Analysis
Department of Mathematics and Statistics
Texas Tech University
We study the asymptotic expansions, as time tends to infinity, of solutions of a system of ordinary differential equations with non-smooth nonlinear terms. The forcing function decays to zero in a very complicated but coherent way. We prove that every decaying solution admits an asymptotic expansion of a new type. This expansion contains a new variable that allows it to be established in a closed-form, but does not affect the meaning and precision of the expansion. Moreover, the expansion is constructed explicitly with the use of the complexification method.
We study the asymptotic expansions, as time tends to infinity, of solutions of a system of ordinary differential equations with non-smooth nonlinear terms. The forcing function decays to zero in a very complicated but coherent way. We prove that every decaying solution admits an asymptotic expansion of a new type. This expansion contains a new variable that allows it to be established in a closed-form, but does not affect the meaning and precision of the expansion. This second talk will focus on the proofs, especially, the construction of the expansions.
Many open problems in the classification of varieties will be described, particularly in relation to the relatively new field of Lipschitz Geometry. The goal is to get natural finite classifications for varieties in a fixed number of variables and fixed degree. (A variety is just the set of solutions of a finite system of polynomial equations.) Along the way, some famous theorems will be described.
Multiple coefficient inverse problems have become influential in mathematical oncology, playing an important role in delineating source regions. In this talk, we present a convexification method to reconstruct both the birth source and the mortality rate in an age-dependent diffusive problem. Using the so-called Fourier-Klibanov basis, this method strongly relies on a new derivation of a coupled nonlinear PDE system with age structure. We then introduce a Tikhonov-like cost functional, weighted by a suitable Carleman function. We will discuss an analysis of the minimization problem, where a new Carleman estimate and a Holder rate of convergence are derived, and present some numerical results.
All studies in Mathematics require the dual notions of null or nowhere, and full or everywhere. These are used to describe mathematical properties that may be hard to find or hard to miss, within a collection of objects. The most familiar examples of these notions are nowhere dense -- dense from topology, and zero measure and full measure from measure theory. These notions enable the formulation of mathematical statements in contexts such as probability, uncertainty, and perturbation and stability. These ideas are however inadequate to describe many common properties found in infinite dimensional vector spaces. To fill these gaps, these notions were generalized to the dual notions of shy—prevalent. The language of "prevalence" has enabled many new discoveries in Analysis, but is still inadequate in some contexts. I shall present a categorical approach to the notion of prevalence. This view of prevalence can offer possible generalizations of this notion.
Newton’s iterative root finding method and the dynamics of the Newton maps of rational functions have been studied in recent years, resulting in the classification of rational functions whose Newton maps are Möbius conjugate to polynomials of degree 1, 2, and 3. In this dissertation, we describe a new approach for obtaining the same results, then classify the sets of rational functions whose Newton maps are Möbius conjugate to polynomials of degree 4, polynomials of degree 5, and polynomials of any finite degree.
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abstract pdf
I will describe a specific connection between geometry and dynamics. More than twenty years ago, G. Perelman posted three papers on arxiv solving the Poincare conjecture, one of the seven Millennium Prize Problems. The key mechanism in his proof is the theory of Ricci flows, smoothly deforming shapes and measurements, and understandings of their singularity models in dimension three. Here we are interested in analogous models with complex structures in real dimension four. That is the geometry. We'll show that such a model can be formulated as an integrable Hamiltonian system, which originally was invented to describe the dynamics of classical mechanics. This striking perspective gives insight into a fundamental conjecture towards generalizing Perelman's work to higher dimensions. This talk is also a trailer for Math 6332, Geometric Mechanics, in the Fall.
Joint seminar with the Geometry, PDE and Mathematical Physics group
In my talk at the Analysis Seminar in the Fall 2024, I discussed several problems on the logarithmic capacity of configurations consisting of $n\ge 3$ disks in $\mathbb{R}^2$ and Newtonian capacity of $n$ balls in $\mathbb{R}^3$. In this talk, I will review main known results in this area, present a few new results and discuss a few problems which remain open. Among these problems are minimization and maximization problems on the logarithmic capacity under certain restrictions on configurations of disks. Similar problems for the hyperbolic capacity of constellations of disks on the hyperbolic plane will be also discussed.
Discretizing wave equations in space typically results in large systems of second-order differential equations. For the time integration of such systems, classical Runge-Kutta-Nyström integrators and their extended variants have been widely used. While effective for small or non-stiff problems, these methods often suffer from stability issues and inefficiency when applied to large, stiff or highly oscillatory systems. To overcome these limitations, we develop and analyze a new class of time integration methods, called exponential Nyström (expN) methods. These methods allow significantly larger time steps without sacrificing accuracy. We establish convergence results up to fifth-order accuracy within the framework of strongly continuous semigroups, with error bounds independent of the stiffness or high frequencies of the system. Our numerical experiments demonstrate that the proposed expN methods outperform existing Nyström-type integrators in both efficiency and accuracy. This is a collaboration with Professor V.T. Luan.