Mathematics Commons^™

Reducing Generalization Error In Multiclass Classification Through Factorized Cross Entropy Loss, Oleksandr Horban

CMC Senior Theses

This paper introduces Factorized Cross Entropy Loss, a novel approach to multiclass classification which modifies the standard cross entropy loss by decomposing its weight matrix W into two smaller matrices, U and V, where UV is a low rank approximation of W. Factorized Cross Entropy Loss reduces generalization error from the conventional O( sqrt(k / n) ) to O( sqrt(r / n) ), where k is the number of classes, n is the sample size, and r is the reduced inner dimension of U and V.

Unveiling The Power Of Shor's Algorithm: Cryptography In A Post Quantum World, Dylan Phares

CMC Senior Theses

Shor's Algorithm is an extremely powerful tool, in utilizing this tool it is important to understand how it works and why it works. As well as the vast implications it could have for cryptography

Bridging Theory And Application: A Journey From Minkowski's Theorem To Ggh Cryptosystems In Lattice Theory, Danzhe Chen

CMC Senior Theses

This thesis provides a comprehensive exploration of lattice theory, emphasizing its dual significance in both theoretical mathematics and practical applications, particularly within computational complexity and cryptography. The study begins with an in-depth examination of the fundamental properties of lattices and progresses to intricate lattice-based problems such as the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP). These problems are analyzed for their computational depth and linked to the Subset Sum Problem (SSP) to highlight their critical roles in understanding computational hardness. The narrative then transitions to the practical applications of these theories in cryptography, evaluating the shift from …

Meromorphic Continuations Of Euler Products With Polynomial Growth Coefficients, Kledis Ahmetaj

Senior Honors Theses and Projects

In this paper, we investigate the meromorphic continuation of Euler products, with a particular focus on those with polynomial growth coefficients. Building upon the classical example of the Riemann zeta function, we derive conditions under which Euler products can be extended beyond their regions of convergence to broader domains of the complex plane, revealing singularities and deepening our understanding of their analytic properties. We begin by establishing a framework for meromorphic continuation and then introduce a general theorem that applies to Euler products of the form

where the coefficients and exponents satisfy specific constraints. We prove that such Euler products …

Cryptographic Algorithms, Cryptocurrencies, And A Predictive Model Of Bitcoin Value By Pls Regression, Paul Kenneth O'Connor

Masters Theses

"With the invention of Bitcoin in 2009, as a seemingly timed response to the ongoing financial crisis, the popularity of the cryptocurrency has since continued to grow. Just this year, the Security Exchange Commission approved Bitcoin for exchange traded funds, allowing major investment firms to begin product trading. With this approval, and during this very moment of writing, Bitcoin has entered a bull market and reached a record value of over 72,000 USD. In addition, the Bitcoin halving event in April of 2024 is expected to increase demand even further. It has been anticipated that Bitcoin and other cryptocurrencies will …

Dynamics And Inverse Problems For Nonlinear Schrödinger Equations, Christopher Hogan

Doctoral Dissertations

"The cubic nonlinear Schrödinger equation (NLS) is a model of interest in the study of physical problems including nonlinear optics and Bose-Einstein condensates. Of particular interest is the study of cubic NLS with inhomogeneities such as localizations of the nonlinearity or terms introducing potential barriers. We first address some preliminaries and techniques useful in the study of the cubic NLS and its variations. We then consider the cubic NLS with a localized nonlinearity in dimensions d ≥ 2. We show that solutions with data given by small-amplitude wave packets accrue a nonlinear phase that determines the X-ray transform of the …

Legendrian Torus And Cable Links, Jennifer Dalton,, John B. Etnyre, Lisa Traynor

Mathematics Faculty Research and Scholarship

We give a classification of Legendrian torus links. Along the way, we give the first classification of infinite families of Legendrian links where some smooth symmetries of the link cannot be realized by Legendrian isotopies. We also give the first family of links that are non-destabilizable but do not have maximal Thurston-Bennequin invariant and observe a curious distribution of Legendrian torus knots that can be realized as the components of a Legendrian torus link. This classification of Legendrian torus links leads to a classification of transversal torus links.

We also give a classification of Legendrian and transversal cable links of …

Improveing F-Beta Score In Classifying Shark Data Into Shark Behaviors, Ibrahim M. Ali

CGU Theses & Dissertations

One metric used to measure classification performance in machine learning is F-beta score. The objective in this thesis is to improve the average F-b score computed in classifying shark data into shark behaviors, namely; Resting, Swimming, Feeding, and Non-Directed Motion (NDM). Synthetic Minority Oversampling Technique (SMOTE) and Adaptive Synthetic Sampling (ADASYN) are utilized to balance the data, from which pre-processed Fast Fourier Transform (FFT), Walsh-Hadamard Transform (WHT), and Autocorrelation (AC) features are extracted then classified using Convolutional Neural Network (CNN) and K-Nearest Neighbors (K-NN). All the combinations of the two balancing techniques, the three feature types, and the two machine …

On Stratified Single-Valued Soft Topogenous Structures, Fahad Alsharari, Yaser Saber, Hanan Alohali, Mesfer H. Alqahtani, Mubarak Ebodey, Tawfik Elmasry, Jafar Alsharif, Amal F. Soliman, Florentin Smarandache, Fahad Sikander

Branch Mathematics and Statistics Faculty and Staff Publications

This paper presents novel concepts including stratified single-valued neutrosophic soft topogenous (stratified svns-topogenous), stratified single-valued neutrosophic soft filter (stratified svns-filter), stratified single-valued neutrosophic soft quasi uniformity (stratified svnsq-uniformity) and stratified single-valued neutrosophic soft quasi proximity (stratified svnsq-proximity). Additionally, we present the idea of single-valued neutrosophic soft topogenous structures, formed by integrating svns-topogenous with svns-filter, and discuss their properties. Furthermore, we explore the connections between these single-valued neutrosophic soft topological structures and their corresponding stratifications.

A Compact Exploration Of Turiyam Neutrosophic Competition Graphs, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Graph theory, a branch of mathematics, examines relationships between entities using vertices and edges. Within this field, Uncertain Graph Theory has emerged to model uncertainties in real-world networks. A notable concept in this area is the competition graph, which captures interactions by connecting vertices that “compete” for the same neighbor, represented by directed edges indicating common neighbors in a digraph. This brief paper introduces the concept of the Generalized Turiyam Neutrosophic Competition Graph and explores its relationships with other graph classes.

Numerical Investigation And Statistical Analysis Of The Flow Patterns Behind Square Cylinders Arranged In A Staggered Configuration Utilizing The Lattice Boltzmann Method, M. Abid, N. Yasin, M. Saqlain, S. Ul-Islam, S. Ahmad

Mathematics & Statistics Faculty Publications

Flow past bluff bodies like square cylinders is important in engineering applications, but flow patterns behind staggered cylinder arrangements remain poorly understood. Existing studies have focused on tandem or side-by-side configurations, while offset orientations have received less attention. The aim of this paper is to numerically investigate flow dynamics and force characteristics behind two offset square cylinders using the single relaxation time lattice Boltzmann method. The effects of changing both the Reynolds number (Re = 1-150) and gap spacing ratio (g* = 0.5-5) between the cylinders are analyzed. Instantaneous vorticity contours, time histories of drag and lift coefficients, power spectra …

Shadow Hands: Solutions For Fermi Questions, September 2024, John Adam

Mathematics & Statistics Faculty Publications

The article titled "Shadow hands: Solutions for Fermi Questions, September 2024" discusses the enigmatic statement made by authors Dave Lynch and Bill Livingston in their book "Color and Light in Nature." The statement explains that the fuzzy outer edge of a shadow, known as the penumbra, is always a constant fraction of about 1/112 of its width divided by its distance from the object casting the shadow. The article provides a solution to Question 1, which asks for an explanation of this statement using elementary geometry. It also offers a solution to Question 2, which asks why the shadow of …

Numerical Solution Of Hybrid Nanofluid And Its Stability Over Permeable Wedge Sheet With Heat Transfer Analysis, Aisha M. Alqahtani, Zeeshan, Waris Khan, Florentin Smarandache, Nidhal Becheikh, Roobaea Alroobaea, Taseer Muhammad

Branch Mathematics and Statistics Faculty and Staff Publications

The inclusion of nanoparticles has the potential to improve the thermal efficiency of the base fluid. The field of nanofluid (NF) dynamics has attracted important attention due to its extensive range of practical uses like fuel cells, solar energy, medication administration, heat transfer, microfabrication, coolant applications, and other related domains. The aim of this study is to scrutinize the impact of Lorentz force, thermal energy, joule heating, heat source and injection parameters, and Brownian and thermoporetic diffusions on the hybrid nanofluid over the moving wedge. The stability inquiry is reported for the existing work in order to confirm the stable …

The Direct And Inverse Scattering Problems For The Third-Order Operator, Ivan Toledo

Mathematics Dissertations - Archive

We consider the full-line direct and inverse scattering problems for the third-order ordinary differential equation containing two potentials decaying sufficiently fast at infinity. The direct scattering problem consists of the determination of the scattering data set when the two potentials are known. The scattering data set is made up of the corresponding scattering coefficients and the bound-state information. On the other hand, the inverse scattering problem involves the recovery of the two potentials when the scattering data set is available. We formulate the inverse scattering problem via a related Riemann--Hilbert problem on the complex plane. We describe the recovery of …

Bicategorical Character Theory, Travis Wheeler

Theses and Dissertations--Mathematics

In 2007, Nora Ganter and Mikhail Kapranov defined the categorical trace, which they used to define the categorical character of a 2-representation. In 2008, Kate Ponto defined a shadow functor for bicategories. With the shadow functor, Dr. Ponto defined the bicategorical trace, which is a generalization of the symmetric monoidal trace for bicategories. How are these two notions of trace related to one another? We’ve used bicategorical traces to define a character theory for 2-representations, and the categorical character is an example.

Properties Of Skew-Polynomial Rings And Skew-Cyclic Codes, Kathryn Hechtel

Theses and Dissertations--Mathematics

A skew-polynomial ring is a polynomial ring over a field, with one indeterminate x, where one must apply an automorphism to commute coefficients with x. It was first introduced by Ore in 1933 and since the 1980s has been used to study skew-cyclic codes. In this thesis, we present some properties of skew-polynomial rings and some new constructions of skew-cyclic codes. The dimension of a skew-cyclic code depends on the degree of its generating skew polynomial. However, due to the skew-multiplication rule, the degree of a skew polynomial can be smaller than its number of roots and hence tricky to …

Adams Operations On The Burnside Ring From Power Operations, Lewis Dominguez

Theses and Dissertations--Mathematics

Topology furnishes us with many commutative rings associated to finite groups. These include the complex representation ring, the Burnside ring, and the G-equivariant K-theory of a space. Often, these admit additional structure in the form of natural operations on the ring, such as power operations, symmetric powers, and Adams operations. We will discuss two ways of constructing Adams operations. The goal of this work is to understand these in the case of the Burnside ring.

Pairs Of Quadratic Forms Over P-Adic Fields, John Hall

Theses and Dissertations--Mathematics

Given two quadratic forms $Q_1, Q_2$ over a $p$-adic field $K$ in $n$ variables, we consider the pencil $\mathcal{P}_K(Q_1, Q_2)$, which contains all nontrivial $K$-linear combinations of $Q_1$ and $Q_2$. We define $D$ to be the maximal dimension of a subspace in $K^n$ on which $Q_1$ and $Q_2$ both vanish. We define $H$ to be the maximal number of hyperbolic planes that a form in $\mathcal{P}_K(Q_1, Q_2)$ splits off over $K$. We will determine which values for $(D, H)$ are possible for a nonsingular pair of quadratic forms over a $p$-adic field $K$.

Dirichlet Problems In Perforated Domains, Robert Righi

Theses and Dissertations--Mathematics

We establish W^1,p estimates for solutions u_ε to the Laplace equation with Dirichlet boundary conditions in a bounded C¹ domain Ω_{ε, η} perforated by small holes in ℝ^d. The bounding constants will depend explicitly on epsilon and eta, where epsilon is the order of the minimal distance between holes, and eta denotes the ratio between the size of the holes and epsilon. The proof relies on a large-scale L^p estimate for ∇u_ε, whose proof is divided into two main parts. First, we show that solutions of an intermediate problem for a …

Slₖ-Tilings And Paths In ℤᵏ, Zachery T. Peterson

Theses and Dissertations--Mathematics

An SLₖ-frieze is a bi-infinite array of integers where adjacent entries satisfy a certain diamond rule. SL₂-friezes were introduced and studied by Conway and Coxeter. Later, these were generalized to infinite matrix-like structures called tilings as well as higher values of k. A recent paper by Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and SL₂-tilings. We extend this result to higher k by constructing a bijection between SLₖ-tilings and certain pairs of bi-infinite strips of vectors in ℤᵏ called paths. The key ingredient in the proof is the relation to Plucker friezes and …