Digital Commons Network^™

Linear Inverse Problems And Neural Networks, Jasjeet Dhaliwal

Master's Theses

We investigate two ideas in this thesis. First, we analyze the results of adaptingrecovery algorithms from linear inverse problems to defend neural networks against adversarial attacks. Second, we analyze the results of substituting sparsity priors with neural network priors in linear inverse problems. For the former, we are able to extend the framework introduced in [1] to defend neural networks against ℓ0, ℓ2,and ℓ∞ norm attacks, and for the latter, we find that our method yields an improvement over reconstruction results of [2].

Blue Red Hackenbush Spiders, Ravi Cho

Master's Theses

One of the goals of Combinatorial Game Theory is to find provable winning strategiesfor certain games. In this paper, we give winning strategies for certain spider positions played using the rules of Blue Red Hackenbush and a variant. Blue Red Hackenbush and its variants are played on a graph of a bLue and Red edges that are connected to a vertex called the ground. We will represent the ground as a horizontal black line. In this paper, we study spider graphs played under two different variants: Blue Red Hackenbush and Reverse Blue Red Hackenbush. Both variants are played by two …

The Role Of Self-Efficacy And Academic Mindset On Middle School Math Achievement For At Promise Youth: An Explanatory Study, Michael Lockwood Mansfield

Dissertations

California’s accountability system has blended its focus on noncognitive factors of achievement as well as evidence of academic growth. School districts are becoming increasingly interested in developing noncognitive factors in their students to help increase student achievement. With the wide variety of mandates and responsibilities schools are shouldering, it will be important to continue developing methods to increase all school accountability metrics in relation with each other. This research aims to explore the relationship between noncognitive factors and mathematics achievement as measured by the Northwest Evaluation Association’s Measure of Academic Progress assessment (MAP). This study surveyed 8th graders throughout a …

Determination Of A Graph's Chromatic Number For Part Consolidation In Axiomatic Design, Jeffery Anthony Cavallaro

Master's Theses

Mechanical engineering design practices are increasingly moving towards a framework called axiomatic design (AD). A key tenet of AD is to decrease the information content of a design in order to increase the chance of manufacturing success. An important way to decrease information content is to fulfill multiple functional requirements (FRs) by a single part: a process known as part consolidation. One possible method for determining the minimum number of required parts is to represent a design by a graph, where the vertices are the FRs and the edges represent the need to separate their endpoint FRs into separate parts. …

Preserving Ideals Of Chemical Reaction Networks, Mark Curiel

Master's Theses

Under the assumption of mass-action kinetics, every chemical reaction network has an associated polynomial dynamical system. Rather than study the dynamics of this system, we shall study the ideal generated by the polynomials in the system called the steady state ideal. In this thesis, we will show that there is a combinatorial way to determine the existence of monomials in the steady state ideal using the underlying structure of the network. This allows us to prove that there is a combinatorial condition that is enough to guarantee the steady state ideal is monomial. We introduce three operations on chemical reaction …

Lenstra-Hurwitz Cliques In Real Quadratic Fields, Daniel S. Lopez

Master's Theses

Let $K$ be a number field and let $\OO_K$ denote its ring of integers. We can define a graph whose vertices are the elements of $\OO_K$ such that an edge exists between two algebraic integers if their difference is in the units $\OO_K^{\times}$. Lenstra showed that the existence of a sufficiently large clique (complete subgraph) will imply that the ring $\OO_K$ is Euclidean with respect to the field norm. A recent generalization of this work tells us that if we draw more edges in the graph, then a sufficiently large clique will imply the weaker (but still very interesting) conclusion …

Knight's Tours And Zeta Functions, Alfred James Brown

Master's Theses

Given an m × n chessboard, we get an associated graph by letting each square represent a vertex and by joining two vertices if there is a valid move by a knight between the corresponding squares. A knight’s tour is a sequence of moves in which the knight lands on every square exactly once, i.e., a Hamiltonian path on the associated graph. Knight’s tours have an interesting history. One interesting mistake regarding Knight’s Tours was made by the famous mathematician Euler. His mistake led to the further study of knight’s tours on 3 × n chessboards. We will explore and …

Drawing Place Field Diagrams Of Neural Codes Using Toric Ideals, Nida K. Obatake

Master's Theses

A neural code is a collection of codewords (0-1 vectors) of a given length n; it captures the co-firing patterns of a set of neurons. A neural code is convexly realizable in dimension two if there exist n convex sets in the plane so that each codeword in the code corresponds to a unique intersection carved out by the convex sets. There are some methods to determine whether a neural code is convexly realizable; however, these methods do not describe how to draw a realization, that is, a place field diagram of the code. In this work, we construct toric …

Geometric Control Theory: Nonlinear Dynamics And Applications, Geoffrey A. Zoehfeld

Master's Theses

We survey the basic theory, results, and applications of geometric control theory. A control system is a dynamical system with parameters called controls or inputs. A control trajectory is a trajectory of the control system for a particular choice of the inputs. A control system is called controllable if every two points of the underlying space can be connected by a control trajectory. Two fundamental problems of control theory include:

1) Is the control system controllable?

2) If it is controllable, how can we construct an input to obtain a particular control trajectory? We shall investigate the first problem exclusively …

Computation In A Localization Of The Free Group Algebra, Olga Zamoruyeva

Master's Theses

The coefficients of a Taylor series expansion of any rational function in one variable satisfy a linear recurrence relation. Our main result is a generalization of this statement for rational functions of multiple non-commutative variables. We show that if such a function is represented in the form of a non-commutative formal power series via Magnus embedding, then the coefficients of this formal power series are determined by a finite set of linear homogeneous recurrence relations. This finite representation of an infinite series allows for efficient computation of operations (multiplication, addition, and in many cases inversion) on non-commutative rational functions.

Preservation Of Periodicity In Variational Integrators, Jian-Long Liu

Master's Theses

Classical numerical integrators do not preserve symplecticity, a structure inherent in Hamiltonian systems. Thus, the trajectories they produce cannot be expected to possess the same qualitative behavior observed in the original system. Pooling recent results from O'Neale and West, we explore a particular class of numerical integrators, the variational integrator, that preserves one aspect of the range of behavior present in Hamiltonian systems, the periodicity of trajectories. We first establish the prerequisites and some key concepts from Hamiltonian systems, particularly symplecticity and action-angle coordinates. Through perturbation theory and its complications manifested in small divisor problems, we motivate the necessity for …

Generic Polynomials, Lucas Spencer Mattick

Master's Theses

In Galois theory one is interested in finding a polynomial over a field that has a given Galois group. A more desirable polynomial is one that parametrizes all such polynomials with that given group as its corresponding Galois group. These are called generic polynomials and we provide detailed proofs of two theorems that give methods for constructing such polynomials. Furthermore, we construct generic polynomials for Sn, C3, V , C4, C6, D3, D4, and D6.

Hittingtime And Pagerank, Shanthi Kannan

Master's Theses

In this thesis, we study convergence of finite state, discrete, and time homogeneous Markov chains to a stationary distribution. Expressing the probability of transitioning between states as a matrix allows us to look at the conditions that make the matrix primitive. Using the Perron-Frobenius theorem we find the stationary distribution of a Markov chain to be the left Perron vector of the probability transition matrix.

We study a special type of Markov chain &mdash random walks on connected graphs. Using the concept of fundamental matrix and the method of spectral decomposition, we derive a formula

that calculates expected hitting times …

Galois Theory And The Hilbert Irreducibility Theorem, Damien Adams

Master's Theses

We study abstract algebra and Hilbert's Irreducibility Theorem. We give an exposition of Galois theory and Hilbert's Irreducibility Theorem: given any irreducible polynomial f(t₁, t₂, …, t_n, x) over the rational numbers, there are an infinite number of rational n-tuples (a₁, a₂, …, a_n) such that f(a₁, a₂, …, a_n, x) is irreducible over the rational numbers.

We take a preliminary look at linear algebra, symmetric groups, extension fields, splitting fields, and the Chinese Remainder Theorem. We follow this by studying …

Geometric Algebra: An Introduction With Applications In Euclidean And Conformal Geometry, Richard Alan Miller

Master's Theses

This thesis presents an introduction to geometric algebra for the uninitiated. It contains examples of how some of the more traditional topics of mathematics can be reexpressed in terms of geometric algebra along with proofs of several important theorems from geometry. We introduce the conformal model. This is a current topic among researchers in geometric algebra as it is finding wide applications in computer graphics and robotics. The appendices provide a list of some of the notational conventions used in the literature, a reference list of formulas and identities used in geometric algebra along with some of their derivations, and …

Mathematical Inequalities, Amy Dreiling

Master's Theses

In this thesis, we discuss mathematical inequalities, which arise in various branches of Mathematics and other related fields. The subject is a vast one, but our focus is on inequalities related to complex analysis, geometry, and matrix theory.

We investigate recently proven trigonometric and hyperbolic inequalities. This includes Katsuura's string of seven inequalities for the sine and tangent functions and Price's Inequality (with new proofs derived by Katsuura and Obaid). We also discuss complex hyperbolic inequalities and inequalities from infinite products.

We then establish geometric inequalities, including those relating parts of the triangle as well as conic sections and their …

Construction And Simplicity Of The Large Mathieu Groups, Robert Peter Hansen

Master's Theses

In this thesis, we describe the construction of the Mathieu group M₂₄ given by Ernst Witt in 1938, a construction whose geometry was examined by Jacques Tits in 1964. This construction is achieved by extending the projective semilinear group PΓL₃(F₄) and its action on the projective plane P²(F₄). P²(F₄) is the projective plane over the field of 4 elements, with 21 points and 21 lines, and PΓL₃(F₄) is the largest group sending lines to lines in P²(F₄).This plane has 168 6-point subsets, hexads, with the …

African American Community College Student Perceptions Of Mathematics Instructor Immediacy Behaviors And Perceived Cognitive Learning, Georgia Lynne Toland

Master's Theses

The purpose of this research is to examine what instructor immediacy behaviors are perceived to most positively affect the perceived cognitive learning of African American community college mathematics students. Data were provided via voluntary student surveys collected in various mathematics classrooms at two community colleges. The survey instrument was modeled from the works of Gorham (1988) and Sanders and Wiseman (1990) and included an experimental survey method of direct questioning. The perceptions of African American community college students regarding the behaviors of their mathematics instructors, along with their stated preferred instructor behaviors, produced a subset of effective strategies similar to …

Elliptic Curves And Cryptography, Senorina Ramos Vazquez

Master's Theses

In this expository thesis we study elliptic curves and their role in cryptography. In doing so we examine an intersection of linear algebra, abstract algebra, number theory, and algebraic geometry, all of which combined provide the necessary background. First we present background information on rings, fields, groups, group actions, and linear algebra. Then we delve into the structure and classification of finite fields as well as construction of finite fields and computation in finite fields. We next explore logarithms in finite fields and introduce the Diffie-Hellman key exchange system. Subsequently, we take a look at the projective and affine planes …

Applications Of Boundary Value Problems, Annie Nguyen

Master's Theses

In this thesis, we solved the Saint-Venant's torsion problem for beams with different cross sections bounded by simple closed curves using various methods. In addition, we solved the flexure problem of beams with certain curvilinear cross sections. The first method was derived by Bassali and Obaid. We focused on cross sections bounded by hyperbolas, circular groves, lemniscate of Booth, and sectorial cross sections. The second method used Tchebycheff polynomials to solve the torsion problem corresponding to the circle and ellipse. The third method used conformal mapping to derive the solution of different cross sections bounded by curvilinear edges. The flexure …

Family Of Circulant Graphs And Its Expander Properties, Vinh Kha Nguyen Tran Nguyen

Master's Theses

In this thesis, we apply spectral graph theory to show the non-existence of an

expander family within the class of circulant graphs. Using the adjacency matrix and its properties, we prove Cheeger's inequalities and determine when the equalities hold. In order to apply Cheeger's inequalities, we compute the spectrum of a general circulant graph and approximate its second largest eigenvalue. Finally, we show that circulant graphs do not contain an expander family.