Contributions To The Teaching And Learning Of Fluid Mechanics, 2021 Montclair State University

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

*Department of Mathematics Facuty 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 and engineering ...

Lecture 05: The Convergence Of Big Data And Extreme Computing, 2021 King Abdullah University of Science and Technology

#### Lecture 05: The Convergence Of Big Data And Extreme Computing, David Keyes

*Mathematical Sciences Spring Lecture Series*

As simulation and analytics enter the exascale era, numerical algorithms, particularly implicit solvers that couple vast numbers of degrees of freedom, must span a widening gap between ambitious applications and austere architectures to support them. We present fifteen universals for researchers in scalable solvers: imperatives from computer architecture that scalable solvers must respect, strategies towards achieving them that are currently well established, and additional strategies currently being developed for an effective and efficient exascale software ecosystem. We consider recent generalizations of what it means to “solve” a computational problem, which suggest that we have often been “oversolving” them at the ...

Lecture 10: Preconditioned Iterative Methods For Linear Systems, 2021 Georgia Institute of Technology

#### 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 in the context of ...

Linear Combinations Of Harmonic Univalent Mappings, 2021 California State University, Stanislaus

#### 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 ...

Gibbs Phenomenon For Jacobi Approximations, 2021 Bard College

#### Gibbs Phenomenon For Jacobi Approximations, Riti Bahl

*Senior Projects Spring 2021*

The classical Gibbs phenomenon is a peculiarity that arises when approximating functions near a jump discontinuity with the Fourier series. Namely, the Fourier series "overshoots" (and "undershoots") the discontinuity by approximately 9% of the total jump. This same phenomenon, with the same value of the overshoot, has been shown to occur when approximating jump-discontinuous functions using specific families of orthogonal polynomials. In this paper, we extend these results and prove that the Gibbs phenomenon exists for approximations of functions with interior jump discontinuities with the two-parameter family of Jacobi polynomials P_{n}^{(a,b)}(x). In particular, we show that ...

The Complex Propagation Of Light Explained Visually: How To Make A Hologram, 2021 Bard College

#### The Complex Propagation Of Light Explained Visually: How To Make A Hologram, Bruno Ray Becher

*Senior Projects Spring 2021*

The complexity of light’s wave nature is shown in the paths that light takes. In this project I will show several useful ways to imagine and predict how light will travel from one place to another. Once light is produced it does not immediately fill a room, instead it undulates through free space as if the space itself was a fluid. Once we understand the way light flows and interacts with its environment not only can we predict and control its shape with a hologram, but also discover clues which give secrets about where the light has been. Telescopes ...

Morphology-Dependent Resonances In Two Concentric Spheres With Variable Refractive Index In The Outer Layer: Analytic Solutions, 2021 Old Dominion University

#### 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 ...

Algebraic, Analytic, And Combinatorial Properties Of Power Product Expansions In Two Independent Variables., 2021 West Virginia University

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

*Graduate Theses, Dissertations, and Problem Reports*

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 ...

On The Construction And Mathematical Analysis Of The Wavelet Transform And Its Matricial Properties, 2020 Universidad Mayor de San Simón

#### On The Construction And Mathematical Analysis Of The Wavelet Transform And Its Matricial Properties, Diego Sejas Viscarra

*Rose-Hulman Undergraduate Mathematics Journal*

We study the properties of computational methods for the Wavelet Transform and its Inverse from the point of view of Linear Algebra. We present a characterization of such methods as matrix products, proving in particular that each iteration corresponds to the multiplication of an adequate unitary matrix. From that point we prove that some important properties of the Continuous Wavelet Transform, such as linearity, distributivity over matrix multiplication, isometry, etc., are inherited by these discrete methods.

This work is divided into four sections. The first section corresponds to the classical theoretical foundation of harmonic analysis with wavelets; it is used ...

Perceiving Mathematics And Art, 2020 University of Arkansas, Fayetteville

#### Perceiving Mathematics And Art, Edmund Harriss

*Mic Lectures*

Mathematics and art provide powerful lenses to perceive and understand the world, part of an ancient tradition whether it starts in the South Pacific with tapa cloth and wave maps for navigation or in Iceland with knitting patterns and sunstones. Edmund Harriss, an artist and assistant clinical professor of mathematics in the Fulbright College of Arts and Sciences, explores these connections in his Honors College Mic lecture.

Harmonic Equiangular Tight Frames Comprised Of Regular Simplices, 2020 Air Force Institute of Technology

#### Harmonic Equiangular Tight Frames Comprised Of Regular Simplices, Matthew C. Fickus, Courtney A. Schmitt

*Faculty Publications*

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin ...

The Direct Scattering Map For The Intermediate Long Wave Equation, 2020 University of Kentucky

#### The Direct Scattering Map For The Intermediate Long Wave Equation, Joel Klipfel

*Theses and Dissertations--Mathematics*

In the early 1980's, Kodama, Ablowitz and Satsuma, together with Santini, Ablowitz and Fokas, developed the formal inverse scattering theory of the Intermediate Long Wave (ILW) equation and explored its connections with the Benjamin-Ono (BO) and KdV equations. The ILW equation\begin{align*} u_t + \frac{1}{\delta} u_x + 2 u u_x + Tu_{xx} = 0, \end{align*} models the behavior of long internal gravitational waves in stratified fluids of depth $0< \delta < \infty$, where $T$ is a singular operator which depends on the depth $\delta$. In the limit $\delta \to 0$, the ILW reduces to the Korteweg de Vries (KdV) equation, and in the limit $\delta \to \infty$, the ILW (at least formally) reduces to the Benjamin-Ono (BO) equation.

While the KdV equation is very well understood, a rigorous analysis of inverse scattering for the ILW equation remains to be accomplished. There is currently no rigorous proof that ...

Polyphase Equiangular Tight Frames And Abelian Generalized Quadrangles, 2019 Air Force Institute of Technology

#### Polyphase Equiangular Tight Frames And Abelian Generalized Quadrangles, Matthew C. Fickus, John Jasper, Dustin G. Mixon, Jesse D. Peterson, Cody E. Watson

*Faculty Publications*

An equiangular tight frame (ETF) is a type of optimal packing of lines in a finite-dimensional Hilbert space. ETFs arise in various applications, such as waveform design for wireless communication, compressed sensing, quantum information theory and algebraic coding theory. In a recent paper, signature matrices of ETFs were constructed from abelian distance regular covers of complete graphs. We extend this work, constructing ETF synthesis operators from abelian generalized quadrangles, and vice versa. This produces a new infinite family of complex ETFs as well as a new proof of the existence of certain generalized quadrangles. This work involves designing matrices whose ...

Fluids In Music: The Mathematics Of Pan’S Flutes, 2019 Montclair State University

#### Fluids In Music: The Mathematics Of Pan’S Flutes, Bogdan Nita, Sajan Ramanathan

*Department of Mathematics Facuty Scholarship and Creative Works*

We discuss the mathematics behind the Pan’s flute. We analyze how the sound is created, the relationship between the notes that the pipes produce, their frequencies and the length of the pipes. We find an equation which models the curve that appears at the bottom of any Pan’s flute due to the different pipe lengths.

A Course In Harmonic Analysis, 2019 Missouri University of Science and Technology

#### A Course In Harmonic Analysis, Jason Murphy

*AOER Course Materials*

These notes were written to accompany the courses Math 6461 and Math 6462 (Harmonic Analysis I and II) at Missouri University of Science & Technology during the 2018-2019 academic year. The goal of these notes is to provide an introduction to a range of topics and techniques in harmonic analysis, covering material that is interesting not only to students of pure mathematics, but also to those interested in applications in computer science, engineering, physics, and so on.

An Alternative Almost Sure Construction Of Gaussian Stochastic Processes In The L2([0,1]) Space, 2019 Bowdoin College

#### An Alternative Almost Sure Construction Of Gaussian Stochastic Processes In The L2([0,1]) Space, Kevin Chen

*Honors Projects*

No abstract provided.

Minimal Principal Series Representations Of Sl(3,R), 2019 William & Mary

#### Minimal Principal Series Representations Of Sl(3,R), Jacopo Gliozzi

*Undergraduate Honors Theses*

We discuss the properties of principal series representations of SL(3,R) induced from a minimal parabolic subgroup. We present the general theory of induced representations in the language of fiber bundles, and outline the construction of principal series from structure theory of semisimple Lie groups. For SL(3,R), we show the explicit realization a novel picture of principal series based on the nonstandard picture introduced by Kobayashi, Orsted, and Pevzner for symplectic groups. We conclude by studying the K-types of SL(3,R) through Frobenius reciprocity, and evaluate prospects in developing simple intertwiners between principal series representations.

Rankin-Cohen Brackets And Fusion Rules For Discrete Series Representations Of Sl(2,R), 2019 William & Mary

#### Rankin-Cohen Brackets And Fusion Rules For Discrete Series Representations Of Sl(2,R), Emilee Cardin

*Undergraduate Honors Theses*

In this paper, we discuss the holomorphic discrete series representations of SL(2,R). We give an overview of general representation theory, from the perspective of both groups and Lie algebras. We then consider tensor products of representations, specifically investigating tensor products of the holomorphic discrete series and their associated algebraic objects, called (g,K)-modules. We then use algebraic techniques to study the fusion rules of the discrete series. We conclude by giving explicit intertwiners, recovering the formula of number-theoretic objects, called Rankin-Cohen brackets.

A General Weil-Brezin Map And Some Applications, 2019 William & Mary

#### A General Weil-Brezin Map And Some Applications, Benjamin Bechtold

*Undergraduate Honors Theses*

We recall a theory generalizing the Heisenberg group on $\R$ to an analogous structure using a locally compact abelian group $G$. Then, using our new, general Heisenberg groups, we generalize the classical Weil-Brezin map, from an operator on $L^2(\R )$ and develop a theory of that generalized Weil-Brezin map on $L^2(G)$ for some locally compact abelian group $G$. We then apply our generalized Weil-Brezin map to recover the Poisson Summation Formula as well as the Plancherel Theorem.

Harmonic Equiangular Tight Frames Comprised Of Regular Simplices, 2019 Air Force Institute of Technology

#### Harmonic Equiangular Tight Frames Comprised Of Regular Simplices, Courtney A. Schmitt

*Theses and Dissertations*

An equiangular tight frame (ETF) is a sequence of equal-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs, which are formed by restricting the characters of a finite abelian group to a difference set. Recently, it was shown that some harmonic ETFs are themselves comprised of regular simplices. In this thesis, we continue the ...