Events
Department of Mathematics and Statistics
Texas Tech University
 | Monday Nov. 11
| | Algebra and Number Theory No Seminar
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A well known result by George Pólya asserts that among all the simply connected domains with fixed area A containing the point z0, the disk of radius p A/π centered at z0 maximizes the conformal radius. In this talk, I will define the geometric mean R2(Ω, E) of the conformal radius R(Ω, a) over the compact subsets E ⊂ Ω with positive area and the geometric mean R1(Ω, γ) of the conformal radius R(Ω, a) over the rectifiable curves γ ⊂ Ω with positive length. Two problems will be introduced and the first problem is about obtaining an upper bound for R2(Ω, E) over all Ω of area A and all compact sets E ⊂ Ω of area 0 < A0 < A and the second problem is about obtaining an upper and lower bound for R1(Ω, γ). Similar problems will be considered for triangles and quadrilaterals. The extremal configurations of these cases will also be discussed. In addition to that, the hyperbolic conformal center of a compact set contained in the unit disk will be introduced and some properties of hyperbolic conformal center will be addressed.
Abstract. In multiscale and multiphysics simulation, a predominant challenge is the accurate coupling of physics of different scales, stiffnesses, and dimensionalities. The underlying problems are usually time dependent, making the time integration scheme a fundamental component of the accuracy. Remarkably, most large-scale multiscale or multiphysics codes use a first-order operator split or (semi-)implicit integration scheme. Such approaches often yield poor accuracy, and can also have poor computational efficiency. There are technical reasons that more advanced and higher order time integration schemes have not been adopted however. One major challenge in realistic multiphysics is the nonlinear coupling of different scales or stiffnesses. Here I present a new class of nonlinearly partitioned Runge-Kutta (NPRK) methods that facilitate high-order integration of arbitrary nonlinear partitions of ODEs. Order conditions for an arbitrary number of partitions are derived via a novel edge-colored rooted-tree analysis. I then demonstrate NPRK methods on novel nonlinearly partitioned formulations of thermal radiative transfer and radiation hydrodynamics, demonstrating orders of magnitude improvement in wallclock time and accuracy compared with current standard (semi-)implicit and operator split approaches, respectively.
This talk may be seen by TTU eraider account holders in the Texas Tech Mediasite Catalog.
 | Thursday Nov. 14 6:30 PM MA 108
| | Mathematics Education Math Circle Aaron Tyrrell Mathematics and Statistics, Texas Tech University
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Math Circle Fall Poster
abstract 2 PM CST (UT-6)
Zoom link available from Dr. Brent Lindquist upon request.