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:** Travis Steele**Advisor: **Khang Tran**Graduation Date:** May 2023**Thesis Title:** Zero Distribution of Polynomials Generated by a Power of a Cubic Polynomial

Abstract: We study the zero distribution of the sequence of polynomials $(P_m^{(\alpha)}(z))_{m=0}^\infty$
generated by $(1+t+zt^3)^{-\alpha}$, where $\alpha$ is an arbitrary fixed positive
real number. Through asymptotic analysis we show that the real interval $(-\infty,-\frac{4}{27})$
contains all but two of the zeros of $P_m^{(\alpha)}(z)$ for $m$ sufficiently large.

**Student:** Eric Montoya**Advisor: **Marat Markin**Graduation Date:** May 2023**Thesis Title:** On Hypercyclicity and Linear Chaos in a Nonclassical Sequence Space and Beyond.**Abstract:** We analyze the hypercyclicity, chaoticity, and spectral structure of (bounded and
unbounded) weighted backward shifts in a nonclassical sequence space. Based on the
weighted backward shifts, we further construct new (bounded and unbounded) linear
hypercyclic and chaotic operators both in the nonclassical sequence space and the
classical space $l_1$.

**Student:** Gabriel Chavez**Advisor:** Yaomingxin Lu**Graduation Date:** May 2023**Thesis Title:** Pedagogy: Its Effects on Non-Cognitive Outcomes for First Semester Calculus I Students **Abstract:** We investigate pedagogical approaches in a first-semester calculus I course at California
State University, Fresno (Fresno State), and their implications for student outcomes
in STEM fields. To address achievement gaps for first-generation, low-income, and
underrepresented minority students (URM), Fresno State redesigned its traditional
calculus course into a reformed calculus curriculum. We develop an analytical framework
to assess the impact of these distinct pedagogical approaches on students’ non-cognitive
achievements, focusing on the role of cooperative learning in fostering non-cognitive
outcomes such as confidence and self-efficacy. Employing a mixed-methods approach,
we conduct a detailed analysis of student surveys, interviews, and classroom observations
to elucidate the specific mechanisms through which pedagogical strategies affect URM
students, incorporating an intersectional lens to better understand the diverse experiences
of these learners. Our findings reveal significant differences in students’ non-cognitive
achievements between traditional and reformed calculus instruction, highlighting the
potential benefits of reformed teaching practices in promoting students’ confidence,
self-efficacy, and overall success in STEM fields.

## Past Theses

**Student:** Gabriel Martinez Lazaro**Advisor:** Marat Markin**Graduation Date:** May 2022**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.

**Student:** Marissa Morado**Advisor: **Comlan De Souza**Graduation Date: **May 2022**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 **Advisor:** Carmen Caprau**Graduation Date:** May 2022**Thesis Title:** Extending the Kauffman 2-Variable Polynomial to Singular Links**Abstract:** In this work we construct invariants of singular links in three-space via the existence
of a classical invariant of framed links known as the Kauffman 2-variable polynomial,
or simply, the Kauffman polynomial. We extend the Kauffman polynomial to singular
links by way of algebras and trace functions. In constructing our invariant, we give
motivation for extending the notions for classical links and braids to those of singular
links and singular braids. Additionally, we define and study the singular Birman-Murakami-Wenzl
algebra and explain how it is used in defining our invariant of singular links. We
then show that there exists an infinite number of invariants that extend the Kauffman
polynomial to singular links. Lastly, we provide an alternative way to construct our
invariants, by defining and studying the Kauffman skein algebra for framed singular
links. This research shows us how we can extend other classical link invariants to
singular links in a similar way.

**Student: **Miguel Baza**Advisor: **Oscar Vega**Graduation Date:** December 2021**Project Title:** The Positively Realizable Groups of H_{8} and G_{3}**Abstract:** Spatial graph theory is the study of graphs embedded in R^{3}. 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 R^{3}. In particular, we study graphs H_{8} and G_{3} 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 H_{8} and G_{3}.

**Student:** Summer Al-Hamdani**Advisor:** Khang Tran**Graduation Date:** May 2021**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**Advisor:** Carmen Caprau**Graduation Date:** May 2021**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**Advisor:** Oscar Vega**Graduation Date:** May 2021**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**Advisor:** Oscar Vega**Graduation Date:** May 2021**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**Advisor:** Oscar Vega**Graduation Date:** May 2021**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.

**Student:** Bradley Scott**Advisor:** Carmen Caprau**Graduation Date:** May 2021**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**Advisor:** Mario Banuelos**Graduation Date:** December 2020**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**Advisor: **Oscar Vega

**December 2019**

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**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**Advisor: **Marat Markin**Graduation Date: **December 2019**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 Siche**lAdvisor: **Marat Markin

**December 2019**

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**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

**Oscar Vega**

**Advisor:****May 2019**

**Graduation Date:****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**Advisor:** Morgan Rodgers**Graduation Date:** May 2019**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).

**Student:** Richard Adams**Advisor:** Khang Tran**Graduation Date:** December 2018**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** Advisor:** Carmen Caprau

**May 2017**

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**A Categorical Model for the Virtual Singular Braid Monoid**

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

**Student: **Nicholas Newsome** Advisor:** Maria Nogin and Adnan H. Sabuwala

**May 2017**

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**An Investigation of Power Sums of Integers**

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**Student: **Jeffrey Park** Advisor:** Tamas Forgacs

**May 2016**

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**Bell Multiplier Sequences**

Thesis Title:

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**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** Advisor:** Carmen Caprau

**May 2016**

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**Polynomial Invariants for Virtual Singular Knots**

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**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** Advisor:** Carmen Caprau

**May 2016**

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**A Study of Projections of 2-Bouquet Graphs**

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**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** Advisor:** Maria Nogin

**May 2016**

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**Investigation of the Topological Interpretation of Modal Logics**

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**Student: **Jennifer Elder** Advisor:** Oscar Vega

**May 2016**

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**Generalizing the Futurama theorem**

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**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** Advisor:** Oscar Vega

**May 2015**

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**The Well-covered Dimension of the Adjacency Graph of Generalized Quadrangles**

Thesis Title:

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**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** Advisor:** Tamas Forgacs

**May 2015**

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**Multiplier Sequences of the Second Kind**

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**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** Advisor:** Oscar Vega

**May 2014**

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**Spreads and Parallelisms**

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**Study of partitions of the lines in a projective plane into lines that partition the points in such plane.**

Abstract:

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**Student: **Katherine Urabe** Advisor:** Carmen Caprau

**May 2014**

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**The Dubrovnik Polynomial of Rational Knots**

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**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.**

Abstract:

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**Student: **Karen Willis** Advisor:** Oscar Vega

**May 2014**

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**Blocking Polygons in Finite Projective Planes**

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**Study of configurations of points in a finite projective plane that do not allow the existence of polygons that are disjoint from such set.**

Abstract:

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