ItemUsage of Python in Differential Equations(Kalamazoo College, 2023-01-01) Paz, Jose; Barth, Eric J., 1964-This paper is a chronological record of the transition from the usage of wxMaxima, a downloadable computer algebra system used in previous iterations of the course to assist students with the computation of differential equations, to Google Colaboratory, a more user-friendly website that uses python-based coding to achieve the same results as wxMaxima, for the Ordinary Differential Equations: Analytical Geometric, Symbolic and Numerical Methods course at Kalamazoo College. I will be referring to Google Colaboratory as “Colab” for simplicity’s sake. This change tackled some of the previous problems that wxMaxima encountered; unlike Maxima, Colab did not require the student to make any downloads and/or changes to their computer, Colab allowed a notebook-like format for all assignments easily accessible in a folder in the user’s Google Drive, Colab updates itself, guaranteeing that all students work on the latest version, and, most importantly, the interface is exactly the same in IOS and Windows. This paper will include the early planning stages for the course, the creation of content, the experience while providing educational content to students (this includes Teaching Assistant tutoring sessions as well as some tutorial videos), the feedback from the students, and areas of improvement to note of. ItemRecognizing Simple Groups by Their Codegree Sets(Kalamazoo College, 2023-01-01) Dolorfino, Mallory; Intermont, Michele, 1967-An arithmetic property of a group G is a property that can be expressed solely in terms of numbers. The restrictions that certain arithmetic properties place on the structure of a group is a well-studied and fruitful area of research; two of the most well-known theorems pertaining to arithmetic properties assert that all groups of order 2 are abelian and that all finite groups of odd order are solvable. While both of these theorems arise from a single arithmetic property, namely the order of a group, strong conjectures have been made concerning sets of arithmetic properties. For example, in 2000, Huppert conjectured that if a finite nonabelian simple group H and a finite group G share the same character degree set, then G ∼= H × A for some abelian group A. More recently, a stronger conjecture, often called the codegree version of Huppert’s conjecture, has been posed as question 20.79 in the Kourovka Notebook: if H is a nonabelian simple group and G is a finite group such that H and G have equal codegree sets, then G ∼= H. In other words, nonabelian simple groups are recognizable by their codegree sets. Huppert’s codegree conjecture has been verified for several simple groups, but it has yet to be completely proved. In this paper, we verify this conjecture for the sporadic groups and the simple alternating groups. ItemMathematics Education Foundations : Developing Math Materials for Students and a System for Peer-Educators(Kalamazoo College, 2023-01-18) Prentice, Noah; Barth, Eric J., 1964-; Love, RachelLearning mathematics is hard, and learning college mathematics is made even harder by textbooks which are primarily used as vessels for homework exercises. These two facts motivated my Senior Integrated Project (SIP) or senior thesis, which consisted of 21 LATEX documents and 5 videos on introductory mathematics topics. The documents were made to be more personalized to the Kalamazoo College curriculum than the all-purpose textbooks often circulated, and with a much larger emphasis on reading comprehension, examples, and conceptual motivations. The thesis also included a system for the creation of more materials in the future, in the hopes that such materials will be made for other subjects and that they will grow as the college does. Following this, I developed a system for the materials’ online and in-person accessibility through the Kalamazoo College Math and Physics Center (MPC). ItemExploring Algebraic Themes in Calculus(Kalamazoo College, 2023-01-01) Bagchi, Tolkien; Intermont, Michele, 1967-This paper outlines two different approaches to studying the algebra underlying the ideas of differentiation and integration typically presented in analysis. In the first section we approach the notion of derivative by defining a derivative to be an endomorphism of an algebra over a commutative ring satisfying the product rule. We prove some general results classifying the structure of derivatives before proceeding to prove some general facts about differential algebras. We conclude this section by studying some of the properties of linear differential operators on differential algebras and their similarities and differences with polynomial rings studied in commutative algebra. In the second section we follow Christopher Heunen’s1 treatment of ℓ2 as a dagger preserving monoidal functor to a similar treatment of L2. ItemAlgorithmic Methods for Computing Monomial Invariants of Abelian Groups(Kalamazoo College, 2022-06-01) Rizzolo, Lucas; Gandini, Francesca; Gandini, FrancescaGiven some arbitrary polynomial ring, an invariant polynomial is a polynomial that is unchanged by the action of a group G. We investigate the ring of invariant polynomials under the action of some abelian groups with the goal of finding generators for this ring. When considering an abelian group, we can always find a basis such that the action is diagonal, so there exists monomial generators mi for the invariant ring. By Noether’s degree bound, the minimal set of generating monomials ⟨m1, . . . ,mk⟩ is finite and the degree of each generating monomial mi is less than |G|. Motivated by the previous work of Gandini and Derksen, we present new theoretical approaches to find invariants of Abelian groups. In combination with the theory, we can apply exact algorithmic approaches and deploy parallel programming methods to create various algorithms for computing the minimal generating set of invariants for the ring of invariants. Finally, we present the Gandini-Ratliff-Rizzolo algorithm, a seeded generation algorithm for computing invariants.