Mathematics Commons^™

On Sharpest Tail Bounds For Functions Of Tail Bounded Random Variables, Stephen Harrison

Mathematics & Statistics ETDs

Consider n real/complex, independent/dependent random variables with respective tail bounds and g a measurable function of the r.v.’s. Consider f the “sharpest” tail bound of g (sharpest in the sense, if f were any less, then for some X1, ..., Xn satisfying the conditions, g(X1, ..., Xn) would not satisfy the tail f). Significant research has been done to approximate f often with high accuracy. These results are often of the form, for g in this family, and tail bounds of Xk in this family, f is bounded by some f′ with high accuracy. However, the question “what would it …

Predictors Of Math Identity In U.S. High School Students, Edwin Flores

Electronic Theses, Projects, and Dissertations

This study explored factors related to high school students’ sense of math identity. Data from the High School Longitudinal Study of 2009 was used which is a nationally representative dataset from the National Center of Education Statistics (NCES). The sample is representative of U.S. high schoolers who began ninth grade in 2009. Multiple regression analysis was performed and factors relating to students’ prior mathematics coursework and attitudes about mathematics were found to predict students’ mathematics identity.

Keywords. Mathematics, identity, high school students

Principal Quandles, Jesse Parrish

Electronic Theses and Dissertations

This thesis concerns principal quandles and, as a special case, Alexander quandles. A principal quandle is a coset quandle, Q(G,H, f), in which the subgroup H is trivial. If in addition the group G is abelian, then the coset quandle is called an Alexander quandle. The isomorphism types of Alexander quandles were classified [16], and some progress was made on describing the isomorphism types of principal quandles in [14] and [13]. The first part of this thesis applies the ideas used in Holmes’ paper to give a classification of the isomorphism types of principal quandles. The second …

Infographic: Pisa Insights – Encouraging Mathematical Thinking, Dominique Russell

Teacher infographics

Each cycle of the global Programme for International Student Assessment (PISA) includes a teacher questionnaire. One questionnaire topic – encouraging mathematical thinking – can help foster critical thinking, enhance problem-solving skills and allow students to see the relevance and connectedness of mathematics in their lives.

Italian Mathematics Prison Teachers’ Perspective On The Teaching/Learning Of Mathematics In Prison School, Andrea Maffia, Luca Decembrotto, Chiara Giberti, Federica Mennuni, Gabriella Pocalana, Silvia Regola, Agnese Telloni

Journal of Prison Education Research

Teaching in the prison context is characterized by a high complexity level; it is a multifaced activity influenced by several factors depending on the specific context. We present a preliminary but overall picture of schooling in Italian prisons by considering teachers’ reflections regarding their experiences in teaching mathematics in correctional facilities. To address the gap in the current literature on this issue, we conducted a SWOT analysis on the reflection provided by a group of nine Italian mathematics teachers who teach in different prisons and with different experiences and backgrounds. Their reflection allowed us to identify and describe six theory-driven …

Universal Centralizers, Morita Abelianization, And Wonderful Models In Lie Theory, Peter Crooks

Funded Research Records

No abstract provided.

Μ-Distributions And Μ-Distributed Sequences, Noah Graham Giddings

Undergraduate Honors Thesis Collection

The project attempts to generalize the notion of a uniform distribution for a broader class of probability measures and illustrate one construction of a sequence that satisfies these properties. A sequence ⟨a_n⟩ is uniformly distributed if the limiting relative frequency of sequence elements in any interval I ⊆ [0,1] corresponds with length(I). We generalize the notion of “uniform distribution” for an atomless Borel probability measure µ. We say that a sequence ⟨a_n⟩ is µ-distributed if the limiting relative frequency of sequence elements in any interval I ⊆ [0,1] equals µ( …

The Unending Constant, Christopher Banyas

Journal of Humanistic Mathematics

No abstract provided.

The Fake Mathematical Mindsets Of Preservice Teachers, Carmen M. Latterell

Journal of Humanistic Mathematics

Many students have a fixed mindset, as do some teachers. Recently I accidentally discovered that most of the preservice elementary teachers in my class were faking their mindset and pretending that they had a growth mindset. This raises a variety of questions, such as, will this follow a fake it until you make it track?

Exploring Commutativity In A Hindi Language Classroom, Kumar Gandharv Mishra

Journal of Humanistic Mathematics

Commutativity is an important concept in mathematics. However, not much emphasis is given to it in the elementary or secondary school curriculum, as there is a lack of interesting noncommutative examples at this level. Since most examples of binary operations provided deal with numbers, commutativity appears trivial. In this article we offer an interesting example of noncommutativity that is familiar to Indian school children from a different context. Students in India study a chapter based on the joining of vowels in Hindi grammar. This process is analogous to producing commutative as well as noncommutative operations. When we bring in mathematical …

Machine Learning: Neural Networking With Relu And Optimization, Aidan Redmond Brownell

Undergraduate Theses, Capstones, and Recitals

At its core, learning is an algorithmic process: it begins with input data, undergoes a series of transformations or computations, and yields an output intended to solve a specific task. This output is then compared against a target or desired result, and the internal mechanisms are updated based on how well the output aligns with expectations. While this feedback-driven process occurs almost effortlessly in humans, it is a far more structured, deliberate, and computationally intensive undertaking for machines.

Basic Theory And Implementations Of Quantum Error Correction, Derek Rodriguez

Undergraduate Theses, Capstones, and Recitals

The introduction of quantum computing has presented algorithmic solutions to computationally difficult challenges that are far more efficient than those of classical computers. These algorithms leverage the properties of quantum mechanics to manipulate the quantum properties of subatomic particles, requiring immense precision and stability. Current quantum hardware, however, is too noisy and introduces too many errors for these algorithms to be useful in practice, necessitating the use of error correction algorithms. This field survey seeks to introduce various principles of quantum mechanics relevant to quantum computing and quantum error correction (QEC), detail the implementation and motivations of a basic QEC …

Some Results In Thermodynamic Formalism, C. Evans Hedges

Electronic Theses and Dissertations

This dissertation investigates several key questions at the intersection of dynamical systems, computability theory, and thermodynamic formalism. In the symbolic setting, we establish novel results regarding the statistical properties of equilibrium states, deriving bounds on probabilities of configurations and relating these bounds to the Gibbs property through the homoclinic relation. Additionally, we examine the computability of thermodynamic quantities such as pressure, ground state energy, and residual entropy. We show that topological pressure is computable from above for general subshifts and computable for strongly irreducible shifts, with similar results extending to ground state energy and residual entropy.

Extending beyond subshifts, we …

The Mckay-Navarro Conjecture For The Prime 2, L. Ruhstorfer, A. A. Schaeffer Fry

Mathematics: Faculty Scholarship

We complete the proof of the McKay-Navarro conjecture (also known as the Galois-McKay conjecture) for the prime 2, by completing the proof of the inductive McKay-Navarro conditions introduced by Navarro-Späth-Vallejo for this prime.

Agent-Based Modeling: Introduction And Actuarial Applications, Rick Gorvett

Mathematics and Economics Faculty Working Papers

Agent-based modeling (ABM) has become an important and valued approach to modeling complex systems. In this paper, I advocate for actuaries to recognize the complex systems-nature of socioeconomic and risk processes and for ABM models to become a regular resource in our actuarial toolkits. These models allow for the observation of potential macro-behavior emerging from the underlying agent-level micro-activity and characteristics. Therefore, ABM models can provide significant insight into the quantification of risk and the identification of optimal strategies. This paper is an introduction and guide to ABM models, and it includes several case studies to illustrate their utility.

Independent Vs. Signed Musicians: Does Thier Status Affect Their Success?, Zachary Vincent

Honors Projects in Mathematics and Economics

As technology has advanced and become more widely available, and as the public learns more about the recording industry and its workings, more musicians are choosing not to sign with record labels, becoming independent musicians. This now raises the question: does a musician’s status of independent or signed determine their level of success? It has been previously found that independent musicians have a more difficult time competing with signed musicians due to the resources provided by record labels and the way labels negotiate with large streaming platforms, giving signed labels more opportunities to be seen in the public eye, but …

Exploration Into Homotopy Equivalence And Vietoris-Rips Complexes, Alexander J. Mobley

Honors Theses

An important question in discrete topology and geometry is how to recover the structure and characteristics of a manifold when only given a finite set of points sampled from that manifold. Thus, if mathematicians have a point cloud of data which induces a discrete metric space, they look for some structure to these points. Understanding the structure of these points often gives needed insight to solve this problem. One such way to determine structure to these points is to use these points to construct a Vietoris-Rips complex. In abstract, Latschev shows that this allows us to recover the homotopy type …

Transformations Of Bound Quivers, Ari Pincus-Kazmar

Mathematics, Statistics, and Computer Science Honors Projects

A quiver Q is a directed multigraph. Representations of Q are assignments of vector spaces and linear maps to the vertices and arrows of Q; these are categorically equivalent to the modules over the path algebra kQ. The indecomposable representations of a given quiver and the irreducible morphisms between them are summarized combinatorially in the Auslander-Reiten quiver which, in the case of algebras of finite representation type, gives almost complete information about the category of representations. We introduce an intuitive transformation from a family of bound quivers to a family of A-type quivers which preserves properties of the Auslander-Reiten quiver.

Equity Vs. Excellence: Dichotomy Or Duo?, Nadjia I. Logans

Honors Program: Senior Projects (Public)

In this thesis, I investigate recent changes in mathematics education and how these developments connect to the principles of equity and excellence. I place a special focus on “active learning” methods. My methodology primarily consisted of interviews with educators, along with analyzing literature from math education researchers. I first define and explain the motivations behind active learning instruction methods. Next, I explore not only how instructors define equity and excellence but how educators perceive the role of the two principles within education. Finally, I challenge the pervasive media portrayal of equity and excellence as being mutually exclusive. Instead, I find …

Bibliography Of Christianity And Mathematics Update, Calvin Jongsma

Faculty Work Comprehensive List

This was a banquet talk given at the Twenty-fourth Biennial Conference of the Association of Christians in the Mathematical Sciences held at Dordt University on May 28-June 1, 2024.

It provides an update on the progress being made in the author’s project of producing a second edition of Bibliography of Christianity and Mathematics. This work is now being developed as a Zotero database with a slightly broader focus on Religious Faith and the Mathematical Sciences and with a much expanded list of books, articles, and blog posts on the topic. The author hopes to make the database public sometime …

Prime Theta-Curves From Torus Knots, Sam Nieman, Jack Calcut

Research Symposium

Classical knot theory is the placement problem for a circle in Euclidean 3-space or in the 3-sphere. That problem has been generalized in numerous ways. Knotted graph theory replaces the circle with a graph. Knotting of certain graphs has been well-studied. That includes the theta-graph consisting of two vertices connected by three edges---a copy of the letter theta. A theta-curve is an embedding of the theta-graph in 3-space or in the 3-sphere. Connected sum of knots yields the notion of a prime knot. Similarly, one may sum two theta-curves at a vertex and one may sum a theta-curve and a …

Basketball Jump Shot Correction Using Computer Vision, Matthew Moran

Mathematics & Computer Science Student Scholarship

Matthew Moran ’25, Computer Science major

Faculty mentor: Dr. Martin Hellwig, Mathematics and Computer Science

Basketball has evolved significantly over the years, with players continuously striving to improve their shooting accuracy and efficiency. As the game becomes more data-driven, advancements in artificial intelligence and computer vision have provided new methods for analyzing and correcting shooting mechanics that allow players to improve their game. Using deep learning, this research aims to fill a gap in these advancements, which is looking at the position of the shooters fingers relative to the ball. Doing so can improve stability and overall shooting performance.

The Significance Of Symmetry: Exploring T-Numbers And Their Role In Virus Survival And Behavior, Jenna Franco, Taylor Ligozio

Mathematics & Computer Science Student Scholarship

Jenna Franco ’25, Biology major

Taylor Ligozio ’25, Biology major

Faculty mentor: Dr. Su Jeong Kang, Mathematics and Computer Science

We studied the symmetry of different structures in common viruses and their T-numbers. We related these T-numbers with characteristics of the virus. Next we researched how the T-number of each virus was optimal for its survival and reproduction in its environment. We can use this information to infer or predict the behavior of viruses in the future based on their T-numbers.

Gerrymandering And How To Find It: A Mathematical Approach, Nicholas Small

Mathematics & Computer Science Student Scholarship

Nicholas Small ’25, Mathematics and Computer Science major

Faculty mentor: Dr. Liam Donohoe, Mathematics and Computer Science

Gerrymandering is a phenomenon that has plagued American politics for over 200 years. This process, which involves redrawing congressional and legislative district boundaries to favor one group over another, has become a fact of life in the past few decades. This project explores the history of gerrymandering and some of the algorithms proposed to combat it. The two main strategies of Gerrymandering are packing—or limiting the other group’s influence by making them the extreme majority in a few districts—and cracking—or splitting up a …

Reinventing Mathematics Learning: An Exploration Of Undergraduate Mathematics Learning Post-Pandemic, Morgan R. Balesano

Honors Scholar Theses

In response to the COVID-19 pandemic, beginning in March 2020 nearly all secondary and undergraduate mathematics students were forced to adapt to new methods of learning and testing. As a result, the current experience for these mathematics students has changed vastly, as have the opinions and preferences of these students in terms of their learning. This study aims to identify instruction and testing resources and methods that students prefer and those that students find less beneficial in their current, post-pandemic educational experience. The past few years have seen a focus on the direct impacts of online learning during the pandemic, …

Exploring Teacher Knowledge In Meeting The Expectations Of Integrated Content Language Development In Mathematics: A Descriptive Case Study, Carrissa Rozelaar-Miller

Doctoral Dissertations and Projects

The purpose of this descriptive case study was to discover how knowledge is applied to meet the expectations of integrated content and language development in mathematics for Title I elementary teachers at an urban school district in the central United States. Halliday’s theory on systemic functional linguistics guided the research as it relates to the function and purpose of language through meaning and social interaction in different genres. The central research question was: How do Title I elementary teachers apply specific integrated content and language development knowledge in mathematics? The participants consisted of 10 teachers with at least three years …

The Importance Of Motivation And Self-Concept For Predicting Performance In Service Mathematics Modules: Replicating And Extending Previous Findings, Catherine Palmer, Maryna Lishchynska, Declan O’Connor, Seán Lacey

Department of Mathematics Publications

Students’ difficulties and lower-than-desired mathematics performance in higher education have been the focus of educational research and practice for a period of time. In a recent study by Lishchynska et al. (2023), learners’ motivation, dispositions towards mathematics and learning strategies were examined as factors potentially affecting performance in service mathematics modules. The main objectives of this study were to replicate the findings of Lishchynska et al. (2023) with a new student cohort; determine if self-concept is a significant contributortomathematicsperformance;andexploreifmathematicalbackgroundremainsan important predictor of mathematics performance for students as they migrate from first year to final year of study. A survey of …

Assessing Maths Modules Online, Blathnaid Sheridan

Case studies: Digital Education

With class sizes increasing and students anxious to know their CA results asap, I began looking at an alternative way of assessing students on their maths modules. Furthermore, in an attempt to ensure better engagement with the content and continuity across maths modules in Years 1 & 2, students are now assessed using online quizzes with immediate results.

Navigating Geometry With Interactive Lessons: A Collection Of Resources In Geogebra That Address Frequently Held Misconceptions, Sara Vance

Honors Program: Senior Projects (Public)

Nearly every high school student in the United States is required to take a geometry class, and for many, it’s unlike any math class they’ve taken before. Additionally, there are certain topics with which past and present geometry students alike tend to struggle. This project was created to address these issues; my goal is that these materials will help students better understand these topics and clear up their frequently held misconceptions about geometry. This project was supported by interviews I conducted with veteran high school geometry teachers with over 75 years of teaching math between them.

This collection of interactive …

Math Is _______: A Study On Prospective Teachers' Perceptions Of Math And Teaching Math, Kayla Mead, Katherine Baker

Journal of Humanistic Mathematics

In this project, we examined one cohort of twenty-eight college students and their perceptions of mathematics at the beginning and end of a semester-long mathematics content course for K-8 prospective teachers (PTs). More specifically, our study explored PTs’ perceptions of what mathematics is and what mathematics teaching is, and how and why those perceptions might change over the semester-long mathematics course. PTs’ perceptions were captured through open-ended surveys and follow-up interviews. Our intention was to better understand the reasonings for changes in perceptions during the course to support teacher educators in their work of preparing prospective teachers.