Math Colloquium: Fri at 2:30pm-3:30pm, CH 443

Title: The inverse conductivity problem

Abstract: The problem of inverse conductivity is both well established as an object of research and yet still a source of interesting and useful physics and mathematics. I aim to give a (fairly rough!) overview of this fascinating topic, from both practical and theoretical perspectives. All levels of mathematicians, physicists and engineers are welcome.
Title: What does the average eigenvalue know?

Abstract:

Inverse spectral theory refers to the use of eigenvalues and related information to understand a linear operator.  Linear operators and matrices are used, for example, to describe such things as vibrating membranes, quantum phenomena, and graphs (in the sense of networks).  The eigenvalues respond to the shape of the membrane, the form of the interaction potential, or respectively the connectedness of the graph, but not in a formulaic way.  The effort to tease out these details from the knowledge of the eigenvalue spectrum was memorably described al little over 50 years ago by Mark Kac in an article entitled "Can one hear the shape of a drum?"   (Incidentally, the answer is "Often, but not always.")

Title: Maximal Monotonicity and Existence of Nonzero Solutions

Abstract:

The theory of maximal monotone operators in Banach spaces plays an important role in the solvability of a large class of partial differential equations, particularly the ones in divergence form.
The talk will begin with an introduction to maximal monotone operators in Hilbert spaces. An extension of the notion to Banach spaces will then be discussed. Among others, examples of maximal monotone operators which appear as the subdifferentials of certain convex functions will be given. Finally, an existence theorem for nonzero solutions of operator equations in Banach spaces will be presented.
Title: On the accuracy and importance of a consistent mass matrix.

Abstract:

The mass matrix is formed by integrating the outer product of two sets of basis functions. A mass matrix can appear in the discretization of an unsteady partial differential equation (PDE) using a finite element method (FEM). The accuracy, stability, and performance of a numerical method is tied to the handling of the mass matrix. In classical FEM, a common technique is to “lump” a mass matrix, summing the off-diagonals to the diagonal, resulting in a diagonal matrix. The simplification results in a trade-off between increased performance of the method and reduced accuracy. In this seminar, Dr. Brazell will discuss his work with mass matrices across three different topics. The first is lumping techniques for high-order streamline-upwind-Petrov-Galerkin (SUPG) methods. The second is psuedo-transient continuation as a globalization technique for the Newton-Rhapson method for solving problems using a Discontinuous Galerkin (DG) method. Lastly, is an explicit DG method with applications in overset meshing and adaptive mesh refinement. Some applications of these methods will include aerodynamics and wind energy.
Title: A posteriori versions of KAM theory: Rigorous results, computations and conjectures

Abstract: KAM theory deals with the existence of quasi-periodic solutions. The original formulation was for systems close to integrable.

In recent times, there have appeared versions of KAM theory without any reference to quasi- integrable systems. One can start from approximate solutions of an invariance equation and then validate them.

This leads easily to some rigorous results such as monogenicity properties and Whitney regularity. Moroever, it allows to validate numerical computations. The proof is based on an iterative method which also leads to very efficient and reliable computations. These computations lead to conjectures on what happens at breakdown and reveal scaling relations. Moreover, there is a very surprising order among scaling relations corresponding to different frequencies.

This is a report on work of many people.

Title: Almost commuting matrices.

Abstract: Suppose that $X$ and $Y$ are two self-adjoint matrices with the commutator $[X,Y]$ of small operator norm. One would expect that $X$ and $Y$ are close to a pair of commuting matrices. Can one provide a distance estimate which only depends on $\|[X,Y]\|$ and not on the dimension? This question was asked by Paul Halmos in 1976 and answered positively by Huaxin Lin in 1993 by indirect C*-algebraic methods, which did not provide any explicit bounds. It was conjectured by Davidson and Szarek that the distance estimate would be of the form $C\|[X,Y]\|^{1/2}$, which is the best possible power. I will explain the main ideas of the proof of this conjecture, obtained jointly with Yuri Safarov, as well as some related results such as the infinite-dimensional version of the question. I will also discuss some recent work in progress on the case of two unitary matrices. In both latter cases, one needs to take care of additional topological obstructions (such as index of winding number) in order for the result to be true.
Title: From the liquid drop model for nuclei to the ionization conjecture for atoms

Abstract:

The liquid drop model is an isoperimetric problem with a competing non-local term. It was originally introduced in the nuclear physics literature in 1930 and has received a lot of attention recently as an interesting problem in the calculus of variations. We discuss some new results and open problems. We show how the insights from this problem allowed us to prove the ionization conjecture in a certain model in density functional theory. The talk will be non-technical and does not assume any knowledge in physics.
Title: A Metrically Defined Uniformization Map of Planar Domains

Abstract:

Lloyd J. Edwards, PhD, Charles R. Katholi, PhD, Byron C. Jaeger, PhD
UAB Department of Biostatistics

Title:
Properties of $R ^2_\beta$ and Tests of Hypotheses for Fixed Effects in the Linear Mixed Model

Abstract:

Properties of the $R^2$ statistic used in linear regression with least squares estimation are well developed. Additionally, the $R^2$ statistic used in linear regression has a corresponding test of hypoyhesis associated with it. Similar properties are developed for $R^2_\beta$, an $R^2$ statistic for fixed effects in the linear mixed model. The central and non-central Beta distributions are used to approximate the distribution of $R^2_\beta$ under corresponding null and alternative test hypotheses. The asymptotic expectation and variance of $R^2_\beta$ are derived. Simulations are used to demonstrate the performance of the Beta approximation. Test statistics are proposed that are based on estimators of the derived expectation and variance. Simulations are used to compare the test statistics to the overall $F$ test for fixed affects in the linear mixed model. Using simulations, the Type I error rate of the proposed $R^2_\beta$ test statistics is shown to be equivalent to the Type I error rate for the overall $F$ test.

Title: Bounds for Preperiodic Points for Maps with Good Reduction

Abstract:

Let $K$ be a number field and let $\phi$ in $K(z)$ be a rational function of degree $d\geq 2$. Let $S$ be the set of places of bad reduction for $\phi$ (including the archimedean places). Let $\Per(\phi,K)$, $\PrePer(\phi, K)$, and $\Tail(\phi,K)$ be the set of $K$-rational periodic, preperiodic, and purely preperiodic points of $\phi$, respectively.
In this talk, we will present two main results. Firstly, assuming that $|\Per(\phi,K)| \geq 4$ (resp.\$|\Tail(\phi,K)| \geq 3$), we prove bounds for $|\Tail(\phi,K)|$ (resp.\$|\Per(\phi,K)|$) that depend only on the number of places of bad reduction $|S|$ (and not on the degree $d$). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when $|\Per(\phi,K)| < 4$ (resp.\$|\Tail(\phi,K)| < 3$).
For the second result, a bound for $|\PrePer(\phi,K)|$ in terms of the number of places of bad reduction $|S|$ and the degree $d$ of the rational function $\phi$ is obtainedThis bound significantly improves a previous result.

Math Colloquium: Fri at 2:30pm-3:30pm, CH 443

Title: A Dynamic Model of Free Fatty Acids, Glucose, and Insulin Metabolism

Abstract:

The role of free fatty acids (FFA) on the progression of type 2 diabetes (T2D) has been widely studied. Prior studies suggest that individuals with shared familiar genetic predispositions to metabolic-related diseases may be vulnerable to dysfunctional regulation of plasma FFA. A vicious cycle may arise when FFA
is not regulated properly leading to the development of insulin resistance, a key indicator for T2D as prolonged insulin resistance results in hyperglycemia. We propose a hypothesis-driven model to quantitatively study the role of FFA on the progression of insulin resistance. The nonlinear dynamics among glucose, insulin, and FFA are modeled using delay differential equations and compared to the well-known minimal model consisting of ordinary differential equations. Model validation and parameter estimation utilizing clinical data of patients who underwent bariatric surgery, serve as the quantitative measures used to evaluate the regulation of FFA production by insulin action within a heterogeneous population (nondiabetic to diabetic). Results show that several metabolic factors for insulin, glucose, and FFA regulation improved post-bariatric surgery, and these results were supported with prior clinical findings.