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Speaker: Dennis Garity
Abstract: This is the first of two talks on certain 3-manifolds that are increasing unions of solid tori. Recently, in a joint paper with Repovs and Wright, we produced infinite classes of such manifolds that were unions of 2 copies of 3-space, and infinite classes that were not such unions. In the first talk, I will introduce some of the background information needed including properties of manifolds that are increasing unions of tori and
some simple characterizations of 3-dimensional Euclidean space.
We explore similarities between the roles of the vorticity in 3D Euler Equations and the perpendicular gradient of the solution to 2D Quasi-Geostrophic Equations. In particular we compare level sets of 2D QGE with vortex lines of 3D Euler, and also compare the integral representations of the velocity and symmetric strain matrix for both equations.
This is a joint work with Kun Wang and Jian Deng.
The Helmholtz equation arises in many problems related to wave propagations, such as acoustic, electromagnetic wave scattering and models in geophysical applications. Developing efficient and highly accurate numerical schemes to solve the Helmholtz equation at large wave numbers is a very challenging scientific problem and it has attracted a great deal of attention for a long time. The difficulties in solving the Helmholtz equations are due to the construction of accurate numerical schemes for the equation and the boundary conditions, and efficient and robust numerical algorithms to solve the resulting indefinite linear systems. Moreover, it is a challenge to derive a numerical scheme which is capable of eliminating or minimizing the pollution effect. The pollution effect is the foremost difficulty which causes a serious problem as the wave number increases. Let $k,h,$ and $n$ denote the wave number, the grid size and the order of a finite difference or finite element approximations, then it is known that the relative error is bounded by ${k}^{\mathrm{\alpha}}(kh{)}^{n}$, where $\alpha =2$ or 1 for a finite difference or finite element method. Consequently, even using a fixed $h$ with $\text{kh}<\mathrm{1,}$ the error increases with $k$ unless we apply a very fine mesh $h$ such that $h<1/{k}^{2}$. However, this will lead to an enormous size of ill-conditioned and indefinite system of linear equations. In this talk, we present a new finite difference scheme with an error estimate given by $k(h{)}^{2n}$ for a one-dimensional problem and the schemes are pollution free. Extension and numerical simulations for multi-dimensional problems will also be reported.
Vershik's adic transformations are defined on the path space of a Bratteli diagram, a nonstationary analogue of the graph of a subshift of finite type. They can be used to model many dynamical systems usually constructed in other ways, such as substitution dynamical systems, cutting and stacking constructions and interval exchange transformations. We classify the invariant measures which are finite for some subdiagram defined by erasing vertices and edges; these measures may be infinite on every open subset of the path space. Ingredients include nonstationary versions of the Frobenius decomposition into communicating states, and a nonstationary Frobenius-Victory theorem. This extends and completes work of Bezuglyi, Kwiatkowski, Medynets and Solomyak. (Joint with Marina Talet of Aix-Marseille University).
This work addresses the problem of adaptive modulation and power in wireless systems with a strict delay constraint. Modulation and transmit power are dynamically adapted to minimize the outage probability for a fixed data rate. A discrete-time stationary Markov chain is used to model the time-varying channel. The problem is formulated as a finite-horizon MDP. The solution is a set of power/modulation allocation policies to be used during the transmission, as a function of the channel and system state. Numerical results show the benefits of such adaptation policies.
The ultimate mobility of a landslide can be strongly contingent on feedback mechanisms in the early stages of motion which influence the frictional resistance of a granular-fluid mixture. An initial failure can lead to runaway acceleration and extensive runout, or, conversely, a stabilizing slump. This distinction is strongly dependent on initial sediment material properties, such as porosity, permeability and fluid content. Traditional slope-stability models attempt to quantify initial force balances of sediment masses, but they do not address the fate of a failing mass. Traditional debris-flow runout models often begin with unrealistic initial force balances and friction coefficients in order to match an observed event. We have developed a mathematical model with the aim of simulating landslides and debris flows, seamlessly from initiation to deposition. The depth-averaged model is a two-phase granular-fluid model borrowing from principles of fluid mechanics, granular mechanics and soil mechanics. The result is a nonconservative hyperbolic system of five PDEs, similar to other St. Venant (shallow water) models for free-surface flows (tsunamis, flooding, etc.). These problems present similar mathematical and computational challenges.
I will describe the mathematical model and computational software that we have developed for these problems. As a case study, simulations of the 2014 Oso, Washington, disaster will be presented. Implications of initial sediment porosity and landslide liquefaction will be described.
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