Department of Mathematics

# Theses

Since Spring 2013, our department has offered a Thesis option to our Masters students. A thesis, unlike a project, consists on original research done by the student. Below are the theses our students have produced.

## Recent Theses

Student: Juhoon Chung
Project Title: Zero Distribution of Generated Taylor Polynomials
Abstract: Our goal in this paper is to study the zero distribution of a sequence of polynomials whose coefficients satisfy a three-term recurrence. Equivalently, these polynomials are Taylor polynomials of a rational function with a polynomial denominator of degree 2. We will use complex analysis to find the bi-variate generating function for that sequence of Taylor polynomials. With this generating function, we prove that the
zeros of these Taylor polynomials lie on one side of the closed disk centered at the origin. The radius of the disk is exactly the reciprocal of the modulus of the smaller zero of the degree two denominator. Finally, we show that the zeros of these Taylor polynomials approach the boundary of
the disk above.

Student: Leila Nabibidhendi
Thesis Title: The Effectiveness of Growth Mindset Activities on Students in Support Classes at Fresno State
Abstract: This project uses mixed methods to study the self-efficacy of students in mathematics support classes (a set of classes designed to provide additional help for underprepared students entering the university).  Also, the quantitative part of the study investigates the effect of mindset
interventions while taking these classes compared to the mindset changes of the students in traditional classes. The Fennema-Sherman mathematics attitude scale and Dweck mindset questionnaire were used to measure the effects of these treatments on the attitudes and beliefs of
participants in this study. The qualitative data analysis was conducted using interviews and student written reflections to gain an insight into how these affected individual students and the whole group.

Student: Gabriel Martinez Lazaro
Thesis Title: On Linear Chaos in The Space of Convergent Sequences
Abstract: We show that linear chaos in the space $c(\N)$ of convergent sequences cannot be arrived at by merely extending the weighted backward shifts in the space $c_0(\N)$ of vanishing sequences. Applying a newly found \textit{sufficient condition for linear chaos}, we furnish concise proofs of the chaoticity of the foregoing operators along with their powers and also itemize their spectral structure. We further construct bounded and unbounded linear chaotic operators in $c(\N)$ as conjugate to the chaotic backward shifts in $c_0(\Z_+)$ via a homeomorphic isomorphism  between the two spaces.

Thesis Title: Solving The Phase Retrieval Problem Using an Artificial Neural Network
Abstract: Previous algorithms have attempted to solve the Phase Retrieval Problem using linear approximations. The Gerchberg Saxton (GS) Algorithm was the first algorithm that created an efficient way to solve the Phase Retrieval Problem by measuring the intensities between the image and Difractionplanes. While initially successful, the algorithm’s drawbacks include a low recovery rate of the phases and stagnated iterations, causing
excessive computation time and unreliable solutions. To improve the success rate of the algorithm, we have created a neural network that combines the reconstructive methods from the Gerchberg Saxton (GS) Algorithm with an artificial neural network (ANN) to produce a more accurate phase retrieval algorithm than the original GS Algorithm. These modifications can be beneficial in applications such as x-ray crystallography, electron microscopy, astronomical imaging, and a multitude of image and signaling problems. Our goal is to provide a practical approach to the Phase Retrieval Problem that provides a feasible solution that is not computationally expensive.

Student: Matthew Nuyten
Thesis Title: Extending the Kauffman 2-Variable Polynomial to Singular Links

Student: Miguel Baza
Project Title: The Positively Realizable Groups of H8 and G3
Abstract: Spatial graph theory is the study of graphs embedded in R3. Given a graph and its automorphism group, we show how studying fixed vertices and edges allows us to determine the possible symmetries which a graph may have in R3. In particular, we study graphs H8 and G3 from the K7 family, demonstrating a new technique to determine that a group cannot be induced by orientation preserving homeomorphisms of the space R . In doing so, we classify the positively realizable groups for the graphs H8 and G3.

## Past Theses

Student: Summer Al-Hamdani
Thesis Title: Zero Distribution of Binomial Combinations of Chebyshev Polynomials of the Second Kind.
Abstract:  We consider the sequence {P_m(z)}_{m=0}^\infty, which is a binomial combination of the well-known Chebyshev polynomials of the second kind; they have all real zeros on the interval (-1,1). We prove that there exists a constant C (independent of m) such that the number of zeros of P_m(z) outside of the interval (-1,1) is at most C for all m in N.

Student: Jagdeeep Basi
Thesis Title: Quandle Coloring Quivers of Torus Knots.
Abstract: Quandles provide an algebraic perspective to the general goal of knot theory of classifying and distinguishing mathematical knots.  We study patterns of dihedral quandles under the lens of quandle colorings and quandle coloring quivers.   This work classifies the full quandle coloring quivers of (p,2)-torus knots as a base case for the full quandle coloring quivers of general (p,q)-torus knots.

Student: Samuel Cleofas
Thesis Title: Hyperplanes Arrangements over Finite Fields.
Abstract: This work studies blocking sets of the complement of a hyperplane arrangement in a vector space over a finite field in a way that it generalizes the work of Settepanella on hyperplane arrangements in a finite space.

Student: Maria Diaz
Thesis Title: The Characteristic Polynomial of a Spread in F_q^4.
Abstract: We introduce the concept of the characteristic polynomial of a spread in F_q^4 and then see what the degree of these polynomials are for a variety of spreads of F_q^4, mostly for small values of q. We focus specially on regular and Andre spreads.

Student: Erick Gonzalez
Thesis Title: The Topological Symmetry Group of Graphs in F_Delta(K_7).
Abstract: We study the embeddings of C_{13} and H_8, and determine which automorphism groups of these graphs can be induced by groups of homeomorphisms of R^3.

Thesis Title: Minimal Generating Sets of Oriented Reidemeister-Type Moves for Knot and Spatial Trivalent Graph Diagrams.
Abstract: It has recently been shown by Polyak that all oriented versions of Reidemeister moves for knot diagrams can be generated by a set of just 4 oriented Reidemeister moves, and no fewer than 4 moves generate them all. We expand upon Polyak's work by proving the existence of an additional 11 minimal generating sets of oriented Reidemeister moves for oriented knot diagrams, and we prove that these 12 sets represent all possible minimal generating sets. Then we consider the Reidemeister-type moves that relate oriented spatial trivalent graph diagrams and prove the minimality of a generating set of 10 such moves.

Student: Erica Sawyer
Thesis Title: Deep Learning Methods for Detecting Structural Variants in Related Individuals
Abstract: We implement neural networks to predict Structural Variants (SVs).  We discuss a model, which incorporates the observed genomic information of two parents and an offspring to predict locations of SVs in the genome of the child.  We discuss a generalization of this model and we investigate the performance of these models under different neural network architectures.

Student: Elizabeth Compton
Oscar Vega
December 2019
Thesis Title: The Power Graph of Split Metacyclic Groups
Abstract: Split metacyclic groups generalize the family of dihedral groups in a natural way. The power graph of a group is a refinement of the group lattice that focuses only on cyclic subgroups.  This thesis characterizes the power graph of split metacyclic groups using exclusively group-theoretical tools.

Student: John Jimenez
Thesis Title: On the Chaoticity of Rolewicz-Type Operators on Function Spaces
Abstract: The chaoticity of Rolewicz-type linear operators on certain function spaces is proved and their spectral structure is revealed.

Student: Edward Sichel
Marat Markin
December 2019
Thesis Title: On Expansive Mappings and Non-Hypercyclicity
Abstract: We take a close look at the nature of expansive mappings on certain metric spaces (compact, totally bounded, and bounded), provide a finer classification for such mappings, and use them to characterize boundedness. We also furnish a simple straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) normal operator in a complex Hilbert space as well as of a certain collection of its exponentials.

Student: Nathan Thom
Thesis Title: The Veldkamp Space of W(p)
Abstract: Characterization of the Veldkamp space of the symplectic generalized quadrangle over a field of prime order. The points of this space are the geometric hyperplanes of W(p), and three points are collinear if all the pairwise intersections coincide.

Student: Anthony Vogt
Thesis Title: Using 2-ovoids to generate independent sets in W(q)
Abstract: Investigation, using tools from geometry, algebra, graph theory and computer software, of how 2-ovoids could be used to construct independent sets in graphs obtained from the generalized quadrangle W(q).

Thesis Title: On hyperbolic polynomials and four-term recurrence with linear coefficients
Abstract: Study necessary and sufficient conditions for the hyperbolicity of polynomials satisfying a four-term recurrence whose first and third power coefficients are linear.

Student: Sarah McGahan
Carmen Caprau
May 2017
Thesis Title:
A Categorical Model for the Virtual Singular Braid Monoid

Abstract: Different approaches are used to study the algebraic structure of the virtual singular braid monoid.

Student: Nicholas Newsome
Maria Nogin and Adnan H. Sabuwala
May 2017
Thesis Title:
An Investigation of Power Sums of Integers

Student: Jeffrey Park
Tamas Forgacs
May 2016
Thesis Title:
Bell Multiplier Sequences

Abstract: An investigation of sequences of real numbers, which - when viewed as linear operators on polynomials - preserve hyperbolicity of polynomials in the Bell basis.

Student: Kelsey Friesen
Carmen Caprau
May 2016
Thesis Title:
Polynomial Invariants for Virtual Singular Knots

Abstract: Well-known polynomial invariants for classical knots are extended to virtual singular knots and the properties of the resulting polynomials are studied.

Student: Elaina Aceves
Carmen Caprau
May 2016
Thesis Title:
A Study of Projections of 2-Bouquet Graphs

Abstract: The concepts of trivializing and knotting numbers are extended from classical knots to spatial graphs and 2-bouquet graphs, in particular. The trivializing and knotting numbers for projections and pseudodiagrams of 2-bouquet spatial graphs are calculated based on the number of precrossings and the placement of the precrossings in the pseudodiagram of the spatial graph.

Student: Bing Xu
Maria Nogin
May 2016
Thesis Title:
Investigation of the Topological Interpretation of Modal Logics

Student: Jennifer Elder
Oscar Vega
May 2016
Thesis Title:
Generalizing the Futurama theorem
Abstract: Every permutation x can be written as a product of cycles that have not been used in the construction of x, as long as two new elements are incorporated in the transpositions. This results generalizes Keeler's Theorem A.K.A. The Futurama Theorem.

Student: Hillary Bese
Oscar Vega
May 2015
Thesis Title:
Abstract: The well-covered space of the adjacency graph of the classical generalized quadrangle W_q is trivial, for every prime power q.

Student: David Heywood
Tamas Forgacs
May 2015
Thesis Title:
Multiplier Sequences of the Second Kind
Abstract: An investigation of sequences of real numbers, which - when viewed as standard diagonal linear operators on polynomials - preserve the real rootedness of polynomials with only real zeroes of the same sign.

Student: Megan Kuneli
Oscar Vega
May 2014
Thesis Title:
Abstract:
Study of partitions of the lines in a projective plane into lines that partition the points in such plane.

Student: Katherine Urabe
Carmen Caprau
May 2014
Thesis Title:
The Dubrovnik Polynomial of Rational Knots
Abstract:
Finding a closed form expression for the Dubrovnik polynomial of a rational knot or link diagram in terms of the entries of its associated vector. The resulting closed form allows a Mathematica program which efficiently computes the Dubrovnik polynomial of rational knots and links.

Student: Karen Willis