Fickian diffusion and anomalous diffusion, their fast numerical simulation
Dr. Hong Wang
Department of MathematicsUniversity of South Carolina
Thursday, December 13, 2012, AB 634, 2:45 - 3:45 pm
Abstract: Diffusion processes arise in nature, sciences, social sciences, and a variety of applications. The fundamental assumption underlying a Fickian diffusion process is that a particle's motion has little or no spatial correlation. The probability of finding a particle somewhere in space can be described by a Gaussian distribution, or equivalently the classical second-order diffusion equation.
Today an increasing number of non-Fickian diffusion processes has been found, ranging from the signaling of biological cells, foraging behavior of animals, to physics, finance, and solute transport in groundwater. In these processes a particle's long walk is not necessarily independent of each other and may have long correlation length, so the processes can have large deviations from the stochastic process of Brownian motion. Recent studies show that fractional diffusion and advection-diffusion equations provide an adequate description of transport processes that exhibit anomalous diffusion, which cannot be modeled properly by classical second-order diffusion equations.
Computationally, an important issue is that the numerical methods for three-dimensional anomalous diffusion processes are prohibitively expensive in terms of computational work and memory requirement. Even the simulation of a three-dimensional linear fractional diffusion equation with a moderate number of grid points can take a state of the art petaflop supercomputer at least hundreds of years to finish.
In this talk we go over classical Fickian diffusion and anomalous diffusion, and their relations with (fractional) calculus. We will also present some fast numerical solution methods for and discuss some mathematical issues on space-fractional diffusion equations.
The Isoperimetric Problem Revisited: Extracting a Short Proof of Sufficiency From Euler's 1744 Proof of Necessity
Professor Richard TapiaCenter for Excellence and Equality EducationRice University
Thursday, December 6, 2012, DMSC 103, 2:30 - 3:45 pm
Abstract: We'll watch a presentation of Dr. Richard Tapia from Rice University, given at the Annual Meeting of SIAM 2012. Slides with synchronized audio, talk was delivered in Minneapolis in July 2012.
Representing a graph on a Hilbert space
Professor David Pask
School of Mathematics and Applied StatistisUniversity of Wollongong
Thursday, November 8, 2012, DMSC 103, 2:30 - 3:45 pm
Abstract: In this presentation, I will show how to represent three different types of graphs on a Hilbert space. In the simplest case the edges and vertices are represented by matrix units in a finite matrix algebra. I will then briefly talk about how the properties of the graph translate into properties of the representation of graph.
Math as I like it
Professor Jin Akiyama
Tokyo University of Science
Thursday, October 25, 2012, DMSC 103, 2:30 - 3:45 pm
Abstract: We'll watch a talk given by Dr. Jin Akiyama from Tokyo University of Science. The presentation was given at International conference "Contemporary Mathematics" in Moscow in June 2009.
Reflections on SIAM, Publishing and the Opportunities before us
Professor Douglas Arnold
School of Mathematics, University of Minnesota
Thursday, October 11, 2012, DMSC 103, 2:30 - 3:45 pm
Presentation given at the Annual Meeting of SIAM 2012. Slides with synchronized audio, talk was delivered in Minneapolis in July 2012.
Homotopies and Liftings
Professor Bruce Blackadar
Department of Mathematics and Statistics, University of Nevada, Reno
Thursday, September 27, 2012, DMSC 103, 2:30 - 3:45 pm
Abstract: Absolute retracts and absolute neighborhood retracts (ANR's) are nice kinds of topological spaces which are very useful in various aspects of topology. One of the most important results about these spaces is the Borsuk Homotopy Extension Theorem.
C*-Algebras were originally algebras of bounded operators on Hilbert spaces, but they are now often thought of as "noncommutative topological spaces." There is good reason for this: commutative C*-algebras are, in a precise sense, really the same thing as (locally compact Hausdorff) topological spaces, and many topological ideas and results can be rephrased nicely in the language of C*-algebras by "turning arrows around" and remain true also for noncommutative C*-algebras. Some powerful machinery has thus come into the theory of operator algebras from topology, and conversely C*-algebras have become an important tool in some aspects of topology, such as the study of dynamical systems.
One can turn the arrows around in the definition of ANR's to obtain a class of C*-algebras called semiprojective C*-algebras, which have proved to be interesting and useful in C*-algebra theory analogous to the uses of ANR's in topology. I recently found a proof of the version of the Borsuk Homotopy Extension Theorem for semiprojective C*-algebras (the commutative proof does not adapt to the noncommutative case).
In the course of the work, I also identified and began to study a property of both spaces and C*-algebras somewhat similar to the ANR property (or semiprojectivity), called e-openness, which appears to be new and interesting even in the commutative case. Outline of the talk:
- What are ANR's and why are they interesting?
- What does the Borsuk Homotopy Extension Theorem say, and how is it proved?
- What do C*-algebras have to do with topology?
- What are semiprojective C*-algebras and why are they interesting?
- What does the version of Borsuk's Theorem for semiprojective C*-algebras say and how is it proved?
- What are e-open and e-closed spaces and why are they interesting?
- A number of examples will be discussed.