Mathematics Commons^™

Spectral Analysis Of Complex Dynamical Systems, Casey Lynn Johnson

CGU Theses & Dissertations

The spectrum of any differential equation or a system of differential equations is related to several important properties about the problem and its subsequent solution. So much information is held within the spectrum of a problem that there is an entire field devoted to it; spectral analysis. In this thesis, we perform spectral analysis on two separate complex dynamical systems. The vibrations along a continuous string or a string with beads on it are the governed by the continuous or discrete wave equation. We derive a small-vibrations model for multi-connected continuous strings that lie in a plane. We show that …

Teaching Introductory Statistics With Datacamp, Benjamin Baumer, Andrew P. Bray, Mine Çetinkaya-Rundel, Johanna S. Hardin

Statistical and Data Sciences: Faculty Publications

We designed a sequence of courses for the DataCamp online learning platform that approximates the content of a typical introductory statistics course. We discuss the design and implementation of these courses and illustrate how they can be successfully integrated into a brick-and-mortar class. We reflect on the process of creating content for online consumers, ruminate on the pedagogical considerations we faced, and describe an R package for statistical inference that became a by-product of this development process. We discuss the pros and cons of creating the course sequence and express our view that some aspects were particularly problematic. The issues …

Aspects Of Quantum Theory In General Relativity And Cosmology, Eamon Mccaughey

Doctoral

After a brief introduction to Cosmology some quantum aspects of General Relativity and cosmology are presented. The radial motion of a massive particles in the ergosphere of the Kerr Black Hole is considered. Screening of Hawking radiation and shielding of the Penrose process is examined in the context of the region of negative energy inside the ergosphere. Tunnelling of such particles between the boundaries of the classically forbidden region will be considered and the transmission coefficient determined. The evolution of Primordial black holes in standard and Loop Quantum Cosmology is reviewed. A stability analysis of Einstein’s universe in both classical …

Dual Quaternion Algebra And Its Derivations, Eyüp Kizil, Yasemi̇n Alagöz

Turkish Journal of Mathematics

It is well known that the automorphism group $Aut(H)$ of the algebra of real quaternions $H$ consists entirely of inner automorphisms $i_{q}:p\rightarrow q\cdot p\cdot q^{-1}$ for invertible $q\in H$ and is isomorphic to the group of rotations $SO(3)$. Hence, $H$ has only inner derivations $D=ad(x),$ $x\in H$. See [4] for derivations of various types of quaternions over the reals. Unlike real quaternions, the algebra $H_{d}$ of dual quaternions has no nontrivial inner derivation. Inspired from almost inner derivations for Lie algebras, which were first introduced in [3] in their study of spectral geometry, we introduce coset invariant derivations for dual …

Cycle Double Covers And Integer Flows, Zhang Zhang

Graduate Theses, Dissertations, and Problem Reports (ETD)

My research focuses on two famous problems in graph theory, namely the cycle double cover conjecture and the integer flows conjectures. This kind of problem is undoubtedly one of the major catalysts in the tremendous development of graph theory. It was observed by Tutte that the Four color problem can be formulated in terms of integer flows, as well as cycle covers. Since then, the topics of integer flows and cycle covers have always been in the main line of graph theory research. This dissertation provides several partial results on these two classes of problems.

Predicting Student Success In Arcadia University’S Math Courses, Chutong Wu, Tong Zhu, Yijin Qiu

Capstone Showcase

This project examines the relative efficacy of Arcadia’s math placement test and math SAT scores in predicting student success, and explores whether SAT scores alone might suffice for certain courses.

An Introduction To Copulas, Yifan Guo, Geng Zhang

Capstone Showcase

Copulas are the mathematical functions that connect the distribution functions of univariate random variables to form multivariate distributions. We define copulas, present some of their key properties, and provide examples of their applications.

Climate Change Models, Lauren Fie

Capstone Showcase

As a result of the changing climate, global temperatures and global mean sea levels (GMSL) have been increasing rapidly. The complex physical systems surrounding this growth make it difficult to form an accurate model. This paper looks at a simplified model proposed and supported by Aral, Guan, and Chang. This model consists of a system of ordinary differential equations that are simplified and solved theoretically, then applied using python to calculate precise values and form predictions.

Stationary Distribution Of Recombination On 4x4 Grid Graph As It Relates To Gerrymandering, Camryn Hollarsmith

Scripps Senior Theses

A gerrymandered political districting plan is used to benefit a group seeking to elect more of their own officials into office. This practice happens at the city, county and state level. A gerrymandered plan can be strategically designed based on partisanship, race, and other factors. Gerrymandering poses a contradiction to the idea of “one person, one vote” ruled by the United States Supreme Court case Reynolds v. Sims (1964) because it values one demographic’s votes more than another’s, thus creating an unfair advantage and compromising American democracy. To prevent the practice of gerrymandering, we must know how to detect a …

Genetic Algorithm Guidance Of A Constraint Programming Solver For The Multiple Traveling Salesman Problem, Jessica M. Rudd, Andrew M. Henshaw, Lauren Staples, Sanjoosh Akkineni, Lin Li, Joe Demaio

Published and Grey Literature from PhD Candidates

This project developed a metaheuristic approach to the Multiple Traveling Salesman Problem that pairs a custom genetic algorithm with a conventional combinatorial optimization solver. This combined approach was used to build an optimal route for two popular radio show hosts to visit each of the 37 Atlanta area Jersey Mike's Subs in one day. This supported a fundraising eort to send children with chronic and terminal illnesses to Disney World through an organization called Bert's Big Adventure. Atlanta-area Jersey Mike's locations donated 100% of proceeds earned on this Day of Giving to Bert's Big Adventure. With the suggested route developed …

Heat Kernel Voting With Geometric Invariants, Alexander Harr

All Graduate Theses, Dissertations, and Other Capstone Projects

Here we provide a method for comparing geometric objects. Two objects of interest are embedded into an infinite dimensional Hilbert space using their Laplacian eigenvalues and eigenfunctions, truncated to a finite dimensional Euclidean space, where correspondences between the objects are searched for and voted on. To simplify correspondence finding, we propose using several geometric invariants to reduce the necessary computations. This method improves on voting methods by identifying isometric regions including shapes of genus greater than 0 and dimension greater than 3, as well as almost retaining isometry.

Phylogenetic Networks And Functions That Relate Them, Drew Scalzo

Williams Honors College, Honors Research Projects

Phylogenetic Networks are defined to be simple connected graphs with exactly n labeled nodes of degree one, called leaves, and where all other unlabeled nodes have a degree of at least three. These structures assist us with analyzing ancestral history, and its close relative - phylogenetic trees - garner the same visualization, but without the graph being forced to be connected. In this paper, we examine the various characteristics of Phylogenetic Networks and functions that take these networks as inputs, and convert them to more complex or simpler structures. Furthermore, we look at the nature of functions as they relate …

Free Semigroups And Identites For A Class Of Monoids, Enrique Salcido

Open Access Theses & Dissertations

The study of words as a mathematical object is a deep and rich field of study. Algebra, Combinatorics, Theoretical Computer Science etc., are major disciplines, which are fully using this study. Combinatorial properties (via Codes, Free Hulls, Infinite Words), and algebraic properties of words are presented in this Thesis. The free semigroup on a set (alphabet) X and finite presentation of semigroups have a central place in the algebraic study of words. The last part of the Thesis is devoted to the study of identities in the alphabet X = {x,y} for a class of monoids. The characterization of such …

Positivity-Preserving Segregate-Flux Method For Infiltration Dynamics In Tumor Growth Models, Gilbert Danso Acheampong

Open Access Theses & Dissertations

We study the positivity preserving property and an incompressibility condition in a recently proposed tumor growth model as well as its numerical simulations. In this model, the biological process is described by a free-boundary problem of hyperbolic equations that govern the in-tumor motion of cancer cells and the infiltration of immune cells. Particularly, due to an assumption that cells take constant volume (the incompressibility condition), the tumor growth/shrinkage is closely correlated to the magnitude of infiltration of immune cells into the tumor.

Despite the fact that previous simulation results largely reproduced experimental data, there remain unanswered questions that are crucial …

Mathematical Modeling Of Microemulsification Processes, Numerical Simulations And Applications To Drug Delivery, Ogochukwu Nneka Ifeacho

Open Access Theses & Dissertations

Microemulsion systems are a great pharmaceutical tool for the delivery of formulations containing multiple hydrophilic and hydrophobic ingredients of varying physicochemical properties. These systems are gaining popularity because of its long shelf life, improved drug solubilisation capacity, easy preparation and improvement of bioavailability. Despite the advantages associated with the use of microemulsion systems in pharmaceutical industries, the major challenge impeding their use has been and continues to be the lack of understanding of these systems.

Microemulsions can be mathematically modeled by an initial boundary value problem involving a sixth order nonlinear time dependent equation. In this Thesis, we present a …

Using Machine Learning On An Imbalanced Cancer Dataset, James Ekow Arthur

Open Access Theses & Dissertations

With an estimated 1.4 million cancer diagnosis worldwide and the increasing death of cancer patients. It is prudent to investigate methods, approaches and smarter ways of predicting and diagnosing of cancer so that a holistic techniques can be used to curb or reduce false predictions , increase exact predictions and also meticulos prognosis information .

Can a feasible technique be developed for the general problem of prognosis and diagnosis of cancer be developed ?

We will show here that this problem of cancer prognosis and diagnosis can be efficiently tackled with the aid of machine learning techniques and the best, …

Planar Motion Control Of A Cube Satellite Using Cold Gas Thrusters, Christian Lozoya

Open Access Theses & Dissertations

This Thesis presents a mathematical model developed for the computational simulation ofCubeSat movement using four thrusters that permit uniaxial translation and rotation. Arbitrary functions are fit to boundary conditions to simulate the force, acceleration, velocity, and displacement of the CubeSat along a plane. The model is used to derive a motion control algorithm assuming constant pressure and mass. A single model describes both translation and rotation. This Thesis also explores the relationship between propellant consumption and the time required to complete a displacement implied by the model.

Stability Of Anisotropic Parabolic Equations Without Boundary Conditions, Huashui Zhan, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

No abstract provided.

Cartan’S Approach To Second Order Ordinary Differential Equations, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

In his work on projective connections, Cartan discusses his theory of second order differential equations. It is the aim here to look at how a normal projective connection can be constructed and how it relates to the geometry of a single second order differential equation. The calculations are presented in some detail in order to highlight the use of gauge conditions

Surviving Mathematics, Nathalie M. Luna

School of Mathematical & Statistical Sciences Faculty Publications

In this essay written in an informal voice, the author shares the ups and downs of her experience in academia. She shares her motivation to study mathematics, her undergraduate experience in Puerto Rico, and her graduate experience in South Texas.

Technological Software In Mathematics, Courtney Kish

Williams Honors College, Honors Research Projects

Technology has been advancing significantly over the years. One area that has been affected is mathematics. Technological software has been developed that has allowed for mathematics to be done using software programs. For example, WebAssign allows students to complete online math homework and example practice problems, as well as watch videos to explain topics in math. However, like all things, this is not without downfalls. While this technology offers students access to online lectures and instant feedback, cheating and costly expenses also come with it.

In this research paper, I will discuss the benefits and shortcomings of different technological software …

Use Of Calculators In High School Mathematics, Alex Mcdaniel

Williams Honors College, Honors Research Projects

For many collegiate level mathematics courses at universities across the country, students are not allowed to use calculators. This is often a huge adjustment for students who have spent their entire high school career using calculators. Whether it is graphing or simple arithmetic, students relied on calculators in high school, but for college, the tool of a calculator may not be available for them. This leads to a multitude of questions: Should calculators be used in high school? Do calculators help students learn or are they simply a tool to get answers? Are calculators beneficial in some courses, but not …

Study Of Grain Growth In Single-Phase Polycrystals, Pawan Vedanti

Wayne State University Dissertations

Materials with random microstructure are characterized by additional thermodynamic parameters, entropy and temperature of microstructure. It has been argued that there is one more law of thermodynamics: entropy of microstructure decays in isolated systems. This assertion has been checked experimentally for the process of grain growth which showed that entropy of grain structure decays indeed as expected. The equation of state for microstructure entropy has also been studied. In general, entropy of grain microstructure is expected to be a function of grain structure energy and the average grain size. Our experiments suggest that in fact, the equation of state degenerates …

Effect Of Hydraulic Resistivity On A Weakly Nonlinear Thermal Flow In A Porous Layer, Dambaru Bhatta, Daniel N. Riahi

School of Mathematical & Statistical Sciences Faculty Publications

Heat and mass transfer through porous media has been a topic of research interest because of its importance in various applications. The flow system in porous media is modelled by a set of partial differential equations. The momentum equation which is derived from Darcy’s law contains a resistivity parameter. We investigate the effect of hydraulic resistivity on a weakly nonlinear thermal flow in a horizontal porous layer. The present study is a realistic study of nonlinear convection flow with variable resistivity whose rate of variation is arbitrary in general. This is a first step for considering more general problems in …

Topological Pressure And Fractal Dimensions Of Cookie-Cutter-Like Sets, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

The cookie-cutter-like set is defined as the limit set of a sequence of classical cookie-cutter mappings. For this cookie-cutter set it is shown that the topological pressure function exists, and that the fractal dimensions such as the Hausdorff dimension, the packing dimension and the box-counting dimension are all equal to the unique zero h of the pressure function. Moreover, it is shown that the h-dimensional Hausdorff measure and the h-dimensional packing measure are finite and positive.

Providing Better Choices: An Exploration Of Solutions In Multi-Objective Optimization And Game Theory Using Variational Analysis, Glenn Matthew Harris

Graduate Research Theses & Dissertations

Multi-objective optimization problems and game theory problems have a wide array of

applications and because of this there are different types of solutions available. This dissertation

explores two areas of optimization and a solution type for each. First, substantial

efficiency (SE) as a type of solution to multi-objective optimization problems that extends

proper efficiency. Secondly, strong Nash equilibria (SNE) as a type of solution to game

theoretic problems that extends Nash equilibria. Substantial efficiency is demonstrated to

be a superior solution to the more rudimentary notion of proper efficiency in solving some

multi-objective financial market and economic problems. Using this …

Machine Learning Analysis To Characterize Phase Variations In Laser Propagation Through Deep Turbulence, Luis Fernando Rodriguez Sanchez

Open Access Theses & Dissertations

The present Dissertation is focused on the analysis of the atmospheric conditions of a turbulent environmental system and its effects on the diffraction of a laser beam that moves through it. The study is based on the optical communication of two labs placed at the summit of two mountains located in Maui, Hawaii. The emitter system is located at the Mauna Loa mountain and the receiver at the Haleakala. The distance between both mountains is 150 km. The emitter system is at a height of 3.1 km and the receiver at 3.4 km. The maritime environment at the location experiences …

The Sigma Chromatic Number Of The Sierpinski Gasket Graphs And The Hanoi Graphs, Agnes Garciano, Reginaldo M. Marcelo, Mari-Jo P. Ruiz, Mark Anthony C. Tolentino

Mathematics Faculty Publications

A vertex coloring c : V(G) → of a non-trivial connected graph G is called a sigma coloring if σ(u) ≠ σ(v) for any pair of adjacent vertices u and v. Here, σ(x) denotes the sum of the colors assigned to vertices adjacent to x. The sigma chromatic number of G, denoted by σ(G), is defined as the fewest number of colors needed to construct a sigma coloring of G. In this paper, we determine the sigma chromatic numbers of the …

Twin Chromatic Indices Of Some Graphs With Maximum Degree 3, Jayson D. Tolentino, Reginaldo M. Marcelo, Mark Anthony C. Tolentino

Mathematics Faculty Publications

Let k ≥ 2 be an integer and G be a connected graph of order at least 3. A twin k-edge coloring of G is a proper edge coloring of G that uses colors from k and that induces a proper vertex coloring on G where the color of a vertex v is the sum (in k ) of the colors of the edges incident with v. The smallest integer k for which G has a twin k-edge coloring is the twin chromatic index of G and is denoted by . In this paper, we determine the …

On Eigenvalue Bounds For The Finite-State Birth-Death Process Intensity Matrix, R.R.P Tan, K Ikeda, Len Patrick Dominic M. Garces

Mathematics Faculty Publications

The paper sets forth a novel eigenvalue interlacing property across the finite-state birth-death process intensity matrix and two clearly identified submatrices as an extension of Cauchy’s interlace theorem for Hermitian matrix eigenvalues. A supplemental proof involving an examination of probabilities acquired from specific movements across states and a derivation of a form for the eigenpolynomial of the matrix through convolution and Laplace transform is then presented towards uncovering a similar characteristic for the general Markov chain transition rate matrix. Consequently, the proposition generates bounds for each eigenvalue of the original matrix, easing numerical computation. To conclude, the applicability of the …