Probabilistic Frames And Concepts From Optimal Transport, 2024 Clemson University

#### 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, 2024 Liberty University

#### 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, 2024 Louisiana State University and Agricultural and Mechanical College

#### 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 dS^{n} 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 dS^{n}. The space of spherical distributions with eigenvalue λ, denoted by D_{λ}^{H'}(dS^{n}), has dimension 2. We construct a basis for the space of …

The Modular Generalized Springer Correspondence For The Symplectic Group, 2024 Louisiana State University

#### 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), 2024 The University of Akron

#### 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, 2023 The Graduate Center, City University of New York

#### 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, 2023 The Graduate Center, City University of New York

#### 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, 2023 Seattle Pacific University

#### 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, 2023 Shri Lemdeo Patil Mahavidyalaya,Mandhal

#### (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, 2023 Louisiana State University and Agricultural and Mechanical College

#### 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, 2023 Università degli Studi di Firenze

#### 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, 2023 Claremont Colleges

#### 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, 2022 Jaypee University of Engineering and Technology

#### (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, 2022 East Tennessee State University

#### 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 …

Commutative C*-Algebras Generated By Toeplitz Operators On The Fock Space, 2022 Louisiana State University and Agricultural and Mechanical College

#### Commutative C*-Algebras Generated By Toeplitz Operators On The Fock Space, Vishwa Nirmika Dewage

*LSU Doctoral Dissertations*

The Fock space $\mathcal{F}(\mathbb{C}^n)$ is the space of holomorphic functions on $\mathbb{C}^n$ that are square-integrable with respect to the Gaussian measure on $\mathbb{C}^n$. This space plays an essential role in several subfields of analysis and representation theory. In particular, it has for a long time been a model to study Toeplitz operators. Grudsky and Vasilevski showed in 2002 that radial Toeplitz operators on $\mathcal{F}(\mathbb{C})$ generate a commutative $C^*$-algebra $\mathcal{T}^G$, while Esmeral and Maximenko showed that $C^*$-algebra $\mathcal{T}^G$ is isometrically isomorphic to the $C^*$-algebra $C_{b,u}(\mathbb{N}_0,\rho_1)$. In this thesis, we extend the result to $k$-quasi-radial symbols acting on the Fock space $\mathcal{F}(\mathbb{C}^n)$. …

Diederich-Fornæss Index On Boundaries Containing Crescents, 2022 University of Arkansas, Fayetteville

#### Diederich-Fornæss Index On Boundaries Containing Crescents, Jason Demoulpied

*Graduate Theses and Dissertations*

The worm domain developed by Diederich and Fornæss is a classic example of a boundedpseudoconvex domains that fails to satisfy global regularity of the Bergman Projection, due to the set of weakly pseudoconvex points that form an annulus in its boundary. We instead examine a bounded pseudoconvex domain Ω ⊂ C2 whose set of weakly pseudoconvex points form a crescent in its boundary. In 2019, Harrington had shown that these types of domains satisfy global regularity of the Bergman Projection based on the existence of good vector fields. In this thesis we study the Regularized Diederich-Fornæss index of these domains, …

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 …

Interpolation And Sampling In Analytic Tent Spaces, 2021 University of Arkansas, Fayetteville

#### Interpolation And Sampling In Analytic Tent Spaces, Caleb Parks

*Graduate Theses and Dissertations*

Introduced by Coifman, Meyer, and Stein, the tent spaces have seen wide applications in harmonic analysis. Their analytic cousins have seen some applications involving the derivatives of Hardy space functions. Moreover, the tent spaces have been a recent focus of research. We introduce the concept of interpolating and sampling sequences for analytic tent spaces analogously to the same concepts for Bergman spaces. We then characterize such sequences in terms of Seip's upper and lower uniform density. We accomplish this by exploiting a kind of Mobius invariance for the tent spaces.

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 …

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 mappings …