Harmonic Analysis and Representation Commons^™

Fundamental Solutions To The Fractional Heat Operator, Jacob Flores

Math Theses

In this thesis, we are interested in showing the existence of a fundamental solution to the fractional heat operator. The fractional heat operator is a nonlocal linear operator used to model the time evolution of anomalous diffusion processes whose applicability arises in a wide variety of fields in the physical sciences, engineering, economics, and finance. Fundamental solutions to a partial differential operator are a class of generalized solutions formulated with rich mathematical analysis grounded in classes of well-behaved smooth functions referred to as test functions and their continuous linear functionals referred to as distributions. A primary tool that we use …

Cdt-1d Cnn Integration With Simpson-Sobolev Regularization For High-Frequency Options Trading: With Fem-Based Heston Option Pricing, Daniel M. Margolis, Johannes Tausch, Arthur K. Selender

Mathematics Theses and Dissertations

This dissertation presents a computational framework for high-frequency options trading that combines Cross-Data-Type 1-D Convolutional Neural Networks (CDT-1D CNN) with Simpson-Sobolev regularization for directional prediction, and finite element methods (FEM) for realistic option pricing during backtesting. The core innovation lies in developing a mathematically rigorous regularization approach that maintains the adaptability of modern deep learning while enabling accurate evaluation through stochastic volatility models. The primary contribution is the Simpson-Sobolev regularization scheme, which extends traditional Sobolev regularization by incorporating Simpson’s rule for numerical integration. This approach achieves higher-order accuracy in approximating the Sobolev norms that control function smoothness. Simpson’s rule attains …

Quiver Of Affine Monoid Of A Vector Space Over Finite Field, James Junie Chen Cleary

Dissertations, Theses, and Capstone Projects

In this paper, we study the quiver of the complex monoid algebra CAFF(n, q). There are n + 1 maximal subgroups of AFF(n, q), each isomorphic to AGL(k, q) for some 0 ≤ k ≤ n. Every irreducible representation of CAFF(n, q) arises from a character of CAGL(k, q) for a suitable k. Thus, we study two different approaches to classifying the characters of CAGL(k, q). Next, we compute the full quiver Q(CAFF(n, q)). Finally, we show that this quiver is a disjoint union of straight-line paths and that its basic algebra has radical square zero. Hence, it has finite …

Constructing Orthonormal Bases With The Residuals Of Successive Approximations, An Introduction To Multiresolution Analysis, Elijah J. Guptill

Master's Theses

Wavelets and wavelet analysis are used in the study of signal processing, quantum field theory, functional analysis, multifractal analysis, and various other areas of mathematics. Multiresolution analysis provides a framework for building a wavelet basis of $\mathcal{L}^{2}(\mathbb{R})$ from a scaling function $\phi$, whose dyadic dilations and translations, $\{2^{j /2}\phi(2^{j}x-k):j,k\in \mathbb{Z}\}$, approximate $\mathcal{L}^{2}(\mathbb{R})$. One of the key properties of  $\phi$ is that it must satisfy $\phi(x)=\sum_{k\in \mathbb{Z}}{p_{k}2^{j /2}\phi(2^{j}x-k)}$ with respect to the norm on $\mathcal{L}^{2}(\mathbb{R})$. This equation is called a two-scale difference equation. Such equations enforce a regularity on the ordinary generating function $2^{-1 /2}\sum_{k\in \mathbb{Z}}{p_{k}z^{k}}$, known as the quadrature condition. …

The Isoperimetric Inequality And Wirtinger’S Inequality, Mason Neal

Honors Theses

In this thesis, we present Hurwitz’s proof of the Isoperimetric Inequality, which roughly states that the area enclosed by a simple closed curve is always less than or equal to the area of a circle with the same perimeter. Hurwitz’s proof relies on Wirtinger’s Inequality. We survey results about periodic functions and Fourier series, and we use them to provide a proof of Wirtinger’s Inequality. We then give a new proof of a variant of Wirtinger’s Inequality due to Alzer and generalize this variant to higher powers.

Numerical And Harmonic Analysis Of Simplex Number Parity, Hunter Dm Hannula

All NMU Master's Theses

The primary object of this thesis is the study of periodicity in the parity of simplex numbers by number-theoretic and harmonic methods. The regular d-simplex numbers are introduced geometrically, arithmetically, and combinatorially. We demonstrate that all sequences indexing even d-simplex numbers are defined by finitely many congruences in a single modulus, and are thus quasiperiodic. We therefore show that these "even index-sequences" are  particular elements in an affine space of functions, providing a natural decomposition result. We then introduce the discrete Fourier transform to construct the periodic parts of each index sequence, enabling the development of explicit forms for the …

Fourier Analysis Of Electronic Synthesizer Waveforms, Jiayan Ling, Jack W. Maseberg

SACAD: Scholarly Activities

We compute Fourier series coefficients for some standard electronic synthesizer waveforms and provide plots of our results.

(Si15-084) Application Of Differentiation Matrices Corresponding To Scaling And Wavelet Functions To Integral Equations, Viresh Kumar, Rakesh Kumar

Applications and Applied Mathematics: An International Journal (AAM)

In the present paper, we have studied the technique by which a differentiation matrix corresponding to scaling and wavelet function is helpful in representing an integral equation in discrete form. In this technique, only finite times continuously differentiable scaling and wavelet functions were used for the differentiation matrix. This method explores the applications of compactly supported wavelets in discretization of Fredholm integral equation and provides us with a bridge for the solution of integral equations via compactly supported finitely differentiable wavelets.

(Si15-010) On Perturbations Of Gabor Frames, Jamkhongam Touthang

Applications and Applied Mathematics: An International Journal (AAM)

Stability plays a crucial role in frame theory and its applications. The present paper studies the interaction between Gabor frames and perturbations, presenting perturbation results related to small changes of the frame parameters and the window functions both in regular and irregular settings. Examples are provided for illustration. Additionally, the paper briefly discusses algorithms pertinent to Gabor frames under perturbations and highlights challenging areas in the field.

Fourier Analyses Of Optical Profilometry As An Inferential Measurement For Impact Coverage., Langdon Feltner, Paul Mort

15th International Conference on Shot Peening

A critical consideration in peening process design is achieving sufficient impact coverage. Conventional methods for assessing coverage rely on manual inspection, which is time-consuming and poorly suited for automated control. In this work, we investigate the use of frequency-domain analysis to quantify surface modification in peened samples using optical profilometry (OP) data. Three-dimensional surface maps of Almen strips were acquired using a high-resolution OP system and analyzed via fast Fourier transform (FFT) to compute spatial power spectral densities (PSDs). PSD maps and radially averaged profiles reveal consistent amplification of harmonic components similar to the nominal particle size, with increasing intensity …

Langlands Reciprocity And The Splitting Behavior Of Primes In Number Fields, Skip E. Moses

Master's Theses

This thesis explores the evolution of reciprocity laws in number theory in order to provide a conceptual bridge between the classical ideas of quadratic reciprocity and the modern framework of the Langlands program. We develop the necessary algebraic background to understand how the splitting behavior of primes in number fields reflects deep arithmetic structure in ℚ. Starting with quadratic fields and cyclotomic extensions, we motivate the development of the Kronecker–Weber theorem and the characterization of abelian extensions of ℚ. We then introduce Artin reciprocity and show how it generalizes quadratic reciprocity through the formalism of Frobenius elements and Artin L-functions. …

Property Testing Ai: An Efficient Frontier, Paul Sopher Lintilhac

Dartmouth College Ph.D Dissertations

In this dissertation, we take a step towards addressing the major problem of a lack of standardized and rigorous approaches to testing and evaluation of AI systems. Taking inspiration from both the fields of Property Testing and Property Based Testing (for programs), we develop a novel taxonomy of partially overlapping classes of properties of AI systems, including simple properties, compound properties, higher order properties, data relation properties, and architecture-utility properties. We argue that this taxonomy categorizes a diverse set of AI traits -- including accuracy, fairness, robustness, monotonicity, point-wise and global privacy properties, sensitivity, and more -- according to the …

Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie

Master's Theses

A Banach algebra is a complex algebra that is simultaneously a Banach space in which the norm is submultiplicative. Notably, $L^1(\mathbb{R})$ with the convolutional product is an Abelian, non-unital Banach algebra that admits an approximate identity. We rectify $L^1(\mathbb{R})$ lacking a unit via the unitization $L^1(\mathbb{R})\times\mathbb{C}$ with identity $(0,1)$. Unitization opens the discussion to the spectrum $\sigma(x)$ of a Banach algebra element, in which the spectrum is a nonempty, compact subset of the complex plane. The spectrum of an Abelian Banach algebra is fully characterized with multiplicative linear functionals, and we prove that the Fourier transform is the unique multiplicative …

The Spectral Asymptotics Of Toeplitz Operators On Hilbert Spaces Of Analytic Functions, Trevor Camper

All Dissertations

Many physical systems, whether they are ocean waves or particles moving through space, can be described using the mathematical language of “partial differ- ential equations.” In many circumstances, it is useful to study how these equations amplify an input to the equation, in which case the amplification factor is called an “eigenvalue.” The usefulness of these amplification factors is that they can be used to describe properties of the physical system. In this dissertation, I have studied this amplification factor for a related set of equations called “Toeplitz operators.” In par- ticular, I have studied eigenvalues using statistical techniques. The …

Stability Criteria For The Generalized El Borhamy-Rashad-Sobhy Equation, Mohamed El-Borhamy Assoc.Prof, Essam Eddin Rashad Prof., Fathi Mousa Dr., Mai Hamouda

Journal of Engineering Research

This article is concerned with the study of stability criteria for one of the generalization form of El Borhamy-Rashad Sobhy equation, which is a linear second-order ordinary differential equation with periodically time varying coefficients. Many engineering applications can be represented by this generalization, for instance, including the modeling of RLC circuit with time varying inductance, resistance and capacitance, and the vibration of a stretched string, whose mass per unit length is periodic, under a periodic motion. An approximate solution is derived by using the Wenhl-Kramers-Brillonin (WKB) approach. A method of constructing Liapunov function is employed to derive extra conditions for …

Local Limit Theorems On Finitely Generated Abelian Groups, Yutong Yan

Honors Theses

In this thesis, we classify the pointwise behavior of finite-range random walks on finitely generated abelian groups in terms of local limit theorems. Random walks are central objects of research in probability theory, and the theory has found applications in statistics, physics, and even card shuffling. One significant topic in this line of study is random walks on finitely generated groups. Starting from the pioneering work of G. Pólya and H. Kesten, random walks on finitely generated groups have been studied extensively. However, many notable results on the subject (local limit theorems, for example) make assumptions about periodicity and irreducibility …

Probabilistic Frames And Concepts From Optimal Transport, Dongwei Chen

All Dissertations

As the generalization of frames in the Euclidean space $\mathbb{R}^n$, a probabilistic frame is a probability measure on $\mathbb{R}^n$ that has a finite second moment and whose support spans $\mathbb{R}^n$. The p-Wasserstein distance with $p \geq 1$ from optimal transport is often used to compare probabilistic frames. It is particularly useful to compare frames of various cardinalities in the context of probabilistic frames. We show that the 2-Wasserstein distance appears naturally in the fundamental objects of frame theory and draws consequences leading to a geometric viewpoint of probabilistic frames.

We convert the classic lower bound estimates of 2-Wasserstein distance \cite{Gelbrich90, …

A Computational Investigation Of Wood Selection For Acoustic Guitar, Jonah Osterhus

Senior Honors Theses

The acoustic guitar is a stringed instrument, often made of wood, that transduces vibrational energy of steel strings into coupled vibrations of the wood and acoustic pressure waves in the air. Variations in wood selection and instrument geometry have been shown to affect the timbre of the acoustic guitar. Computational methods were utilized to investigate the impact of material properties on acoustic performance. Sitka spruce was deemed the most suitable wood for guitar soundboards due to its acoustic characteristics, strength, and uniform aesthetic. Mahogany was deemed to be the best wood for the back and sides of the guitar body …

Analytic Wavefront Sets Of Spherical Distributions On The De Sitter Space, Iswarya Sitiraju

LSU Doctoral Dissertations

In this work, we determine the wavefront set of certain eigendistributions of the Laplace-Beltrami operator on the de Sitter space. Let G′ = O_1,n(R) be the Lorentz group, and let H′ = O_1,n−1(R) ⊂ G′ be its subset. The de Sitter space dSⁿ is a one-sheeted hyperboloid in R^1,n isomorphic to G′/H′. A spherical distribution is an H′-invariant eigendistribution of the Laplace-Beltrami operator on dSⁿ. The space of spherical distributions with eigenvalue λ, denoted by D_λ^H'(dSⁿ), has dimension 2. We construct a basis for the space of …

The Modular Generalized Springer Correspondence For The Symplectic Group, Joseph Dorta

LSU Doctoral Dissertations

The Modular Generalized Springer Correspondence (MGSC), as developed by Achar, Juteau, Henderson, and Riche, stands as a significant extension of the early groundwork laid by Lusztig's Springer Correspondence in characteristic zero which provided crucial insights into the representation theory of finite groups of Lie type. Building upon Lusztig's work, a generalized version of the Springer Correspondence was later formulated to encompass broader contexts.

In the realm of modular representation theory, Juteau's efforts gave rise to the Modular Springer Correspondence, offering a framework to explore the interplay between algebraic geometry and representation theory in positive characteristic. Achar, Juteau, Henderson, and Riche …

Modeling Inflation Using A Fast Fourier Transform (Fft), Blake Smith

Williams Honors College, Honors Research Projects

This paper utilizes a Fast Fourier Transform (FFT) algorithm to construct a trigonometric interpolant for the Consumer Price Index (CPI), which is then differentiated and used to obtain a continuous function for “instantaneous” (i.e., month-wise) inflation, as opposed to a 12-month percent-change. Fourier coefficients are analyzed to investigate underlying periodicities in the newly constructed function. This metric does not hold significant predictive value but it may prove helpful in retroactive analysis of inflation trends.

Fuglede's Conjecture In Some Finite Abelian Groups, Thomas Fallon

Dissertations, Theses, and Capstone Projects

This dissertation thoroughly examines Fuglede's Conjecture within some discrete settings, shedding light on its intricate details. Fuglede's Conjecture establishes a profound connection between the geometric property of being a tiling set and the analytical attribute of being a spectral set. By exploring the conjecture on various discrete settings, this thesis delves into the implications and ramifications of the conjecture, unraveling its implications within the field.

On The Spectrum Of Quaquaversal Operators, Josiah Sugarman

Dissertations, Theses, and Capstone Projects

In 1998 Charles Radin and John Conway introduced the Quaquaversal Tiling. A three dimensional hierarchical tiling with the property that the orientations of its tiles approach a uniform distribution faster than what is possible for hierarchical tilings in two dimensions. The distribution of orientations is controlled by the spectrum of a certain Hecke operator, which we refer to as the Quaquaversal Operator. For example, by showing that the largest eigenvalue has multiplicity equal to one, Charles Radin and John Conway showed that the orientations of this tiling approach a uniform distribution. In 2008, Bourgain and Gamburd showed that this operator …

Understanding The Beauty Of Mathematics By Composing Claude Debussy's Syrinx Into Mathematical Equations, Mackenzi Mehlberg

Honors Projects

Mesmerizing melodies and narrative storytelling are exemplified in Claude Debussy's Syrinx. As a well-known piece of solo flute literature, it is considered beautiful. Conversely, mathematics is seen as logical, and by implication not beautiful. Using Fourier Analysis, Syrinx can be represented in a different context: a series of mathematical equations. These mathematical equations can then be played as a different interpretation of Syrinx. With this interpretation, we see that mathematics is beautiful.

(R2028) A Brief Note On Space Time Fractional Order Thermoelastic Response In A Layer, Navneet Lamba, Jyoti Verma, Kishor Deshmukh

Applications and Applied Mathematics: An International Journal (AAM)

In this study, a one-dimensional layer of a solid is used to investigate the exact analytical solution of the heat conduction equation with space-time fractional order derivatives and to analyze its associated thermoelastic response using a quasi-static approach. The assumed thermoelastic problem was subjected to certain initial and boundary conditions at the initial and final ends of the layer. The memory effects and long-range interaction were discussed with the help of the Caputo-type fractional-order derivative and finite Riesz fractional derivative. Laplace transform and Fourier transform techniques for spatial coordinates were used to investigate the solution of the temperature distribution and …

Analytic Continuation Of Toeplitz Operators And Commuting Families Of C*-Algebras, Khalid Bdarneh

LSU Doctoral Dissertations

In this thesis we consider the Toeplitz operators on the weighted Bergman spaces and their analytic continuation. We proved the commutativity of the $C^*-$algebras generated by the analytic continuation of Toeplitz operators with special class of symbols that are invariant under suitable subgroups of $SU(n,1)$, and we showed that commutative $C^*-$algebras with symbols invariant under compact subgroups of $SU(n,1)$ are completely characterized in terms of restriction to multiplicity free representations. Moreover, we extended the restriction principal to the analytic continuation case for suitable maximal abelian subgroups of the universal covering group $\widetilde{SU(n,1)}$, and we obtained the generalized Segal-Bargmann transform, where …

Music: Numbers In Motion, Graziano Gentili, Luisa Simonutti, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

Music develops and appears as we allow numbers to acquire a dynamical aspect and create, through their growth, the various keys that permit the richness of the musical texture. This idea was simply adumbrated in Plato’s work, but its importance to his philosophical worldview cannot be underestimated. In this paper we begin by discussing what is probably the first written record of an attempt to create a good temperament and then follow the Pythagoreans approach, whose problems forced musicians, over the next several centuries up to the Renaissance and early modern times, to come up with many different variations.

Long Increasing Subsequences, Hannah Friedman

HMC Senior Theses

In my thesis, I investigate long increasing subsequences of permutations from two angles. Motivated by studying interpretations of the longest increasing subsequence statistic across different representations of permutations, we investigate the relationship between reduced words for permutations and their RSK tableaux in Chapter 3. In Chapter 4, we use permutations with long increasing subsequences to construct a basis for the space of ��-local functions.

(Si10-068) Performance Analysis Of Cosine Window Function, Vikas Misra, Narendra Singh, M. Shukla

Applications and Applied Mathematics: An International Journal (AAM)

This paper reviews the mathematical functions called the window functions which are employed in the Finite Impulse Response (FIR) filter design applications as well as spectral analysis for the detection of weak signals. The characteristic properties of the window functions are analyzed and parameters are compared among the known conventional cosine window (CW) functions (Rectangular, Hamming, Hanning, and Blackman) and the variable Kaiser window function. The window function expressed in the time domain can be transformed into the frequency domain by taking the Discrete Fourier Transform (DFT) of the time domain window function. The frequency response of the window function …

Bbt Acoustic Alternative Top Bracing Cadd Data Set-Norev-2022jun28, Bill Hemphill

STEM Guitar Project’s BBT Acoustic Kit

This electronic document file set consists of an overview presentation (PDF-formatted) file and companion video (MP4) and CADD files (DWG & DXF) for laser cutting the ETSU-developed alternate top bracing designs and marking templates for the STEM Guitar Project’s BBT (OM-sized) standard acoustic guitar kit. The three (3) alternative BBT top bracing designs in this release are
(a) a one-piece base for the standard kit's (Martin-style) bracing,
(b) 277 Ladder-style bracing, and
(c) an X-braced fan-style bracing similar to traditional European or so-called 'classical' acoustic guitars.

The CADD data set for each of the three (3) top bracing designs includes …