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Harmonic Analysis and Representation Commons

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Thin Blue Seam, Sebastian Anton-Ojeda 2017 Bard College

Thin Blue Seam, Sebastian Anton-Ojeda

Senior Projects Spring 2017

This performance began as an exploration of liminal spaces in the context of a post-modern world. The lines between a suburban, consumptive society, fraught with binaries and thresholds while on the fringes of nature, were what interested me the most. However, this line of thought quickly took me to far more ancient places, to a view of nature driven by animism and pervaded by spiritual introspection. In Celtic Ireland, to give an example, every stone and river is witness to a myth. The Celts of the old world as described by anthropologist Marie-Louise Sjoestedt are constantly straddling the line between ...


Compactness Of Isoresonant Potentials, Robert G. Wolf 2017 University of Kentucky

Compactness Of Isoresonant Potentials, Robert G. Wolf

Theses and Dissertations--Mathematics

Bruning considered sets of isospectral Schrodinger operators with smooth real potentials on a compact manifold of dimension three. He showed the set of potentials associated to an isospectral set is compact in the topology of smooth functions by relating the spectrum to the trace of the heat semi-group. Similarly, we can consider the resonances of Schrodinger operators with real valued potentials on Euclidean space of whose support lies inside a ball of fixed radius that generate the same resonances as some fixed Schrodinger operator, an ``isoresonant" set of potentials. This isoresonant set of potentials is also compact in the topology ...


Explicit Formulae And Trace Formulae, Tian An Wong 2016 The Graduate Center, City University of New York

Explicit Formulae And Trace Formulae, Tian An Wong

All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

In this thesis, motivated by an observation of D. Hejhal, we show that the explicit formulae of A. Weil for sums over zeroes of Hecke L-functions, via the Maass-Selberg relation, occur in the continuous spectral terms in the Selberg trace formula over various number fields. In Part I, we discuss the relevant parts of the trace formulae classically and adelically, developing the necessary representation theoretic background. In Part II, we show how show the explicit formulae intervene, using the classical formulation of Weil; then we recast this in terms of Weil distributions and the adelic formulation of Weil. As an ...


On The Free And G-Saturated Weight Monoids Of Smooth Affine Spherical Varieties For G=Sl(N), Won Geun Kim 2016 The Graduate Center, City University of New York

On The Free And G-Saturated Weight Monoids Of Smooth Affine Spherical Varieties For G=Sl(N), Won Geun Kim

All Graduate Works by Year: Dissertations, Theses, and Capstone Projects

Let $X$ be an affine algebraic variety over $\mathbb{C}$ equipped with an action of a connected reductive group $G$. The weight monoid $\Gamma(X)$ of $X$ is the set of isomorphism classes of irreducible representations of $G$ that occur in the coordinate ring $\mathbb{C}[X]$ of $X$. Losev has shown that if $X$ is a smooth affine spherical variety, that is, if $X$ is smooth and $\mathbb{C}[X]$ is multiplicity-free as a representation of $G$, then $\Gamma(X)$ determines $X$ up to equivariant automorphism.

Pezzini and Van Steirteghem have recently obtained a combinatorial characterization of the weight ...


Limiting Forms Of Iterated Circular Convolutions Of Planar Polygons, Boyan Kostadinov 2016 CUNY New York City College of Technology

Limiting Forms Of Iterated Circular Convolutions Of Planar Polygons, Boyan Kostadinov

Publications and Research

We consider a complex representation of an arbitrary planar polygon P centered at the origin. Let P(1) be the normalized polygon obtained from P by connecting the midpoints of its sides and normalizing the complex vector of vertex coordinates. We say that P(1) is a normalized average of P. We identify this averaging process with a special case of a circular convolution. We show that if the convolution is repeated many times, then for a large class of polygons the vertices of the limiting polygon lie either on an ellipse or on a star-shaped polygon. We derive a ...


Extension Theorems On Matrix Weighted Sobolev Spaces, Christopher Ryan Loga 2016 University of Tennessee, Knoxville

Extension Theorems On Matrix Weighted Sobolev Spaces, Christopher Ryan Loga

Doctoral Dissertations

Let D a subset of Rn [R n] be a domain with Lipschitz boundary and 1 ≤ p < ∞ [1 less than or equal to p less than infinity]. Suppose for each x in Rn that W(x) is an m x m [m by m] positive definite matrix which satisfies the matrix Ap [A p] condition. For k = 0, 1, 2, 3;... define the matrix weighted, vector valued, Sobolev space [L p k of D,W] with

[the weighted L p k norm of vector valued f over D to the p power equals the sum over all alpha with order less than k of the integral over D of the the pth power ...


Nonlinear Harmonic Modes Of Steel Strings On An Electric Guitar, Joel Wenrich 2016 Linfield College

Nonlinear Harmonic Modes Of Steel Strings On An Electric Guitar, Joel Wenrich

Senior Theses

Steel strings used on electric and acoustic guitars are non-ideal oscillators that can produce imperfect intonation. According to theory, this intonation should be a function of the bending stiffness of the string, which is related to the dimensions of length and thickness of the string. To test this theory, solid steel strings of three different linear densities were analyzed using an oscilloscope and a Fast Fourier Transform function. We found that strings exhibited more drastic nonlinear harmonic behavior as their effective length was shortened and as linear density increased.


Unions Of Lebesgue Spaces And A1 Majorants, Greg Knese, John E. McCarthy, Kabe Moen 2016 Washington University in St. Louis

Unions Of Lebesgue Spaces And A1 Majorants, Greg Knese, John E. Mccarthy, Kabe Moen

Mathematics Faculty Publications

We study two questions. When does a function belong to the union of Lebesgue spaces, and when does a function have anA1majorant? We provide a systematic study of these questions and show that they are fundamentally related. We show that the union ofLwp(ℝn)spaces withw∈Apis equal to the union of all Banach function spaces for which the Hardy–Littlewood maximal function is bounded on the space itself and its associate space.


Tessellations: An Artistic And Mathematical Look At The Work Of Maurits Cornelis Escher, Emily E. Bachmeier 2016 University of Northern Iowa

Tessellations: An Artistic And Mathematical Look At The Work Of Maurits Cornelis Escher, Emily E. Bachmeier

Honors Program Theses

The purpose of this study was to learn more about the mathematics of tessellations and their artistic potential. Whenever I have seen tessellations, I always admire them. It is baffling how such complicated shapes can be repeated infinitely on the plane. This research aimed to increase the amount of tessellation information and activities available to secondary mathematics teachers by connecting the tessellations of Maurits Cornelis Escher with their underlying mathematics in order to use them for teaching secondary mathematics. The overarching premise of this study was to create something that would also be beneficial in my career as a secondary ...


Filters And Matrix Factorization, Myung-Sin Song, Palle E. T. Jorgensen 2015 Southern Illinois University Edwardsville

Filters And Matrix Factorization, Myung-Sin Song, Palle E. T. Jorgensen

SIUE Faculty Research, Scholarship, and Creative Activity

We give a number of explicit matrix-algorithms for analysis/synthesis

in multi-phase filtering; i.e., the operation on discrete-time signals which

allow a separation into frequency-band components, one for each of the

ranges of bands, say N , starting with low-pass, and then corresponding

filtering in the other band-ranges. If there are N bands, the individual

filters will be combined into a single matrix action; so a representation of

the combined operation on all N bands by an N x N matrix, where the

corresponding matrix-entries are periodic functions; or their extensions to

functions of a complex variable. Hence our setting ...


Wave Packet Transform Over Finite Fields, Arash Ghaani Farashahi 2015 Faculty of Mathematics, University of Vienna

Wave Packet Transform Over Finite Fields, Arash Ghaani Farashahi

Electronic Journal of Linear Algebra

In this article we introduce the notion of finite wave packet groups over finite fields as the finite group of dilations, translations, and modulations. Then we will present a unified theoretical linear algebra approach to the theory of wave packet transform (WPT) over finite fields. It is shown that each vector defined over a finite field can be represented as a coherent sum of finite wave packet group elements as well.


Reproducing Kernel Hilbert Space Vs. Frame Estimates, Palle E. T. Jorgensen, Myung-Sin Song 2015 The University of Iowa

Reproducing Kernel Hilbert Space Vs. Frame Estimates, Palle E. T. Jorgensen, Myung-Sin Song

SIUE Faculty Research, Scholarship, and Creative Activity

We consider conditions on a given system F of vectors in Hilbert space H, forming a frame, which turn Hinto a reproducing kernel Hilbert space. It is assumed that the vectors in F are functions on some set Ω . We then identify conditions on these functions which automatically give H the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.


The Structure And Unitary Representations Of Su(2,1), Andrew J. Pryhuber 2015 Bowdoin College

The Structure And Unitary Representations Of Su(2,1), Andrew J. Pryhuber

Honors Projects

No abstract provided.


Discrete Fourier Analysis And Wavelets Applications To Signal And Image Processing, Sean Broughton, Kurt Bryan 2014 Rose-Hulman Institute of Technology

Discrete Fourier Analysis And Wavelets Applications To Signal And Image Processing, Sean Broughton, Kurt Bryan

S. Allen Broughton

Discrete Fourier Analysis and Wavelets presents a thorough introduction to the mathematical foundations of signal and image processing. Key concepts and applications are addressed in a thought-provoking manner and are implemented using vector, matrix, and linear algebra methods. With a balanced focus on mathematical theory and computational techniques, this self-contained book equips readers with the essential knowledge needed to transition smoothly from mathematical models to practical digital data applications.(from Wiley website)


A New Subgroup Chain For The Finite Affine Group, David Alan Lingenbrink Jr. 2014 Harvey Mudd College

A New Subgroup Chain For The Finite Affine Group, David Alan Lingenbrink Jr.

HMC Senior Theses

The finite affine group is a matrix group whose entries come from a finite field. A natural subgroup consists of those matrices whose entries all come from a subfield instead. In this paper, I will introduce intermediate sub- groups with entries from both the field and a subfield. I will also examine the representations of these intermediate subgroups as well as the branch- ing diagram for the resulting subgroup chain. This will allow us to create a fast Fourier transform for the group that uses asymptotically fewer opera- tions than the brute force algorithm.


Fast Algorithms For Analyzing Partially Ranked Data, Matthew McDermott 2014 Harvey Mudd College

Fast Algorithms For Analyzing Partially Ranked Data, Matthew Mcdermott

HMC Senior Theses

Imagine your local creamery administers a survey asking their patrons to choose their five favorite ice cream flavors. Any data collected by this survey would be an example of partially ranked data, as the set of all possible flavors is only ranked into subsets of the chosen flavors and the non-chosen flavors. If the creamery asks you to help analyze this data, what approaches could you take? One approach is to use the natural symmetries of the underlying data space to decompose any data set into smaller parts that can be more easily understood. In this work, I describe how ...


On Local Fractional Continuous Wavelet Transform, Yang Xiaojun 2013 D. Baleanu

On Local Fractional Continuous Wavelet Transform, Yang Xiaojun

Xiao-Jun Yang

We introduce a new wavelet transform within the framework of the local fractional calculus. An illustrative example of local fractional wavelet transform is also presented.


Local Fractional Discrete Wavelet Transform For Solving Signals On Cantor Sets, Yang Xiaojun 2013 Y. Zhao

Local Fractional Discrete Wavelet Transform For Solving Signals On Cantor Sets, Yang Xiaojun

Xiao-Jun Yang

The discrete wavelet transform via local fractional operators is structured and applied to process the signals on Cantor sets. An illustrative example of the local fractional discretewavelet transformis given.


Characterization Of The Drilling Via The Vibration Augmenter Of Rotary-Drills And Sound Signal Processing Of Impacted Pipe As A Potential Water Height Assessment Tool, Nicholas Morris 2013 California Polytechnic State University, San Luis Obispo

Characterization Of The Drilling Via The Vibration Augmenter Of Rotary-Drills And Sound Signal Processing Of Impacted Pipe As A Potential Water Height Assessment Tool, Nicholas Morris

STAR (STEM Teacher and Researcher) Presentations

The focus of the internship has been on two topics: a) Characterize the drilling performance of a novel percussive augmenter – this drill was developed by the JPL’s Advanced Technologies Group and its performance was characterized; and b) Examine the feasibility of striking a pipe as a means of assessing the water height inside the pipe. The purpose of this investigation is to examine the possibility of using a simple method of applying impacts to a pipe wall and determining the water height from the sonic characteristic differences including damping, resonance frequencies, etc. Due to multiple variables that are relevant ...


Mathematical Aspects Of Heisenberg Uncertainty Principle Within Local Fractional Fourier Analysis, Yang Xiaojun 2013 D. Baleanu

Mathematical Aspects Of Heisenberg Uncertainty Principle Within Local Fractional Fourier Analysis, Yang Xiaojun

Xiao-Jun Yang

In this paper, we discuss the mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis. The Schrödinger equation and Heisenberg uncertainty principles are structured within local fractional operators.


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