# Mathematics Senior Integrated Projects

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This collection includes Senior Integrated Projects (SIP's, formerly known as Senior Individualized Projects) completed in the Mathematics Department.Abstracts are generally available to the public, but PDF files are available only to current Kalamazoo College students, faculty, and staff.If you arenota current K College student, faculty, or staff member, email us at dspace@kzoo.edu to request access to this material.

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Item An Exploration of Happy Ending Problem in Higher Dimensions(Kalamazoo College, 2024-03-01) Miller, Carter Ray; Oloo, StephenIn this paper we will generalize the happy-ending problem proved by Szekeres and Erdos in 1935 to n-dimensional spaces; we prove that a set of n+3 points in general position in Rn must have a subset of n+2 points that is the vertex set of some convex polytope.Item The Linear Algebraic Properties of Complete Graphs and their Connections(Kalamazoo College, 2024-01-01) Torres, Derik; Intermont, Michele, 1967-This paper explores properties that are in complete graphs. In this, we discover complete graphs, once translated into an adjacency matrix, are always invertible. The adjacency matrix gives us quite a bit of information that we then use to help find the eigenvalues and the eigenvectors of the matrix. We attempt to take it a step further by looking at those same properties once we start to combine these graphs through single links and no links. Once again, we make use of representing our connected complete graphs as a matrix of blocks and playing around with the properties of a block matrix to help find the eigenvalues and the eigenvectors of our disjoint and connected adjacency matrix of complete graphs.Item Icosahedral Virus Transitions: Computational Techniques(Kalamazoo College, 2024-03-04) Silva, Xavier; Oloo, Stephen; Wilson, DavidIcosahedral viruses have the symmetries of an icosahedron, which involves 2-fold, 3-fold, and 5-fold rotational symmetries. We can characterize these virus capsids with finite sets of points (called a point array) which we realize in 6D (not just 3D) for the purpose of crystallography: our 6D point arrays naturally fit inside 6D icosahedral lattices. There are 55 standard point arrays (called one-base) from which we build all the others. We model virus maturation by 6D linear transformations (transitions) of point arrays that preserve some or all of icosahedral symmetry. To find these transitions (preserving either the full icosahedral group symmetry or one of its maximal subgroups A4,D10, or D6) we solve matrix equations of the form TB0 = B1 for T, where T, the transition matrix, is a 6×6 matrix that depends on either 2, 4, 6, or 8 real variables and the matrices B0 and B1 are representations of the point arrays. We employ parallel computation techniques to efficiently find transitions between these point arrays. We are able to reproduce previously discovered transitions for the Cowpea Chlorotic Mottle Virus that preserve D6 symmetry, and we create a comprehensive list of what symmetries can be preserved between any possible combination of the 55 standard point arrays.Item Usage 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.Item Recognizing 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.