Mathematics Commons^™

Bifurcations And Resultants For Rational Maps And Dynatomic Modular Curves In Positive Characteristic, Colette Lapointe

Dissertations, Theses, and Capstone Projects

No abstract provided.

Matrix Approximation And Image Compression, Isabella R. Padavana

Master's Theses

This thesis concerns the mathematics and application of various methods for approximating matrices, with a particular eye towards the role that such methods play in image compression. An image is stored as a matrix of values with each entry containing a value recording the intensity of a corresponding pixel, so image compression is essentially equivalent to matrix approximation. First, we look at the singular value decomposition, one of the central tools for analyzing a matrix. We show that, in a sense, the singular value decomposition is the best low-rank approximation of any matrix. However, the singular value decomposition has some …

On An Equivalence Of Divisors On M̄0,N From Gromov-Witten Theory And Conformal Blocks, Linda Chen

Mathematics & Statistics Faculty Works

We consider a conjecture that identifies two types of base point free divisors on M̄_0,n. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated with simple Lie algebras in type A. Here we reduce this conjecture on M̄_0,n to the same statement for n = 4. A reinterpretation leads to a proof of the conjecture on M̄_0,n for a large class, and we give sufficient conditions for the non-vanishing of these divisors.

New Examples Of Self-Dual Near-Extremal Ternary Codes Of Length 48 Derived From 2-(47,23,11) Designs, Sanja Rukavina, Vladimir Tonchev

Michigan Tech Publications

In a recent paper (Araya and Harada, 2023), Araya and Harada gave examples of self-dual near-extremal ternary codes of length 48 for 145 distinct values of the number A12 of codewords of minimum weight 12, and raised the question about the existence of codes for other values of A12. In this note, we use symmetric 2-(47,23,11) designs with an automorphism group of order 6 to construct self-dual near-extremal ternary codes of length 48 for 150 new values of A12.

(R2067) Solutions Of Hyperbolic System Of Time Fractional Partial Differential Equations For Heat Propagation, Sagar Sankeshwari, Vinayak Kulkarni

Applications and Applied Mathematics: An International Journal (AAM)

Hyperbolic linear theory of heat propagation has been established in the framework of a Caputo time fractional order derivative. The solution of a system of integer and fractional order initial value problems is achieved by employing the Adomian decomposition approach. The obtained solution is in convergent infinite series form, demonstrating the method’s strengths in solving fractional differential equations. Moreover, the double Laplace transform method is employed to acquire the solution of a system of integer and fractional order boundary conditions in the Laplace domain. An inversion of double Laplace transforms has been achieved numerically by employing the Xiao algorithm in …

(R2074) A Comparative Study Of Two Novel Analytical Methods For Solving Time-Fractional Coupled Boussinesq-Burger Equation, Jyoti U. Yadav, Twinkle R. Singh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a comparative study between two different methods for solving nonlinear timefractional coupled Boussinesq-Burger equation is conducted. The techniques are denoted as the Natural Transform Decomposition Method (NTDM) and the Variational Iteration Transform Method (VITM). To showcase the efficacy and precision of the proposed approaches, a pair of different numerical examples are presented. The outcomes garnered indicate that both methods exhibit robustness and efficiency, yielding approximations of heightened accuracy and the solutions in a closed form. Nevertheless, the VITM boasts a distinct advantage over the NTDM by addressing nonlinear predicaments without recourse to the application of Adomian polynomials. …

(R2087) Modal Operators On Bipolar Intuitionistic Fuzzy Matrices, T. Muthuraji, P. Punitha Elizabeth

Applications and Applied Mathematics: An International Journal (AAM)

Bipolar intuitionistic fuzzy sets are currently a robust area in several industries. Additionally, bipolar intuitionistic fuzzy matrices are a highly regarded topic in many fields including psychology, engineering, qualitative reasoning and multi-criteria decision making. A civilization must deal with negative and positive problems. In this article, we discussed some algebraic properties of modal operators on bipolar intuitionistic fuzzy matrices. Finally, we derive some results of necessity and possibility operators along with Max-Min product.

Hyperbolic Groups And The Word Problem, David Wu

Master's Theses

Mikhail Gromov’s work on hyperbolic groups in the late 1980s contributed to the formation of geometric group theory as a distinct branch of mathematics. The creation of hyperbolic metric spaces showed it was possible to define a large class of hyperbolic groups entirely geometrically yet still be able to derive significant algebraic properties. The objectives of this thesis are to provide an introduction to geometric group theory through the lens of quasi-isometry and show how hyperbolic groups have solvable word problem. Also included is the Stability Theorem as an intermediary result for quasi-isometry invariance of hyperbolicity.

Parabolic And Non-Parabolic Surfaces With Small Or Large End Spaces Via Fenchel-Nielsen Parameters, Michael Antony Pandazis

Dissertations, Theses, and Capstone Projects

We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface X that determine whether or not a surface X is parabolic. Fix a geodesic pants decomposition of a surface and call the boundary geodesics in the decomposition cuffs. For a zero or half-twist flute surface, we prove that parabolicity is equivalent to the surface having a covering group of the first kind. Using that result, we give necessary and sufficient conditions on the Fenchel-Nielsen parameters of a half-twist flute surface X with increasing cuff lengths such that X is parabolic. As an application, we determine whether or not each …

Sigma_N-Correct Forcing Axioms, Benjamin P. Goodman

Dissertations, Theses, and Capstone Projects

I introduce a new family of axioms extending ZFC set theory, the Sigma_n-correct forcing axioms. These assert roughly that whenever a forcing name a' can be forced by a poset in some forcing class Gamma to have some Sigma_n property phi which is provably preserved by all further forcing in Gamma, then a' reflects to some small name such that there is already in V a filter which interprets that small name so that phi holds. Sigma_1-correct forcing axioms turn out to be equivalent to classical forcing axioms, while Sigma_2-correct forcing axioms for Sigma_2-definable forcing classes are consistent relative to …

Explicit Composition Identities For Higher Composition Laws In The Quadratic Case, Ajith A. Nair

Dissertations, Theses, and Capstone Projects

The theory of Gauss composition of integer binary quadratic forms provides a very useful way to compute the structure of ideal class groups in quadratic number fields. In addition to that, Gauss composition is also important in the problem of representations of integers by binary quadratic forms. In 2001, Bhargava discovered a new approach to Gauss composition which uses 2x2x2 integer cubes, and he proved a composition law for such cubes. Furthermore, from the higher composition law on cubes, he derived four new higher composition laws on the following spaces - 1) binary cubic forms, 2) pairs of binary quadratic …

Cellular Automata Modeling Approach Of Addiction, Ruba Hameed

Theses

The purpose of this thesis includes the application of a mathematical model, as well as cell automata simulation, to shed light on the behavior of addiction spread in a population. The model describes the key determinants of drug addiction which includes peer influence, availability of substances, support networks, and awareness campaigns. The research aims to gain insights into the underlying mechanisms and patterns of addiction spread to develop effective prevention and intervention strategies.

The study begins by creating a discrete dynamical system that captures the transitions between 4 different states of substance use, including non-users (N), experimental users (E), recreational …

Mathematical Modeling Of An Epidemic In Scalefree Network With Imperfect Vaccination, Heba Hameed

Theses

In light of the recent COVID-19 pandemic, the mathematical epidemiological model has proven to be essential for understanding the disease dynamic and finding the best control tool to help contain the disease and minimize its impact. This thesis investigates the dynamics of an infectious disease spread with latent infection and vaccination. The aim is to study this model’s dynamic in a network that reflects the heterogeneity of the environment where the disease spreads. The vaccination is assumed to be not perfect, which means that vaccinated persons are likely to be infected and that the vaccinated person could lose their immunity, …

Topological Indices And Their Applications In Designing Drugs, Fedaa Ismail Abunawa

Theses

This research delves into the use of indices, in the field of drug design with a specific focus on anticancer medications. Topological indices, values derived from representations of chemical structures provide meaningful connections to the physical and chemical characteristics of molecules. These Indices act as tools for predicting behavior playing a vital role in crafting therapeutic drugs. The study primarily delves into topological indices like the Sombor index, Randi´c index, and Atom Bond Connectivity (ABC) index, which are calculated for structures and their relationships with physical properties such as molar volume, refractive index, and flash point are explored using statistical …

Pt-Symmetry And Eigenmodes, Tamara Gratcheva

University Honors Theses

Spectra of systems with balanced gain and loss, described by Hamiltonians with parity and time-reversal (PT) symmetry is a rich area of research. This work studies by means of numerical techniques, how eigenvalues and eigenfunctions of a Schrodinger operator change as a gain-loss parameter changes. Two cases on a disk with zero boundary conditions are considered. In the first case, within the enclosing disk, we place a parity (P) symmetric configuration of three smaller disks containing gain and loss media, which does not have PT-symmetry. In the second case, we study a PT-symmetric configuration …

A Microgenetic Learning Analysis Of Contextuality In Reasoning About Exponential Modeling, Elahe Allahyari

Dissertations

This work explores the complex cognitive processes students engage in when addressing contextual tasks requiring linear and exponential models. Grounded within Piagetian constructivism and the Knowledge in Pieces (KiP) epistemological perspective (diSessa, 1993, 2018), this empirical study in a clinical setting develops a Microgenetic Learning Analysis (MLA) of the reasoning of 14 students from an Algebra II course. It reveals the critical role of cognitive disequilibrium as an essential cognitive state for conceptual development and the process of reorganizing knowledge systems. The study uncovers the fluctuations in students’ reasoning patterns and the significant impact on students’ reasoning patterns of task-specific …

On Near-Linear Cellular Automata Over Near Spaces, Abdul-Rahman M. Nasser

Dissertations

Cellular Automata can be considered as examples of massively parallel machines. They are computational mathematical objects consisting of a grid of cells, each of which can exist in a finite number of states. These cells evolve over discrete time steps according to a set of predefined rules based on the states of neighboring cells. The notion of cellular automata was first introduced by Ulam and von Neumann and then popularized by John H. Conway in the 1970s with one of the most famous examples being The Game of Life.

This research builds on and generalizes the work of Tullio Ceccherini-Silberstein …

Dehn's Problems And Geometric Group Theory, Noelle Labrie

Master's Theses

In 1911, mathematician Max Dehn posed three decision problems for finitely

presented groups that have remained central to the study of combinatorial

group theory. His work provided the foundation for geometric group theory,

which aims to analyze groups using the topological and geometric properties

of the spaces they act on. In this thesis, we study group actions on Cayley

graphs and the Farey tree. We prove that a group has a solvable word problem

if and only if its associated Cayley graph is constructible. Moreover, we prove

that a group is finitely generated if and only if it acts geometrically …

Representation Theory And Its Applications In Physics, Max Varverakis

Master's Theses

Representation theory, which encodes the elements of a group as linear operators on a vector space, has far-reaching implications in physics. Fundamental results in quantum physics emerge directly from the representations describing physical symmetries. We first examine the connections between specific representations and the principles of quantum mechanics. Then, we shift our focus to the braid group, which describes the algebraic structure of braids. We apply representations of the braid group to physical systems in order to investigate quasiparticles known as anyons. Finally, we obtain governing equations of anyonic systems to highlight the differences between braiding statistics and conventional Bose-Einstein/Fermi-Dirac …

A Thesis, Or Digressions On Sculptural Practice: In Which, Concepts & Influences Thereof Are Explained, Set Forth, Catalogued, Or Divulged By Way Of Commentaries To A Poem, First Conceived By The Artist, Fed Through Chatg.P.T., And Re-Edited By The Artist, To Which Are Added, Annotated References, Impressions And Ruminations Thereof, Also Including Private Thoughts & Personal Accounts Of The Artist, Jaimie An

Masters Theses

This thesis is an exercise in, perhaps a futile, attempt to trace just some of the ideas, stories, and musings I might meander through in my process. It’s not quite a map, nor is it a neat catalogue; it is a haphazard collection of tickets and receipts from a travel abroad, carelessly tossed in a carry-on, only to be stashed upon returning home. These ideas are derived from much greater thinkers and authors than myself; I am a mere collector or a translator, if that, and not a very good one, for much is lost. I do not claim comprehensive …

A Brief Introduction To General Topology, Richard P. Millspaugh

Open Educational Resources

The material in this text is intended to be accessible to undergraduates who have had an introduction to elementary set theory and proof techniques. It includes sufficient material from general topology to prove the two main topological results found in a standard first semester calculus course: the Intermediate Value Theorem and the Extreme Value Theorem. This material can be found in Chapters 2 through 6 and makes up the bulk of the text. Rather than approaching these topics through use of the standard euclidean metric, it defines the standard topology on R in terms of the usual order on R. …

Linear Ode Systems Having A Fundamental Matrix Of The Form F(Mt), Kevin L. Shirley, Vicky W. Klima

CODEE Journal

We interweave scaffolded problem statements with exposition and examples to support the reader as they explore specific linear systems of differential equations with variable coefficients, $\vec{x}'(t)=A(t)\vec{x}(t)$ and initial value $\vec{x}(0)$. We begin with a constant $n\times n$ matrix $M$ and a real or complex-valued function $f$, analytic at the eigenvalues of $M$ with $f(0) = 1$, and construct a linear system of differential equations with solutions $x(t)=f(Mt)\vec{x}(0)$, where $t$ is a parameter in some interval including zero. In general, the solutions to the resulting non-autonomous system are more difficult to analyze than solutions to the constant coefficient case. However, some …

A Novel Consumer-Centric Metric For Evaluating Hearing Device Audio Performance, Vinaya Manchaiah, Steve Taddei, Abram Bailey, De Wet Swanepoel, Hansapani Rodrigo

School of Mathematical & Statistical Sciences Faculty Publications

Background and Aim: The emergence of direct-to-consumer hearing devices has introduced confusion in making appropriate choices, highlighting the need for users to be well-informed for optimal device selection. Currently, no established metric offers insights into the sound performance of these devices. This study aimed to introduce and assess a novel consumer-centric metric (i.e., SoundScore) for hearing device audio performance.

Method: The SoundScore metric was created based on five dimensions of hearing device audio performance (i.e., speech benefit in quiet and moderate, speech benefit in loud, own voice perception, feedback control, streamed music sound quality). Tests were conducted under lab conditions …

A Combinatorial Proof Of A Partition Perimeter Inequality, Hunter Waldron

Michigan Tech Publications

The partition perimeter is a statistic defined to be one less than the sum of the number of parts and the largest part. Recently, Amdeberhan, Andrews, and Ballantine proved the following analog of Glaisher’s theorem: for all m ≥ 2 and n ≥ 1, there are at least as many partitions with perimeter n and parts repeating fewer than m times as there are partitions with perimeter n with parts not divisible by m. In this work, we provide a combinatorial proof of their theorem by relating the combinatorics of the partition perimeter to that of compositions. Using this technique, …

Composition-Theoretic Series And False Theta Functions, William Keith, Robert Schneider, Andrew V. Sills

Michigan Tech Publications

Many natural partition-theoretic series can be equally readily interpreted as compo-sition-theoretic series, but this viewpoint seems to have not been much employed in either theory. We consider some of the consequences of this viewpoint. As examples, we give results concerning the reciprocals of Ramanujan’s theta functions and of the false theta functions of L. J. Rogers, and raise an array of questions related to these. Part of this study may be considered a natural dual of the truncated pentagonal number theorem of Andrews and Merca.

Balanced N-Color Compositions, Jesus Omar Sistos Barron, Hua Wang

Mathematical Sciences: Faculty Publications

We introduce the notion of balanced n-color compositions, and consider the related enumeration problems. We also establish bijections with four other combinatorial objects, namely two types of Motzkin lattice paths, pairs of regular compositions with restricted parts, and regular compositions with a centered maximum.

On A Generalization Of A Theorem Of Ibukiyama To Evaluate Three Imprimitive Character Sums, Brad Isaacson

Publications and Research

In a previous paper, we expressed three families of character sums by certain generalized Bernoulli functions which in turn were expressed by generalized Bernoulli numbers via a complicated and indirect process. In this paper, we generalize a theorem of Ibukiyama to directly express these generalized Bernoulli functions by generalized Bernoulli numbers. As a result, we can express the three families of character sums by generalized Bernoulli numbers in a more elegant fashion than was done before.

On The T-Degree Of Anderson Thakur Polynomials In The Case Of Fields Of Order P Where P Is A Prime, Joshua Bush

Master's Theses

The Anderson Thakur polynomials are of great interest in the investigation of zeta values. Here, we address the question of their degree in finite fields of the form Fp. We define a graph that is closely connected to Anderson Thakur polynomials and which represents a polynomial we denote as Anxn−1. Collections of terms of this polynomial are equating with the vertices of the graph and are vanishing or nonvanishing. The vanishing structure of the graph is analyzed and we conclude with a simple degree formula.

Engaging Students In Partial Differential Equations Through Modeling Fourier's Law Of Conduction, Justin G. Trulen, Kayla Keller, John Sinclair, Lauren Meagher

CODEE Journal

In an effort to make active learning exercises in a partial differential equations course, we present a student activity modeling Fourier's law of conduction under the framework of the heat equation. An overview of the heat equation, including several avenues of study, is provided. Then we give an intuitive way of constructing the heat equation from a couple of fundamental properties of physics including Fourier's law of conduction. We outline an experiment that can be run to collect their own data to model Fourier's law of conduction as well as provide data we collected. We conclude with a student activity …

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 …