Events
Department of Mathematics and Statistics
Texas Tech University
 | Tuesday
Sep. 29
3:30 PM
MATH 017
|
| Real-Algebraic Geometry
Spec I
David Weinberg
Department of Mathematics and Statistics, Texas Tech University
|
 | Wednesday
Sep. 30
3:00 PM
Online
|
| Algebra and Number Theory
No Seminar
|
We use the Einstein random-walk paradigm for diffusion to derive a degenerate nonlinear parabolic equation and study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity function that is a (non-negative) polynomial of the dependent variable of the problem. We motivate these types of equations using Einstein's random walk paradigm, but instead of Taylor expansion we used Caratheodory theorem unlike Einstein's original work leading to a partial differential equation in non-divergence form. On the other hand, using conservation
principles leads to a partial differential equation in divergence form. A transformation is derived to handle both cases. We investigate a qualitative properties of the solution using maximum principle and energy method, in order to obtain bounds above and below for the time-evolution of the solution to the non-linear diffusion problem. Having thus sandwiched, we prove that, unlike the case of degenerate diffusion, the solution converges onto the linear diffusion solution at long times. Select numerical examples support the mathematical theorems and illustrate the convergence process.In this talk, I will discuss Section 2.2 from the recent paper “Proper Orbifold Cohomology” by Sati and Schreiber in which the concept of groups and group actions are formulated for infinity-toposes. Externally, these structures are known as grouplike E_n-algebras, but can be constructed internally in a more natural way. I will define groups, group actions, principal bundles, and fiber bundles.The agenda for the Financial Math seminar is:
1. Yuan Hu: crypto-currency portfolios and option valuation
2. Zari Rachev: Vol-of-vol indices
3. Jason Bailey: Early warning signals for real estate bubbles