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Harmonic Analysis and Representation Commons

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Linear Combinations Of Harmonic Univalent Mappings, Dennis Nguyen 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 ...


A Mathematical Programming Approach For Integrated Multiple Linear Regression Subset Selection And Validation, Seokhyun Chung, Young-Woong Park, Taesu Cheong 2020 University Of Michigan

A Mathematical Programming Approach For Integrated Multiple Linear Regression Subset Selection And Validation, Seokhyun Chung, Young-Woong Park, Taesu Cheong

Information Systems and Business Analytics Publications

Subset selection for multiple linear regression aims to construct a regression model that minimizes errors by selecting a small number of explanatory variables. Once a model is built, various statistical tests and diagnostics are conducted to validate the model and to determine whether the regression assumptions are met. Most traditional approaches require human decisions at this step. For example, the user may repeat adding or removing a variable until a satisfactory model is obtained. However, this trial-and-error strategy cannot guarantee that a subset that minimizes the errors while satisfying all regression assumptions will be found. In this paper, we propose ...


On The Construction And Mathematical Analysis Of The Wavelet Transform And Its Matricial Properties, Diego Sejas Viscarra 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, Edmund Harriss 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, Matthew C. Fickus, Courtney A. Schmitt 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, Joel Klipfel 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, Matthew C. Fickus, John Jasper, Dustin G. Mixon, Jesse D. Peterson, Cody E. Watson 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 ...


A Course In Harmonic Analysis, Jason Murphy 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, Kevin Chen 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), Jacopo Gliozzi 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), Emilee Cardin 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, Benjamin Bechtold 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, Courtney A. Schmitt 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 ...


Fourier Series Expansion Methods For The Heat And Wave Equations In Two And Three Dimensions On Spherical Domains, Matthew Eller 2019 University of Nebraska at Omaha

Fourier Series Expansion Methods For The Heat And Wave Equations In Two And Three Dimensions On Spherical Domains, Matthew Eller

Student Research and Creative Activity Fair

Description: The Fourier series expansion method is an invaluable approach to solving partial differential equations, including the heat and wave equations. For homogeneous heat and wave equations, the solution can readily be found through separation of variables and then expansion of the solution in terms of the eigenfunctions. Solutions to inhomogeneous heat and wave equations through Fourier series expansion methods were not readily available in the literature for two- and three-dimensional cases. In my previous paper, I developed an approach for solving inhomogeneous heat and wave equations on cubic domains using Fourier series expansion methods. I shall extend my general ...


Operator Algebras Generated By Left Invertibles, Derek DeSantis 2019 University of Nebraska - Lincoln

Operator Algebras Generated By Left Invertibles, Derek Desantis

Dissertations, Theses, and Student Research Papers in Mathematics

Operator algebras generated by partial isometries and their adjoints form the basis for some of the most well studied classes of C*-algebras. Representations of such algebras encode the dynamics of orthonormal sets in a Hilbert space.We instigate a research program on concrete operator algebras that model the dynamics of Hilbert space frames.

The primary object of this thesis is the norm-closed operator algebra generated by a left invertible $T$ together with its Moore-Penrose inverse $T^\dagger$. We denote this algebra by $\mathfrac{A}_T$. In the isometric case, $T^\dagger = T^*$ and $\mathfrac{A}_T$ is a representation ...


Global Existence And Asymptotic Behaviors For Some Nonlinear Partial Differential Equations., Ismahan Dhaw Binshati 2019 West Virginia University

Global Existence And Asymptotic Behaviors For Some Nonlinear Partial Differential Equations., Ismahan Dhaw Binshati

Graduate Theses, Dissertations, and Problem Reports

We study global existence and asymptotic behavior of the solutions for two-fluid compressible isentropic Euler-Maxwell equations by the Fourier transform and energy method. We discuss the case when the pressure for two fluids is not identical and we also add the friction between two fluids. In addition, we discuss the rates of decay of $L^{p}-L^{q}$ norms for a linear system. Moreover, we use the result for $L^{p}-L^{q}$ estimates to prove the decay rates for the nonlinear systems. In addition, we prove existence of heteroclinic orbits for the nonlinear Vlasov and the one-dimensional Vlasov-Poisson systems ...


Markov Chains And Generalized Wavelet Multiresolutions, Myung-Sin Song, Palle Jorgensen 2018 Southern Illinois University Edwardsville

Markov Chains And Generalized Wavelet Multiresolutions, Myung-Sin Song, Palle Jorgensen

SIUE Faculty Research, Scholarship, and Creative Activity

We develop some new results for a general class of transfer operators, as they are used in a construction of multi-resolutions. We then proceed to give explicit and concrete applications. We further discuss the need for such a constructive harmonic analysis/dynamical systems approach to fractals.


Equiangular Tight Frames That Contain Regular Simplices, Matthew C. Fickus, John Jasper, Emily J. King, Dustin G. Mixon 2018 Air Force Institute of Technology

Equiangular Tight Frames That Contain Regular Simplices, Matthew C. Fickus, John Jasper, Emily J. King, Dustin G. Mixon

Faculty Publications

An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. A regular simplex is a special type of ETF in which the number of vectors is one more than the dimension of the space they span. In this paper, we consider ETFs that contain a regular simplex, that is, have the property that a subset of its vectors forms a regular simplex. As we explain, such ETFs are characterized as those that achieve equality in a certain well-known bound from the theory of compressed sensing. We then consider the so-called binder of such an ...


Rediscovering The Interpersonal: Models Of Networked Communication In New Media Performance, Alicia Champlin 2018 University of Maine

Rediscovering The Interpersonal: Models Of Networked Communication In New Media Performance, Alicia Champlin

Electronic Theses and Dissertations

This paper examines the themes of human perception and participation within the contemporary paradigm and relates the hallmarks of the major paradigm shift which occurred in the mid-20th century from a structural view of the world to a systems view. In this context, the author’s creative practice is described, outlining a methodology for working with the communication networks and interpersonal feedback loops that help to define our relationships to each other and to media since that paradigm shift. This research is framed within a larger field of inquiry into the impact of contemporary New Media Art as we experience ...


R Program For Estimation Of Group Efficiency And Finding Its Gradient. Stochastic Data Envelopment Analysis With A Perfect Object Approach, Alexander Vaninsky 2018 CUNY Hostos Community College

R Program For Estimation Of Group Efficiency And Finding Its Gradient. Stochastic Data Envelopment Analysis With A Perfect Object Approach, Alexander Vaninsky

Publications and Research

The data presented here are related to the research article “Energy-environmental efficiency and optimal restructuring of the global economy” (Vaninsky, 2018) [1]. This article describes how the world economy can be restructured to become more energy-environmental efficient, while still increasing its growth potential. It demonstrates how available energy-environmental and economic information may support policy-making decisions on the atmosphere preservation and climate change prevention. This Data article presents a computer program in R language together with examples of input and output files that serve as a means of implementation of the novel approach suggested in publication[1]. The computer program utilizes ...


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