Paul X. Uhlig

Bio: Paul X. Uhlig is an academic researcher from St. Mary's University. The author has contributed to research in topics: Dirichlet eigenvalue & Drum. The author has an hindex of 6, co-authored 8 publications receiving 247 citations.

Curriculum Guidelines for Undergraduate Programs in Data Science

Abstract: The Park City Math Institute 2016 Summer Undergraduate Faculty Program met for the purpose of composing guidelines for undergraduate programs in data science. The group consisted of 25 undergraduate faculty from a variety of institutions in the United States, primarily from the disciplines of mathematics, statistics, and computer science. These guidelines are meant to provide some structure for institutions planning for or revising a major in data science.

Curriculum Guidelines for Undergraduate Programs in Data Science

Abstract: The Park City Math Institute (PCMI) 2016 Summer Undergraduate Faculty Program met for the purpose of composing guidelines for undergraduate programs in Data Science. The group consisted of 25 undergraduate faculty from a variety of institutions in the U.S., primarily from the disciplines of mathematics, statistics and computer science. These guidelines are meant to provide some structure for institutions planning for or revising a major in Data Science.

Where Best to Hold a Drum Fast

Abstract: Allowed to fasten, say, one-half of a drum's boundary, which half produces the lowest or highest bass note? The answer is a natural limit of solutions to a family of extremal Robin problems for the least eigenvalue of the Laplacian. We produce explicit extremizers when the drum is a disk, while, for general shapes, we establish existence and necessary conditions, and build and test a pair of numerical methods.

On the optimal insulation of conductors

Abstract: We coat a conductor with an insulator and equate the effectiveness of this procedure with the rate at which the body dissipates heat when immersed in an ice bath. In the limit, as the thickness and conductivity of the insulator approach zero, the dissipation rate approaches the first eigenvalue of a Robin problem with a coefficient determined by the shape of the insulator. Fixing the mean of the shape function, we search for the shape with the least associated Robin eigenvalue. We offer exact solutions for balls; for general domains, we establish existence and necessary conditions and report on the results of a numerical method.

Where Best to Hold a Drum Fast

Abstract: If we are allowed to fasten, say, one half of a drum's boundary, which half produces the lowest or highest bass note? The answer is a natural limit of solutions to a family of extremal Robin problems for the least eigenvalue of the Laplacian. We produce explicit extremizers when the drum is a disk while for general shapes we establish existence and necessary conditions, and we build and test a pair of numerical methods.

Computational Thinking 計算論的思考

Abstract: It represents a universally applicable attitude and skill set everyone, not just computer scientists, would be eager to learn and use.

The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem

Abstract: We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue $\lambda_2$ of the Laplacian of the weighted graph. In this paper we consider the problem of assigning transition rates to the edges so as to maximize $\lambda_2$ subject to a linear constraint on the rates. This is the problem of finding the fastest mixing Markov process (FMMP) on the graph. We show that the FMMP problem is a convex optimization problem, which can in turn be expressed as a semidefinite program, and therefore effectively solved numerically. We formulate a dual of the FMMP problem and show that it has a natural geometric interpretation as a maximum variance unfolding (MVU) problem, , the problem of choosing a set of points to be as far apart as possible, measured by their variance, while respecting local distance constraints. This MVU problem is closely related to a problem recently proposed by Weinberger and Saul as a method for “unfolding” high-dimensional data that lies on a low-dimensional manifold. The duality between the FMMP and MVU problems sheds light on both problems, and allows us to characterize and, in some cases, find optimal solutions.

Optimizing the fundamental Neumann eigenvalue for the Laplacian in a domain with small traps

Abstract: An optimization problem for the fundamental eigenvalue λ0 of the Laplacian in a planar simply-connected domain that contains N small identically-shaped holes, each of radius e � 1, is considered. The boundary condition on the domain is assumed to be of Neumann type, and a Dirichlet condition is imposed on the boundary of each of the holes. As an application, the reciprocal of the fundamental eigenvalue λ0 is proportional to the expected lifetime for Brownian motion in a domain with a reflecting boundary that contains N small traps. For small hole radii e, a two-term asymptotic expansion for λ0 is derived in terms of certain properties of the Neumann Green’s function for the Laplacian. Only the second term in this expansion depends on the locations xi ,f ori =1 ,...,N , of the small holes. For the unit disk, ring-type configurations of holes are constructed to optimize this term with respect to the hole locations. The results yield hole configurations that asymptotically optimize λ0 .F or ac lass of symmetric dumbbell-shaped domains containing exactly one hole, it is shown that there is a unique hole location that maximizes λ0. For an asymmetric dumbbell-shaped domain, it is shown that there can be two hole locations that locally maximize λ0. This optimization problem is found to be directly related to an oxygen transport problem in skeletal muscle tissue, and to determining equilibrium locations of spikes to the Gierer–Meinhardt reactiondiffusion model. It is also closely related to the problem of determining equilibrium vortex configurations within the context of the Ginzburg–Landau theory of superconductivity.

The Method of Fundamental Solutions Applied to the Calculation of Eigenfrequencies and Eigenmodes of 2D Simply Connected Shapes

Abstract: In this work we show the application of the Method of Fundamental Solutions(MFS) in the determi- nation of eigenfrequencies and eigenmodes associated to wave scattering problems. This meshless method was al- ready applied to simple geometry domains with Dirich- let boundary conditions (cf. Karageorghis (2001)) and to multiply connected domains (cf. Chen, Chang, Chen, and Chen (2005)). Here we show that a particular choice of point-sources can lead to very good results for a fairly general type of domains. Simulations with Neumann boundary condition are also considered. keyword: Eigenfrequencies, Eigenmodes, Acoustic waves, Method of fundamental solutions