Mathematics Commons^™

The Set Card Game And Partitions Into Maximal Caps: A Partial Difference Sets Approach, John Polhill

Journal of Humanistic Mathematics

More specifically, we revisit the problem of determining the maximum number of cards that can be dealt in the SET® card game without any matches, which is well known to be equivalent to the problem of determining the size of a maximal cap in the affine space AG(4,3). We present a new approach to the problem by using partial difference sets, a method that is equivalent to using strongly regular graphs with a regular automorphism group. The paper includes several possible avenues for future investigations on how partial difference sets might be used in related problems.

Manipulating Mathematics, Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

No abstract provided.

Front Matter

Journal of Humanistic Mathematics

No abstract provided.

Sums Of Consecutive Integers And Sequence Of Divisors, Mushrat Fatema, Abdulkadir Hassen

STEM Student Research Symposium Posters

This work is concerned with two problems in number theory. The first part is to find the maximum number of sequence of integers such that two consecutive numbers in the list are multiple or factor of the other, and integers that can be expressed as a sum of consecutive integers.

Boundary Integral Method For A Modified Mullins–Sekerka System, Ly Le

Mathematics & Statistics ETDs

This thesis develops a boundary integral method for a modified Mullins–Sekerka system arising as the sharp-interface limit of a nonreciprocal Cahn–Hilliard model. Nonreciprocal coupling changes the classical Cahn–Hilliard structure by introducing an additional conserved field, leading to coupled elliptic and parabolic dynamics at the interface. Using matched asymptotic expansions, we formally derive the modified Mullins–Sekerka model and then apply the boundary integral method to rewrite it on the moving interface. The elliptic component is represented using the periodic Green’s function for the Laplace equation, while the parabolic component is represented using the periodic heat kernel. This method reduces the bulk …

Nerve Constructions And Mapper, Alexander Bram Fritschi

Mathematics & Statistics ETDs

Mapper is a data visualization tool commonly used in topological data analysis to study large, often high-dimensional datasets. Mapper operates through the selection of a lens function, a clustering algorithm, and a cover. The Mapper graph is constructed using the nerve of the cover after the clustering algorithm is performed; it is therefore useful to study nerves to better understand Mapper. In this thesis, we will utilize the properties of nerves to find the minimal point set that produces a given graph. We will then extend this to Mapper to determine what Mapper graphs may be constructed over a given …

Interpretable Case-Control Inference Through Log-Linear General Location Models, Zacharia Stuart

Mathematics & Statistics ETDs

This dissertation analyzes one of the few publicly available NFL injury datasets to study field type and non-contact lower-limb injuries. Field type is studied jointly with other risk factors to understand how these factors interact to affect injury risk. The data were gathered through a case-control sampling scheme, which limits direct inference on absolute injury probabilities. While not the most common approach for case-control data, this dissertation models the retrospective distribution directly through Log-Linear General Location Models (Log-Linear GLOMs). Through a log-linear structure placed on a log-odds-ratio reparameterization, the model provides directly interpretable marginal and interaction contributions to injury log-odds …

The Uncertainty Principles, Lee Michael Felicetti

Mathematics & Statistics ETDs

The Heisenberg uncertainty principle is a central aspect of quantum mechanics, but also illustrates an essential quality of the Fourier transform. After Heisenberg, a variety of uncertainty inequalities emerged in the fields of physics and mathematics. In this thesis we will analyze the Heisenberg uncertainty principle in both the setting of quantum mechanics and Fourier analysis. We will then look at how the work of Heisenberg has been expanded upon in both physics and mathematics. Particularity, we will see how uncertainty principles can be applied to signal recovery and explore current research in this field.

Error-Tolerant Metric Dimension, Leander Ten Hoff

Master's Theses

Metric dimension is a graph parameter that measures the smallest distance-based unique coordinate system on a graph. Fault-tolerant metric dimension is a variant that requires redundancy. Inspired by this idea, we propose and investigate a new variant we call error-tolerant metric dimension. We first prove some fundamental results to gain familiarity with this more complex variant. Then, we use these tools to study relative behaviors of error-tolerant metric dimension. We prove explicit computations for simple graph families, including bipartite complete graphs and paths. Finally, we extend extremal results from previous papers on metric dimension, including a full correction of an …

The Relationship Between Foliations Of The Plane, Kaplan Diagrams, And Non-Hausdorff 1-Manifolds, Monique Justine Howe

Master's Theses

In this paper we seek to explain the relationship between foliations of the plane, Kaplan diagrams, and simply connected non-Hausdorff 1-manifolds with countable basis that are orientable with an ordering on branch points. We will walk the reader through the definitions of all of these objects, and provide examples with a focus on the motivating example of the Reeb foliation. This paper will describe and define the known bijection from the set of foliations of the plane, F, to the set of Kaplan diagrams, K, its inverse, and the one-to-one map from F to the set of simply connected non-Hausdorff …

Rewriting The Calculus 3 Workshop At San José State University, Moorea Lippert

Master's Theses

Calculus 3 at San José State University is a very important course for any student pursuing a STEM degree. However, the course is very challenging, so the university offers a supplemental workshop course for Calculus 3 students. Unfortunately, this workshop has not aligned with modern research in math education. This thesis documents efforts to recreate the workshop to better align with this modern research.

Monoids With K-Strictly Local Geodesic Languages, William M. Hong

Master's Theses

A paper by Gilman, Hermiller, Holt and Rees (2011) prove that a group G finitely generated by X is virtually free if and only if the language of geodesics words in G over X is k-strictly local if and only if the word problem is context-free. If M is a monoid finitely generated by X, we define Γu(M,X) to be the underlying, undirected graph of Γ(M,X). We prove that if the language of geodesics in Γu(M,X) based at the identity is k-strictly local, then the monoid and semi-group word problems are context-free.

Fermat's Last Theorem Over Quadratic Number Fields, Richie Tay

Master's Theses

Fermat’s Last Theorem states that the equation xn + yn = zn has no nontrivial integer solutions for n ≥ 3. While the rational case is completely solved, the situation over quadratic number fields is not as straightforward. For some exponents nontrivial solutions exist, while for others they do not. This thesis studies Fermat-type equations over quadratic number fields, combining classical methods with more modern techniques from the theory of elliptic curves. The thesis first reviews the rational cases n = 2, n = 3, n = 4, then develops the necessary background on quadratic number fields, including rings of …

Math 141 Calculus I (Queens College), Anisha Clarke

Open Educational Resources

This syllabus provides the outline for a calculus one course using OpenStax Calculus Volume I and no graphing calculator.

Deep Neural Networks: A Formulation Via Non-Archimedean Analysis, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified by using numbers in the ring of integers of non-Archimedean local fields. These rings have a natural hierarchical organization as infinite rooted trees. Natural morphisms on these rings allow us to construct finite multilayered architectures. The new DNNs are robust universal approximators of real-valued functions defined on the mentioned rings. We also show that the DNNs are robust universal approximators of real-valued square-integrable functions defined in the unit interval.

Probabilistic Metric Dimension, Kyle Worley

Master's Theses

The metric dimension of a graph G is the smallest number of unique vertices (often called “marked vertices” or “landmarks”) such that each vertex has a unique distance to the marked vertices. If a set of vertices S ⊆ V (G) when marked gives all vertices unique distances, it is called a resolving set. Hence if G is a graph with a resolving set S, then for all v, u ∈ V (G) there exists s ∈ S such that d(v, s) ̸= d(u, s). The metric dimension dim(G) = min(|S|). Two widely studied variants exist, one where edges are …

Rational Non-Euclidean Triangles And Elliptic Curves, Tyler Morales

Master's Theses

We say a triangle △ is rational if its side lengths are rational. For a geometry of constant curvature κ, we say △ is a rational triangle if the generalized tangents of its side lengths are rational. We are able to parameterize rational triangles having the same inradius and semiperimeter on an plane curve, write a bijection between the rational points on this curve and triples of side lengths, and show that this curve is an elliptic curve. Then, we write a general transformation for our plane curve to short Weierstrass form to compute ranks and add points easily using …

Fundamental Solutions To The Fractional Heat Operator, Jacob Flores

Math Theses

In this thesis, we are interested in showing the existence of a fundamental solution to the fractional heat operator. The fractional heat operator is a nonlocal linear operator used to model the time evolution of anomalous diffusion processes whose applicability arises in a wide variety of fields in the physical sciences, engineering, economics, and finance. Fundamental solutions to a partial differential operator are a class of generalized solutions formulated with rich mathematical analysis grounded in classes of well-behaved smooth functions referred to as test functions and their continuous linear functionals referred to as distributions. A primary tool that we use …

A Professional Development Course On Data-Driven Dynamical Systems At A Primarily Undergraduate Institution - Part B: Professional Development Framework And Institutional Implementation, Alessandro M. Selvitella, Jeffrey R. Anderson

CODEE Journal

This paper presents the professional-development and institutional-implementation component of a three-paper program on introducing data-driven dynamical systems into undergraduate ordinary differential equations instruction at a primarily undergraduate institution. The companion paper \cite{SelvitellaAnderson2026} develops the mathematical and computational content, including regression, regularization, Dynamic Mode Decomposition, Sparse Identification of Nonlinear Dynamics, and a Van der Pol oscillator activity. The present paper asks a complementary implementation question: what departmental structures, graduate teaching assistant roles, professional-development activities, and course-support pathways are needed for such materials to become teachable within local undergraduate settings?

We describe a three-phase professional-development model for graduate teaching assistants in the …

A Dynamics Viewpoint On The Approximation Capabilities Of Infinite-Depth Neural Networks, Andreus Brammer

Electronic Theses and Dissertations 2020 - Present

Neural Ordinary differential equations(Neural ODEs) provide a continuous-time framework for learning dynamical systems by representing the evolution of a state variable through a parameterized differential equation. While most existing methods rely on numerical time stepping methods during training such as RK4, this work investigates an alternate spectral framework where both the solution and the time-dependent network parameters are represented by global polynomial expressions.

Two methods are employed: the first uses Taylor series representations, and the second approach uses Chebyshev polynomial expansions together with fast spectral transforms and efficient polynomial multiplication techniques. In each formulation, the Neural ODE is transformed into …