Harmonic Analysis and Representation Commons^™

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

Department of Mathematics Faculty 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, 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.

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

UNO 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 …

Operator Algebras Generated By Left Invertibles, Derek Desantis

Department of Mathematics: Dissertations, Theses, and Student Research

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 of the Toeplitz algebra. Of particular interest …

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

Graduate Theses, Dissertations, and Problem Reports (ETD)

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. In the nonlinear Vlasov case with …