Harmonic Analysis and Representation Commons^™

Contributions To The Teaching And Learning Of Fluid Mechanics, Ashwin Vaidya

Department of Mathematics Faculty Scholarship and Creative Works

This issue showcases a compilation of papers on fluid mechanics (FM) education, covering different sub topics of the subject. The success of the first volume [1] prompted us to consider another follow-up special issue on the topic, which has also been very successful in garnering an impressive variety of submissions.

As a classical branch of science, the beauty and complexity of fluid dynamics cannot be overemphasized. This is an extremely well-studied subject which has now become a significant component of several major scientific disciplines ranging from aerospace engineering, astrophysics, atmospheric science (including climate modeling), biological and biomedical science …

Interpolation And Sampling In Analytic Tent Spaces, Caleb Parks

Graduate Theses and Dissertations

Introduced by Coifman, Meyer, and Stein, the tent spaces have seen wide applications in harmonic analysis. Their analytic cousins have seen some applications involving the derivatives of Hardy space functions. Moreover, the tent spaces have been a recent focus of research. We introduce the concept of interpolating and sampling sequences for analytic tent spaces analogously to the same concepts for Bergman spaces. We then characterize such sequences in terms of Seip's upper and lower uniform density. We accomplish this by exploiting a kind of Mobius invariance for the tent spaces.

Lecture 10: Preconditioned Iterative Methods For Linear Systems, Edmond Chow

Mathematical Sciences Spring Lecture Series

Iterative methods for the solution of linear systems of equations – such as stationary, semi-iterative, and Krylov subspace methods – are classical methods taught in numerical analysis courses, but adapting these methods to run efficiently at large-scale on high-performance computers is challenging and a constantly evolving topic. Preconditioners – necessary to aid the convergence of iterative methods – come in many forms, from algebraic to physics-based, are regularly being developed for linear systems from different classes of problems, and similarly are evolving with high-performance computers. This lecture will cover the background and some recent developments on iterative methods and preconditioning …

Linear Combinations Of Harmonic Univalent Mappings, Dennis Nguyen

Rose-Hulman Undergraduate Mathematics Journal

Many properties are known about analytic functions, however the class of harmonic functions which are the sum of an analytic function and the conjugate of an analytic function is less understood. We wish to find conditions such that linear combinations of univalent harmonic functions are univalent. We focus on functions whose image is convex in one direction i.e. each line segment in that direction between points in the image is contained in the image. M. Dorff proved sufficient conditions such that the linear combination of univalent harmonic functions will be univalent on the unit disk. The conditions are: the mappings …

The Influence Of The Psychophysical Assessment Paradigm On Pitch Discrimination For Adults (And A Pilot Sample Of Children), Ashley G. Flagge, Lucile Puranen, Madhuri S. Mulekar

Ashley Flagge Publications

Pitch discrimination ability has been of research interest due to its potential relationship to language and literacy. However, assessment protocols for pitch discrimination have varied widely. Prior studies with both children and adults have produced conflicting performance findings across different pitch discrimination research paradigms, though they have consistently shown that discrimination accuracy is based on the psychophysical assessment method applied. In the present study, we examined pitch discrimination performance among convenience samples of 19 adult women and ten female children across six different adaptive psychophysical measurement conditions. We found pitch discrimination performance in both groups to be impacted by the …

Algebraic, Analytic, And Combinatorial Properties Of Power Product Expansions In Two Independent Variables., Mohamed Ammar Elewoday

Graduate Theses, Dissertations, and Problem Reports (ETD)

Let $F(x,y)=I+\hspace{-.3cm}\sum\limits_{\substack{p=1\\m+n=p}}^{\infty}\hspace{-.3cm}A_{m,n}x^my^n$ be a formal power series, where the coefficients $A_{m,n}$ are either all matrices or all scalars. We expand $F(x,y)$ into the formal products $\prod\limits_{\substack{p=1\\m+n=p}}^{\infty}\hspace{-.3cm}(I+G_{m,n}x^m y^n)$, $\prod\limits_{\substack{p=1\\m+n=p}}^{\infty}\hspace{-.3cm}(I-H_{m,n}x^m y^n)^{-1}$, namely the \textit{ power product expansion in two independent variables} and \textit{inverse power product expansion in two independent variables} respectively. By developing new machinery involving the majorizing infinite product, we provide estimates on the domain of absolute convergence of the infinite product via the Taylor series coefficients of $F(x,y)$. This machinery introduces a myriad of "mixed expansions", uncovers various algebraic connections between the $(A_{m,n})$ and the $(G_{m,n})$, and uncovers various algebraic …

Morphology-Dependent Resonances In Two Concentric Spheres With Variable Refractive Index In The Outer Layer: Analytic Solutions, Umaporn Nuntaplook, John A. Adam

Mathematics & Statistics Faculty Publications

In many applications constant or piecewise constant refractive index profiles are used to study the scattering of plane electromagnetic waves by a spherical object. When the structured media has variable refractive indices, this is more of a challenge. In this paper, we investigate the morphology dependent resonances for the scattering of electromagnetic waves from two concentric spheres when the outer shell has a variable refractive index. The resonance analysis is applied to the general solutions of the radial Debye potential for both transverse magnetic and transverse electric modes. Finally, the analytic conditions to determine the resonance locations for this system …