Nonlinear Systems Seminar
Michael Kovalyov, University of Alberta, Canada

In the talk I discuss a class of rather interesting solutions of integrable systems that I call harmonic breathers. These solutions arise as nonlinear analogues of exp[i(kx - ω(k)t))], in the sense that a huge class of solutions of integrable systems corresponding to the continuous spectrum of the associated Lax pair can be decomposed into these solutions just like solutions of linear evolution PDEs are decomposable into exp[i(kx - ω(k)t)] by means of the Fourier transform. Unlike the Fourier decomposition, the decomposition into harmonic breathers though is nonlinear with a whole bunch of new properties. Harmonic breathers also provide an example of wave-particle duality different from that of Heisenberg. Last but not least harmonic breathers exhibit behavior observed in real life situations.

Stochastic Systems Seminar

Forest simulation models have been proven remarkably effective at capturing the dynamics of real forests. In mathematical terms, individual-based simulators are spatial stochastic processes that predict properties of populations and communities by simulating the fate of every plant throughout its life cycle. Unfortunately, non-linear spatial stochastic processes are notoriously intractable, which limits the usefulness of forest simulators to basic scientists, and, also, they require too much computer power to be used at large scale, such as in global models; one cannot simulate every tree on the Earth. To solve the twin problems of computational intensiveness and mathematical intractability, what is needed is a way to predict a forest's community dynamics using only individual-level information, but without simulating every plant. This requires so-called macroscopic equations for variables of interest to ecologists, such as the mean density and size structure of each species and how these change though time.

In physical systems, macroscopic equations for the dynamics of fluids can be derived from stochastic models of the random collisions and transformations of individual molecules. Using similar approach we have developed a new spatial individual-based forest model that is based on a new approximation for the plasticity of crown shape. Its structure allows us to derive an accurate approximation to the individual-based model for the means of the stochastic process in a forest simulator that predicts the mean densities and size structures in the simulator using the same parameter values and functional forms, and, also, is analytically tractable. The approximation is represented by a system of Von Foerster partial differential equations coupled with an integral equation that we call the Perfect Plasticity Approximation (PPA). We have derived a series of analytical results including equilibrium abundances for trees of different crown shapes, stability conditions, transient behaviors, and coexistence conditions.

For more information please contact:

Nicolai Strigul
Research Assistant Professor
Phone: 609.258.437
Fax: 201.216.8321

nstrigul@stevens.edu

Stochastic Systems Seminar
Andreas Hamel, Princeton University

Duality for extended real-valued convex functions is a well-studied, even classical subject based on works of Fenchel, Moreau, Rockafellar, among many others. A corresponding satisfying theory for functions mapping into the power set of a partially ordered locally convex space is still missing. Such a theory seems to be very desirable since it has already been observed e.g. by Luc in 1989 that the dual of a convex vector optimization problem is set-valued in nature. Moreover, the concept of convex set-valued risk measures has been defined recently in financial mathematics which asks for a corresponding dual representation theory.

We shall present a duality concept that is based on a new notion of a ne minorants for set-valued functions and show that almost every concept (e.g. properness, sublinearity, conjugates, inf-convolution) and result (e.g. biconjugation and Fenchel-Rockafellar duality theorems) known in the scalar case can be formulated within this new framework.

A special feature of the methodology is that proofs do not rely on the corresponding scalar theory as in almost every duality theory for vector optimization problems. On the other hand, every main result can equivalently be expressed as a result for a family of scalar problems.

Finally, we shall show the theory at work when applied to linear vector optimization problems and to set-valued risk measures.

Stochastic Systems Seminar

This talk addresses the issue of cooperation versus selfishness in wireless networks, and discusses possible performance gains, and implementation tradeoffs. The talk merges three different topics under the same umbrella: pricing for enforcing cooperation in slotted Aloha; adaptive channel allocation spectrum etiquette for cognitive radios; and interference aware routing for near-far effect mitigation. The common element is we show that in all three cases, cooperation can significantly increase the overall network performance. 

For more information please contact:

Dr. Cristina Comaniciu
Associate Professor & EE Graduate Program Director
Burchard Building
Room 211
Phone: 201.216.5606
Fax: 201.216.8246

ccomanic@stevens-tech.edu

Algebraic Cryptography Center - Workshop