Mathematics Commons^™

Path Odd-Covers Of Graphs, Steffen Borgwardt, Calum Buchanan, Eric Culver, Bryce Frederickson, Puck Rombach, Youngho Yoo

Mathematics: Faculty Scholarship

We introduce and study “path odd-covers,” a weakening of Gallai's path decomposition problem and a strengthening of the linear arboricity problem. The path odd-cover number  of a graph G is the minimum cardinality of a collection of paths whose vertex sets are contained in  and whose symmetric difference of edge sets is. We prove an upper bound on  in terms of the maximum degree Δ and the number of odd-degree vertices  of the form . This bound is only a factor of 2 from a rather immediate lower bound of the form . We also investigate some natural relaxations of …

Mckinsey-Tarski Algebras And Raney Extensions, G. Bezhanishvili, R. Raviprakash, A. L. Suarez, Joanne Walters-Wayland

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce the notion of Raney morphism between MT-algebras and show that the resulting category is equivalent to the category of Raney extensions. This is done by generalizing the construction of the Funayama envelope of a frame. The resulting notion of the T₀-hull of a Raney extension generalizes that of the T_D-hull of a frame.

Unitary Rational Functions: The Scaled Quaternion Case, Daniel Alpay, Ilwoo Choo, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

We develop the theory of minimal realizations and factorizations of rational functions where the coefficient space is a ring of the type introduced in our previous work, the scaled quaternions, which includes as special cases the quaternions and the split quaternions. The methods involved are not a direct generalization of the complex or quaternionic settings, and in particular, the adjoint is not the classical adjoint and we use properties of real Hilbert spaces. This adjoint allows to define the counterpart of unitarity for matrix-rational functions, and we develop the corresponding theories of realizations and unitary factorizations. We also begin a …

Does Sequence Matter? Impact Of Redesigning Sequential Calculus Course On Students’ Learning Outcomes, Chantal Levesque-Bristol Dr., Wonki Lee Dr., Emily M. Bonem, Benjamin C. Wiles, Jennifer D. Moss, Wilella D. Burgess, Weiling Li

International Journal of Teaching and Learning in Higher Education

As colleges and universities increasingly transform their STEM courses through the adoption of more active, student-centered pedagogies, there is a growing need to understand the impact of these educational innovations on student outcomes. This study is conducted in the context of a university-wide faculty development and course redesign project, IMPACT (Instruction Matters, Purdue Academic Course Transformation). IMPACT supports faculty in implementing student-centered pedagogical practices and creating equitable and inclusive learning environments. This study focuses on a sequence of two introductory calculus courses (Calculus 1 and Calculus 2, hereafter CALC 1 and CALC 2) that were transformed as part of the …

Toward Completeness Theorem For Guarded Kleene Algebra With Tests, Hung Pham

Honors Theses

Code refactoring is a fundamental practice in software engineering, in which a program is restructured without changing the actions it performs and the results it produces. To carry out refactoring with confidence, one requires a formal method for verifying that two programs are equivalent. Guarded Kleene Algebra with Tests (GKAT) provides such a framework, an algebraic system designed to reason about a natural class of programs, namely those in which every branch and loop is governed by a Boolean condition, such as if–else and while statements. Central to GKAT is a finite set of algebraic axioms for deriving program equivalences. …

Extensions Between Modules Defined By Lattice Paths In The Preprojective Algebra, Chloe Napier

Theses and Dissertations--Mathematics

In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach. The main goal is to find a combinatorial description for when certain cluster variables are compatible, or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc, and Schröer that answer this question in the case of …

Catalan And Hyper-Catalan Numbers: Combinatorial Applications To Polynomial Equations, Leilani Natale

Williams Honors College, Honors Research Projects

In this paper, we study Catalan numbers and their generalization, hyper-Catalan numbers, and explore how these sequences arise naturally in the context of solving polynomial equations using infinite power series. We begin by introducing the Catalan numbers through their combinatorial interpretation as triangulations of convex polygons. Using this geometric definition, we derive a relation whose recursive structure leads to a quadratic functional equation. Interpreting this relation as a formal power series equation allows us to express solutions to quadratic equations as infinite power series whose coefficients are given by the Catalan numbers. This framework is then extended by allowing polygon …

Representations Of Finite Groups And Diagrammatic Algebras, Hudson Yeend

CMC Senior Theses

Representation theory allows mathematicians to study abstract mathematical objects using the powerful and concrete tools of linear algebra. This thesis aims to present some foundational concepts in representation theory and apply these concepts to specific groups and algebras. We begin by examining representations of finite groups, culminating with a proof of Maschke's theorem. We then use the correspondence between a group and its group algebra to segue into a study of representations of diagrammatic algebras, where we introduce analogous notions of decomposition. We end with a study of quiver representations, noting that Gabriel's theorem and the kQ-modular structure transcend …

Mat 1500 Calculus I Syllabus, Tian Cai

Open Educational Resources

No abstract provided.

Mat 1600 Syllabus Ii Syllabus, Tian Cai

Open Educational Resources

No abstract provided.

Studies On The Depth Formula And On Reducing Dimensions, Brian Mccourt Laverty

Graduate Theses, Dissertations, and Problem Reports (ETD)

This dissertation presents the author’s recent research, conducted under the supervision of Professor Olgur Celikbas, and based on two articles—one published and one in progress. These works develop two closely related research directions in commutative algebra. Together, they contribute to the subject by addressing aspects of existing conjectures, establishing new results, and introducing methods for studying homological invariants.

The first research direction concerns the depth formula, namely the equality \[ \depth_R(M)+\depth_R(N)=\depth(R)+\depth_R(M\otimes_RN) \] where $M$ and $N$ are finitely generated $R$-modules. A classical result of Auslander \cite{Aus} shows that the depth formula holds provided that either $M$ or $N$ has finite …

Lie-Galois Theory, Giovanni Reed

Honors Undergraduate Theses

Differential equations are a much-studied topic in the field of mathematics, as well as other sciences, such as engineering, economics, and biology. While much is known concerning these, there is still a large gap in our knowledge about such equations. It is important therefore, both to mathematics and other sciences, that we gain a more complete knowledge of differential equations, in particular the nature of their solutions. In this research, we investigate the solution space of linear ordinary differential equations (ODEs) from the standpoint of differential algebra. Differential algebra allows an ODE to be treated similarly to a polynomial, allowing …

Test Retakes In Introductory Math Courses, John Hird, Chris Mcclain

2026 Scholarly Teaching Conference: Poster Session Papers

In this poster, we describe the implementation of an exam retake model using specifications grading for math courses taken by non-STEM majors. This system was implemented by two faculty members over three years in two sequential courses. During that time, we tried several versions of allowed retakes, with varying restrictions on partial credit. Some of the challenges that we faced were scaling the system for use by different faculty members and with different courses, managing faculty workload on writing and grading multiple exams, and managing student expectations.

When The Gorenstein Projective Modules Are Precovering: A Survey On The Existence Of The Gorenstein Projective Precovers, Alexis L. Mcburney

College of Graduate Studies: Theses & Dissertations

This thesis surveys known results on the existence of Gorenstein projective precovers and studies conditions under which the class of Gorenstein projective modules is precovering. In classical homological algebra, projective resolutions are used to study modules, while relative homological algebra replaces projective modules with a different class that must be precovering for resolutions to exist. Gorenstein homological algebra extends these ideas using Gorenstein projective modules. We review key developments by Jorgensen, Murfet and Salarian, Estrada, Iacob, and Yeomans, and Cortés and Saroch. We discuss the strongest known existence result, showing that Gorenstein projective precovers exist under suitable conditions; in particular, …

Fuchs' Problem For Quasi-Cyclic Groups, Dalen F. Elliott

Theses and Dissertations

Fuchs’ problem asks which groups can arise as the group of units of a ring. Although the finite cyclic case has been completely classified, much less is known in the infinite setting. This thesis contributes to this problem by investigating quasi-cyclic. (Pr¨ufer) groups and their finite direct products. We show that for every odd prime p, there is no commutative ring R such that R×∼= Cp∞. This obstruction arises from characteristic restrictions and the algebraic structure of finite fields. More generally, we prove that any group in which every element has order a power of an odd prime p and …

Fostering A Growth Mindset In Mathematics: Faculty And Student Experiences, Yolanda G. Rush

Theses and Dissertations

According to the Center for Community College Student Engagement (2019), many students attending two-year institutions need productive persistence strategies, including the development of a growth mindset. Although some growth mindset interventions have been effective in improving academic achievement among students (Boaler, 2016; Canning et al., 2024) and persistence (Lewis, 2019) among students, especially those with developmental needs (Suh et al., 2019) and those in mathematics, little is known about the experiences of students and teachers (i.e., students’ perceptions of teachers’ intentions and implementation) as teachers work to foster a growth mindset culture (Murphy et al., 2021). In this dissertation, I …

Classifying Mathieu-Zhao Subspaces In Products Of Cyclic Rings, Sarah A. Huber

Theses and Dissertations

Mathieu-Zhao subspaces are a generalization of ideals in an algebra and were introduced by Wenhua Zhao in connection to the Jacobian conjecture and its variants. These subspaces have interesting properties, and often the problem of classification is hard. In this thesis, we investigate the structure of Mathieu-Zhao subspaces of the cartesian product of integers modulo powers of a prime p, Zpr × Zps . We will give a complete classification of the subgroups, maximal subgroups, Mathieu-Zhao subspaces, and maximal Mathieu-Zhao subspaces in these rings.

Free Probability Theory Over The Scaled Hyperbolic Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we introduce a notion of free probability over the scaled hyperbolic numbers. Scaled hypercomplex numbers {D_t} _t∈R are constructed as sub-structures of scaled hypercomplex numbers {H_t} _t∈R under the scales (or, the moments) of the set R of real numbers.We show that if t < 0, then the classical free probability theory covers our free probability on {D_t} _{t< 0}; if t > 0, then our free probability on {D_t} _t>0 is represented by the free probability over the classical hyperbolic numbers D = D₁; and if t = 0, then the free probability on D₀ is actually over the …

(R2127) On 2-Absorbing Hesitant Primary Fuzzy Ideals Of Rings, B. Anitha, M. Vidhya

Applications and Applied Mathematics: An International Journal (AAM)

By presenting 2-absorbing hesitant primary fuzzy ideals, we begin the investigation of a generalisation of hesitant primary fuzzy ideals (HPRFI) in rings in this research. The concepts of a weakly completely 2-absorbing hesitant primary fuzzy ideal (WC2-AHPRFI) and a Weakly completely 2-absorbing hesitant fuzzy ideal (WC2-AHFI) are developed, and their structural features and attributes are examined. We introduce the idea of a 2-absorbing hesitant K-fuzzy ideal (2-AHK-FI), 2-absorbing hesitant K-primary fuzzy ideal (2-AHK-PRFI) and examine a few of its characteristics.

(R2094) On Mds Bisymmetric Rhotrices Using Self-Dual Bases And Conjugate Elements Of Finite Fields, Shalini Gupta, Ruchi Narang, Manpreet Singh

Applications and Applied Mathematics: An International Journal (AAM)

Bisymmetric matrices have wide range of applications in statistics, engineering problems, information theory and computer science including coding theory and cryptography. In cryptography, a rhotrix being a couple matrix doubles the security of the cryptosystem. Here, we construct maximum distance separable (MDS) bisymmetric rhotrices using self-dual bases and conjugate elements of finite fields. MDS rhotrices are very crucial for the designing of block ciphers and hash functions in cryptography.

A Leslie System For A Demographic Simulation: From An Actuarial Point Of View, David Kings

Electronic Theses and Dissertations

This thesis develops a discrete stochastic linear systems interpretation of age–stage demographic evolution grounded in Leslie operators and realized in a discrete-event simulation implemented with salabim. The central claim is that one annual cycle of the simulation constitutes a cone-preserving, stochastic affine transformation on a high- dimensional population state vector indexed by age, sex, marital status, household type, employment, and education, and that the composition of yearly operators yields a random matrix product whose top Lyapunov exponent is the stochastic counterpart of the Perron–Frobenius growth rate (Caswell, 2001; Tuljapurkar, 1997)[1, 2]. The actuarial bridge is constructed by mapping simulated survival …

The Kaczmarz Algorithm In Hilbert C*-Modules, Daniel Alpay, Chad Berner, Eric S. Weber

Mathematics, Physics, and Computer Science Faculty Articles and Research

The Kaczmarz algorithm in Hilbert spaces is a classical iterative method for stably recovering vectors from inner product data. In this paper, we extend the algorithm to the setting of Hilbert C*-modules and establish analogues of its effectiveness in both finite-dimensional and stationary cases. Consequently, we demonstrate that continuous families of elements in a Hilbert space can be uniformly recovered using the Kaczmarz algorithm. Additionally, we develop a normalized Cauchy transform for continuous families of measures and use it to provide sufficient conditions under which standard frames in Hilbert C(X)-modules can be generated by the Kaczmarz …

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 …

Invitation To Polynomiography Via Chatgpt: For Teachers, Students And Artists, Bahman Kalantari

LASER Journal

Teachers, students, and artists in the United States and abroad who have encountered polynomiography, in lectures, demos, or software, consistently appreciate its educational value and artistic potential. While dedicated polynomiography programs require upkeep as systems evolve, AI chatbots now offer a practical, accessible alternative. This article invites readers to explore polynomiography with ChatGPT, broadening access beyond specialized tools. While the approach will not match the full range or polish of advanced software, it provides a powerful and flexible entry point with many possibilities.

At its core, polynomiography transforms polynomial equations, each encoding a finite set of points in the complex …

(Si15-064) Permutation Pentanomials Over Finite Fields With Even Characteristic, Shalini Gupta, Sushil Kumar, Ashima .

Applications and Applied Mathematics: An International Journal (AAM)

Permutation polynomials over finite fields constitute an active area of research and play an important role in diverse domains, including finite geometry, combinatorial design, coding theory, and cryptography. The study of these polynomials has a long history, and many results have been obtained in recent years. This paper presents new classes of permutation pentanomials based on permutation over the unit circle of finite fields with even characteristic that contribute to the theoretical development of permutation polynomials.

Dimension Of Subalgebras Of Fomin-Kirillov Algebras, Taro Ikeda, Sirous Homayouni

Annual Student Research Poster Session

One of the central open problems concerning the Fomin-Kirillov algebras was whether their dimension is finite or infinite. This question was recently resolved in one research by C. Barligea, where it was shown that FK(n) is infinite-dimensional for all n ≥ 6. In our work, we investigate the dimension of a certain subalgebra of FK(6).

How Many Symmetries Of The Regular N-Gon Are Even?, Thi Mai Khoi Ha

Rose-Hulman Undergraduate Mathematics Journal

One of the simplest classes of finite groups used as a source of counterexamples in a first course of modern algebra is the class of finite dihedral groups. Among the subgroups of dihedral group, finding subgroups of index 2 is of interest in part because these subgroups are normal subgroups. In this article, we use the representations of the symmetries of the dihedral groups as permutations of the vertices and determine concretely all its subgroups of index 2. Under this representation or embedding, the article determines the intersection of the dihedral group with the corresponding alternating groups when they are …

Sugar: A Sequence Unfolding Based Transformer Model For Group Activity Recognition, Yash U. Gondkar

Graduate Masters Theses

Large Language Models have improved significantly in the past couple of years due to the adoption of transformers. However, transformers still find it challenging to process videos due to limited context size caused by their quadratic computing cost. Therefore, we studied a booming field in machine learning which powers applications like social scene analysis and video surveillance systems called Group Activity Recognition (GAR). We found that recent models were able to achieve more than 90% accuracy on popular datasets like the Volleyball dataset, however, it turned out that even they relied on transformers.

Therefore, in this work, we developed a …

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

Funded Research Records

No abstract provided.

Career: Algebra And Representation Theory For Non-Semisimple Topological Field Theory, Matthew Young

Funded Research Records

No abstract provided.