What equation does a diffusing particle obey?

Speaker: 

Professor Janek Wehr

Institution: 

University of Arizona

Time: 

Friday, February 12, 2010 - 11:00am

Location: 

RH 306

Motion of a Brownian particle in a force field is described in the Smoluchowski-Kramers approximation by a stochastic differential
equation---Langevin equation.
If the diffusion coefficient depends on the particle's position, this equation is ambiguous due to several possible interpretations
of the stochastic differential. Two most often used interpretations are those of Ito and Stratonovitch, so the problem
is sometimes called the Ito-Stratonovitch dilemma. I will discuss the results of a recent experiment, which determine what
is the correct interpretation of the Langevin equation and show how they are consistent mathematically with the
Smoluchowski-Kramers approximation. Possible implications for studying a class of stochastic differential equations will
be mentioned.

Facial structure in the unit ball of a JB*-triple

Speaker: 

Professor Antonio Peralta

Institution: 

University of Granada, Spain

Time: 

Friday, March 12, 2010 - 2:00pm

Location: 

RH 340N

We shall present a solution of a problem which has been open for over twenty years.

In 1992, C. Akemann and G.K. Pedersen described the structure of the norm-closed faces of the unit ball of a C*-algebra A in terms of the compact partial isometries in A**. Three years earlier, C.M. Edwards and G.T.
Ruttimann gave a complete description of the weak*-closed faces of the unit ball of a JBW*-triple, and in particular, in a von Neumann algebra. However, the question whether the norm-closed faces of the unit ball in a JB*-triple E are determined by the compact tripotents in E** has remained open.

We shall survey the positive answer established by
C.M. Edwards, F.J. Fernndez-Polo, C. Hoskin and the speaker in a recent paper.

Mathematical models of biochemical reaction networks

Speaker: 

Gheorghe Craciun

Institution: 

U Wisconsin, Madison

Time: 

Monday, April 5, 2010 - 4:00pm

Location: 

RH 306

Mathematical models of biochemical reaction networks give rise to deterministic or stochastic dynamical systems that are usually high dimensional, nonlinear, and have many unknown parameters. Nevertheless, it is often possible to draw strong conclusions on the dynamics of such systems based on graph-theoretical properties of the reaction network. Moreover, we show that these results can be generalized to yield criteria for global injectivity for large classes of nonlinear maps. We also explain how these results relate to other problems, such as the Jacobian Conjecture in algebraic geometry and Bezier self-intersection in computer graphics

On the rate of best approximation

Speaker: 

Professor Timur Oikhberg

Institution: 

U.C. Irvine

Time: 

Friday, February 19, 2010 - 4:00pm

Location: 

MSTB 120

We consider the the problem of approximating a given object x (say, a function) by a sequence (x_n), whose terms belong to the prescribed family of sets (A_n)$ (for instance, A_n may be
the space of polynomials of degree less than n). For each n, compute the distance E_n from x to A_n. How does the sequence (E_n) behave? What are the connections between its rate of
decrease and the properties of x? Can we discern any patterns in the sequence (E_n)? We attempt to answer these questions for different families (A_n).

Limit of the infinite horizon discounted Hamilton-Jacobi equation

Speaker: 

Professor Hector Sanchez-Morgado

Institution: 

Universidad Nacional Autnoma de Mxico

Time: 

Tuesday, February 2, 2010 - 4:00pm

Location: 

RH 306

Motivated by the infinite horizon discounted problem, we study the convergence of solutions of the Hamilton Jacobi equation when the discount vanishes. If the Aubry set consists in a
finite number of hyperbolic critical points, we give an explicit expression for the limit.

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