Michael W. Stewart Monday 02 May 2022 3:30 pm Keller, room 302
Title: “A numerical method for solving the eigenvalue problem associated with neutron diffusion inside nuclear reactor cores”
Abstract: “In this talk we will explore a mathematical tool that might assist in tackling one problem in the design of nuclear reactors, namely a numerical method for finding solutions to the neutron diffusion equation during steady state operations. After a brief look at the physics of nuclear fission and the physical aspects of nuclear reactors we will derive a partial differential equation that can be used to model such systems in a time independent steady state. The use of finite element discretization allows us to find weak solutions to the eigenvalue problem which emerges, and so we review weak solutions, the discretization of continuous problems, and what is known about the linear systems that such discretization produces. The solutions of such a discretized problem will naturally differ from the exact solution, and so we set bounds on the errors that are introduced. A possible algorithm to solve the generalized eigenvalue problem, and some computational experiments will then be reviewed.”
Jason L Greuling Tuesday, May 3rd, 3pm Keller 302
Title: A Discrete Regge Complex
This paper is interested in a differential complex that arises in the study of finite element methods for certain partial differential equations from Riemannian geometry involving curvature. Specifically, we derive a two dimensional Regge complex that includes the linearized curvature operator, curlTcurl, and relate it to a complex that includes the linearized discrete notion of curvature on a triangulation. We establish a correspondence between the two complexes, giving a relationship between the two linearized maps.
Hien Ha Thursday May 12, 3pm Keller 302
Tile: p-adic numbers
Abstract: The field of real numbers is completed from the field of rational numbers with respect to the distance metric. The heart of the completion process is the limit of Cauchy sequences. Recall that a metric space (X, d) is complete if every Cauchy sequence in X converges to a point in X. We know that Q is not complete with respect to the distance metric. For example, the Cauchy sequence of rational numbers 3 , 31/10 , 314 /100 , 3141/ 1000 …. is converging to π which is not in Q . Filling all convergent points of Cauchy sequences we get R. The field of p-adic numbers Qp is also completed from Q with respect to a different metric called the p-adic metric which is induced from the p-adic norm. We also make use of the convergence of the Cauchy sequences for this process. In this project we will see how Qp is completed from Q with respect to the p-adic norm. We also describe two trees associated with Qp: the trees of balls in the field Qp and the trees of lattices in the vector space Qp × Qp.