential attractors in Banach spaces }\\ \\
%author
{\bf Alain Miranville }\\
%affiliation
Universite de Poitiers, France \\
%e-mail
email:Alain.Miranville@mathlabo.univ-poitiers.fr
%coauthor
%{\bf Chengkui Zhong } \\Math Depart. at Lanzhou University , P.R.China \\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Our aim in this talk is to give a construction of exponential
attractors in Banach spaces and present some applications and
consequences.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Exponential attractors in Banach spaces }\\ \\
%author
{\bf Basil Nicolaenko }\\
%affiliation
Arizona State University, USA \\
%e-mail
email:byn@math.la.asu.edu
%coauthor
%{\bf Chengkui Zhong } \\Math Depart. at Lanzhou University , P.R.China \\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We extend the theory of exponential attractors from the Hilbert
space setting to the Banach spaces context. No squeezing
conditions are needed, in contrast to previously known
constructions. The only requirements are for the semiflow to be
differentiable in some absorbing ball; and for the linearized
semiflow at every point inside the absorbing ball to split into
the sum of a compact operator plus a contraction. This turns out
to be equivalent to the weakest conditions known for a semiflow to
possess a global classical attractor of finite fractal dimension
and seems to indicate that exponential attractors are as general
as global attractors of finite dimension. Applications are made to
3D Navier-Stokes equations with fast rotation. This is a joint
work with Le Dung ( University of Texas in San Antonio). Ref.
"Exponential Attractors in Banach Spaces",L.Dung and B.
Nicolaenko, J. of Dynamics and Diff. Equations, Vol. 13,
4,791-806(2001).
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Uniform attractors for dynamical systems with memory }\\ \\
%author
{\bf V. Pata }\\
%affiliation
Politecnico di Milano, Italy \\
%e-mail
email:pata@mate.polimi.it
%coauthor
%{\bf Chengkui Zhong } \\Math Depart. at Lanzhou University , P.R.China \\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We are concerned with a special class of dissipative infinite
dimensional dynamical system which has been recently investigated;
namely, evolution systems with memory subject to time dependent
external forces. These models arise in the description of several
phenomena like, e.g., heat conduction in special materials,
viscoelasticity, phase transitions. In order to pursue a global
analysis, the ``past history" of the system is viewed as a
supplementary variable, ruled by an equation of hyperbolic type.
This choice, however, entails a loss of compactness which has to
be circumvented in order to get a satisfactory asymptotic
analysis.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Exponential attractor for the wave equation with nonlinear damping }\\ \\
%author
{\bf Dalibor Prazak }\\
%affiliation
Charles University, Prague, Czech Republic \\
%e-mail
email:prazak@karlin.mff.cuni.cz
%coauthor
%{\bf Chengkui Zhong } \\Math Depart. at Lanzhou University , P.R.China \\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The evolution of the equation \[ u_{tt} + g(u_t) - \triangle {u} +
f(u) = 0 \] is studied in bounded two or three-dimensional domain
under the zero boundary condition. We show that the equation has
an exponential attractor provided that the nonlinearities $f$ and
$g$ are functions of certain polynomial growth.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Infinite-dimensional exponential attractors for
reaction-diffusion equations
in unbounded domains and their approximation }\\ \\
%author
{\bf Sergey Zelik }\\
%affiliation
Universite de Poitiers, France \\
%e-mail
email:zelik@mathlabo.univ-poitiers.fr
%coauthor
%{\bf David Hoff} \\ Indiana University \\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider the following reaction-diffusion system problem in an
unbounded domain $\Omega\subset\R^n$:
$$
\left. \begin{array}{l}
\Dt u=a\Dx u-(\vec L,\Nx)u-f(u)+g,\ \ x\in\Omega, \\
t\ge0,\ \ u\tto=u_0, \ \ u\DOM=0. \end{array} \right. \eqno (1) $$
Here $u=u(t,x)=(u^1,\cdots,u^k)$ is an unknown vector-valued
function, $\Dx$ is the Laplacian with respect to the variable $x$,
$\vec L\in C^1_b(\Omega)$ is a given divergent free vector field
in $\Omega$, $a\in { L(\R^k,\R^k)}$ is a given constant diffusion
matrix with positive symmetric part ($a+a^*>0$) and $f(u)$ is a
given nonlinear interaction function which is assumed to satisfy
the following conditions: $$ \begin{array}{l} 1.\ \ f\in C^2(\R^k,\R^k),\\
2.\ \ f(v).v\ge -C+\alpha|v|^2 ,\ \ f'(v)\ge-K,\\ 3.\ \ |f(v)|\le
C(1+|v|^p), \end{array} $$ where $u.v$ denotes the standard inner
product in $\R^k$, $\alpha>0$ and the fixed exponent $p\ge1$
satisfies the growth restriction $p<1+\frac4{n-4}$ if $n>4$.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Exponential attractors for the compressible Navier-Stokes equations }\\ \\
%author
{\bf Mohammed Ziane }\\
%affiliation
University of Southern California, USA \\
%e-mail
email:ziane@math.usc.edu \\
%coauthor
{\bf David Hoff}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The existence of exponential attractors for the Navier-Stokes
equations of compressible flow will be presented. As a by product
of the analysis, the existence of the uniform attractor with a
finite fractal dimension follows.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Ginzburg-Landau Equation in Superconductivity and Related
Topics
}\\
Organizer: Yoshihisa Morita, Ryukoku University
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Non-linear surface supeconductivity for type II superconductors in the large domain limit}\\ \\
%author
{\bf Yaniv Almog }\\
%affiliation
Technion, Israel \\
%e-mail
email:almog@math.technion.ac.il
%coauthor
%{\bf Boris Belinskiy}\\
%abstract
\\
The Ginzburg-Landau model for superconductivity is considered in
two-dimensions. We show, for smooth bounded domains, that
superconductivity remains concentrated near the surface when the
applied magnetic field is decreased below $H_{C_3}$ as long as it
is greater than $H_{C_2}$. We demonstrate this result in the large
domain limit, i.e, when the domain's size tends to infinity.
Additionally, we prove that for applied fields greater than
$H_{C_2}$, the only solution in $\Bbb{R}^2$ satisfying normal
state condition at infinity is the normal state. The above results
have been proved in the past for the linear case. Here we prove
them here for non-linear problems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Triality and primal-dual algorithm for Ginzburg-Landau equation }\\ \\
%author
{\bf David Gao }\\
%affiliation
Virginia Tech and Louisiana Tech, USA\\
%e-mail
email:gao@vt.edu \\
%coauthor
{\bf Ping Lin }
%abstract
\\
The Ginzburg-Landau Equation is central to material science, which
has been subjected to a substantial study during the last twenty
years. Since the total potential energy associated with this
equation is a nonconvex (double-well) functional, traditional
direct analysis and related numerical methods for solving this
nonconvex variational problem are difficult. Based on the
canonical dual transformation method proposed recently by Gao
(2000), we will show that the Ginzburg-Landau Equation is
equivalent to the so-called differential algebraic equations
(DAEs). Therefore, a primal-dual algorithm is developed for
solving the nonconvex Ginzburg-Landau boundary value problem. This
method can also be used to control the phase transformation in
superconductivity. Application are illustrated by a two
dimensional example. Reference: Gao, D.Y. (2000). Duality
Principles in Nonconvex Systems: Theory, Methods and Applications.
Kluwer Academic Publishers, Dordrecht/London/Boston, xviii+454pp.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Ginzburg-Landau equation and existence of stable solutions with domain dependency }\\ \\
%author
{\bf Shuichi Jimbo }\\
%affiliation
Hokkaido University, Japan \\
%e-mail
email:jimbo@math.sci.hokudai.ac.jp
%coauthor
%{\bf Ping Lin }\\
%abstract
\\
I consider critical points of GL functional (solutions of GL
equations) with magnetic effect (in 2-dim and 3-dim cases) and
existence of local minimizers and its dependency of geometrical
property of the domain. One may naturally understand if the domain
is simple, only simple situation could occur while complicated
domains will have interesting phenomena. I talk about this
direction of the study of GL equations.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Direct numerical simulations of the time-dependent Ginzburg-Landau }\\ \\
%author
{\bf Masahiko Machida }\\
%affiliation
Japan Atomic Energy Research Institute, Japan \\
%e-mail
email:mac@koma.jaeri.go.jp
%coauthor
%{\bf Ping Lin }\\
%abstract
\\
Recently, the superconducting vortex matter has been intensively
investigated in the physics community in order to understand
anomalous behaviors of the vortex systems in High-Tc
superconductors, and consequently, many novel concepts have been
established in the present decade. The most prominent idea is the
melting of the Abrikosov vortex lattice and the new phase called
the vortex liquid state is added to the phase diagram of the
superconducting states on the magnetic field and the temperature.
Generally, the transition and the new phase have been mainly
studied by using Monte Calro techniques on the simplified systems,
i.e., XY model, interacting vortex line model and so on. In this
paper, we approach to the melting and the vortex liquid phase by
using the time-dependent Ginzburg-Landau equation coupled with the
Maxwell equation under the thermal noise. The way is the most
fundamental and intriguing because all fluctuating components are
coupled. We demonstrate that vortex liquid phases grow as multiple
droplets and percolate over the whole region at the melting
transition point while we compare the melting with that in the
other systems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Ginzburg-Landau equation in a thin domain }\\ \\
%author
{\bf Yoshihisa Morita }\\
%affiliation
Ryukoku University, Japan \\
%e-mail
email:morita@rins.ryukoku.ac.jp
%coauthor
%{\bf Ping Lin }\\
%abstract
\\
The Ginzburg-Landau equation in a thin domain is a model of the
superconductivity in a thin material (film) and it is an
Euler-Lagrange equation of the Ginzburg-Landau energy functional.
The convergence of a (global) minimizer of the energy functional
as the thinness tends to zero was studied by Rubinstein-Schatzman
and Chapman-Du-Gunzburger. Here we show that there exist stable
solutions to the equation, that is, local minimizers of the
functional, with vortices in the absence of a forcing magnetic
field for an appropriate 3-dimensional thin domain.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Surface superconductivity and boundary effect }\\ \\
%author
{\bf Xingbin Pan }\\
%affiliation
National Singapore University, Singapore\\
%e-mail
email:matpanxb@nus.edu.sg
%coauthor
%{\bf Ping Lin }\\
%abstract
\\
In this talk we shall discuss the asymptotic behavior of the
minimal solutions of Ginzburg-Landau system, and describe the
concentration phenomena of surface superconductivity for type 2
superconductors in applied magnetic fields lying in between
critical fields $H_{C_2}$ and $H_{C_3}$. Our main concern is the
effect of the sample geometry on the value of critical fields and
on the profile of superconducting sheath at surface.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Thermal effects in superconductivity }\\ \\
%author
{\bf Daniel Phillips }\\
%affiliation
Purdue University, USA \\
%e-mail
email:phillips@math.purdue.edu
%coauthor
%{\bf Ping Lin }\\
%abstract
\\
We analyze a model for non-isothermal superconductivity derived
independently by G. Maugan, and S. Zhou and K. Miya. The model is
described by a parabolic system based on the time dependent
Ginzburg-Landau equation(TGLE),the Maxwell equation, and an energy
equation such that the Clausius-Duhem inequality holds. We prove
existence, regularity and large time behavior results for
solutions to initial value boundary value problems for this
system. The sensitivity of superconducting materials to thermal
variations is a major obstacle in their applications. In practice
one sees that vortex motion in current patterns generates thermal
energy, producing "hot spots" which results in suppressing
superconductivity. Our principal qualitative result is that we
exhibit this phenomenon in solutions. Prior analytic work on
vortex motion centered on the isothermal model of the TGLE and
Maxwell equations. This setting is completely kinematic and
produces markedly different evolutions. This work is joint with
Eunjee Shin (Penn State).
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A nonstiff Euler discretization of the complex Ginzburg-Landau equation in one space dimension}\\ \\
%author
{\bf Peter Takac }\\
%affiliation
Universitaet Rostock, Germany \\
%e-mail
email:peter.takac@mathematik.uni-rostock.de
%coauthor
%{\bf Ping Lin }\\
%abstract
\\
A nonstiff discretization method is applied to the complex
Ginz\-burg-Lan\-dau equation with periodic boundary conditions,
\begin{eqnarray*} & \partial_t u = (1+i\nu) \partial_{xx}^2 u + Ru - (1+i\mu)|u|^2 u ; \\ & u(x + 2\pi,t) = u(x,t) \quad\mbox{ for } -\infty < x < \infty ,\; t\geq 0 . \end{eqnarray*}
This parabolic equation is discretized first in space by a
truncated Fourier series (that is, a finite Fourier sum) and then
in time by an explicit Euler method. The exponential decay of
Fourier modes in both, the original complex Ginzburg-Landau
equation and the resulting equation for the truncated Fourier
series (independently from the number of Fourier modes in the
truncation) is used in an essential way to prove the nonstiffness.
Also dissipativity of the discretized equation is established and
numerical results are discussed.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Micromagnetic soliton and Landau-Lifshitz equation}\\ \\
%author
{\bf Jian Zhai }\\
%affiliation
Zhejiang University, China \\
%e-mail
email:zhai@math.zju.edu.cn
%coauthor
%{\bf Ping Lin }\\
%abstract
\\
We plan to study the properties of the weak solutions to nonlinear
partial differential equations arisen from ferromagnets,
especially the hyper-surface moving by its curvature, the dynamics
of topological magnetic solitons and the behaviour of the
solutions to Landau-Lifshitz equation.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Learning Theory, Dynamic System and Neural Networks
}\\
Organizer: Kayvan Najarian, University of North Carolina at
Charlotte
\\
\hspace{0.85in}Mirsad Hadzikadic, University of North Carolina at
Charlotte
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf A learning theory approach to the construction of predictor models }\\ \\
%author
{\bf Marco Campi }\\
%affiliation
University of Brescia, Italy \\
%e-mail
email:campi@bsing.ing.unibs.it
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
We present a new framework for the identification of predictor
models from data, which builds on recent developments of
statistical learning theory. The three key elements of our
approach are: i) an unknown mechanism that generates the observed
data; ii) a family of predictor models, among which we select our
predictor model based on observations; iii) an optimality
criterion that we want to minimize. A major departure from
standard identification theory is taken in that we consider
interval models for prediction (that is models that returns output
intervals, as opposed to output values). Moreover, we introduce a
consistency criterion (the model is required to be consistent with
observations) which act as a constraint in the optimization
procedure. In this framework, the model has not to be interpreted
as a faithful description of reality, but, rather, as an
instrument to perform prediction. To the optimal model, we attach
a certificate of reliability, that is a statement of probability
that the computed model will be consistent with future unknown
data.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Learning theory and neural networks for signal processing and control }\\ \\
%author
{\bf M. Misaghian }\\
%affiliation
J.C. Smith University, USA \\
%e-mail
email:knajaria@uncc.edu \\
%coauthor
{\bf Kayvan Najarian }
%abstract
\\
Dynamic feedback neural networks are known to present powerful
tools in modelling of complex dynamic models. Since in many real
applications, the stability of such models (specially in presence
of noise) is of great importance, it is essential to address
stochastic stability of such models. In this paper, sufficient
conditions for stochastic stability of some families of dynamic
neural model (including two families of sigmoid neural networks)
are presented. These stability conditions are set on the weights
of the networks and can be easily tested.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Learning theory and neural networks for signal processing and control }\\ \\
%author
{\bf Kayvan Najarian }\\
%affiliation
UNC Charlotte, USA \\
%e-mail
email:knajaria@uncc.edu \\
%coauthor
{\bf M. Hadzikadic }
%abstract
\\
The learning theory has provided the tools to evaluate the
accuracy and the statistical confidence of the models and
classifiers developed from the sample training data. In addition,
the need to generate the least complex models to provide
pre-specified levels of accuracy and confidence has been addressed
in the learning theory from different standpoints. In this paper,
first the main ideas of the learning theory (specially the
Probably Approximately Learning theory) are briefly described, and
then some important applications and impacts of this theory in
different areas of signal processing and control are explained.
The paper also describes different practical techniques that apply
the ideas of the learning theory to minimize the complexity of the
models developed from the training data.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Neural networks for length-dependent and length-independent functional classification and prediction of small sequences of lipase, protease, and isomerase }\\ \\
%author
{\bf Dean Warren }\\
%affiliation
UNC Charlotte, USA \\
%e-mail
email:dswarren@uncc.edu\\
%coauthor
{\bf Kayvan Najarian }
%abstract
\\
We use neural networks to classify proteins according to their
functionality. Primary structures of small lipase, protease, and
isomerase proteins (containing between 100 and 200 amino acids)
were used for length-dependent and length-independent
classification. In length-dependent classification, the main
feature is either amino acid label (integer code), amino acid
hydrophobic values, or the solubility values defined here. The
classification performances of these length-dependent measures
were assessed using neural network classifiers for the three
enzyme classes. In addition, a set of length-independent features
related to signal complexity is introduced and used for neural
classification of the above-mentioned families of proteins. Each
neural network was developed for the given feature sets, trained,
and tested for suitability in classifying enzymes. The results of
length-dependent and length-independent neural classifiers are
compared with each other.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Application of learning theory to protein sequence classification }\\ \\
%author
{\bf Dean Warren }\\
%affiliation
UNC Charlotte, USA \\
%e-mail
email:dswarren@uncc.edu
%coauthor
%{\bf M. Hadzikadic }\\
%abstract
\\
Recently, Valiant's Probably Approximately Correct (PAC) learning
theory has been extended to learning m-dependent data. With this
extension, training data set size for sigmoid neural networks have
been bounded without underlying assumptions for the distribution
of the training data. These extensions allow learning theory to be
applied to training sets which are definitely not independent
samples of a complete input space. In our work, we are developing
length independent measures as training data for protein
classification. This paper applies these learning theory methods
to the problem of training a sigmoid neural network to recognize
protein biological activity classes as a function of protein
primary structure. Specifically, we explore the theoretical
training set sizes for classifiers using the full amino acid
sequence of the protein as the training data and using length
independent measures as the training data. Results show bounds for
training set sizes given protein size limits for the full sequence
input compared to bounds for input that is sequence length
independent.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Symmetries and Differential Equations in Physics and Other
Applications
}\\
Organizer: Weiqing Xie, Cal Poly Pomona\\
\hspace{0.8in}M. Nakashima, Cal Poly Pomona
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf A reducible representation of the generalized symmetry group of a quasiperiodic flow }\\ \\
%author
{\bf Lennard Bakker }\\
%affiliation
Franklin and Marshall College, USA \\
%e-mail
email:l$\_$bakker@fandm.edu
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
The generalized symmetry group of a flow on a manifold is the
group theoretic normalizer, within the group of diffeomorphisms of
the manifold, of the one parameter abelian group of
diffeomorphisms generated by the flow. Up to conjugacy, the
generalized symmetry group of a quasiperiodic flow on an n-torus
is determined by a system of uncoupled first order partial
differential equations. New types of symmetries (other than the
classical types of symmetries and time-reversing symmetries) may
exist depending on certain algebraic relationships being satisfied
by pair wise ratios of the frequencies of the quasiperiodic flow.
These new types of symmetries, when they exist, are a dominant
feature of a reducible linear representation of the generalized
symmetry group in the de Rham cohomology of the n-torus.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
%\title
\noindent {\bf Boundary values in de Sitter space }\\ \\
%author
{\bf Martin Nakashima }\\
%affiliation
Cal Poly Pomona, USA
%e-mail
%email:ccao@cnls.lanl.gov \\
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
The theory of Dirac Singletons, as developed by Flato and
Fronsdal, is a massless gauge theory in de Sitter space. Although
noted primarily for its physical content and connections to group
theory, it may be of further interest to mathematicians as it
suggests some problems in differential equations. This talk will
present some of these issues.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Second order dynamical systems used for generating
"practical" test functions for filtering and sampling procedures
}\\ \\
%author
{\bf Cristian I. Toma }\\
%affiliation
Politehnica University, Bucharest, Romania \\
%e-mail
email: cgtoma@physics1.physics.pub.ro
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
As it is known, in averaging procedures the user is interested in
the mean value of the received signal over a certain time
interval. Usually this operation is performed by an integration of
the signal on this time interval (considered to be constant) the
result of the integration being proportional to the mean value of
the signal. However, such structures are very sensitive at random
variations of the integration period (generated by the switching
phenomena at the end of the integration). For this reason, a
multiplication of the received signal with a test-function (a
function which differs to zero only on this time interval and with
continuous derivatives of any order on the whole real axis) is
recommended. This paper presents some invariance properties of
differential equations, which can be used for generating a
"practical" test-function on this time interval, and it presents
also the properties of second order oscillating systems
(considered as generating "practical" test functions) in filtering
and sampling procedures.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A mathematical model from stress driven diffusion}\\ \\
%author
{\bf Weiqing Xie }\\
%affiliation
Cal Poly Pomona, USA \\
%e-mail
email: wxie@csupomona.edu
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
This talk concerns the study of a system of differential equations
involving stress-driven diffusion which occurs in materials
science and technology and its applications. We will explain and
analyze the mathematical model and present mathematical analysis
for the problem.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Continuous Media and Optimal Design }\\
Organizer: Pablo Pedregal, Universidad de Castilla-La Mancha
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Homogenization and localization with an interface }\\ \\
%author
{\bf Yves Capdeboscq }\\
%affiliation
Rutgers University, USA \\
%e-mail
email:ycrc@math.rutgers.edu \\
%coauthor
{\bf Gregoire Allaire } and {\bf Andrey Piatnitski}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider the homogenization of a spectral problem for a
singularly perturbed periodic medium. This is a model of a
reaction-diffusion equation used for determining the power
distribution in a nuclear reactor core. In this context,
homogenization results are at the basis of many multiscale type
methods for computing solutions. We suppose that the domain is
composed of two periodic medium separated by a planar interface.
Three different situations arise as the period tends to zero.
First, there is a global homogenized problem as if there were no
interface. Second, the limit is made of two homogenized problems
with a Dirichlet boundary condition on the interface. Third, there
is an exponential localization near the interface of the first
eigenfunction.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Bounds for properties of composites from expanding (smart) materials }\\ \\
%author
{\bf A. Cherkaev }\\
%affiliation
University of Utah, USA \\
%e-mail
email:cherk@math.utah.edu
%coauthor
%{\bf Gregoire Allaire }\\Ecole Polytechnique \\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
An actuator is a cantilever that moves its free end in a
prescribed fashion when it is activated; for example, the
bi-metallic plate is bent when heated. It is an example of a
device made of an active material that changes its shape
responding to an activating signal. To design such devices, one
considers composites from thermo-elastic materials or from
expandable (``smart'') materials that experience phase
transformation in response to the activating signal. It is assumed
that the materials in the composition have different tensors of
elastic moduli and different anisotropic expansion tensors that
show the change of the shape of the pure materials. The (generally
anisotropic) composite is also characterized by an effective
tensor of elastic moduli and an effective expansion tensor; both
depend on the microstructure of the composite. In the talk, we
discuss new coupled bounds for these two tensors. To derive the
bounds, we adapt and modify the technique of the translation
method as well as the technique of Shapery. The result generalizes
the previously obtained bounds for isotropic thermal expansion
coefficient by Shapery, Hashin and Rozen, Gibiansky and Torquato.
Using the derived bounds, a numerical algorithm is developed that
enables to demonstrate bounds and coupling of various components
of the effective tensors. In particular, we are checking
optimality of numerically obtained anisotropic structures with
extremal expansion in a given direction and constrained elastic
properties. The talk describes a part of the collaborative
research with Ole Sigmund (Danish Technical University).
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Effective dynamics in thin ferromagnetic films }\\ \\
%author
{\bf Carlos Garcia-Cervera }\\
%affiliation
University of California, Santa Barbara, USA \\
%e-mail
email:cgarcia@math.ucsb.edu \\
%coauthor
{\bf E Weinan}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In a ferromagnetic material, the dynamics of the relaxation
process is affected by the presence of a strong shape or material
anisotropy. We systematically explore this fact to derive the
effective dynamical equation for a soft ferromagnetic thin film.
We show that as a consequence of the interplay between shape
anisotropy and damping, the gyromagnetic term is effectively also
a damping term for the in-plane components of the magnetization
distribution. We validate our result through numerical simulation
of the original Landau-Lifshitz equation and our effective
equation.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf The yield set of perfectly plastic polycrystals }\\ \\
%author
{\bf Guillermo Goldsztein }\\
%affiliation
Georgia Institute of Technology, USA \\
%e-mail
email:ggold@math.gatech.edu
%coauthor
%{\bf Weinan E Princeton University } \\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Metals are usually found in the form of polycrystals, that is,
large collections of bonded grains. The atoms of each grain form a
periodic lattice and thus, each grain is a single crystal. The
yield set of single crystals (i.e. the set of stresses that the
single crystal can withstand) is anisotropic. Thus, the yield set
of polycrystals depends not only on the yield set of its grains,
but also on the polycrystalline texture (i.e. shape, orientation
and spatial distribution of the grains). The evaluation of the
yield set of polycrystals leads to a constraint optimization
problem in which one of the constraints is a linear PDE. In this
talk I will discuss this problem.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Optimal design of composite conductors with weakly discontinuous objective functionals }\\ \\
%author
{\bf Y. Grabovsky }\\
%affiliation
Temple University, USA \\
%e-mail
email:yury@euclid.math.temple.edu
%coauthor
%{\bf Weinan E Princeton University } \\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Consider a body in space that is to be occupied by two different
isotropic conductors subject to a resource constraint: the volume
fraction occupied by the better conductor is bounded above by a
fixed amount. The body contains a given source density. The
problem is to design a layout of the two materials minimizing a
given objective functional depending explicitly on the gradient of
the resulting electrostatic potential. If one attempts to solve
this problem numerically one might discover that the layout
oscillates on the numerical mesh size scale. The solution to this
problem is to expand the design space allowing composite materials
as structural elements. It is also necessary to change the
original functional to a new one that takes into account the
information of possible oscillations of the electric field on
infinitely small scale because these oscillations cannot be
resolved numerically. In my talk I will overview the fundamental
dichotomy between weakly continuous and weakly discontinuous
functionals and will describe recent attempts to deal with the
harder weakly discontinuous problems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Homogenized failure criteria and the design of composite structures for strength and stiffness }\\ \\
%author
{\bf R. Lipton }\\
%affiliation
Lousiana State University , USA \\
%e-mail
email:lipton@math.lsu.edu
%coauthor
%{\bf Weinan E Princeton University } \\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
New bounds for the effective strength domain for multi phase
linearly elastic-perfectly plastic composites are obtained. These
bounds are given in terms of derivatives of effective elastic
tensors and incorporate the effects of microscopic stress
concentrations. The results hold in the general homogenization
context provided by the theory of G-convergence of elliptic
operators and apply to functionally graded materials. These
results are used to develop a numerical scheme for the design of
composite structures for maximum strength and stiffness.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Numerical modelling in grain growth: multiscale approach }\\ \\
%author
{\bf Irene Livshits }\\
%affiliation
University of Central Arkansas, USA \\
%e-mail
email:irenel@mail.uca.edu
%coauthor
%{\bf Weinan E Princeton University } \\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Grain growth, as any physical phenomena, exists in general on
different scales and appears differently depending on the scale of
observation. The microscopic behavior is determined by
interactions between particles and influences macroscopic
properties of the considered system. However, the connection
between microscopic and macroscopic is often difficult to
establish. In this talk, I will discuss a hierarchy of numerical
models, deterministic and stochastic, that describe grain growth
on different scales. Each such model inherits its properties from
a more detailed predecessor. As the final result of such grain
coarsening, we deduce partial differential equations for the
evolution of general properties of grain growth -- distribution
functions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Optimal design from a variational perspective }\\ \\
%author
{\bf Pablo Pedregal }\\
%affiliation
Universidad de Castilla-La Mancha, Spain \\
%e-mail
email:Pedregal@uclm.es
%coauthor
%{\bf Weinan E Princeton University } \\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
I will analyze a typical optimal design problem in conductivity
from a variational point of view. Starting from a general cost
functional depending explicitly on derivatives of the electric
potential, we will see how far we can go in finding the
appropriate convex hull leading to relaxation of the initial
problem. I will also discuss how this information translates into
optimal designs for the original optimal design problem.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Mathematical modelling of the myocardial fiber organization }\\ \\
%author
{\bf Annie Raoult }\\
%affiliation
Laboratoire de Modélisation et Calcul et Laboratoire TIMC, Université Joseph Fourier, Grenoble, France \\
%e-mail
email:annie.raoult@imag.fr
%coauthor
%{\bf Weinan E Princeton University } \\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
It is commonly believed that the myocardium design and structure
allow maximal mechanical efficiency in the systole and diastole
processes. The long-term purpose of our multidisciplinary approach
is to try and propose a mathematical model for the mechanical
behaviour of the myocardium. Such a model has to take into account
the specific fiber organization in such a muscle, whose
organization, geometrical, and mechanical properties are different
from those of a skeletal muscle. We recall that the dissection or
peeling techniques are not precise enough, since apparently
preferred fibre directions can be inferred by the experimental
process. Here, we use data provided by measurements by means of
polarized light microscopy developed in Grenoble. All mathematical
and numerical strategies we have used on these data back up the
conjecture by Streeter which states that myocardial fibers run as
geodesics on a nested set of surfaces. Work under progress is
devoted to writing an appropriate p.d.e's model and rechecking on
this model the geodesic hypothesis.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Dynamics of damage in two-dimensional structures with waiting elements }\\ \\
%author
{\bf Liya Zhornitskaya }\\
%affiliation
University of Utah, USA \\
%e-mail
email:zhornits@math.utah.edu \\
%coauthor
{\bf Andrej Cherkaev}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider the dynamics of the damage of a bridge-like $2-D$
structure made from specially constructed waiting elements. Each
element consists of two elastic links of different equilibrium
lengths. Whenever the elongation of the shorter link exceeds some
critical value, it undergoes an irreversible damage process which
eventually leads to the breakage. Thereafter the second (longer)
link assumes the stress. Let $\alpha$ be a portion of material
used for the first link and $1-\alpha$ be a portion of material
used for the second link. We compare the waiting element model
with the usual structure, consisting of only one link (of shorter
length). In the waiting element model the usual structure
corresponds to $\alpha = 1$. By performing various numerical
experiments when the structure is impacted by a projectile
modelled as an "elastic ball", we show that in some cases the
waiting element structure can spread the damage over a large area
and therefore withstand larger stresses than the usual structure.
Several movies will be shown to illustrate this phenomenon. We
also address the question of the optimum choice of parameter
$\alpha$.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Mathematical Neuroscience
}\\
Organizer: Mary Pugh, University of Toronto
\end{center}
\vskip .2in
\begin{multicols}{2}
% ----------------------------------------------------------------
%\title
\noindent {\bf Dynamics in responses to auditory motion }\\ \\
%author
{\bf Alla Borisyuk }\\
%affiliation
Courant Institute, NYU, USA \\
%e-mail
email:borisyuk@cims.nyu.edu \\
%coauthor
{\bf John Rinzel }
%abstract
\\
We are interested in the cellular mechanisms shaping the neuronal
responses to auditory motion stimuli. In this framework, we study
how the presence of biologically realistic features influences the
performance of a periodically driven system. We show that the
presence of the cellular mechanisms of firing rate adaptation and
the post-inhibitory rebound is sufficient to account for
experimentally observed phenomena, such as rebound responses,
hysteresis and phase shift in response to periodic forcing. We
quantify various physiologically relevant features of the response
and make testable predictions. We also compare the performance of
spiking and firing-rate-type models in the context of the auditory
motion processing.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Biophysical mechanisms for frequency encoding in a model sensory system }\\ \\
%author
{\bf Sharon Crook }\\
%affiliation
University of Maine, USA \\
%e-mail
email:crook@math.umaine.edu
%coauthor
%{\bf John Rinzel }\\
%abstract
\\
It is well known that the passive membrane properties of neurons
function as a low-pass filter for synaptic inputs. Current inputs
arriving at low frequencies yield large voltage responses, but
high frequency inputs are attenuated or blocked. Neurons can also
exhibit bandpass filtering properties with large responses when
driven by inputs near their resonant frequencies and smaller
responses at other frequencies. This mechanism occurs due to the
interactions between active and passive membrane properties. For
example, currents that actively oppose changes in membrane voltage
and also activate slowly relative to the membrane time constant
will produce resonance. We examine the effects of morphology,
passive membrane properties, and active channels on frequency
tuning in a model sensory system. Theoretical studies and
simulations show that the low-pass filtering observed in some
neurons can be explained by the passive electrotonic structure of
the dendritic arbor and the dynamic sensitivity of the spike
initiation zone. In contrast, neurons that exhibit bandpass tuning
at higher frequencies have dendritic structures with fewer
branching structures and larger diameters that are more
electrotonically compact. This morphology causes less attenuation
of higher frequencies; however, ion channels that resonate in the
desired frequency range are also required.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Chaotic phase synchronization in systems with small phase diffusion }\\ \\
%author
{\bf Kresimir Josic }\\
%affiliation
Boston University, USA \\
%e-mail
email:josic@math.bu.edu \\
%coauthor
{\bf Margaret Beck }
%abstract
\\
The geometric theory of phase locking between periodic oscillators
as proposed by Winfree and Kuramoto has been used with much
success in the past. I will discuss extensions of this theory to
phase coherent chaotic systems. This approach explains the
qualitative features of phase locked chaotic systems and provides
an analytical tool for a quantitative description of the phase
locked states. Moreover, this geometric viewpoint allows for the
identification of obstructions to phase locking even in systems
with negligible phase diffusion, and provides sufficient
conditions for phase locking to occur. These techniques were
applied to the R\"{o}ssler system and a phase coherent electronic
circuit and good agreement was found between numerical results,
experiments and theoretical predictions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Temporal synchronization of pyramidal cells by high-frequency, depressing inhibition }\\ \\
%author
{\bf Steve Kunec }\\
%affiliation
New Jersey Institute of Technology, USA\\
%e-mail
email:sak0232@njit.edu
%coauthor
%{\bf Margaret Beck }\\
%abstract
\\
The sharp wave-associated ripple is a high-frequency,
extracellular recording observed in the rat hippocampus during
periods of immobility. During the ripple, pyramidal cells
synchronize over a short period of time despite the fact that
these cells have sparse recurrent connections. The timing of
synchronized pyramidal cell spiking is critical for encoding
information that is passed onto post-hippocampal targets. Both the
synchronization and precision of pyramidal cells is believed to be
coordinated by inhibition provided by a vast array of
interneurons. We consider a minimal model consisting of a single
interneuron which synapses onto a network of uncoupled pyramidal
cells. We show that fast decaying, high-frequency, depressing
inhibition is capable of rapidly synchronizing the pyramidal cells
and modulating spike timing. These mechanisms are robust in the
presence of intracellular noise. We prove the existence and
stability of synchronous, periodic solutions using geometric
singular perturbation techniques. The effects of synaptic
strength, synaptic recovery, and inhibition frequency are
discussed. In contrast to prior work which suggests that the
ripple is produced by homogeneous populations of either pyramidal
cells or interneurons, our results suggest that cooperation
between interneurons and pyramidal cells is necessary for ripple
genesis.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Stabilization of ``bumps''by noise }\\ \\
%author
{\bf Carlo Laing }\\
%affiliation
University of Ottawa, Canada \\
%e-mail
email:claing@science.uottawa.ca \\
%coauthor
{\bf Andre Longtin }
%abstract
\\
Spatially localized regions of active neurons (``bumps'') have
been proposed as a mechanism for working memory, the head
direction system, and feature selectivity in the visual system.
Stationary bumps are ordinarily stable, but including spike
frequency adaptation in the neural dynamics causes a stationary
bump to become unstable to a moving bump through a supercritical
pitchfork bifurcation in bump speed. Adding spatiotemporal noise
to the network supporting the bump can cause the average speed of
the bump to decrease to almost zero, reversing the effect of the
adaptation and ``restabilizing'' the bump. This restabilizing can
be understood by examining the effects of noise on the normal form
of the pitchfork bifurcation where the variable involved in the
bifurcation is bump speed. This noise--induced stabilization is a
novel example in which moderate amounts of noise have a beneficial
effect on a system, specifically, stabilizing a spatiotemporal
pattern. Determining which aspects of our model system (integral
rather than diffusive coupling, a slow variable, travelling
structures that appear through a pitchfork bifurcation in speed)
are necessary for this type of behavior remains an open problem.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Dynamics of neurons connected by inhibitory and electrical coupling }\\ \\
%author
{\bf Timothy Lewis }\\
%affiliation
New York University, USA \\
%e-mail
email:tl14@nyu.edu \\
%coauthor
{\bf John Rinzel }
%abstract
\\
Recent findings suggest that many inhibitory cell networks in the
brain are connected through both inhibitory and electrical
coupling. However, it is unclear how the interaction of these two
coupling modes affects the dynamics of these networks. To begin
addressing this issue, we use the theory of weakly coupled
oscillators to study the influence of coupling parameters on
synchronization patterns in a model of intrinsically oscillating
cells connected by both inhibition and electrical coupling.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Characterizing coupled neurons with white noise analysis }\\ \\
%author
{\bf Duane Nykamp }\\
%affiliation
UCLA, USA \\
%e-mail
email:nykamp@math.ucla.edu
%coauthor
%{\bf John Rinzel }\\
%abstract
\\
We present an asymptotic analysis of two coupled linear-nonlinear
neurons. By measuring the spike times of both neurons in response
to a white noise stimulus, one can characterize the neurons'
properties and their mutual connections. The linear-nonlinear
model used in the analysis is similar to a widely used
phenomenological model of a neuron in response to sensory
stimulation. Moreover, we demonstrate that the results of the
analysis also work with more realistic neuron models. Thus, this
analysis may help characterize neural circuitry in sensory brain
regions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Phase plane analysis of neural decoding in the Rodent Whisker-Barrel System }\\ \\
%author
{\bf David Pinto }\\
%affiliation
Brown University, USA \\
%e-mail
email:dpinto@bu.edu
%coauthor
%{\bf John Rinzel }\\
%abstract
\\
In the rodent whisker-to-barrel pathway, populations of thalamic
neurons encode information from the periphery in terms of changes
in the population firing rate. Correspondingly, cortical neurons
respond preferentially to rapid changes in the rate of firing
among input neurons from the thalamus. Previous computational
models based on known features of cortical circuitry have captured
this and other aspects of the thalamocorical response
transformation in the rodent whisker system. In this presentation,
I will examine these models using a modified version of phase
plane analysis in order to understand the mechanisms that underlie
cortical sensitivity to thalamic input timing. The analysis
reveals that cortical processing in our model depends on strong
inhibition that renders the net effect of cortical connections to
be {\em damping}. This distinguishes it from previous models of
cortical microcircuits, in which the net effect of cortical
connections is {\em amplifying}. I will conclude with a brief
comparison of the two proposed mechanisms of cortical processing
and suggest a possible experimental means for distinguishing
between them.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Lateral inhibition is neither necessary nor sufficient for sustained, patterned activity }\\ \\
%author
{\bf Jonathan Rubin }\\
%affiliation
University of Pittsburgh, USA \\
%e-mail
email:rubin@math.pitt.edu
%coauthor
%{\bf John Rinzel }\\
%abstract
\\
Evidence suggests that sustained, localized neuronal activity, or
bumps, may play a role in working memory or representation of
internal states, such as head direction. Previous (and ongoing)
theoretical work has demonstrated that a synaptic architecture
featuring recurrent excitation and long-range inhibition, together
known as lateral inhibition, can lead to activity bumps in
neuronal network models. However, this architecture is absent in
some areas of the brain where such activity may be relevant.Here
we show how a rate-based integrodifferential equation model can
support bump solutions without recurrent excitation. We also
provide conditions under which no spatial patterns exist despite
the presence of lateral inhibition.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Modulating model inhibitory neuronal networks }\\ \\
%author
{\bf Frances Skinner }\\
%affiliation
Toronto Western Research Institute, Canada\\
%e-mail
email:fskinner@uhnres.utoronto.ca
%coauthor
%{\bf John Rinzel }\\
%abstract
\\
Synchronous oscillatory activity in networks of interneurons
connected by inhibitory synapses play critical roles in brain
function. Differences in the kinetics of the inhibitory response
observed with anesthetics can affect this activity. We use
theoretical insights to suggest a new mechanism of anesthesia. In
particular, we suggest that the different behavioral effects of
different anesthetic drugs might lie in the different ways in
which these drugs modulate inhibitory network coherence.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Dynamics of the visual cortex }\\ \\
%author
{\bf Andrew Sornborger }\\
%affiliation
Mt. Sinai School of Medicine, USA \\
%e-mail
email:ats@camelot.mssm.edu \\
%coauthor
{\bf Ehud Kaplan }
%abstract
\\
We present multivariate harmonic analysis methods for the analysis
of dynamical optical imaging data; discuss the various components
of the dynamical signal resulting from the harmonic analysis; and
present an overall strategy for constructing a population model of
visual cortex, with results from one functional component of the
model.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A neuronal network model of the macaque primary visual cortex }\\ \\
%author
{\bf Louis Tao }\\
%affiliation
Courant Institute of Mathematics, USA \\
%e-mail
email:tao@cims.nyu.edu \\
%coauthor
{\bf Michael Shelly} and {\bf David McLaughlin}
%abstract
\\
Our objective is a realistic theory of the visual cortex that can
explain the visual selectivity, dynamics, and the diversity of
visual properties in cortical cell populations. To do this, we
have studied a large-scale computational model of Macaque V1
[McLaughlin et al. 2000 PNAS] based on anatomy and physiology.
Cells in the model are classified as Simple or Complex by the same
index of linearity of spatial summation that has been used in
physiology experiments. Previously we offered an explanation of
how Simple cells could exist in the model despite the
non-linearity of the LGN input and of cortico-cortical excitation
[Wielaard et al 2001 J. NS.] Now we report that Complex cells
arise in the model by allowing for randomness in synaptic coupling
strengths, which can increase the importance of network
excitation, and randomness in the strength of LGN input. My work
suggests that the Simple-Complex classification reflect different
synaptic balances within the same basic model circuit. Since the
dichotomy of `Simple' and `Complex' behavior is seen in other
areas of visual processing, the basic mechanism may be widely
operating.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Dynamical patterns in the basal ganglia and related neuronal networks }\\ \\
%author
{\bf Alice Yew }\\
%affiliation
The Ohio State University, USA \\
%e-mail
email:yew@math.ohio-state.edu
%coauthor
%{\bf Michael Shelly }\\{\bf David McLaughlin}\\
%abstract
\\
A conductance-based network model was developed (in collaboration
with J. Rubin, D. Terman, and C. Wilson) to describe neural
interactions in the basal ganglia, with the aim of testing
hypotheses on the origin of activity states associated with
disorders such as Parkinson's disease. Computer simulations reveal
that the system exhibits a variety of spatiotemporal patterns,
including episodic synchronous oscillations, clustered rhythms,
travelling waves, and irregular uncorrelated spiking. Using a
dynamical systems approach, we analyze how synaptic coupling
interacts with intrinsic neuronal properties to generate these
types of behavior. We also investigate how transitions between
patterns are effected as parameters are varied, and make
comparisons with other similarly wired neuronal networks.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Difference Equations and Their Applications
}\\
Organizer: Youssef Raffoul , University of Dayton
\end{center}
\vskip .2in
\begin{multicols}{2}
\vskip .2in
% ----------------------------------------------------------------
%\title
\noindent {\bf Oscillatory properties of third order neutral delay differential equations }\\ \\
%author
{\bf John Graef }\\
%affiliation
University of Tennessee at Chattanooga, USA \\
%e-mail
email:john-graef@utc.edu \\
%coauthor
{\bf R. Savithri} and {\bf E. Thandapani}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The authors consider the third order neutral delay differential
equation $$ \left(a(t)\left(b(t)\left(y(t) +
py(t-\tau)\right)'\right)'\right)' + q(t)f(y(t-\sigma)) = 0
\eqno(*) $$ where $a(t) > 0$, $b(t) > 0$, $q(t) \ge 0$, $0 \le p
<1$, $\tau > 0$, and $\sigma > 0$. Criteria for the oscillation of
all solutions of ($\ast$) are obtained. Examples illustrating the
results are included.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
%----------------------------------------------------------
%\title
\noindent {\bf Oscillation properties of an Emden-Fowler type equation on discrete time scales }\\ \\
%author
{\bf Joan Hoffacker }\\
%affiliation
University of Georgia, USA \\
%e-mail
email:johoff@math.uga.edu \\
%coauthor
{\bf Elvan Akin-Bohner}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this talk we explore the oscillation properties of
$$u^{\Delta^2}(t) + p(t)u^{\gamma}(\sigma(t)) = 0$$ on a time
scale $\mathbb{T}$ with only isolated points, where $p(t)$ is
defined on $\mathbb {T}$ and $\gamma$ is a quotient of odd
positive integers. We define oscillation in this setting, and
generate conditions on the integral of $p(t)$ which guarantee
oscillation and no oscillatory solutions. In addition we consider
the case when solutions of this equation has asymptotically
positively bounded differences.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Exponential stability in nonlinear difference equations }\\ \\
%author
{\bf Muhammad Islam }\\
%affiliation
University of Dayton, USA \\
%e-mail
email:muhammad.islam@notes.udayton.edu
%coauthor
%{\bf R. Savithri Periyar University, India \\E. Thandapani Periyar
%University, India }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We employ non-negative definite Lyapunov functionals to obtain
conditions that guarantee exponential stability and uniform
exponential stability of the zero solution of the nonlinear
discrete system $$x(n+1) = f(n, x(n)), x(n_{0}) = x_{0}, \mbox{
for } n \geq n_{0}.$$ The theory is illustrated with several
examples.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Singular conjugate boundary value problems on a time scale }\\ \\
%author
{\bf Eric Kaufmann }\\
%affiliation
University of Arkansas, USA \\
%e-mail
email:erkaufmann@ualr.edu
%coauthor
%{\bf R. Savithri Periyar University, India \\E. Thandapani Periyar
%University, India }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Let ${\mathbb T}_1$ be a time scale symmetric about $1/2$. Let
$1/2 \in {\Bbb T}$ be dense and define ${\mathbb T} = {\mathbb
T}_1 \bigcap [0,1]$. The conjugate nonlinear boundary value
problem,
\begin{eqnarray*}
-u^{\Delta\Delta}(t) = a(t)f(u(t)), t \in {\mathbb T}\setminus\{0,1\}\\
u(0) = u(1) = 0,
\end{eqnarray*}
where $a(t)$ is singular at $t = 1/2$ and $f$ satisfies certain
growth conditions, is shown to have infinitely many solutions
using Krasnosel'ski\u{\i}'s fixed point theorem.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
%\title
\noindent {\bf Stability properties of linear Volterra discrete systems with nonlinear perturbation }\\ \\
%author
{\bf Touhid Khandaker }\\
%affiliation
Southern Illinois University at Carbondale, USA\\
%e-mail
email:khandaker@math.siu.edu
%coauthor
%{\bf Elvan Akin-Bohner University of Nebraska-Lincoln }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider a Volterra discrete system with nonlinear perturbation
$$x(n+1)= A(n)x(n) + \sum^{n}_{s=0}B(n,s)x(s) + g(n,x(n))$$ and
obtain necessary and sufficient conditions for stability
properties of the zero solution employing the resolvent equation
coupled with the variation of parameters formula.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Accurate estimates for the solutions of difference equations }\\ \\
%author
{\bf Rigoberto Medina }\\
%affiliation
Universidad de Los Lagos, Chile \\
%e-mail
email:rmedina@ulagos.cl
%coauthor
%{\bf R. Savithri Periyar University, India \\E. Thandapani Periyar
%University, India }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We will be concerned with a non-autonomous perturbed linear
discrete dynamical system. A well known result of Perron, which
dates back to 1929 (see Ortega(1973) and LaSalle(1976)), states
that these kind of systems are asymptotically stable if the matrix
$A$ of the linear part is stable, i.e., the spectral radius of $A$
is less than one, and the perturbation satisfies an asymptotic
property. We can see that this kind of results are purely local
results which gives no information about the size of the region of
asymptotic stability nor the norm of the solutions. In this
report, estimates for the norms of the solutions and the size of
the regions of stability of non-autonomous perturbed linear
difference equations are derived. The methodology is based on the
''freezing'' method and on the recent estimates for the powers of
a constant matrix. Finally, we will illustrate our main results by
considering partial difference equations which model reaction and
diffusion processes.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A second-order self-adjoint dynamic equation on a time scale }\\ \\
%author
{\bf Kirsten Messer }\\
%affiliation
University of Nebraska, Lincoln, USA\\
%e-mail
email:kmesser1@aol.com
%coauthor
%{\bf Elvan Akin-Bohner University of Nebraska-Lincoln }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Recently, the study of dynamic equations on time scales has seen
an increased level of interest from researchers seeking to study
both differential and difference equations in a more general
setting. In this paper, we are concerned with the dynamic equation
$[p(t)x^{\Delta}]^{\nabla}+ q(t)x = f(t)$. Relatively little
research has been done on this equation which combines both
"delta" and "nabla" derivatives in a single dynamic equations. We
will discuss various results concerning this equation, including
results on zeros of solutions, disconjugacy, and factorizations.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Positivity and discrete models for the Lotka-Volterra equations }\\ \\
%author
{\bf Ronald Mickens }\\
%affiliation
Clark Atlanta University, USA \\
%e-mail
email:rohrs@math.gatech.edu
%coauthor
%{\bf R. Savithri Periyar University, India \\E. Thandapani Periyar
%University, India }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The first nontrivial mathematical model for predator-prey
interactions was the Lotka-Volterra equations [1]. The two
coupled, first-order ODE's has biological relevant solutions in
the first quadrant, i.e., $x(0) > 0$ and $y(0) > 0$, lead to $x(t)
> 0$ and $y(0) > 0$. Further, there is a single non-negative
fixed-point, around which all solutions periodically oscillate. We
consider the application of the nonstandard finite-difference
techniques of Mickens [2] to formulate corresponding discrete time
models of these equations. In particular, enforcement of the
positivity condition is made by the use of nonlocal discrete
representations for both the linear and quadratic terms appearing
in the Lotka-Volterra differential equation. Our studies indicate
that one must be careful in carrying out this procedure. Both
linear stability analysis and numerical work will be used to
illustrate our results. References [1] J. D. Murray 1989
Mathematical Biology (Springer-Verlag, Berlin); section 3.1. [2]
R. E. Mickens 1994 Nonstandard Finite Difference Models of
Differential Equations (World Scientific, Singapore).
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
%\title
\noindent {\bf The nabla exponential function }\\ \\
%author
{\bf Allan Peterson }\\
%affiliation
University of Nebraska-Lincoln, USA \\
%e-mail
email:apeterso@math.unl.edu
%coauthor
%{\bf Elvan Akin-Bohner University of Nebraska-Lincoln }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We will develop the nabla exponential function and give several of
its properties.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Positive periodic solutions of nonlinear functional difference equations}\\ \\
%author
{\bf Youssef Raffoul }\\
%affiliation
University of Dayton, USA \\
%e-mail
email:youssef.raffoul@notes.udayton.edu
%coauthor
%{\bf R. Savithri Periyar University, India \\E. Thandapani Periyar
%University, India }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this paper, we apply a cone theoretic fixed point theorem and
obtain sufficient conditions for the existence of multiple
positive periodic solutions to the nonlinear functional difference
equations $$ x(n+1) = a(n)x(n)\pm \lambda h(n) f(x(n-\tau(n))).$$
where $a(n), h(n), \lambda, f(x)$ and $\tau$ are positive and
periodic of period $T$.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Comparison of eigenvalues for Sturm-Liouville boundary value problems on a measure chain }\\ \\
%author
{\bf Denise Reid }\\
%affiliation
Valdosta State University, USA \\
%e-mail
email:dtreid@valdosta.edu
%coauthor
%{\bf Elvan Akin-Bohner University of Nebraska-Lincoln }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Under consideration is a class of even ordered linear differential
equations with Sturm-Liouville boundary conditions. The
differential equation is, in fact, a general dynamic equation
containing delta-derivatives whose solution is defined on a
measure chain. For a pair of eigenvalue problems for this dynamic
equation, we first verify the existence of a smallest possible
eigenvalue and then establish a comparison between the smallest
eigenvalues of each eigenvalue problem.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Existence of periodic solutions of nonlinear discrete second order equations }\\ \\
%author
{\bf Jesus Rodriguez }\\
%affiliation
North Carolina State University, USA\\
%e-mail
email:rodrigu@math.ncsu.edu
%coauthor
%{\bf R. Savithri Periyar University, India \\E. Thandapani Periyar
%University, India }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this paper we study the existence of $T$-periodic solutions of
equations of the form \[
\begin{array}{lll} x(t+2)+bx(t+1)+cx(t)+f(t, x(t))=h(t)
\end{array} \] \noindent where $f$ is nonlinear, smooth,
$f(t+T,x)=f(t,x)$ for each $(t,x)$ and $h(t+T)=h(t)$ for all $t$.
Under the assumption of the existence of a $T$-periodic solution
for a specific forcing term $h$, we aim to find conditions on the
nonlinear term $f$ which will allow us to establish the existence
of $T$-periodic solutions to a system of the form \[
\begin{array}{lll} x(t+2)+bx(t+1)+cx(t)+f(t, x(t))=h(t)+g(t)
\end{array} \] \noindent where $g$ is $T$-periodic and
``relatively large." We formulate our problem as an operator
equation in a space of $T$-periodic sequences and use a homotopy
argument that eventually connects the existence of $T$-periodic
solutions of the discrete equation to a differential equation on a
sequence space.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Equations with partial derivatives and differential equations used for simulating acausal pulses in mathematical physics }\\ \\
%author
{\bf Cristian Toma }\\
%affiliation
Politehnica University, Romania \\
%e-mail
email:cgtoma@physics1.physics.pub.ro \\
%coauthor
{\bf Paul Sterian }
%University, India }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Some phenomena in physics (such as the phenomenon of photonic
echo) appears for an external observer as non-causal pulses
suddenly emerging from an active medium (prepared by some other
optical pulses). Such a pulse is very hard to be simulated without
using physical quantities corresponding to the internal state of a
great number of atoms. The only mathematical possibility of
simulating such pulses without using a great number of variables
consists in the use of test-functions. It is shown that such
functions can be put in correspondence with acausal pulses in
physics. This study shows that the wave-equation considered on the
length interval (0, 1) (an open set), starting at the initial
moment of time from null initial conditions, can possess as
possible solution a test-function represented by a propagating
direct wave coming from outside the length-interval. For
explaining the reason why such acausal pulses do not appear in
real circumstances some methods from statistical physics are
used. While at the zero moment of time all derivatives of the
amplitude of the “real” string are equal to zero, it is shown that
we may consider the zero moment of time as a bifurcation point
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
%\title
\noindent {\bf On the periodic nature of the solutions of the reciprocal difference equations with maximum }\\ \\
%author
{\bf Hristo Voulov }\\
%affiliation
Southern Illinois University Carbondale, USA \\
%e-mail
email:hvoulov@siu.edu
%coauthor
%{\bf Elvan Akin-Bohner University of Nebraska-Lincoln }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We prove that every positive solution of the difference equation
$x[n]=\mbox{max} \{A/x[n-1],B/x[n-2],C/x[n-3]\}$ is eventually
periodic of (not necessarily prime) period T, which is explicitly
determined in terms of the coefficients A,B and C.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
% ----------------------------------------------------------------
\end{multicols}
\begin{center}
{\Large \bf Representations of Dynamical Systems
}\\
Organizer: Marc Rouff, University of Caen Basse Normandie \\
\hspace{1.35in} Michel Cotsaftis, Ecole Centrale d'Electronique,
Paris France
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Asymptotic vs. projective representation for solutions of dynamical system equations }\\ \\
%author
{\bf Michel Cotsaftis }\\
%affiliation
LTME/ECE, Paris, France \\
%e-mail
email:mcot@ece.fr
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
The problem of dynamical evolution of a system exhibiting internal
branchings toward bifurcated states is considered by observing
that as a consequence the number of "relevant" state space
dimensions where power flows is varying with time. Classical
projection method to represent the solution does no longer apply,
and more adapted asymptotic method taking advantage of branched
states characteristic time and space scales is discussed. Analytic
expressions of the solution are given, allowing to construct the
modified state space dynamics due to these branchings and to
design a correct controller which self-consistently accounts for
the internal power flow created by the opening of internal
branched modes. Application is made to compliant actuated
deformable one-link system.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Towards functional optimal control using $C^k$ spline functions }\\ \\
%author
{\bf Zakaria Lakhdari }\\
%affiliation
Laboratoire Universitaire des Sciences Appliqu\'ees de Cherbourg - UCBN -EIC, France \\
%e-mail
email:zakaria.lakhdari@chbg.unicaen.fr
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
Differential nonlinear optimal control is known as difficult
numerical problem because of the integration of the adjoint
vector. We show in this paper that functional approach of this
problem directly by $C^k$ spline functions \cite{1}-\cite{2},
avoid the notion of adjoint vector and leads to more suitable
algorithmics.
\begin{thebibliography}{99}
\bibitem{1} M. Rouff : The computation of $C^k$ spline functions,
{\it Computers and Mathematics with Applications}, {\bf 23}, (1),
pp 103-110, 1992.
\bibitem{2} M. Rouff et W.L. Zhang : $C^k$ spline functions and linear operators, {\it
Computers and mathematics with applications}, {\bf 28}, (4), pp
51-59, 1994.
\end{thebibliography}
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A New discretization method using $C^k$ spline functions }\\ \\
%author
{\bf Philippe Makany }\\
%affiliation
Laboratoire Universitaire des Sciences Appliqu\'ees de Cherbourg - UCBN -EIC, France \\
%e-mail
email:p.makany.iut@chbg.unicaen.fr
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
We know that in $H_k(\Omega)$, the Sobolev space generated by
$C^k$ spline functions \cite{1}-\cite{2} on an open set $\Omega$,
every differential relations of order $k$ or less, are written as
arbitrarily precise and simple algebraic relations [3]-[4]-[5]. In
this sense, and for every kind of differential equations, total or
partial, linear or nonlinear, explicit or implicit, and always for
a finite small numbers of terms, and for an arbitrarily precision,
$H_k(\Omega)$ the Sobolev space generated by the set of $C^k$
spline functions, plays the same role then Fourier Spaces for
linear total or partial differential equations.\vskip0.1cm This
fact opens the way to a new functional and invariant calculus in a
suitable space defining by the all differential invariants and
symmetries of the problem.\vskip0.1cm In Automatic Control
Scientists and Engineers always study behaviors or controllers of
a defined process trough a state equation, defined by Kalman in
the early sixties, or through their nonlinear implicit or explicit
extensions, i.e. only on ordinary differential equations
(ODE's).\vskip0.1cm Then the control of Partial Differential
Equations (PDE's) remains a difficult or unsolved problem in many
cases.\vskip0.1cm We present here a $C^k$ discretization method
for Extended Lagrangian systems mixing the time defined on $\R$,
the space of the real numbers, and $H_k(\Omega)$ where $\Omega$ is
the open set of the non-time variables. We show by using the
example of vibrations control in a one link robotic arm \cite{6},
that in $D_k(\Omega)\equiv\R\times H_k(\Omega)$, the space of
discretization generated by this method, the resulting state
equations represent the behavior of the system with a localized
small ball of errors depending on $I$ the number discretization
points on $\Omega$.
\begin{thebibliography}{99}
\bibitem{1} M. Rouff : The computation of $C^k$ spline functions,
{\it Computers and Mathematics with Applications}, {\bf 23}, (1),
pp 103-110, 1992.
\bibitem{2} M. Rouff et W.L. Zhang : $C^k$ spline functions and linear operators, {\it
Computers and mathematics with applications}, {\bf 28}, (4), pp
51-59, 1994.
\bibitem{3} M. Rouff : $C^k$ spline functions in applied differential problem : the first algorithmic structures, {\it Systems Analysis Modelling and Simulations}, Gordon \& Breach science Publishers, {\bf 26}, pp 197-205, 1996.
\bibitem{4} M. Rouff et M. Alaoui : Computation of dynamical electromagnetic problems using Lagrangian Formalism and multidimensional $C^k$ spline functions, {\it Zeitschrift für Angewandte Mathematik und Mechanik}, Akademie Verlag Berlin, {\bf 76}, (1), pp 513-514, 1996.
\bibitem{5} M. Rouff et M. Alaoui : 2D nonlinear dynamics of magnetic domain wall motion in ferromagnetic material such as Crystalline Fe-Si, {\it The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, COMPEL}, {\bf 19}, (3), pp 866-877, 2000.
\bibitem{6} M. Rouff et M. Cotsaftis : Invariance properties in deformable flexion-torsion loaded bodies spectra, MTNS'2000, International symposium on Mathematical Theory of Networks and Systems, Perpignan, France, 19-23 juin 2000.
\end{thebibliography}
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Towards a new invariant and functional calculus w$C^k$ spline functions }\\ \\
%author
{\bf Marc Rouff }\\
%affiliation
Laboratoire Universitaire des Sciences Appliquees de Cherbourg - UCBN -EIC, France\\
%e-mail
email:marc.rouff@chbg.unicaen.fr\\
%coauthor
%{\bf Edriss S. Titi }\\
%abstract
\\
Functional expansions based on $C^k$ spline Functions
\cite{1}-\cite{2}, lead to the remarkable property that the
coefficients of these functional expansions are the only set of
the whole total or partial derivatives up to $k$ of the considered
approximated functions themselves at each point of discretization
on the open set $\Omega$.\vskip0.1cm This property leads to
include in $H_k(\Omega)$ the Sobolev space generated by the $C^k$
spline functions, the differential constraints, invariants,
symmetries and the equations of evolution themselves as exact
algebraic relations, redefining by the same way the suitable
functional space of representation associated to the considered
phenomenon [3]-[4]-[5].\vskip0.1cm The Fourier Transforms of $C^k$
spline functions are $C^k$ wavelets, with the obvious property
that these kind of spectra generated by $C^k$ wavelets have as
coefficients the set of the whole total or partial derivatives up
to $k$ of the associated dual function at each point of
discretization of the dual space. This property opens the way to
many computations for example, new definitions of the entropy of a
signal.\vskip0.1cm At Last duality of direct and Fourier spaces
are used to present a self consistent relation of existence, which
open the way to a large amount of applications, for example the
definition with an arbitrarily good precision of non entire
derivatives partial or total, or non entire integrals simple or
multiple. \sl But today the most interesting applications of
these $C^k$ spline functions calculus are the ability to replace
classical differential and integral calculus by a functional and
invariants calculus which can be exact ( Formal calculus ) on $\Q$
\begin{thebibliography}{99}
\bibitem{1} M. Rouff : The computation of $C^k$ spline functions,
{\it Computers and Mathematics with Applications}, {\bf 23}, (1),
pp 103-110, 1992.
\bibitem{2} M. Rouff et W.L. Zhang : $C^k$ spline functions and linear operators, {\it
Computers and mathematics with applications}, {\bf 28}, (4), pp
51-59, 1994.
\bibitem{3} M. Rouff : $C^k$ spline functions in applied differential problem : the first algorithmic structures, {\it Systems Analysis Modelling and Simulations}, Gordon \& Breach science Publishers, {\bf 26}, pp 197-205, 1996.
\bibitem{4} M. Rouff et M. Alaoui : Computation of dynamical electromagnetic problems using Lagrangian Formalism and multidimensional $C^k$ spline functions, {\it Zeitschrift für Angewandte Mathematik und Mechanik}, Akademie Verlag Berlin, {\bf 76}, (1), pp 513-514, 1996.
\bibitem{5} M. Rouff et M. Alaoui : 2D nonlinear dynamics of magnetic domain wall motion in ferromagnetic material such as Crystalline Fe-Si, {\it The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, COMPEL}, {\bf 19}, (3), pp 866-877, 2000.
\end{thebibliography}
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Stochastic Analysis and Applications
}\\
Organizer:S. Sathananthan ,Tenessee State University
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Large-scale stochastic hybrid parabolic systems under jump Markovian perturbations-I: convergence and stability via Lyapunov functions }\\ \\
%author
{\bf M. Anabtawi }\\
%affiliation
Tennessee State University, USA \\
%e-mail
email:anabtawi@coe.tsuniv.edu \\
%coauthor
{\bf S. Sathananthan }
%abstract
\\
In this paper, the qualitative properties of the jump Markovian
perturbations caused by the interactions among the states of a
stochastic hybrid parabolic partial differential system are
investigated. The concept of vector Lyapunov-like functions
coupled with the decomposition-aggregation techniques are utilized
to develop a comparison principle and sufficient conditions are
established for various types of convergence and stability in the
$p$-th moment and probability of the equilibrium state of the
system under jump Markovian perturbations. This frame work of
decomposition-aggregation is ideally suited for reducing the
dimensionality problem arising in testing large-scale systems for
the concept of convergence and stability. In addition, an example
is given to illustrate the significance of the presented results.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Stochastic perturbation of solutions in optical fibers }\\ \\
%author
{\bf Anjan Biswas }\\
%affiliation
Tennessee State University, USA\\
%e-mail
email:ABiswas@tnstate.edu
%coauthor
%{\bf S. Sathananthan }\\
%abstract
\\
The soliton perturbation theory is used to study and analyze the
stochastic perturbation of optical solitons in addition to the
deterministic perturbations that are governed by the nonlinear
Schrodinger's equation. The corresponding Langevin equations are
derived and analyzed. The deterministic perturbations that are
considered here are both Hamiltonian as well as of non-Hamiltonian
type. Finally, the soliton mean drift velocity is calculated in
presence of these perturbation terms.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Stability of singularly perturbed\\stochastic systems }\\ \\
%author
{\bf Janus Golec }\\
%affiliation
Fordham University, USA\\
%e-mail
email:Golec@fordham.edu
%{\bf S. Sathananthan }\\
%abstract
\setcounter{equation}{0}
\\
We consider the singularly perturbed stochastic system:
\begin{equation}
\begin{array} {rclcr}
dx^{\epsilon}_t & = & f_1 (\epsilon,x^\epsilon)_t,y^{\epsilon}_t)dt & + & \sigma_1
(\epsilon,x^{\epsilon}_t,y^{\epsilon}_t)dw_t \\
\epsilon dy^{\epsilon}_{t} & = & f_2
(\epsilon,x^{\epsilon}_{t},y^{\epsilon}_t)dt & + &
\sqrt{\epsilon}\sigma_2
(\epsilon,x^{\epsilon}_t,y^{\epsilon}_t)dw_t
\end{array}
\end{equation}
where $\epsilon>0$ is a small parameter, $w_t$ the standard Wiener
process. We give sufficient conditions for stability of system
(1). They are formulated in terms of Liapunov functions for the
reduced-order and the boundary layer system associated with (1).
In this way we analyze the full order system by means of its lower
order components and their interconnecting structure.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Stability boundedness and tightness of stochastic flows}\\ \\
%author
{\bf D. Kannan}\\
%affiliation
University of Georgia,USA \\
%e-mail
email: kannan@arches.uga.edu\\
%{\bf S. Sathananthan }\\
TBA
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Occupation time large deviations of two dimensional symmetric simple exclusion process }\\ \\
%author
{\bf Tzong-Yow Lee }\\
%affiliation
University of Maryland at College Park, USA \\
%e-mail
email:tyl@math.umd.edu
%coauthor
%{\bf S. Sathananthan }\\
%abstract
\\
We prove a large deviations principle for the occupation time of a
site in the two dimensional symmetric simple exclusion process.
The decay probability rate is of order $t/log t$ and the rate
function is given by $\Upsilon_\alpha (\beta) = (\pi/2)
\{\sin^{-1}(2\beta -1)-\sin^{-1}(2\alpha -1) \}^2$. The proof
relies on a large deviations principle for the polar empirical
measure which contains an interesting $\log$ scale spatial
average. A contraction principle permits to deduce the occupation
time large deviations from the large deviations for the polar
empirical measure.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Nearly optimal impulsive controls for reflected wideband width process }\\ \\
%author
{\bf K. Ramachandran }\\
%affiliation
University of South Florida, USA \\
%e-mail
email:ram@chuma.cas.usf.edu
%coauthor
%{\bf S. Sathananthan }\\
%abstract
\\
Near optimal control problem for a wideband noise process with
impulsive controls and constrained to a bounded region will be
considered. The method developed will show that sequence of
physical processes converges (weakly) to a reflected controlled
diffusion process with impulses as the ''approximating parameter''
goes to zero. The cost functional of the wideband system will also
converge to the corresponding cost functional of the limit
problem. Due to the reflection at the boundary, pseudo path
topology will be used in the weak convergence analysis. This
method will be applied to study control of a heavy traffic queuing
system.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Practical stability criteria for nonlinear stochastic systems by decomposition and aggregation}\\ \\
%author
{\bf S. Sathananthan }\\
%affiliation
Tennessee State University, USA\\
%e-mail
email:satha@coe.tsuniv.edu
%coauthor
%{\bf S. Sathananthan }\\
%abstract
\\
In this talk, the concept of practical stability is investigated
for the large-scale stochastic systems of Ito-Doob type. The
concept of vector-Lyapunov like functions coupled with the
decomposition-aggregation techniques are utilized to develop a
comparison principle and, sufficient conditions are established
for various types of practical stability criteria in the p-th mean
and in probability. This framework of decomposition-aggregation is
ideally suited for reducing the dimensionality problem arising in
testing large-scale systems for the concepts of convergence and
stability.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\newpage
\begin{center}
{\Large \bf Computational and Theoretical Issues in Fluid Dynamics
}\\
\hspace{-0.6in} Organizer: Juan Lopez, Arizona State University
\\\hspace{1.6in} Jie Shen, University of Central Florida and Purdue University
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Instabilities in swirling flows with walls: numerical modelling, simulations and analysis }\\ \\
%author
{\bf Patrick Bontoux }\\
%affiliation
Marseille, France\\
%e-mail
email:bontoux@l3m.univ-mrs.fr\\
%coauthor
{\bf Olivier Czarny} and {\bf Emilia Crespo}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
A direct numerical solver based on the spectral Chebyshev-Fourier
approximation with a projection method is proposed and used for
the analysis of a wide range of fluid dynamics instabilities
involving rotation, walls and heat transfer. Some particular
structures are emphasized in the Taylor-Couette system such as
spiral and wavy vortex regimes. In the case of the rotor-stator
configuration, different phenomena can be investigated depending
on the aspect ratio, as transition to turbulence in the Bödewadt
layer near the stator, or vortex breakdown along the rotation
axis. New results concern the Küppers-Lortz instability due to the
interaction of thermal convection and slow rotation effects.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On projection method }\\ \\
%author
{\bf Benyu Guo }\\
%affiliation
Shanghai Normal University, China\\
%e-mail
email:byguo@guomai.sh.cn \\
%coauthor
{\bf Jun Zou }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The projection method was introduced by Chorin and Teman as an
efficient algorithm for numerical solution of the Navier-Stokes
equations. It is based on time splitting discretization which
decouples the computations of the velocity and the pressure. The
main advantages of this method are that it saves computational
cost and preserves the incompressibility. The convergence analysis
of projection method was initiated by Chorin and Teman. Shen
obtained some results on the convergence rate. E and Liu
considered two-dimensional, semi-periodic Navier-Stokes equations.
The purpose of this work is to analyze the convergence rate of the
fully discrete projection method and to improve the accuracy of
the numerical solution by the pressure correction for the n
dimensional, periodic problem of Navier-Stokes equations.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Control of Lagrangian coherent structures }\\ \\
%author
{\bf George Haller }\\
%affiliation
MIT, USA\\
%e-mail
email:ghaller@mit.edu
%coauthor
%{\bf Jun Zou \\Chinese University of Hong Kong }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
It has recently been realized that the kinematics of turbulent
mixing is governed by distinguished material lines that attract or
repel fluid particles exponentially. These Lagrangian coherent
structures are generalizations of stable and unstable manifolds
known from steady and time-periodic flows. Because Lagrangian
structures have a decisive impact on stretching, folding, and
transport in the flow, controlling them locally can lead to global
changes in mixing. In this talk I explore this idea via two
examples. First, I describe passive control of coastal pollution
spread using radar data of Monterey Bay. Second, I discuss active
control of mixing in the wake of a bluff body flame holder.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A particle method and adaptive treecode for vortex sheet motion in 3-D flow }\\ \\
%author
{\bf Robert Krasny }\\
%affiliation
University of Michigan, USA \\
%e-mail
email:krasny@umich.edu \\
%coauthor
{\bf Keith Lindsay }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
A Lagrangian particle method is presented for computing vortex
sheet motion in 3-D flow. The particles are advected using the
Rosenhead-Moore form of regularized Biot-Savart kernel and the
velocities are evaluated using a particle-cluster treecode based
on Taylor approximation in Cartesian coordinates. The Taylor
coefficients of the regularized kernel are computed using a
recurrence relation. Several adaptive techniques are implemented
to reduce the CPU time including variable order approximation,
nonuniform rectangular cells, and a run-time choice between Taylor
approximation and direct summation. The method is applied to
simulate the azimuthal instability of a vortex ring and the merger
of two vortex rings.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf High order schemes for Navier-Stokes equations with spectral methods: numerical investigation of their comparative properties }\\ \\
%author
{\bf Gerard Labrosse }\\
%affiliation
University Paris-Sud XI, France \\
%e-mail
email:labrosse@limsi.fr \\
%coauthor
{\bf Leriche E., Perchat E.} and {\bf Deville M.O. }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Since the first numerical experiments were performed with the
incompressible Navier-Stokes equations, solving the linear Stokes
system has been of major concern with respect to its stability,
accuracy and computational efficiency once the non-linear terms
are explicitly treated as a given source. The question remains how
to decouple the pressure and velocity fields in order to get an
easily tractable numerical system with given stability and
space-time accuracy properties. Many strategies have been
proposed.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Multiple scale models in complex fluids }\\ \\
%author
{\bf Chun Liu }\\
%affiliation
Penn State University, USA \\
%e-mail
email:liu@math.psu.edu
%coauthor
%{\bf Keith Lindsay\\ National Center for Atmospheric Research }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this talk, several dynamical systems modelling specific types
of complex fluids are introduced. These models all involve
multiple spatial scale effects. The relation between these and
other existing models will be discussed. We will also study the
relations between the variational procedure; the basic energy law;
stability; and the higher order energy estimates. The different
non-Newtonian properties such systems exhibit is of particular
interest. We will discuss some analytical, as well as modelling
problems in these models.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Complex dynamics in a short Taylor-Couette annulus}\\ \\
%author
{\bf Juan Lopez }\\
%affiliation
Arizona State University, USA \\
%e-mail
email:lopez@math.asu.edu \\
%coauthor
{\bf F. Marques} and {\bf J. Shen}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Nonlinear dynamics of the flow in a short annulus driven by the
rotation of the inner cylinder and bottom endwall is considered.
The shortness of the annulus enhances the role of mode
competition, and the associated dynamics are found to be organized
by a number of local codim-2 bifurcations as well as global
homoclinic bifurcations. The dynamics are explored using a 3D
Navier-Stokes solver, which is also implemented in a number of
invariant subspaces in order to follow unstable solutions and
obtain a fairly complete bifurcation diagram of the mode
competitions. (for special session 32).
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf 3D Navier-Stokes equations with initial data characterized by uniformly large vorticity }\\ \\
%author
{\bf Alex Mahalov }\\
%affiliation
Arizona State University, USA \\
%e-mail
email:alex@taylor.la.asu.edu
%coauthor
%{\bf F. Marques\\ Universitat Politecnica de Catalunya \\ J.Shen\\University of Central Florida}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We prove existence on infinite time intervals of regular solutions
to the 3D Navier-Stokes Equations for fully three-dimensional
initial data in $R^3$ characterized by uniformly large vorticity
and infinite energy; smoothness assumptions for initial data are
the same as in local existence theorems. The global existence is
proven using techniques of fast singular oscillating limits and
the Littlewood-Paley dyadic decomposition. Infinite time
regularity is obtained by bootstrapping from global regularity of
the limit equations and strong convergence theorems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Dynamics in viscous vortex cores }\\ \\
%author
{\bf Monika Nitsche }\\
%affiliation
University of New Mexico, USA \\
%e-mail
email:nitsche@math.unm.edu
%coauthor
%{\bf F. Marques\\ Universitat Politecnica de Catalunya \\ J.Shen\\University of Central Florida}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Vortex blob simulations of vortex sheet roll-up have shown that
the flow becomes chaotic in the vortex core, displaying resonance
phenomena characteristic of slightly perturbed Hamiltonian
systems. The resonance is induced by small self-sustained
oscillations in the core vorticity, which are in turn attributed
to the self-induced strainfield of the vortex. In this talk I will
present the extent to which the chaotic dynamics are present in
viscous vortices, and the dependence on the dimensional parameters
of the flow. The results are based on 4th order finite difference
simulations of the vortex evolution.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A new class of velocity-correction schemes for the incompressible Navier-Stokes equations }\\ \\
%author
{\bf Jie Shen }\\
%affiliation
Purdue University and University of Central Florida, USA \\
%e-mail
email:shen@math.purdue.edu
%coauthor
%{\bf F. Marques\\ Universitat Politecnica de Catalunya \\ J.Shen\\University of Central Florida}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We'll introduce a new class of velocity-correction projection
schemes in the standard form and in rotational form. We will show
that the velocity-correction schemes lead to slightly better error
estimates compared with the usual pressure-correction schemes, but
more importantly, the high-order versions of the
velocity-correction schemes in rotational form are stable.
Furthermore, we will show that the splitting-up schemes proposed
by Orszag, Israeli and Deville (1986) and Kaniadakis, Israeli and
Orszag (1991) can be recast as a velocity-correction scheme in
rotational form. We will also present some numerical simulations
of 3-D rotating flows.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Residence times, chaotic advection and thrombosis in anatomically accurate arterial flows }\\ \\
%author
{\bf Yiannis Ventikos }\\
%affiliation
Laboratory for Thermodynamics in Emerging Technologies,Technology, Swiss Federal Institute of Technology, Zurich, Switzerland \\
%e-mail
email:Yiannis.Ventikos@ethz.ch
%coauthor
%{\bf F. Marques\\ Universitat Politecnica de Catalunya \\ J.Shen\\University of Central Florida}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We are focusing on the application of dynamical systems theory
concepts and tools in biothermofluidics. More specifically, we are
investigating the mixing properties of blood when flowing within
arterial segments of the cerebrum that suffer from multiple
saccular aneurysms. When computational fluid dynamics techniques
are applied on anatomically accurate reconstructions of such
vessels, kinematic considerations yield novel evidence of
spatiotemporal chaos. We show that residence time maps exhibit
strong non-uniformity, linked to the entry patterns of the blood
in the aneurismal sacs, but also to the strong divergence of flow
lines within the sacs. Similar trends are observed when the basins
of attraction of single or multiple aneurismal sacs are examined.
A direct connection between the understanding that dynamical
systems can offer for traditionally medical concepts, like
thrombosis and pharmacokinetics, is made and quantitative results
involving the thrombotic risk of particular malformations are
provided, based on the above notions. The impact of such findings
in connection with thrombosis and pharmacokinetics, but also when
interventional planning is considered, can be quite significant,
especially when similar procedures, capable of yielding such
results, can be embedded in the daily clinical protocols.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Accurate local time stepping algorithm for solving PDEs }\\ \\
%author
{\bf Jianping Zhu }\\
%affiliation
University of Akron, USA\\
%e-mail
email:jzhu@uakron.edu
%coauthor
%{\bf Jun Zou \\Chinese University of Hong Kong }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this presentation, we will discuss an accurate implicit local
time stepping algorithms. The basic idea is to use proper
interpolation and extrapolations to match solutions calculated
using different time steps, and to advance solutions at all
spatial grid points to the same time level. We will show that for
systems of first
order hyperbolic equations, an interpolation or extrapolation of order p-1 is needed
to maintain consistency and accuracy of a time integration algorithm of order p.
Numerical results will also be discussed in the presentation.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Zeta Functions of Graphs and Related Topics
}\\
Organizer: Audrey Terras, U. of California
\end{center}
\vskip .2in
\begin{multicols}{2}
%\title
\noindent {\bf Ramanujan type graphs and bigraphs}\\ \\
%author
{\bf Cristina Ballantine}\\
%affiliation
Dartmouth College, USA \\
%e-mail
email:Cristina-Maria.Ballantine@Dartmouth.EDU
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We will show that quotients of the
Bruhat-Tits building of $SL_2(\mathbb{Q}_p)$ form an infinite
family of graphs which are almost Ramanujan. We will also
investigate the Bruhat-Tits tree associated with
$U_3(\mathbb{Q}_p)$ and show why one should be able to estimate
its spectrum.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Zeta functions of infinite graphs}\\ \\
%author
{\bf Bryan Clair}\\
%affiliation
Saint Louis University, USA\\
%e-mail
email: bryan@SLU.EDU
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Start with a finite graph $X$. Take an infinite regular covering
$Y$ of $X$, with covering group $\Gamma$. Using the trace on the
von Neumann algebra associated to $\Gamma$, there is an $L^2$ zeta
function for $Y$ which enjoys many of the properties of the
Ihara-Hashimoto-Bass zeta function for finite graphs, including a
version of the rationality formula. For families of finite graphs
covering $X$, the normalized zeta functions of the finite graphs
converge to the $L^2$-zeta function for $Y$.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Zeta functions of graphs, Shimura varieties, and
dynamical systems
}\\ \\
%author
{\bf Jerome W. Hoffman}\\
%affiliation
Louisiana State University,USA\\
%e-mail
email: hoffman@math.lsu.edu
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
This talk surveys the relations, known or conjectural, connecting
zeta functions defined for three classes of objects -
(hyper)graphs, modular varieties, and dynamical systems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Ramanujan hypergraphs
}\\ \\
%author
{\bf Wen-Ching Winnie Li}\\
%affiliation
Penn State University, USA\\
%e-mail
email: wli@math.psu.edu
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
A hypergraph is a higher dimensional generalization of a graph. In
this talk we introduce the concept of Ramanujan hypergraphs and
discuss explicit construction of such hypergraphs. These are
extensions of Ramanujan graphs.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Random walks on knot diagrams}\\ \\
%author
{\bf Xiaosong Lin}\\
%affiliation
University of California, Riverside, USA\\
%e-mail
email: xl@math.ucr.edu
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We will discuss some features of the study of a model of random
walks on knot diagrams. This model relates the Alexander and Jones
polynomials in knot theory with the Ihara-Selberg type zeta
function in number theory and graph theory.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Laplacians related to Zeta functions and an
application to cogrowth of graphs.
}\\ \\
%author
{\bf Sam Northshield}\\
%affiliation
State U. of New York, Plattsburgh, USA\\
%e-mail
email: samuel.northshield@plattsburgh.edu
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The concept of cogrowth of groups goes back to Ol'shanskii's
settling of the von Neumann conjecture in 1984. Ol'shanskii
constructed a group which was non-amenable (equivalently, its
cogrowth constant was strictly less than the growth constant of
its corresponding free covering) but was not an extension of a
free group. The notion of amenability for finitely generated
groups has been extended to arbitrary graphs and, by a suitable
definition of cogrowth constant, we prove that a graph which is
the cover of a finite graph is amenable if and only if its
cogrowth constant equals the growth constant of its free cover.
The proof uses harmonic functions with respect to operators of the
form (I-uA+$u^2$Q) which are, by Bass' theorem, related to zeta
functions on graphs.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Isospectrality conditions for regular graphs}\\ \\
%author
{\bf Gregory Quenell}\\
%affiliation
Mt. Holyoke College, USA\\
%e-mail
email: quenell@mtholyoke.edu
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In 1994, Hubert Pesce proved that two compact hyperbolic surfaces
are (strongly) isospectral if and only if their covering groups
are representation equivalent as subgroups of the automorphism
group of hyperbolic 2-space. The proof depends on a certain length
spectrum that can be described geometrically on the surface or
algebraically in the covering group.We discuss the analogous
length spectrum on regular graphs, looking at it combinatorially
on the graph and algebraically in the covering group. We also note
that our length spectrum appears in the Ihara zeta function.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf L-functions and the Selberg trace formula for semiregular bipartite graphs}\\ \\
%author
{\bf Iwao Sato}\\
%affiliation
Oyama National College of Technology, Japan\\
%e-mail
email: isato@oyama-ct.ac.jp\\
%coauthor
{\bf Hirobumi Mizuno}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We give a decomposition formula for the L-function of a
semiregular bipartite graph G. Furthermore, we present the Selberg
trace formula for the above L-function of G.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A new kind of Zeta function: when number theory meets graph theory}\\ \\
%author
{\bf Audrey Terras}\\
%affiliation
U. of California, San Diego, USA\\
%e-mail
email:aterras@ucsd.edu
%coauthor
%{\bf Hirobumi Mizuno}\\Meisei University
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The tree of zeta functions has many branches including those from
number theory (Riemann and Dedekind zeta functions), spectral
geometry of manifolds (Selberg's zeta function), and graph theory
(Ihara's zeta function). It is also possible to mix in group
representations and obtain L-functions. Applications include
analogues of the prime number theorem and analogues of the work on
what is now called quantum chaos - the statistics of energy levels
of various non-classical physical systems. For example, the poles
of the Ihara zeta function of a connected regular graph satisfy
the Riemann hypothesis if and only if the graph is a Ramanujan
graph (meaning that the second largest eigenvalue of the adjacency
matrix, in absolute value, is in some sense smallest possible). In
this talk I will compare the various sorts of zetas and
investigate properties and applications of 3 types of graph zeta
functions (the vertex or Ihara zeta, the edge and the path zetas)
considered in Stark and Terras, Advances in Math., Vol. 121 (1996)
and Vol. 154 (2000).
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Chaotic properties of quotients of trees (joint
work with Audrey Terras)
}\\ \\
%author
{\bf Dorothy I. Wallace}\\
%affiliation
Dartmouth College, USA\\
%e-mail
email:Dorothy.I.Wallace@Dartmouth.EDU
%coauthor
%{\bf Hirobumi Mizuno}\\Meisei University
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The geodesic flow on the k-regular tree produces induces a
trajectory on the graph corresponding to a quotient of the tree.
We look at the self correlation of the induced flow and show that
it exhibits chaotic properties.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Statistical Description of the Dynamics of
Large and Disordered Systems}\\
\hspace{-1.6in} Organizer: Ilya Timofeyev, Courant Institute, NYU \\
Peter R Kramer, Rensselaer Polytechnic Institute(RPI)
\end{center}
\vskip .2in
\begin{multicols}{2}
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Normal modes on average in stochastic systems }\\ \\
%author
{\bf Carlo Alabiso }\\
%affiliation
Dipartimento di Fisica, Universita' di Parma, Italy \\
%e-mail
email:alabiso@fis.unipr.it \\
%coauthor
{\bf Mario Casartelli}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In studying non linear Hamiltonian systems, the main attention has
been generally posed on the range of validity of the near
integrable behaviour, as outlined by the KAM and Nekhoroshev
theories. Our approach, tested mainly on Fermi-Pasta-Ulam model
but also on Lennard-Jones and Toda systems, consists in looking
for the survival of some other kind of near integrable behaviour
in the regime of motion where KAM theorem is no more applicable,
i.e. above the so called strong stochastically threshold. We
explore the persistence of a pseudo-harmonic spectrum (not
connected to a perturbative approach), and the properties of some
geometrical quantities as the Frenet-Serret curvatures, and their
relations to dynamical properties. The stability of our results in
thermodynamic limit has been numerically checked, always starting
from generical initial conditions. The type of phase-space
chaoticity in presence of pseudo-harmonicity has been studied by
usual spectral methods on physically meaningful time series.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
%\title
\noindent {\bf Stages of energy transfer in the Fermi-Pasta-Ulam system }\\ \\
%author
{\bf Joseph Biello }\\
%affiliation
Rensselaer Polytechnic Institute, USA \\
%e-mail
email:biellj@rpi.edu \\
%coauthor
{\bf Peter R. Kramer} and {\bf Yuri Lvov}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The (alpha) Fermi-Pasta-Ulam model of
weakly coupled nonlinear oscillators was originally introduced to
study the dynamical relaxation of a system toward thermodynamic
equilibrium. The original simulations fell in a regime which
famously failed to exhibit this convergence. We revisit the
original question for larger systems at energies sufficiently
large to allow the system to exhibit relaxation toward thermal
equilibrium. Direct numerical simulations reveal several
interesting stages in this process as the energy evolves from a
large-scale excitation toward equipartition among all modes. We
focus on characterizing these intermediate stages of energy
transfer in physical terms.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Power-law spectra for the damped and driven spectral truncation of the Burgers-Hopf model }\\ \\
%author
{\bf David Cai }\\
%affiliation
Courant Institute, USA \\
%e-mail
email:cai@cims.nyu.edu \\
%coauthor
{\bf E. Vanden-Eijnden }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Damped and driven Fourier-truncated Burgers-Hopf equations will be
considered. The damping and white-noise forcing setup the inertial
range in the problem and the power-law spectrum emerges. The
asymptotic theory will be presented which allows to predict the
spectrum in the limit of infinitely small damping and forcing.
Numerical simulations confirm the analytical predictions with
surprising accuracy.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Statistically relevant conserved quantities for the truncated Burgers-Hopf equation }\\ \\
%author
{\bf Gregor Kovacic }\\
%affiliation
Rensselaer Polytechnic Institute, USA \\
%e-mail
email:kovacg@rpi.edu \\
%coauthor
{\bf Rafail V. Abramov} and {\bf Andrew J. Majda}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The importance of the cubic conserved quantity (Hamiltonian) for
the equilibrium statistical mechanics will be discussed. By direct
numerical simulation, corrections to energy equipartition will be
computed that are due to the atypical values of Hamiltonian, which
is different from the energy. Statistical-mechanical arguments
will be presented explaining that for randomly-chosen initial
conditions, these corrections are negligible. An alternative
computation of the corrections to energy equipartition via a
purely statistical-mechanical Monte-Carlo simulation will also be
discussed.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Application of weak turbulence theory to Fermi-Pasta-Ulam system }\\ \\
%author
{\bf Peter Kramer }\\
%affiliation
Rensselaer Polytechnic Institute, USA \\
%e-mail
email:kramep@rpi.edu \\
%coauthor
{\bf Joseph Biello } and {\bf Yuri Lvov}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Weak turbulence theory was developed primarily to predict the
properties of the stationary energy spectrum for weakly
interacting turbulent waves in fluids, but the mathematical
formalism can be applied to general weakly nonlinear Hamiltonian
systems with diffuse coupling among modes. We adapt weak
turbulence theory to obtain dynamical scaling predictions for the
shape of the energy spectrum in the Fermi-Pasta-Ulam model at
various intermediate stages of its evolution, and the time scales
required to achieve these stages. To do this, we introduce some
generalizations to weak turbulence theory which have broader
application.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Remarkable statistical behavior for truncated Burgers-Hopf Dynamics }\\ \\
%author
{\bf Ilya Timofeyev }\\
%affiliation
Courant Institute, USA \\
%e-mail
email:ilyat@cims.nyu.edu \\
%coauthor
{\bf A. Majda }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
The Fourier-Galerkin spectral truncations of the inviscid
Burgers-Hopf equation are introduced as simple one-dimensional
models with a well-defined mathematical structure, intrinsic chaos
and scale separation. Energy-based statistical mechanical
predictions of energy equipartition and a correlation scaling
theory that involves eddy viscosity will be presented.
Computational results will be shown that confirm the predictions
of these theories, and also shed further light on global
phase-space properties of the Burgers-Hopf system, such as a
number of exactly-obtainable invariant subspaces.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A priori tests of a stochastic mode reduction strategy }\\ \\
%author
{\bf Eric Vanden-Eijnden }\\
%affiliation
Courant Institute, USA \\
%e-mail
email:eve2@cims.nyu.edu \\
%coauthor
{\bf I. Timofeyev} and {\bf A. Majda }\\
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
A recently developed mode-elimination theory will be discussed,
where the spectral truncation of the Burger-Hopf equation serves
as a stochastic heat bath in various coupled prototype models. The
non-linear self-interactions of the bath are first replaced by
stochastic terms and then the fast variables are completely
eliminated form the model allowing for the low dimensional reduced
equations for slow variables. Numerical simulations of the
original coupled model, intermediate stochastic model and reduced
equations will be presented which verify the applicability of the
mode-reduction strategy.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\end{multicols}
\begin{center}
{\Large \bf Delay Differential Equations
}\\
Organizer: Hans-Otto Walther, Universitat Giesen, Germany
\end{center}
\vskip .2in
\begin{multicols}{2}
% ----------------------------------------------------------------
%\title
\noindent {\bf A class of evolution equations of Volterra type}\\ \\
%author
{\bf O. Arino}\\
%affiliation
%Department of Mathematics, The University of Kansas \\
%e-mail
email:Ovide.Arino@univ-pau.fr \\
%coauthor
{\bf M. El Massoud}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this talk we consider equations of Volterra type in
which the term under the integral is of the form $Au(s)$ where $A$
is an unbounded operator. In his monograph on the subject, J.
Pruess has collected some old results and also established some
new ones about the Cauchy problem associated to that equation. In
the linear case there is an analogue of the Hille-Yosida theorem
for semigroups. The class of equations we are dealing with arises
from the study of the shallow water equation. While it proved
impossible for us to check the existence theorem stated by Pruess,
we are able, using some properties of the equation at hand to
derive the result with a completely different method. After a
quick motivation for this equation, the talk will focus on
explaining the method.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Stability and approximation of linear retarded and neutral systems }\\ \\
%author
{\bf Richard Fabiano }\\
%affiliation
University of North Carolina at Greensboro, USA \\
%e-mail
email:fabiano@uncg.edu \\
%coauthor
{\bf Janos Turi }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider the linear delay equation $$\frac{d[x(t) +
Cx(t-r)]}{dt} = Ax(t) + Bx(t-r),$$ where $A$, $B$ and $C$ are
constant $n\times n$ matrices. Using techniques from linear
semigroup theory, we discuss conditions on the matrices $A$, $B$,
$C$ which guarantee exponential stability of the solution
semigroup associated with this equation. We also discuss the
solution semigroups which arise from certain finite dimensional
semidiscrete approximation schemes for this equation.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On reaction-diffusion systems with time delays }\\
\\
%author
{\bf Wei Feng }\\
%affiliation
University of North Carolina at Wilmington, USA \\
%e-mail
email:fengw@uncwil.edu \\
%coauthor
{\bf Xin Lu }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this talk we give an overview
about our recent results on reaction-diffusion systems modelling
the dynamics of single or interacting populations with time delay
effects. For various models with constant or periodic parameters,
competitive interactions and environmental toxicant, the issues of
global stability, permanence, and asymptotic periodicity are
discussed.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Dynamics of delay equations with sine-like nonlinearity}\\ \\
%author
{\bf B. Lani-Wayda}\\
%affiliation
Math. Inst. Univ. Giessen, Germany \\
%e-mail
email:Bernhard.Lani-Wayda@math.uni-giessen.de \\
%coauthor
{\bf Roman Srzednicki}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Delay equations $\dot x(t) = f(x(t-1))$ with a function $f$
similar to the sine function are models for the control of high
frequency oscillators. The variable $x$ stands for the difference
between desired and actual phase of the oscillator. Complicated
dynamical behavior of such systems is observed in physical and
numerical experiments. In joint work with Roman Srzednicki, we
provide analytical proofs for such behavior in two different
situations.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On global stability and chaos in a family of scalar
functional differential equations}\\ \\
%author
{\bf E. Liz}\\
%affiliation
Universidad de Vigo, Spain \\
%e-mail
email:eliz@dma.uvigo.es \\
%coauthor
{\bf Sergei Trofimchuk} and {\bf Victor Tkachenko }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider a family of scalar delay differential equations
$x'(t)=-\delta x(t)+f(t,x_t)$, with a nonlinearity $f$ satisfying
a sort of negative feedback condition combined with a boundedness
condition. Some well-known delay differential equations as the
Mackey- Glass equation, Nicholson's blowflies equation, and
equations with maxima are kept within our considerations. \par We
present a global stability criterion for this family when
$\delta\geq 0$, which in particular unifies the celebrated
$3/2$-conditions given for the Yorke and the Wright type
equations. The sharp character of our condition is proved
analyzing differential equations with maxima. For an appropriate
choice of the parameters, we show the existence of solutions with
chaotic behavior in this type of equations.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Smooth invariant manifolds for differential
equations with state-dependent delay}\\ \\
%author
{\bf T. Krisztin}\\
%affiliation
University of Szeged, Hungary \\
%e-mail
email:krisztin@math.u-szeged.hu
%coauthor
%{\bf Boris Belinskiy}
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider a class of delay differential equations where the
right hand sides are not smooth in the usual sense, they are
smooth only on spaces of smooth functions. Equations with
state-dependent delay satisfy these conditions. We prove the
existence and smoothness of local unstable and center manifolds
near equilibrium points.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Max-plus operators and differential-delay
equations}\\ \\
%author
{\bf R.D. Nussbaum}\\
%affiliation
Rutgers University, USA \\
%e-mail
email:nussbaum@math.rutgers.edu
%coauthor
%{\bf Kumar V. Adhikarala Mississippi State University }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\setcounter{equation}{0}
\\
We shall discuss joint work with Professor John Mallet-Paret in
which we consider the nonlinear differential-delay equation
\begin{equation}
ax'(t)=f(x(t),x(t-r)), r:=r(x(t)).
\end{equation}
Here $f$ and $r$ are given functions and $a>0$. A simple-looking
but non-trivial example to which our theory applies is provided by
\begin{equation}
ax'(t)=-x(t)-kx(t-r), r:=1+cx(t),
\end{equation}
where $a>0$, $c>0$, and $k>1$. Under appropriate assumptions on
$f$ and $r$, we know that eq. (1) has, for all sufficiently small
$a>0$, a ''slowly oscillating periodic solution'' which depends on
$a$. We are interested in the limiting shape of the graphs of such
solutions as $a$ approaches 0. In answering this question, we are
led to the study of max-plus-equations:
\begin{equation}
x(t)+p=
\end{equation}
\[\max\{k(s,t)+x(t):c(s)\le t\le d(s),0\le s\le L\}\]
In (3), $k$, $c$ and $d$ are given continuous functions, and one
seeks an ''additive eigenvalue $p$'', and an ''additive
eigenvector $x(t)$'' which give a solution of (3) on $[0,L]$. The
presence of functions $c$ and $d$ makes eq. (3) much more subtle
than in the case $c(t):=0$ and $d(t):=L$.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Convergence to equilibria in nonquasimonotone delay differential equations}\\ \\
%author
{\bf M. Pituk}\\
%affiliation
Department of Mathematics and Computing, University of Veszpr\'em, Hungary \\
%e-mail
email:pitukm@almos.vein.hu
%coauthor
%{\bf Kumar V. Adhikarala Mississippi State University }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
Consider the scalar autonomous functional differential equation $$
\dot x(t)=f(x_t), $$ where $f:C\rightarrow\bold R$ is continuous.
Here $C=C([-r,0],\bold R)$, $r>0$ and $x_t\in C$ is defined by
$x_t(s)=x(t+s)$ for $s\in[- r,0]$. Assuming that the solutions of
the above equation generate a strongly order preserving semiflow
with respect to the exponential ordering introduced by H. L. Smith
and H. Thieme, we present necessary and sufficient conditions for
the generic and global convergence of the solutions.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Dynamics of oscillations in multi-dimensional
discontinuous systems with delay}\\ \\
%author
{\bf E. Shustin}\\
%affiliation
Tel Aviv Univeristy, Israel \\
%e-mail
email:shustin@tau.ac.il
%coauthor
%{\bf Kumar V. Adhikarala Mississippi State University }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider a system of delay differential equations
\[\dot x_i(t)=F_i(x_1(t),...,x_n(t),t)-\mbox{sign}\,\,x_i(t-h_i), i=1,...,n,\]
with positive constant delays $h_1,...,h_n$ and perturbations
$F_1,...,F_n$ absolutely bounded by a constant less than 1. This
system models a negative feedback controller of relay type
intended to bring the system to the origin. Non-zero delays do not
allow such a stabilization, but cause oscillations around zero
level in any variable. We introduce integral-valued densities of
zeroes of the solution components, and show that they always
decrease to some limit values. We show that, for any prescribed
limit zero densities, there exists at least an $n$-parametric
family of solutions, and we establish certain conditions under
which all slowly oscillating solutions are stable, and fastly
oscillating solutions are not. Finally, we show that a
modification of relay controllers (with arbitrary delays) can
exponentially quench oscillations.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Forward-backward functional differential equations, holomorphic factorization and applications}\\ \\
%author
{\bf S. Verduyn Lunel}\\
%affiliation
Universiteit Leiden, The Netherlands \\
%e-mail
email:verduyn@math.leidenuniv.nl \\
%coauthor
{\bf J. Mallet-Paret }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
In this lecture we introduce and motivate the forward-backward
functional differential equation
\[\dot{x}(t)=ax(t)-bx(t-1)+cx(t+1)\]
defined on the real axis. Such equations arise naturally in
various contexts, for example, in the study of travelling waves in
discrete spatial media such as lattices.
Since the mixed-type equation is not an initial value problem it
is our goal to decompose solutions of this equation as sums of
''forward'' solutions and ''backward'' solutions. We show that the
set of all forward solutions defines a semigroup which can be
realized by a retarded functional differential equation except
for possibly finitely many modes, and similarly for the set of
backward solutions as an advanced functional differential
equation. Holomorphic factorizations play a crucial role in our
results.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Smooth solution operators for differential equations with state-dependent delay}\\ \\
%author
{\bf H.O. Walther}\\
%affiliation
Mathematisches Institut der Universitat Gieben, Germany \\
%e-mail
email:Hans-Otto.Walther@math.uni-giessen.de
%coauthor
%{\bf Kumar V. Adhikarala Mississippi State University }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
It is shown how to obtain continuously differentiable solution
operators for differential equations with state-dependent delay.
This permits a geometric theory for such equations, beginning with
smooth local invariant manifolds.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A simple delayed neural network with large capacity for associative memory}\\ \\
%author
{\bf Jianhong Wu}\\
%affiliation
York University, Canada\\
%e-mail
email:wujh@mathstat.yorku.ca
%coauthor
%{\bf Kumar V. Adhikarala Mississippi State University }
%abstract
%\begin{center} {\bf Abstract} \end{center}
\\
We consider both continuous and discrete models for a network of
two neurons with delay feedback, and describe the multistability
in the form of either multiple stable periodic orbits or unstable
periodic orbits with large domains of attraction. We will compare
the dynamics of discrete and continuous models, and address the
issue of the lower bound for possible stable periodic orbits of an
order-preserving and contractive map. We will also discuss
potential applications in associative memory and periodic pattern
recognition.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Topological horseshoes and delay differential equations}\\ \\
%author
{\bf Piotr Zgliczynski}\\
%affiliation
Jagiellonian University, Institute of Mathematics, Poland \\
%e-mail
email:zgliczyn@im.uj.edu.pl \\
%coauthor
{\bf K. Wojcik }
%abstract
%\begin{center} \setcounter{equation}{0} {\bf Abstract} \end{center}
\setcounter{equation}{0}
\\
Consider a delay differential equation and an ordinary
differential equation
\begin{eqnarray}
x'(t)&=&f(x(t-\tau)) \label{eq:dode} \\
x'&=&f(x) \label{eq:ode}
\end{eqnarray}
where $x \in {\Bbb R}^n$,and $f\in {C}^\infty $. \\We study the
following question: Assume that equation (\ref{eq:ode}) has
chaotic solutions, i.e. a suitable Poincar\'{e} map has a Smale's
horseshoe. Will the corresponding delay equation (\ref{eq:dode})
also have chaotic solutions for small delays $\tau
>0$? We present a proof that the answer to this question is
positive, namely for small delays a suitable Poincar\'{e} map,
$P_\tau$, for delay equation does have a topological horseshoe. We
prove that this implies an existence of an invariant set $S_\tau$
for $P_\tau$, such that $P_\tau$ on $S_\tau$ can be semiconjugated
with a Bernoulli shift. Moreover we obtain an infinite number of
periodic orbits with unbounded periods.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
\end{multicols}
\begin{center}
{\Large \bf Transport Phenomena }\vskip -0.25in \
$$\begin{array}{rl}
\mbox{Organizer:}& \mbox{Liqiu Wang, The University of Hong Kong}
\end{array} $$
\end{center}
\vskip .2in
\begin{multicols}{2}
% ----------------------------------------------------------------
%\title
\noindent {\bf Transport of solutions in porous media: statement of the problem, numerical method and applications }\\ \\
%author
{\bf Leonid Bronfenbrener }\\
%affiliation
University of Leuven, The Netherlands \\
%e-mail
email:vera@cbs.gov.il
%coauthor
%{\bf P. Perera Texas Tech University }
%abstract
%\begin{center}{\bf Abstract} \end{center}
\\
The analysis of the transport of chemical reacting solutions in
porous media and its crystallization (dissolution) is of
fundamental importance. The interest in this problem and
development of various kinds of the models in recent years arise
from practical requirements. In our opinion it is important to
study the solutions transport in porous building materials under
wetting and drying process conditions. It is also interesting to
consider the kinetics of crystallization that influence on the
solid zone formation. In the present study we consider the
instantaneous kinetics. Thus, the purpose of this work is to
investigate numerical method for the transport of solutions in
porous building materials under wetting and drying process
conditions. The statement of the problem and numerical method
based on the implicit finite-difference scheme are presented. The
convergence criterion of the iteration process was obtained. It is
considered also the approximation error for the non-uniform mesh
domain. The theoretical studies of the drying and wetting
processes indicate the stability and high efficiency of the
method. As application the transport of solution in ceramic brick
is considered. The concentration distributions, evolution of the
phase front and crystallization process are presented.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf PA link between chaos and vortex dynamics in a transitional boundary layer }\\ \\
%author
{\bf C.B Lee }\\
%affiliation
Peking University, China \\
%e-mail
email:cblee@mech.pku.edu.cn
%coauthor
%{\bf P. Perera Texas Tech University }
%abstract
%\begin{center}{\bf Abstract} \end{center}
\\
A model of the dynamic physical processes that occur in a
transitional boundary layer flow is described. The CS-solitons,
the closed vortex, the secondary closed vortex and the chain of
ring-like vortices are postulated to be the basic flow structures
of the transitional boundary layer as well as the turbulent
boundary layer. It is argued that the central features of the
transitional and developed turbulent boundary layer flows can be
explained in terms of how the series vortices interact with each
other, and with the CS-solitons. The physical process that leads
to the regeneration of the new closed vortex along the border of
the CS-soliton is described, as well as the processes of the
evolution and the interaction of the vortices to high frequency
vortices. The model is supported by recent important developments
in the theory of unsteady surface layer separation and a number of
‘kernel’ experiments which serves to both transitional and
developed turbulent boundary layer. An important aspect of the
model is that it has been formulated to be consistent with
accepted rational mechanics concepts that are known to provide a
proper physical description of other flows. By the way, a link
between chaos and the transitional dynamic system is established.
The result shows that fractal dimensions can be used to describe
all the transitional processes as a necessary factor such as
Reynolds number.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On the appearance of film ruptures in plain thick layers of moving liquid }\\ \\
%author
{\bf Valery Nosov }\\
%affiliation
National Polytechnical Institute of Mexico, Mexico \\
%e-mail
email:v$\_$nosov@hotmail.com \\
%coauthor
{\bf Julio Gomez-Mancilla }
%abstract
%\begin{center}{\bf Abstract} \end{center}
\\
The film rupture (which closely related to the cavitation) can
appear in the thick layers of moving liquid even for sufficiently
small Reynolds numbers. The cavitation is the phenomenon of liquid
boiling and/or escape of dissolved gases from the liquid. These
two
types of cavitation are commonly referred as vapor and gaseous cavitation.
The gaseous cavitation is connected with the fact that the real liquid
usually contains up to 10\% by volume of dissolved air or other gases.
If the pressure in liquid decreases sufficiently then the
dissolved gas comes out and forms gas bubbles or cavities. This
phenomenon encountered in such part of moving liquid where the
liquid pressure drops. The gaseous cavitation changes the
parameters of movement, but does not damage moving surfaces. In
contrary, vapor cavitation is an origin of fatigue-type damaging
effect on moving surfaces. The physical origin and effects of
these types of cavitations are different, but for the following we
do not distinguish its. In both cases the one-phase liquid flow
became two-phase flow in which liquid and gas phases are separated
by some surfaces. The transition from one-phase flow to two-phase
flow must satisfy general physical conservation laws
–specifically, laws of mass and energy conservation. We consider
the appearance of film rupture in the plain layers of moving
liquid been an isothermal processes, where the heating of liquid
is ignored since heat transfer is a sufficiently slow process in
comparison to the rapid appearance of film rupture. Alternatively,
we consider steady state operating conditions. The thickness of
liquid layer is assumed to be so small that it is possible to
consider a liquid movements as a two-dimensional one. Four typical
two-dimensional cases of the thick layers of moving liquid are
considered. In these cases the thick layer of liquid is assumed to
be between : an inclined plane moving over an immobile plane, a
circle rotating over an immobile plane, a rotating circle no
coaxial with another immobile circle, two rotating circles. Based
on energy and mass balances the conditions under which liquid film
rupture (or cavitation) can or cannot occur in the thick layers of
moving liquid are derived for four cases mentioned above.
Expressions depending on geometrical parameters and on parameters
of movements where film rupture might appears, are given. Some
applications to the plain journal bearings are also presented.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf A critical review of turbulence modelling: physical constraints and physics-preserving models }\\ \\
%author
{\bf Liqiu Wang }\\
%affiliation
The University of Hong Kong, Hong Kong \\
%e-mail
email:lqwang@hkucc.hku.hk \\
%coauthor
{\bf Julio Gomez-Mancilla }
%abstract
%\begin{center}{\bf Abstract} \end{center}
\\
The present review, which includes some new material, consists of
two parts: physical constraints in turbulence modelling and
physics-preserving turbulent closure models that preserve the
frame indifference and satisfy both the principle of material
frame indifference (PMFI) and the second law of thermodynamics.
The former commences with careful definition of the average
operation, the Reynolds stress, the turbulent heat flux and the
turbulent mass flux. The remainder of this part is on the
developments, to date, of three physical constraints imposed by
the invariance, the realizability and the PMFI. In particular, two
sufficient conditions are discussed for the Reynolds stress tensor
and the turbulent heat/mass flux vector to be frame indifferent.
The application of the second law of thermodynamics to a thermally
isolated system and an irreversible process concludes that
realizability inequalities in turbulent flows follow logically
from the second law of thermodynamics. How system rotations affect
flow fields is critically examined from a basic theoretical
standpoint. Also critically reviewed is the literature on the
range of validity of the PMFI to the turbulence modelling. The
latter is devoted to the progress, to date, of physics-preserving
closure models of the Reynolds stress and the turbulent heat/mass
flux. In particular, both necessary and sufficient conditions are
developed in a systematic, rigorous way for turbulent closure
models to satisfy the three constraints reviewed in the first
part. The results have either confirmed some intuitive arguments
or offered new insights into turbulence modelling, and are of
significance in clarifying some controversies in the literature,
examining how well existing models preserve the physics, and
developing new models. Also developed are a linear theory, a
quadratic theory and a flow decomposition theorem to simplify and
guide the work of developing specific physics-preserving models.
This part ends with a further constraint on the physics-preserving
models by the disappearance of Reynolds stress and turbulent
heat/mass flux at a vanishing value of the mean velocity. Most
methods and results in the present review are also valid for the
higher-order correlations and the subgrid-scale (SGS) modelling in
the large eddy simulation (LES).
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Burgers' Weak Turbulence by Dynamical Systems Approach }\\ \\
%author
{\bf Mingtian Xu }\\
%affiliation
University of Hong Kong, Hong Kong \\
%e-mail
email:lqwang@hkucc.hku.hk
%coauthor
%{\bf Julio Gomez-Mancilla }\\
%abstract
%\begin{center}{\bf Abstract} \end{center}
\\
In the present work, the Green's function of the one-dimensional
heat conduction equation is employed to convert Burgers' equation
into the infinite dynamical system described by an integration
equation system with respect to the time variable. The infinite
dimensional system is truncated to finite dimensional dynamical
systems. These systems are solved by a numerical method proposed
in the present paper. The calculated results not only describe the
scenario of the route to the Burgers' turbulence, but also reveal
a new explanation for the mechanism of the occurrence of the
intermittence in the turbulence. In addition, these results
demonstrate a similar behavior for the dynamical systems with
dimensions ranging from 30 to 450 that are utilized to approximate
the Burgers' equation. This shows some validity of using finite
dynamical systems to approach the transition process from laminar
flow to the turbulence.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Solution filtering technique for solving Burgers' equation }\\ \\
%author
{\bf Tianliang Yang }\\
%affiliation
University of Kentucky, USA \\
%e-mail
email:tlyang@engr.uky.edu \\
%coauthor
{\bf J. M. McDonough }
%abstract
\\
Burgers' equation is a one-dimensional (1-D) analogue of the
Navier-Stokes (N.--S.) equations; it embodies all the main
mathematical features of the N.--S. equations. In the present
study, we test the solution filtering technique by solving
Burgers' equation numerically. The solution filtering technique is
a new approach proposed for dealing with aliasing of underresolved
solution as arise in large eddy simulation (LES) of turbulence.
The idea underlying the solution filtering technique is that
filtering aliased solutions is far simpler than dealing with the
consequences of filtering nonlinear differential equations. The
basic approach of the solution filtering technique is that the
governing equations (Navier--Stokes equations) are not filtered
and are solved directly on a grid system that is much coarser than
required by direct numerical simulation; then the solution at each
time step is filtered. Our present studies show that such an
approach works quite well for Burgers' equation. In spite of the
fact that the research carried out in the present studies employs
Burgers' equation rather than the full N.--S. equations, from
previous studies we believe that the results obtained from
Burgers' equation will apply to the N.--S. equations, at least in
a general way. Therefore, it is expected that the solution
filtering technique will possess significant potential in solving
the practical turbulent problem governed by the three-dimensional
(3-D) N.--S. equations.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Shear force distribution and heat transfer in
laminar boundary layer flows for power law fluid }\\ \\
%author
{\bf Liancun Zheng }\\
%affiliation
University of Science and Technology, Beijing, China \\
%e-mail
email:zliancun@263.net \\
%coauthor
{\bf Xinxin Zhang }
%abstract
\\
Analytical and numerical solutions are presented for the momentum
and energy laminar boundary layer equations in power law fluid
utilizing a similarity transformation and the shooting technique.
The results indicated that for power law exponents $0< n \leq 1$ ,
the skin friction $\sigma$ decreases with increasing $n$ , and
the dimensionless shear force decreases with increasing in
dimensionless velocity $t$ . When $N_{PT}=1$, the velocity
distribution in the viscous boundary layer is the same as the
temperature distribution in the thermal boundary layer and
$\delta=\delta_T$ . For $N_{PT}>1$, the increase of the viscous
diffusion exceeds that of thermal diffusion with increasing
$N_{PT}$, i.e. $\delta_T(t)<\delta(t)$. The thermal diffusion
ratio increases with increasing $n(0 0, \eqno(1) $$ $$
u_{\big\vert_{z=0}}=\varphi(x,y);~x\in{\bf R}^n,y\ge 0, \eqno(2)
$$ $$ {\partial u\over\partial y}_{\Big\vert_{y=0}}=0;~x\in{\bf
R}^n,z>0, \eqno(3) $$ where $g$ is continuous, $\varphi$ is
continuous and bounded and $B_{k,y}$ denotes singular Bessel
operator $\displaystyle{{1\over y^k}{\partial \over \partial y}
\big(y^k {\partial \over \partial y}\big)}$ with a positive
parameter $k.$ \noindent We prove the following assertions: {\bf
Theorem 1}: There exists a unique bounded solution of (1)-(3).
{\bf Theorem 2}: Let $A\in(-\infty,+\infty),\,u(x,y,z)$ be the
bounded solution of problem~(1)-(3). Then for any $x$ from ${\bf
R}^{n}$ and any positive $y$ $$
\begin{array}{l}
u(x,y,z){\buildrel{z\to\infty}\over{\longrightarrow}}A\Longleftrightarrow
\vspace{0.1in}
\\ {(n+k+1)\Gamma\big({n+k+1\over 2}\big)\over\pi^{n\over 2}
\Gamma\big({k+1\over 2}\big)r^{n+k+1}}\!\!
\int\limits_{B_+(r)}\!\!\!\eta^kf[\varphi(\xi,\eta)]d\xi d\eta
{\buildrel{r\to\infty}\over{\longrightarrow}}f(A), \end{array}$$
where $f(s)=\int_0^s {\exp(\int_0^xg(\tau)d\tau) dx.}$
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Relative equilibria and relaxation oscillation of point vortices in a plane }\\ \\
%author
{\bf Tatsuyuki Nakaki }\\
%affiliation
Kyushu University, Japan \\
%e-mail
email:nakaki@math.kyushu-u.ac.jp
%coauthor
%{\bf Per Sandholdt Sauer-Danfoss }\\{\bf Ningning Song }\\
%abstract
\\
The motion of assembly of point vortices on the two-dimensional
Euler fluid is discussed. The vortices drift away on the fluid due
to an interaction between the vortices, and this phenomenon can be
described by the ordinary differential equation in Hamiltonian
form. This equation is analyzed for a long time. When two
vortices is on the fluid, the motion can be easily analyzed and we
can find the results in textbooks on fluid dynamics. Three point
vortices problem is analyzed by Aref~(1979). Aref and
Pomphrey~(1982) show four point vortices which exhibit chaotic
motion. \par In this talk, we consider five point vortices under
some initial configuration. In this problem, an relative
equilibrium is stable in some sense. By numerical simulations we
find that a solution located near the equilibrium exhibits an
relaxation oscillation, and that another solution near the
equilibrium does not show typical behavior. We already know that,
under different initial configuration, some five point vortices
also exhibit an relaxation oscillation (Nakaki~(1999)). We can
observe that these two oscillations are belonging to a different
category. In this talk we shall discuss the oscillations from
numerical and mathematical points of view.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Dynamics of local principal configurations of surfaces
immersed in ${\bf R}^4$ with isolated simple umbilical points }\\ \\
%author
{\bf Matias Navarro }\\
%affiliation
Facultad de Ciencias, UASLP, Mexico \\
%e-mail
email:matias@galia.fc.uaslp.mx
%coauthor
%{\bf Per Sandholdt Sauer-Danfoss }\\{\bf Ningning Song }\\
%abstract
\\
The N-principal configuration of a surface M in 4-Euclidean space
is the set formed by the umbilical points and the lines of maximal
and minimal curvature with respect to a unitary smooth vector
field N normal to M. We describe here the bifurcation diagram of
N-principal configurations, where N is parameterized in the space
of 1-jets of normal vector fields which define an isolated
umbilical point.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Almost periodic solutions of linear equations in Hilbert spaces }\\ \\
%author
{\bf Gaston N'Guerekata }\\
%affiliation
Morgan State University, USA \\
%e-mail
email:gnguerek@morgan.edu
%coauthor
%{\bf Per Sandholdt Sauer-Danfoss }\\{\bf Ningning Song }\\
%abstract
\\
We discuss sufficient conditions to ensure almost periodicity of
solutions of linear equations of the form $x"(t)=A\,x(t)$ and
$x"(t)=A\,x(t)$ in Hilbert spaces.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Asymptotic stability of planar waves for multidimensional
viscous conservation laws in half space }\\ \\
%author
{\bf Shinya Nishibata }\\
%affiliation
Tokyo Institute of Technology, Japan \\
%e-mail
email:shinya@is.titech.ac.jp
%coauthor
%{\bf Per Sandholdt Sauer-Danfoss }\\{\bf Ningning Song }\\
%abstract
\\
The purpose of the present talk is to show the asymptotic
stability of one dimensional planar waves to multidimensional
viscous conservation laws in the half space:
\begin{equation}\label{1} u_t + f(u)_x +
\sum_{i=1}^{n-1}g(u)_{y_i}=u_{xx} + \Delta_y u, \quad x>0,
\end{equation} where $(x,y)=(x, y_1, y_2, \cdots, y_{n-1})\in{\bf
R}^n$ ($n \ge 2$). Here, the flux function $f$ is assumed to be
uniformly convex. It is assumed that $u_{b0},\ y\in {\bf
R}^{n-1}|u(x,y,t)-\phi(x,t)| \le C(1+t)^{-\frac{n}{2}+\varepsilon}
$ is obtained, where $\varepsilon$ is an arbitrarily small
positive constant .
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Applications of Epsilon-Pseudo-Orbits }\\ \\
%author
{\bf Douglas Norton }\\
%affiliation
Villanova University, USA \\
%e-mail
email:douglas.norton@villanova.edu
%coauthor
%{\bf Per Sandholdt Sauer-Danfoss }\\{\bf Ningning Song }\\
%abstract
\\
One recent trend in Dynamical Systems is for inherently
quantitative techniques to yield qualitative information about the
systems under investigation. Epsilon-pseudo-orbits for epsilon of
fixed size can be related to approximations for discrete dynamical
systems, yielding implications for observability of attractors and
other sets defined by the dynamics. In particular, the Conley
Decomposition of a space can be approximated by an epsilon-coarse
Conley Decomposition. Epsilon-pseudo-orbits yield models not only
of computer models themselves but also models of real-world
phenomena in which the epsilon-jumps are key ingredients of the
behavior of the system. One such system is neural activity of the
brain. Basic results on epsilon-pseudo-orbits and some
applications will be presented.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Some convex and monotone skew-product semiflows }\\ \\
%author
{\bf Sylvia Novo }\\
%affiliation
Universidad de Valladolid, Spain \\
%e-mail
email:sylnov@wmatem.eis.uva.es \\
%coauthor
{\bf Rafael Obaya} and {\bf Ana M. Sanz }
%abstract
\\
We study the topological and ergodic structure of a class of
convex and monotone skew-product semiflows. We assume the
existence of two strictly ordered minimal subsets $K_1$ and $K_2$,
we obtain a ergodic representation of their upper Lyapunov
exponents and we prove that they vanish or not simultaneously. In
the case of null upper Lyapunov exponents, we provide a complete
description of the ergodic structure of the skew-product semiflow
in a positively invariant region $K$ defined by the minimal
subsets. Finally, we study the behavior of the trajectories on $K$
in the hyperbolic case. Some examples of skew-product semiflows
generated by differential equations and satisfying the assumptions
of monotonicity and convexity are also presented.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Bounded trajectories set of a scalar convex differential equation }\\ \\
%author
{\bf Rafael Obaya }\\
%affiliation
Universidad de Valladolid, Spain \\
%e-mail
email:rafoba@wmatem.eis.uva.es \\
%coauthor
{\bf Ana I. Alonso }
%abstract
\\
We study the topological and ergodic structure of the set of
bounded trajectories of the flow defined by a scalar convex
differential equation. We characterize the minimal subsets, the
ergodic measures concentrated on them and study the long time
behaviour of the bounded trajectories in terms of the linearized
equations. We obtain continuity properties with respect to the
coefficients in different topologies.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Abstract finite time extinction problems }\\ \\
%author
{\bf Mark Oxley }\\
%affiliation
Air Force Institute of Technology, USA \\
%e-mail
email:Mark.Oxley@afit.edu
%coauthor
%{\bf Ana I. Alonso }\\
%abstract
\\
Let $X$ be a Banach space and $C$ be an ordering cone in $X$. Let
$A$ be the generator of a $C_0$ semigroup $\{S(t):t>0\}$. We pose
an (abstract) extinction problem as follows. Let $0\neq z \in C$
and assume the function $u(t)=S(t)z$ is a strong solution to the
initial-value problem $$ u_t = A\,u~,~~~u(0)=z~. $$ We say $u$
extinct in finite time if there exists a time $T\in (0,\infty)$
such that $u(t)=0$ for all $t \geq T$. We discuss necessary and
sufficient conditions on the generator $A$ such that $u$ extinct
in finite time. Applications to systems of nonlinear
reaction-diffusion systems will be given.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Conservation laws and invariant solutions for soil
water equations
}\\ \\
%author
{\bf G. S. Pai}\\
%affiliation
University of North West, South Africa \\
%e-mail
email:cmkmat@unibo.uniwest.ac.za \\
%coauthor
{\bf C. M. Khalique}
%abstract
\\
A mathematical model was developed to simulate soil water
infiltration, redistribution,and extraction in a bedded soil
profile overlaying a shallow water table and irrigated by a line
source drip irrigation system. The governing partial differential
equation can be written as $$
C(\psi)\psi_t=\left(K(\psi)\psi_x\right)_x+
\left(K(\psi)(\psi_z-1)\right)_z-S(\psi), \eqno(1) $$ where $\psi$
is soil moisture pressure head, $C(\psi)$ is specific water
capacity, $K(\psi)$ is unsaturated hydraulic conductivity,
$S(\psi)$ is a sink or source term, $t$ is time, $x$ is the
horizontal and $z$ is the vertical axis which is considered
positive downward. See Vellidis, G., Smajstrla, A. G., "Modelling
soil water redistribution and extraction patterns of
drip-irrigated tomatoes above a shallow water table", {\it
Transactions of the American Society of Agricultural Engineers},
35 (1), 1992, 183-191. We generate conservation laws for certain
soil water equations and determine the conserved vectors by direct
method and then using a theorem due to Kara and Mahomed
("Relationship between Symmetries and Conservation Laws", {\it
International Journal of Theoretical Physics,} Vol. 39, No. 1,
2000) we will then check if the condition for which point
symmetries of soil water equations associate with conservation
laws is satisfied. Invariant solutions of equation (1) for some
particular types of the coefficients $C(\psi), K(\psi)$ and
$S(\psi)$ when an extension of the principal Lie algebra $L_p$
occurs will also be calculated.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Small oscillations in infinite dimensional resonant systems }\\ \\
%author
{\bf Simone Paleari }\\
%affiliation
Universita' di Milano Bicocca, Italy \\
%e-mail
email:simone@mat.unimi.it
%coauthor
%{\bf C M Khalique }\\
%abstract
\\
We present some abstract result concerning the existence of
(cantor-like) families of periodic solutions in some nonlinear
PDEs; we also provide some applications to concrete examples, like
nonlinear string equations and nonlinear plate equations. These
results can be seen as a partial extension, to the infinite
dimensional and resonant setting, of the classical Lyapunov center
theorem. The techniques used involve a novel interaction between
averaging theory and the Lyapunov-Schmidt decomposition method.
References: D.Bambusi, S.Paleari --- Journal of Nonlinear
Sciences, 11, 69--87 (2001) S.Paleari, D.Bambusi, S.Cacciatori ---
ZAMP, 52, 1033--1052 (2001) D.Bambusi, S.Paleari --- CPAA, to
appear (2002).
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Dynamical approach in analysis:
functional equations, boundary problems for PDE, integral geometry
}\\ \\
%author
{\bf Boris Paneah }\\
%affiliation
Technion, Haifa, Israel \\
%e-mail
email:peter@techunix.technion.ac.il
%coauthor
%{\bf C M Khalique }\\
%abstract
\setcounter{equation}{0}
\\
This talk is devoted to solution of several new problems in the
three independent fields of Analysis : functional equations,
boundary problems for higher order hyperbolic differential
equations in bounded domains and integral geometry. All these
problems at first sight does not give even a merest hint about
some dynamical systems connected with them. Nevertheless it turned
out that when solving these quite different problems an essential
part of information can be obtained with the help of dynamical
methods. To apply these methods we introduce a semigroup $\phi$
of maps in an interval I generated by {\itshape two} maps
$\alpha$ and $\beta$ in I which are closely connected with
the problems in question. On a side the language of orbits of this
semigroup enables to formulate easily conditions (sometimes
necessary and sufficient)of solvability of these problems. On the
other side one of the most essential technical elements in the
proof of the main statements is a searching of some specific
attractors of the noncommutative dynamic system generated by
semigroup $\phi$ . Even in functional equations with their long
history our approach allows to obtain completely new results which
have nothing in common with what was known earlier. And this is
without any hard analytical work. In particular we solve at the
first time an {\itshape nonhomogeneous} Cauchy equation on a
curve and improve significantly what was known about homogeneous
one. Note that purely dynamical part of the talk seems to contain
setting of new problems related to noncommutative dynamical
systems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Stability of linear functional equations in Banach
modules }\\ \\
%author
{\bf Chun-Gil Park }\\
%affiliation
Chungnam National University, Korea \\
%e-mail
email:cgpark@math.cnu.ac.kr
%coauthor
%{\bf C M Khalique }\\
%abstract
\setcounter{equation}{0}
\\
Let $A$ be a unital $C^*$-algebra, and let ${_A}{\mathcal B}$ and
${_A}{\mathcal C}$ be left Banach $A$-modules. Let $f : {_A}
{\mathcal B} \to {_A}{\mathcal C}$ be a mapping with $f(0)=0$ such
that \begin{eqnarray}{l} p^{n}f(\frac{x_1+\cdots
+x_{p^{n}}}{p^{n}}) \vspace{0.1in}\nonumber\\ \qquad + (pk-p)
\sum_{i=1}^{p^{n-1}}f(\frac{x_{pi-p+1} + \cdots + x_{pi}}{p}) \vspace{0.1in}\nonumber\\
\qquad = k\ \sum_{i=1}^{p^n} f(\frac{x_{i}+ \cdots +x_{i+k-1}}{k})
\end{eqnarray} for all $x_1=x_{p^n+1}, \cdots,x_{k-1}=x_{p^n+k-1},
x_{k}, \cdots, x_{p^n} \in {_A}{\mathcal B}$. In this paper, we
prove the Hyers-Ulam-Rassias stability of the functional equation
(1) in Banach modules over a unital $C^*$-algebra.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
\noindent {\bf Homogenized transport by a spatiotemporal mean flow
with
small-scale periodic fluctuations }\\ \\
%author
{\bf Grigorios Pavliotis }\\
%affiliation
Department of Mathematical Sciences, RPI, USA \\
%e-mail
email:pavlig@rpi.edu \\
%coauthor
{\bf Peter R. Kramer }
%abstract
\\
The transport of a pollutant in the ocean or atmosphere is
influenced strongly by both the prevailing large-scale mean flow
structure and disordered turbulent motion prevalent on smaller
scales. To obtain some insight into the effects of turbulent
transport, various authors over the last decade have studied the
transport of passive tracers in model flows which have a periodic
structure. For such flows, one can develop a rigorous
homogenization theory to describe their effective transport on
large scales and long times. We will present an extension of these
homogenization studies to a class of model flows which consist of
a superposition of a large-scale mean flow with a small-scale
periodic structure (both of which can depend on space and time).
We rigorously derive homogenized equations for these models, in
which the mean flow and periodic structure are shown to interact
nonlinearly. Different kinds of homogenized equations can emerge,
depending upon the spatiotemporal relationships between the mean
flow and the periodic fluctuations and the relative magnitude of
the molecular diffusivity. It is shown that the small-scale
structure is responsible for an enhancement in the diffusivity as
well as for the presence of an effective drift, both of which are
functions of space and time, with values depending upon the local
properties of the mean flow. The physical manifestations of the
interaction between the mean flow and the periodic fluctuations
are illustrated through some simple examples.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On homoclinic tangencies in maps with entropy-carrying horseshoes }\\ \\
%author
{\bf Steven Pederson}\\
%affiliation
Morehouse College, USA \\
%e-mail
email:spederso@morehouse.edu
%coauthor
%{\bf Peter R. Kramer }\\
%abstract
\\
We study sufficient conditions under which a generic interval map
with non-constant topological entropy and an entropy-carrying
invariant set has a homoclinic tangency.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Global well-posedness and stability of a partial integro-differential equation }\\ \\
%author
{\bf Hana Petzeltova }\\
%affiliation
Institute of Mathematics AV CR, Czech Republic \\
%e-mail
email:petzelt@math.cas.cz
%coauthor
%{\bf Peter R. Kramer }\\
%abstract
\setcounter{equation}{0}
\\
In this contribution we consider the equation
\begin{eqnarray}
u_{tt}(t,x)=\int_0^ta(t-s)u_{txx}(s.x)ds+ \\
\int_0^t b(t-s)(g(u_x(s,x))_x ds s+f(t,x). \nonumber
\end{eqnarray} Here $a(t)$ and $b(t)$ are creep kernels which
behave like
$$a(t)\sim c_a t^{\alpha-1},\quad b(t)\sim c_b t^{\beta-1},$$
$$\quad \mbox{as } t\rightarrow 0,\ \alpha,\ \beta \in (0,1],\
\alpha<\beta.$$ The nonlinearity $g$ is supposed to be as
$g(r)\sim c r |r|^{m-1}$ at $\infty$, but its derivative is
allowed to change sign. We require the problem to be {\em
subcritical} in the sense that $ \frac{m-1}{m+1}< 2
\frac{\beta-\alpha}{1+\alpha}.$ Then the equation (1) is globally
well-posed and the $L_\infty$-norm of $u_x$ stays globally bounded
in time. Furthermore we show global asymptotic stability of the
trivial solution, provided the steady state problem $$g(u_x)_x=0$$
admits only the trivial solution and the corresponding
linearization is asymptotically stable. A multidimensional problem
will also be discussed.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Topological-numerical approach to the existence of periodic trajectories }\\ \\
%author
{\bf Pawel Pilarczyk }\\
%affiliation
Jagiellonian University and Georgia Tech, USA \\
%e-mail
email:Pawel.Pilarczyk@ii.uj.edu.pl
%coauthor
%{\bf Peter R. Kramer }\\
%abstract
\\
Periodic trajectories are objects of basic interest in the
investigation of dynamical properties of solutions to ODEs.
Although such orbits can often be easily detected in numerical
simulations, rigorous proof of their existence in concrete
examples may be difficult. In this talk we introduce a method
which may be used to prove the existence of a periodic solution to
an ODE if numerical simulations indicate the existence of a
hyperbolic periodic trajectory. The method is based on the Conley
Index theory for discrete and continuous dynamical systems. It
uses recently developed algorithms for efficient computation of
rigorous enclosures of solutions to the initial value problem over
a compact time interval, as well as algorithms for relative
homology computation of representable sets and maps. The goal of
the method is achieved by verifying the assumptions of a theorem
by McCord, Mischaikow and Mrozek, from which one can conclude the
existence of a periodic trajectory. A few examples of application
of this method to some concrete differential equations are
discussed. This method is a significant improvement of our
previously developed method which was valid only for attracting
periodic trajectories.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Remarks on Derrick's equation }\\ \\
%author
{\bf Lorenzo Pisani }\\
%affiliation
University of Bari, Italy \\
%e-mail
email:pisani@dm.uniba.it \\
%coauthor
{\bf Teresa D'Aprile }
%abstract
\\
In 1964 C.H. Derrick proved a well known non-existence result and
proposed several classes of field equation in order to avoid it.
In the static case one of these model equations takes the
following form $$- \Delta_p u +V'(u)=0,\eqno(1).$$ with $p>n$ and
$V(\xi)\geq V(0)=0$. If $u$ is a scalar field, in several cases
equation (1) has no nontrivial solutions. We have also studied a
vector-valued version of (1): $u:{\bf R}^n \to {\bf R}^{n+1}$ and
$V$ diverges for $\xi\to\bar\xi\ne 0$. In this situation the
fields having finite energy are characterized by a topological
constraint, the charge. So we can prove the existence of
nontrivial solutions; at least on solution is a true minimizer of
the energy.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Analytic Continuation {\it INTO THE FUTURE} }\\ \\
%author
{\bf David Pravica }\\
%affiliation
East Carolina University, USA \\
%e-mail
email:pravicad@mail.ecu.edu \\
%coauthor
{\bf The ECU Math Modelling Group East Carolina University }
%abstract
\\
Contrary to the title, analytic continuation into the future is
not possible, however a theory of advanced differential equations
is presented. To explain, consider the well-defined delay
differential equation, $y^\prime (t)=y(t/q)$, $y(0)=1$, for some
$q>1$. The solution $y(t)=exq(t)$ is an entire function that is
oscillatory for $t<0$. However, the advanced differential equation
$y^\prime (t)=y(qt)$ is not well-defined and the Taylor series has
0 radius of convergence. By generalizing the theory of Gevrey
series, formal solutions lead to analytic functions defined on an
open sector domain containing $t>0$ and with vertex at $t=0$. The
kernel of the corresponding Laplace transform is an infinite
series of exponential functions. Properties of this kernel are
discussed.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf The global dynamics of isothermal chemical systems with critical nonlinearity }\\ \\
%author
{\bf Yuanwei Qi }\\
%affiliation
HKUST, Hong Kong\\
%e-mail
email:maqi@ust.hk \\
%coauthor
{\bf Yi Li }
%abstract
\\
In this paper, we study the Cauchy problem of a cubic
autocatalytic chemical reaction system $$u_{1, t} = u_{1, xx} -
u^{ \alpha}_1 u^{\beta}_2, u_{2, t} = du_{2, xx} + u^{ \alpha}_1
u^{\beta}_2$$ with non-negative initial data, where the exponents
$ \alpha, \beta$ satisfy $ 1 < \alpha, \beta < 2$, $ \alpha +
\beta = 3$ and the constant $ d > 0$ is the Lewis number. Our
purpose is to study the global dynamics of solutions under mild
decay of initial data as $ | x| \rightarrow \infty$. We show the
exact large time behaviour of solutions which is universal.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Random representations for viscous fluids and their uniform approximation by ordinary differential equations }\\ \\
%author
{\bf Diego Rapoport }\\
%affiliation
University of Buenos Aires, Argentina \\
%e-mail
email: drapo@unq.edu.ar
%coauthor
%{\bf Yi Li }\\
%abstract
\\
We retake our contribution to the Kennesaw Conference on the
random representations for the Navier-Stokes (NSE) and kinematic
dynamo (KDE) equations, on smooth manifolds, from the point of
view of the random extension of the classical development method
in differential geometry due to E.Cartan, i.e. stochastic
differential geometry. We show that evolution equations on smooth
manifolds support a uniform approximation by ordinary differential
equations on the same manifolds. We apply these constructions to
derive the random representations for NS and KDE, and their
realization by o.d.e´s. In contrast with the usual classical
dynamical systems approach to turbulence, by projection into
chosen modes, this approach is low-dimensional, since the
approximations are defined on the same manifold as the
infinite-dimensional case. We further present a pure-noise
representation, in terms of which the deformation tensor is
incorporated into a new invariant diffusion tensor. This
construction is valid for any dimension other than one. We discuss
this latter approach in terms of the usual distinction between 2
and 3-dimensional fluids, and the analytical random
representations presented at Kennesaw.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Random representations for viscous fluids on smooth boundary manifolds }\\ \\
%author
{\bf Diego Rapoport }\\
%affiliation
University of Buenos Aires, Argentina \\
%e-mail
email: drapo@unq.edu.ar
%coauthor
%{\bf Yi Li }\\
%abstract
\\
We present the implicit analytical random representations for
Navier-Stokes equations, on smooth boundary manifolds, with
reflecting vorticity at the boundary. This is an invariant
extension of the random vortex method on 2d, in computational
fluid dynamics. We derive these representations by application of
the methodology of stochastic differential geometry. We specialize
to the flat case. We discuss, on continuing with our previous
talk, the pure-noise representation, and the uniform approximation
by classical dynamical systems. continuing with our previous talk,
the pure-noise representation, and the uniform approximation by
classical dynamical systems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Spectrally determined growth for creeping flow of the upper convected Maxwell fluid }\\ \\
%author
{\bf Michael Renardy }\\
%affiliation
Virginia Tech, USA \\
%e-mail
email: renardym@math.vt.edu
%coauthor
%{\bf Yi Li }\\
%abstract
\\
While it is well known that the stability of Newtonian flows is
determined by the eigenvalues of a linearized equation, there are
no general results of this type for non-Newtonian fluids. In this
lecture, it is shown that linear stability of steady creeping
flows of the upper convected Maxwell fluid is indeed determined by
the spectrum of the linearized operator. The proof uses the theory
of evolution semigroups over dynamical systems.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Existence of square integrable solutions for nonlinear differential equations }\\ \\
%author
{\bf Yuri Rogovchenko }\\
%affiliation
Eastern Mediterranean University, Turkey \\
%e-mail
email:yuri.rogovchenko@emu.edu.tr \\
%coauthor
{\bf Octavian Mustafa }
%abstract
\\
Using various approaches, we discuss existence and nonexistence of
square-integrable solutions for several classes of second and
higher order nonlinear differential equations.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Preservation of belief: A new view of deceptive signalling }\\ \\
%author
{\bf Jonathan Rowell }\\
%affiliation
Cornell University, USA \\
%e-mail
email: rowler@cam.cornell.edu
%coauthor
%{\bf Yi Li }\\
%abstract
\\
The current view of behavior-signalling dynamics, typically from a
game theoretic stand point, is that the introduction of deceptive
signalling inevitably destabilizes the entire signalling system.
The only exception to this general rule is when there is a
differential cost associated with the production of signals. This
is known as the Handicap Principle. However, the real world is
filled with examples of signalling systems that contain no
identifiable costs to deceptive signals, such as predator
warnings. As such, the Handicap Principle is not applicable, and
the general rule of deception-driven instability should hold. In
this talk, I will challenge this long-held view of signalling
dynamics and demonstrate that it is possible for belief to persist
without employing the Handicap Principle. I will present a dynamic
model in which selective pressures can maintain belief while also
permitting a limited level of deception in the communication
system.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On certain elliptic system with nonlinear self-cross diffusions }\\ \\
%author
{\bf Kimun Ryu }\\
%affiliation
Korea University,South Korea \\
%e-mail
email: ryukm@gauss.korea.ac.kr \\
%coauthor
{\bf Inkyung Ahn }
%abstract
\\
We discuss the coexistence of positive solutions to certain
elliptic systems with strongly coupled nonlinear self-cross
diffusions under Robin boundary conditions. Three different
interactions between two species are considered. The existence of
positive solutions to self-cross diffusive system can be expressed
in terms of the spectral property of differential operators of
nonlinear Schrodinger type which reflect the influence of the
domain and nonlinearity in the system. Therefore the coexistence
of positive steady-state of self-cross diffusive system shares
certain common features with the systems in which the diffusions
are constants.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Global stability results for N-dimensional Lotka-Volterra delay differential systems }\\ \\
%author
{\bf Yasuhisa Saito }\\
%affiliation
Osaka Prefecture University, Japan \\
%e-mail
email: saito@ms.osakafu-u.ac.jp
%coauthor
%{\bf Inkyung Ahn }\\
%abstract
\setcounter{equation}{0}
\\
This paper presents global stability results for $n-$dimensional
Lotka-Volterra systems with time delays. Consider \begin{equation}
\left\{ \begin{aligned} x'_1(t) = & \, x_1(t)[r_1-ax_1(t)+b_1
x_1(t-\tau_1) \\ & -b_2 x_2(t-\tau_2)] \\ x'_k(t) = & \,
x_k(t)[r_k-ax_k(t)+b_k x_{k-1}(t-\tau_{k-1})\\ & +b_1
x_k(t-\tau_k)-b_{k+1}x_{k+1}(t-\tau_{k+1})] \\ & \qquad \qquad
k=2, \cdots, n-1
\\ x'_n(t) = & \, x_n(t)[r_n-ax_n(t)+b_n
x_{n-1}(t-\tau_{n-1}) \\ & +b_1x_n(t-\tau_n)]
\end{aligned}\right.
\end{equation} where $a$, $b_i$, $\tau_i$ $(i = 1, 2, \cdots, n)$
are constants with $a >0$ and $\tau_i \geq 0$. It is shown that
the positive equilibrium of (1) is globally asymptotically stable
for all delays $\tau_i \geq 0$ $(i = 1, 2, \cdots, n)$, if
$\sqrt{b_1^2 + b_2^2 + \cdots + b_n^2} \le a$ holds. This result,
for $n=2$, corresponds to a result which gives a necessary and
sufficient condition for global stability (see [Saito, Hara, and
Ma, {\it J. Math. Anal. Appl.} {\bf 236} (1999), 534-556]). The
global stability result for (1) is also extended to an
$n-$dimensional Lotka-Volterra system with distributed delays.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Homogeneous $C^*$-algebras generated by idempotents }\\ \\
%author
{\bf Mikhail Shchukin }\\
%affiliation
Belarusian State University, Belarus \\
%e-mail
email: ms@appsys.net
%coauthor
%{\bf Inkyung Ahn }\\
%abstract
\\
Every homogeneous $C^{*}$-algebra corresponds to the algebraic
fibre bundle. $C^{*}$-algebra is called non-trivial if the
corresponding algebraic fibre bundle is non-trivial. All
$C^{*}$-algebras generated by idempotents that studied before
corresponded to the trivial algebraic fibre bundles. In the work
was showed that every homogeneous separable non-commutative
$C^{*}$-algebra can be generated by three idempotents. It follows
from here that we need to study the topology properties of
$C^{*}$-algebras generated by idempotents to describe such
algebras properly. Also in the work we found the minimal number of
idempotent generators for every homogeneous $C^*$-algebra $A$ with
the set of maximal ideals homeomorphic to the sphere $S^2$.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Contact with adhesion of a membrane }\\ \\
%author
{\bf Meir Shillor }\\
%affiliation
Oakland University, USA \\
%e-mail
email: shillor@oakland.edu \\
%coauthor
{\bf K. T. Andrews }
%abstract
\\
We describe a new model for the dynamic adhesive contact between a
membrane and a rigid obstacle. We present the classical and
variational formulations of the model, state the existence and
uniqueness result and indicate the ideas of the proof. Then, we
shortly describe the quasistatic problem, which is an interesting
version of the classical obstacle problem for the Laplace
operator. In this problem, in addition to existence and
uniqueness, we obtained error estimates on the numerical
approximations and we present some numerical simulations.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents }\\ \\
%author
{\bf Elves Silva }\\
%affiliation
Universidade de Brasilia, Brazil \\
%e-mail
email: elves@mat.unb.br
%coauthor
%{\bf K. T. Andrews }\\
%abstract
\\
The main results of this paper establish, via the variational
method, the multiplicity of solutions for quasilinear elliptic
problems involving critical Sobolev exponents under the presence
of symmetry. The concentration-compactness principle allows to
prove that the Palais-Smale condition is satisfied below a certain
level.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Heaviside analytical solutions of nonlocal wave equations in inhomogeneous, dispersive media }\\ \\
%author
{\bf Valentino Simpao }\\
%affiliation
Mathematical Consultant Services, USA \\
%e-mail
email: mcs007@muhlon.com
%coauthor
%{\bf K. T. Andrews }\\
%abstract
\\
Heaviside operational methods are used to obtain analytical
solutions of various nonlocal wave equations, arising in the
context of wave propagation studies in inhomogeneous, dispersive
media. The results obtain for various auxiliary condition
scenarios in the direct and inverse problem cases. For example, by
employing the results in conjunction with meromorphic function
construction techniques, various material properties[e.g., the
phenomenological inhomogeneous dispersive index of refraction] may
be calculated explicitly from experimental data.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Heaviside realisation of formal differential operators: quantum implications }\\ \\
%author
{\bf Valentino Simpao }\\
%affiliation
Mathematical Consultant Services, USA \\
%e-mail
email: mcs007@muhlon.com
%coauthor
%{\bf K. T. Andrews }\\
%abstract
\\
By means of Heaviside operational schemes, formal differential
operators[i.e., Arbitrary functions of the derivative symbols] are
realised as convolution operators on the prescribed class of
operands. This connection allows an alternative framework for
momentum/energy dependence in quantum operators. As an example,
the foundation of a new first-order momentum/energy quantum
relativistic wave operator is presented.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Well-posedness of partial difference equations }\\ \\
%author
{\bf Pavel E. Sobolevskii }\\
%affiliation
Institute of Mathematics,Israel \\
%e-mail
email: pavels@math.huji.ac.il
%coauthor
%{\bf K. T. Andrews }\\
%abstract
\\
It is well-known in the theory of differential equations that the
coercive inequality approach appeared to be very useful for the
investigation of general boundary value problems for elliptic and
parabolic differential equations. The coercive inequalities hold
also for various difference analogous of such problems. The main
role of the coercive inequalities for difference problems lies in
that they present a special type of stability, which permits the
existence of exact, i.e. two-sides estimates of the rate of
convergence approximate solutions (with respect to the
corresponding coercive norms). As it turns out, there are
situations when the difference problems are well-posed, but their
limit variants - differential problems - are ill-posed.
Established exact (with respect to steps of difference schemes)
coercive inequalities gives the possibility of finding (almost)
exact estimates of convergence rate of approximate solutions in
cases, when differential problems are ill-posed.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Robust decentralized control of formation flight }\\ \\
%author
{\bf Dusan Stipanovic }\\
%affiliation
Stanford University, USA \\
%e-mail
email:dusko@stanford.edu \\
%coauthor
{\bf Haitham A. Hindi } and {\bf Rodney Teo}
%abstract
\\
In this talk we consider robust decentralized control of formation
flight for Unmanned Aerial Vehicles (UAVs). Each UAV is modeled as
a set of nonlinear differential equations. After applying input to
state feedback linearization, we obtain a dynamic system which is
represented as a perturbed linear system where the perturbation is
a nonlinear sector bounded function. The dynamic model of the
formation is treated as an interconnected system where the
subsystems are local dynamic models for each UAV. Our goal is to
robustly stabilize the system using decentralized control, that
is, control under information structure constraints. We are
interested in local stability, that is, we consider the
stabilization problem in the vicinity of the nominal regime of
operation for the formation of UAVs. We derive conditions for
local stability: it is known that the sector which bounds the
perturbation for local stability is in general smaller than the
sector that bounds the perturbation for global stability, and
therefore the results are less restrictive. The stabilization
problem is formulated as a convex optimization problem in terms of
Linear Matrix Inequalities (LMIs). This feature offers an
efficient way for design of local controllers that stabilize the
overall system.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Traveling waves in two phase fluid flow }\\ \\
%author
{\bf Jianzhong Su }\\
%affiliation
University of Texas at Arlington, USA \\
%e-mail
email:su@uta.edu \\
%coauthor
{\bf B. Tran }
%abstract
\\
Fluid fingering phenomena arise from various physical problems
such as oil recovery and phase transition. The interface between
the two different fluids evolves according to the physical laws
and its motion is governed by a partial differential equation with
free boundaries. In this talk, we will first provide some physical
background of fluid fingering problems, then we will discuss the
Saffman-Taylor finger solutions of Hele-Shaw equation. These
finger solutions are travelling wave solutions whose finger-shaped
interfaces are moving along a certain direction at a constant
speed. The existence of symmetric 3-dimensional finger solutions
in is shown through a fixed point argument of the boundary
integral equation. The process of finger splitting is also shown
through a family of finger solutions with cusp angles at their
tips.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf The entropy topology and true laminations for complex H\'{e}non maps }\\ \\
%author
{\bf Meiyu Su }\\
%affiliation
Long Island University, USA \\
%e-mail
email:msu@liu.edu
%coauthor
%{\bf B. Tran }\\
%abstract
\\
The simplest holomorphic dynamical systems which display
interesting behavior are polynomials and rational maps of $C$. The
simplest invertible holomorphic dynamical systems with interesting
dynamical behavior are probably the polynomial diffeomorphisms of
$C\sp 2$ that are conjugate to the generalized H\`{e}non maps.
Many studies and interesting results have been developed. For
example, Bedfold, Lyubich, and Smillie showed that there exists a
unique invariant and ergodic measure $\mu$ of maximal entropy with
a support containing the Julia set $J$ which has the projective
measures to the stable and unstable manifolds given by the
positive $(1,1)$-currents $\mu^{\pm}=dd^cG^{\pm}$, where $G^{\pm}$
are the plorisubharmonic Green functions of the sets $K^{\pm}$
(see~\cite{kn:bls1}, \cite{kn:bls2}, \cite{kn:bls3},
\cite{kn:bls4}, \cite{kn:bls5}, \cite{kn:ho}, and \cite{kn:wu},
etc.). We, in this paper, are specially interested in the
lamination structure for the stable and unstable partitions
composed of global stable and unstable manifolds. We prove that
for every generalized H\'{e}non map $f$ of $C\sp 2$ of degree
$d>1$, there are true $f$-invariant expanding and contracting
solenoidal laminations--injected into $C\sp 2$, which fill up the
measure $\mu$ and whose leaves are conformally isomorphic to the
complex plane $C$. Holonomy maps preserve the transversal measures
and hence these laminations yield measured solenoidal Riemann
surface laminations describing the currents $\mu+$ and $\mu^-$
(See \cite{kn:su} for the general definition of measured
solenoidal Riemann surface laminations). The new ingredient is a
$\sigma$-finite topology on transversals defined by logarithms of
measures obtained by conditioning $\mu$, the `entropy topology'.
\begin{thebibliography}{Dillo 83} \bibitem{kn:bls1} E Bedford and
J Smillie, {{\it Polynomial diffeomorphisms of $C\sp 2$: currents,
equilibrium measure and hyperbolicity}, Invent. Math., {\bf 103}
(1991) 69-99.} \bibitem{kn:bls2} E Bedford and J Smillie, { { \it
Polynomial Diffeomorphisms of $C\sp 2$. II: Stable manifolds and
recurrence}, Indiana Univ. Math. J., {\bf 40} (1991) 657-679.}
\bibitem{kn:bls3} E Bedford and J Smillie, { { \it Polynomial
diffeomorphisms of $C\sp 2$. III: Ergodicity, exponents and
entropy of the equilibrium measure}, Math. Ann., {\bf 249} (1992)
395-420.}
\bibitem{kn:bls4} E Bedford, M Lyubich, and J Smillie, {{\it
Polynomial diffeomorphisms of $C\sp 2$ IV: The measure of maximal
entropy and laminar currents}, Invent. Math., {\bf 112} (1993)
77-125.} \bibitem{kn:bls5} E Bedford and J Smillie, { { \it
Polynomial diffeomorphisms of $C\sp 2$. V: Critical points,
Lyapunov exponents and solenoidal mappings}, Preprint, May 10,
1995.}
\bibitem{kn:ho} J. Hubbard $\&$ R. Oberste-Vorth, {{\it H\'{e}non
mappings in the complex domain I $\&$ II}, Preprint, IHES, Paris.}
\bibitem{kn:su} M Su, { {\it Measured solenoidal Riemann surfaces
and halomorphic dynamics}, J Diff. Geo., {\bf 47} (1997) 170-195.}
\bibitem{kn:wu} H Wu, { {\it Complex stable manifold theorems and
generalized complex H\'{e}non mappings}, Thesis, ISAS/SISSA
(1991-92).} \end{thebibliography}
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Existence and stability of travelling curved fronts in the Allen-Cahn equation }\\ \\
%author
{\bf Masaharu Taniguchi }\\
%affiliation
Tokyo Institute of Technology, Japan \\
%e-mail
email:masaharu.taniguchi@is.titec.ac.jp
%coauthor
%{\bf B. Tran }\\
%abstract
\\
This is a joint work with Dr. H. Ninomiya of Ryukoku University.
We study stability of travelling curved fronts for the Allen-Cahn
equation in the two-dimensional Euclidean space. First, we study
the existence and stability of travelling curved fronts in a
generalized mean curvature flow. Then constructing supersolutions
and subsolutions, we study the existence, uniqueness and stability
of travelling curved fronts for the Allen-Cahn equation.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Cellular neural networks, mosaic patterns and spatial chaos }\\ \\
%author
{\bf Larry Turyn }\\
%affiliation
Wright State University, USA \\
%e-mail
email:larry.turyn@wright.edu
%coauthor
%{\bf B. Tran }\\
%abstract
\\
We consider a Cellular Neural Network (CNN), with a bias term, on
the integer lattice $\mathbb Z^2$ in the plane. Three kinds of
space-dependent, asymmetric coupling (template) appropriate for
CNN in the hexagonal lattice on are studied. We characterize the
mosaic patterns and study their spatial entropy. It appears that
for this problem, asymmetry of the template has a more robust
effect on the spatial entropy than does the sign of one particular
parameter in the templates.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf On the solvability of periodically perturbed linear equations at resonance in the principal eigenvalue }\\ \\
%author
{\bf Antonio Urena }\\
%affiliation
Universidad de Granada, Spain\\
%e-mail
email:ajurena@ugr.es
%coauthor
%{\bf B. Tran }\\
%abstract
\setcounter{equation}{0}
\\
In this talk we deal with selfadjoint, elliptic boundary value
problems of the type \begin{equation}\label{maineq} \begin{cases}
-div(A(x)\nabla u)-b(x) u+g(u)=h_a(x),& x\in\Omega\\
u(x)=0,& x\in\partial\Omega \end{cases}
\end{equation} Where $ h_a(x)=\tilde h(x)+a\varphi(x)$ and $\Omega$ is a bounded, convex, smooth domain
in $\mathbb R^N$, $(N\geq 2)$, and $A:\Omega\to\mathcal
M_N(\mathbb R)$, $b:\Omega\to\mathbb R$ are $C^\infty$ mappings on
$\bar\Omega$. $A(x)$ is assumed to be symmetric and positive
definite for any $x\in\Omega$ and we impose the linear operator
$\mathcal L(u):=-div(A(x)\nabla u)-b(x) u,\ u\in H_0^1(\Omega)$,
to be uniformly elliptic and resonant on its principal eigenvalue.
Let $\varphi\in C^\infty(\bar\Omega)$ be a generator of
$\ker\mathcal L$. Both $\tilde h:\Omega\to\mathbb R$ and
$g:\mathbb R\to\mathbb R$ are assumed to be H\"{o}lder continuous
functions with $g\not\equiv 0$ periodic and with zero mean
-observe that this latter hypothesis is not restrictive-,
$\int_\Omega\tilde h(x)\varphi(x)dx=0$, and we write, for any
$a\in\mathbb R$, $h_a:=\tilde h+a\varphi$. This problem has a long
history that goes back to the pioneering work of Dancer
\cite{dan}. Here, the {\em ordinary} boundary value problem
(\ref{maineq}) $[N=1]$, was first explored in detail for the case
$g(u)=\Lambda\sin(u)$. In this framework, it was shown (Theorem 4,
pp. 182) that, for any given $\tilde h$, there exists
$\epsilon_0=\epsilon_0(\tilde h)>0$ such that problem
$(\ref{maineq})$ has solution for any $|a|\leq\epsilon_0$.
Further, the problem was seen to have infinitely many solutions
for $a=0$. Many subsequent efforts were devoted to extend the
Theorem above to general periodic nonlinearities $g$ and higher
dimensions $N$, (see \cite{cos-jeg-sch-sch, sch-sch0,
sch-sch1,sch-sch2,war}), and nowadays the problem has been well
understood in the cases of space dimension $1$ or $2$. However, a
basic question like the following: {\em Do Dancer's results remain
true for space dimension $N\geq 3$?} \noindent seems to have
remained unsolved up to date (see \cite{cos-jeg-sch-sch,sch-sch2}
for related numerical experiments). We generalize Dancer's results
for space dimension $N=3$. Most of them do not remain true for
higher dimensions. \begin{thebibliography}{8}
\bibitem{cos-jeg-sch-sch} D. Costa, H. Jeggle, R. Schaaf, K.
Schmitt, {\em Oscillatory perturbations of linear problems at
resonance}, Results in Math., Vol. {\bf 14} (1998), pp. 275-287.
\bibitem{dan} E. N. Dancer, {\em On the use of asymptotics in
nonlinear boundary value problems}, Ann. Mat. Pura Appl., Vol.
{\bf 131} (1982), pp. 167-185. \bibitem{sch-sch0} R. Schaaf, K.
Schmitt, {\em A class of nonlinear Sturm-Liouville problems with
infinitely many solutions}, Trans. Amer. Math. Soc., Vol. {\bf
306}, No. 2 (1998), pp. 853-859. \bibitem{sch-sch1} R. Schaaf, K.
Schmitt, {\em Periodic perbations of linear problems at resonance
on convex domains}, Rocky Mount. J. Math., Vol. {\bf 20}, 4
(1990), pp. 1119-1131. \bibitem{sch-sch2} R. Schaaf, K. Schmitt,
{\em On the number of solutions of semilinear elliptic problems:
Some numerical experiments}, Lectures in Appl. Math. (AMS), Vol.
{\bf 26} (1990), pp. 541-559. \bibitem{war} J.R. Ward, {\em A
boundary value problem with a periodic nonlinearity}, Nonl. Anal.,
Vol. {\bf 10}, No. 2, (1986), pp. 207-213. \end{thebibliography}.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Resonance interaction between charged particles and monochromatic waves }\\ \\
%author
{\bf Dmitri Vainchtein }\\
%affiliation
University of California Santa Barbara, USA \\
%e-mail
email:dmitri@engineering.ucsb.edu \\
%coauthor
{\bf Anatoly I.Neishtadt } and {\bf Lev M.Zelenyi }
%abstract
\\
We investigate the influence of monochromatic electromagnetic wave
on the charged particles' motion in the configurations
characteristic of the magnetic field reversals (e.g. in the
Earth's magnetotail). The particles dynamics is described by a
Hamiltonian system with two and a half degrees of freedom. The
smallness of some dimensionless physical parameters allows us to
solve this problem in the frame of the perturbation theory and
reduces the problem of resonant wave-particle interaction to the
analysis of slow passages of a particle through the resonance. We
show that the resonant processes result in chaotization of
particles and also may lead to the significant acceleration of the
particles and even, for some values of the parameters of a wave,
to the almost free acceleration. We calculate the characteristic
times of the chaotization of particles due to resonant effects and
separatrix crossings and discuss the relative importance of these
phenomena.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Travelling waves for solid-gas reaction-diffusion systems under conditions of constant gas flux }\\ \\
%author
{\bf Stella Vernier-Piro }\\
%affiliation
University of Cagliari, Italy \\
%e-mail
email: svernier@unica.it \\
%coauthor
{\bf Cornelis van der Mee }
%abstract
\\
We study travelling wave solutions of a model describing the
conversion of a porous solid as it reacts irreversibly with a gas
moving through its pores. The model consists of a coupled
parabolic system for the concentrations $C$ of the gas and $S$ of
the solid as functions of position and time. The coupling occurs
only in a separated nonlinear term of the form $F(S)G(C)$, while
the diffusion and flux appear only in the equation for $C$. We
prove the existence of a unique travelling wave profile and give
conditions for the existence of conversion and penetration fronts.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Longterm dynamics for equations modelling nonuniform deformable bodies with heavy rigid attachments }\\ \\
%author
{\bf Patrick Wilber }\\
%affiliation
Texas A $\&$ M University, USA \\
%e-mail
email: jwilber@math.tamu.edu
%coauthor
%{\bf Cornelis van der Mee }\\
%abstract
\\
An important basic problem in solid mechanics is describing the
forced motions of nonuniform deformable bodies with heavy rigid
attachments. I present recent research on the longterm dynamics of
degenerate nonlinear partial differential equations that govern
such motions. My main result is proving that the dynamical system
generated by a discretization of the governing equations has an
absorbing ball whose size is independent of the order of the
discretization. This result implies the existence of an absorbing
ball for the infinite-dimensional dynamical system corresponding
to the original degenerate partial differential equations and
thereby serves as a critical step for establishing the existence
of global attractors. I also address the interesting mechanical
question of how nonuniformity complicates the longterm dynamics of
the coupled mechanical systems I consider.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Boundary conditions for hyperbolic relaxation problems }\\ \\
%author
{\bf Wen-Qing Xu }\\
%affiliation
UMass Amherst, USA \\
%e-mail
email: xu@math.umass.edu
%coauthor
%{\bf Cornelis van der Mee }\\
%abstract
\\
We study the initial-boundary value problem (IBVP) for the Jin-Xin
relaxation model in arbitrary space dimensions. The main interest
is to understand the boundary layer behavior of the solution of
the relaxation IBVP and to establish its asymptotic convergence to
the underlying equilibrium system of hyperbolic conservation laws
in the limit of small relaxation rate. The key is to determine the
appropriate structural stability conditions, particularly, those
on boundary conditions such that the relaxation IBVP is stiffly
well-posed or uniformly well-posed independent of the relaxation
rate. We derive, in an explicit and easily checkable form, a stiff
version of the classical Uniform Kreiss Condition, and hence
referred to as Stiff Kreiss Condition. The Stiff Kreiss Condition
is shown to be necessary and sufficient for the stiff
well-posedness and the asymptotic convergence of the relaxation
IBVP.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Multiple solutions of superquadratic second order dynamical systems }\\ \\
%author
{\bf Xiangjin Xu }\\
%affiliation
Johns Hopkins University, USA \\
%e-mail
email:xxu@math.jhu.edu
%coauthor
%{\bf Cornelis van der Mee }\\
%abstract
\\
In this paper we consider the second order dynamical system $$
A\ddot{x}=-\nabla V(x)$$ where $x\in{\mathbb R}^N$, $A$ is a
nonsingular $N\times N$ symmetric matrix, but not necessary to be
positive definite. the existence of infinitely many distinct
$T$-periodic solutions for the superquadratic second order
dynamical system and some perturbed systems is proved.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf The robustness of the schistosomiasis transmission can be explained by immunity }\\ \\
%author
{\bf Hyun Yang }\\
%affiliation
UNICAMP, Brazil \\
%e-mail
email:hyunyang@ime.unicamp.br
%coauthor
%{\bf Cornelis van der Mee }\\
%abstract
\\
In order to analyze the robustness of the overall transmission of
schistosomiasis, we compare mathematical modelling considering the
acquired immunity and the contacts pattern with infested water.
The model's assumptions are the simplest possible to enhance the
differences between these two hypotheses. With respect to the
human host, it is assumed the mounting of an immune response after
elapsing a fixed period of time L from the first infection, which
is partially protective and never lost. With respect to the
contacts pattern with infested water, it is assumed a decreasing
age-related function constrained to the infective parameter. The
results obtained from both models are compared between them and
with the results obtained from a purely random process (Poisson)
model, which is taken as the basic model. The robustness is
assessed by analyzing the range of the variation of two
environment related parameters.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf Analysis on dynamic investment strategy }\\ \\
%author
{\bf Ming Yang }\\
%affiliation
Huazhong University of Science and Technology, China \\
%e-mail
email:yangming$\_$hust@263.net \\
%coauthor
{\bf Li Chulin }
%abstract
\\
The article provides a framework of valuing of investment
opportunity (VIO) on a research and development $R\&D$ project by stochastic
differential equation. The stochastic dynamic setting presents a situation
in which a firm has an investment opportunity with a competitive rival firm.
On one hand, uncertainty and irreversibility imply an option value of
waiting and therefore greater hesitancy in each firm's investment decisions.
On the other hand, both firms have stochastic time completing the project;
the fear of preemption by rival suggests acting quickly. The firms must
trade-off between them. The paper analyzes VIO and the optimal investment rule using real
option approach combined with dynamic game theory. The VIO is modeled as a
solution of stochastic differential equation (SDE) with free boundary P*.
The optimal investment rule is exerting investment when P>P* and waiting
when P 0$ we construct an
approximate inertial manifold (AIM) which contains all the steady
states of the system and has an attractive neighborhood of
thickness $\eta.$ The dependence of AIMs on the delay time is
investigated.
\[ \longrightarrow \infty \diamond \infty \longleftarrow \]
\vskip .1in
% ----------------------------------------------------------------
%\title
\noindent {\bf The two phase problem for viscous flows: On the existence and regularity }\\ \\
%author
{\bf Rodolfo Salvi }\\
%affiliation
Department of Mathematics - Politecnico di Milano, Italy \\
%e-mail
email: rodsal@mate.polimi.it
%coauthor
%{\bf Inkyung Ahn }\\
%abstract
\\
Flow problems with moving boundary are very interesting phenomena
which are often found in the nature and engineering applications.
These problems consist of the free surface flows, flows with free
interface and flows in a moving domain with an assigned law. In
this paper we are concerned with the two phase flow with free
interface. \medskip Let $ \Omega $ be a bounded domain of $ R^3 $
with the boundary $ \partial \Omega $. The domain $ \Omega $
consists of two time-dependent subdomains $ \Omega^i(t) , i =1,2
,$ which are filled by the fluids (1) and (2) ( also the vacuum),
respectively:
$$ \Omega = \Omega^1(t) \cup \Omega^2(t) $$ for $ 0\leq t < T.$
\noindent The interface $ \Gamma(t) $ between two fluids is
expressed by
$$ \Gamma(t) = (\partial \Omega^1(t) \cup \partial \Omega^2(t))\setminus \partial \Omega $$
for $ 0 \leq t < T$. On the interface $ \Gamma(t) $, we assume the
standard transmission conditions. Using the characteristic
function of $ \Omega^1 $, we transform the two phase free problem
into one fluid flow in a cylindrical domain $ Q_T = \Omega \times
(0,T) $. We prove the existence of a weak solution and the local
existence of a strong solution.