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Series: CDSNS Colloquium

Gas-liquid transition is one of the most basic problem to study in
equilibrium phase transitions. In the pressure-temperature phase
diagram, the gas-liquid coexistence curve terminates at a critical point
C, also called the Andrews critical point. It is, however, still an
open question why the Andrews critical point exists and what is the
order of transition going beyond this critical point. To answer this
basic question, using the Landau's mean field theory and the Le
Chatelier principle, a dynamic model for the gas-liquid phase
transitions is established. With this dynamic model, we are able to
derive a theory on the Andrews critical point C: 1) the critical point
is a switching point where the phase transition changes from the first
order with latent heat to the third order, and 2) the liquid-gas phase
transition going beyond Andrews point is of the third order. This
clearly explains why it is hard to observe the liquid-gas phase
transition going beyond the Andrews point. In addition, the study
suggest an asymmetry principle of fluctuations, which appears also in
phase transitions in ferromagnetic systems.
The analysis is based on the dynamic transition theory we have developed
recently with the philosophy to search the complete set of transition
states. The theory has been applied to a wide range of nonlinear
problems. A brief introduction for this theory will be presented as
well. This is joint with Tian Ma.

Series: CDSNS Colloquium

We consider a shift transformation and a Gibbs measure and
estimate the drop in entropy caused by deleting an arbitrarily small
(cylinder) set. This extends a result of Lind. We also estimate the speed
at which the Gibbs measure escapes into the set, which relates to recent
work of Bunimovich-Yurchenko and Keller-Liverani. This is joint with
Andrew Ferguson.

Series: CDSNS Colloquium

Based on a sequence of discretized American option price processes under the multinomial model proposed by Maller, Solomon and Szimayer (2006), the sequence converges to the counterpart under the original L\'{e}vy process in distribution for almost all time. We prove a weak convergence in this case for American put options for all time. By adapting Skorokhod representation theorem, a new sequence of approximating processes with the same laws with the multinomial tree model defined by Maller, Solomon and Szimayer (2006) is obtained. The new sequence of approximating processes satisfies Aldous' criterion for tightness. And, the sequence of filtrations generated by the new approximation converges to the filtration generated by the representative of L\'{e}vy process weakly. By using results of Coquet and Toldo (2007), we give a complete proof of the weak convergence for the approximation of American put option prices for all time.

Series: CDSNS Colloquium

There are many interesting patterns observed in activator-inhibitor systems.
A well-known model is the FitzHugh-Nagumo system. In conjunction with
calculus of variations, there is a close relation between the stability
of a steady state and its relative Morse index. We give a sufficient
condition in diffusivity for the existence of standing wavefronts joining
with Turing patterns.

Series: CDSNS Colloquium

In this lecture, I will discuss a class of multidimensional maps with one nonlinearity,
often called discrete-time Lurie systems. In the 2-D case, this class includes Lozi map and
Belykh map.
I will derive rigorous conditions for the multidimensional maps to have a generalized
hyperbolic attractor
in the sense of Bunimovich-Pesin. Then, I will show how these chaotic maps can be embedded
into the flow,
and I will give specific examples of three-dimensional piece-wise linear ODEs, generating
this class of hyperbolic attractors.

Series: CDSNS Colloquium

We improved a computational model of leukemia development from stem cells to terminally differentiated cells by replacing the probabilistic, agent-based model of Roeder et al. (2006) with a system of deterministic, difference equations. The model is based on the relatively recent theory that cancer originates from cancer stem cells that reside in a microenvironment, called the stem cell niche. Depending on a stem cell’s location within the stem cell niche, the stem cell may remain quiescent or begin proliferating. This emerging theory states that leukemia (and potentially other cancers) is caused by the misregulation of the cycle ofproliferation and quiescence within the stem cell niche.Unlike the original agent-based model, which required seven hours per simulation, our model could be numerically evaluated in less than five minutes. The results of our numerical simulations showed that our model closely replicated the average behavior of the original agent-based model. We then extended our difference equation model to a system of age-structured partial differential equations (PDEs), which also reproduced the behavior of the Roeder model. Furthermore, the PDE model was amenable to mathematical stability analysis, which revealed three modes of behavior: stability at 0 (cancer dies out), stability at a nonzero equilibrium (a scenario akin to chronic myelogenous leukemia), and periodic oscillations (a scenario akin to accelerated myelogenous leukemia).The PDE formulation not only makes the model suitable for analysis, but also provides an effective mathematical framework for extending the model to include other aspects, such as the spatial distribution of stem cells within the niche.

Series: CDSNS Colloquium

We introduce a change of coordinates allowing to capture in a fixed reference frame the profile of travelling wave solutions for nonlinear parabolic equations. For nonlinearities of bistable type the asymptotic travelling wave profile becomes an equilibrium state for the augmented reaction-diffusion equation. In the new equation, the profile of the asymptotic travelling front and its propagation speed emerge simultaneously as time evolves. Several numerical experiments illustrate the effciency of the method.

Series: CDSNS Colloquium

We present a new theory on Hamiltonian PDE. The linear theory
solves an old spectral problem on boundedness of L infinity
norm of the eigenfunctions of the Schroedinger operator on the
2-torus. The nonlinear theory develops Fourier geometry, eliminates the convexity
condition on the (infinite dimension)
Hamiltonian and is natural for PDE.

Series: CDSNS Colloquium

Many dynamical systems may be subject to stochastic excitations, so to find an efficient method to analyze the stochastic system is very important. As for the complexity of the stochastic systems, there are not any omnipotent methods. What I would like to present here is a brief introduction to quasi-non-integrable Hamiltonian systems and stochastic averaging method for analyzing certain stochastic dynamical systems. At the end, I will give some examples of the method.

Series: CDSNS Colloquium

Stable sets and unstable sets of a dynamical system with positive entropy
are investigated. It is shown that in any invertible system with positive entropy,
there is a measure-theoretically ?rather big? set such that for any point from the
set, the intersection of the closure of the stable set and the closure of the
unstable set of the point has positive entropy.
Moreover, for several kinds of specific systems, the lower bound of Hausdorff
dimension of these sets is estimated. Particularly the lower bound of the Hausdorff
dimension of such sets appearing in a positive entropy diffeomorphism on a smooth
Riemannian manifold is given in terms of the metric entropy and of Lyapunov exponent.