Mathematics Commons^™

Local-Global Results On Discrete Structures, Alexander Lewis Stevens

Electronic Theses and Dissertations

Local-global arguments, or those which glean global insights from local information, are central ideas in many areas of mathematics and computer science. For instance, in computer science a greedy algorithm makes locally optimal choices that are guaranteed to be consistent with a globally optimal solution. On the mathematical end, global information on Riemannian manifolds is often implied by (local) curvature lower bounds. Discrete notions of graph curvature have recently emerged, allowing ideas pioneered in Riemannian geometry to be extended to the discrete setting. Bakry- Émery curvature has been one such successful notion of curvature. In this thesis we use combinatorial …

Banach Spaces On Topological Ramsey Structures, Cheng-Chih Ko

Electronic Theses and Dissertations

A Banach space T₁(d, θ) with a Tsirelson-type norm is constructed on the top of the topological Ramsey space T₁ defined by Dobrinen and Todorcevic [6]. Finite approximations of the isomorphic subtrees are utilised in constructing the norm. The subspace on each “branch” of the tree is shown to resemble the structure of an ℓ_∞ⁿ⁺¹ -space where the dimension corresponds to the number of terminal nodes on that branch. The Banach space T₁(d, θ) is isomorphic to (∑_n∊ℕ⊕ℓ_∞ⁿ⁺¹)_p , where d ∈ ℕ with d ≥ 2, …

Topics In Moufang Loops, Riley Britten

Electronic Theses and Dissertations

We will begin by discussing power graphs of Moufang loops. We are able to show that as in groups the directed power graph of a Moufang loop is uniquely determined by the undirected power graph. In the process of proving this result we define the generalized octonion loops, a variety of Moufang loops which behave analogously to the generalized quaternion groups. We proceed to investigate para-F quasigroups, a variety of quasigroups which we show are antilinear over Moufang loops. We briefly depart from the context of Moufang loops to discuss solvability in general loops. We then prove some results on …

Stroke Clustering And Fitting In Vector Art, Khandokar Shakib

Senior Independent Study Theses

Vectorization of art involves turning free-hand drawings into vector graphics that can be further scaled and manipulated. In this paper, we explore the concept of vectorization of line drawings and study multiple approaches that attempt to achieve this in the most accurate way possible. We utilize a software called StrokeStrip to discuss the different mathematics behind the parameterization and fitting involved in the drawings.

The Infinity Conundrum: Understanding Topics In Set Theory And The Continuum Hypothesis, Sabrina Grace Helck

Senior Independent Study Theses

This project is concerned with articulating the necessary background in order to understand the famous result of the undecidability of the continuum hypothesis. The first chapter of this independent study discusses the foundations of set theory, stating fundamental definitions and theorems that will be used throughout the remainder of the project. The second chapter focuses on ordinal and cardinal numbers which will directly relate to the final chapter. First, there is a clear explanation of the notion of order and what it means for a set to be well-ordered. Then ordinal numbers are defined and some properties are listed and …

Highlights Generation For Tennis Matches Using Computer Vision, Natural Language Processing And Audio Analysis, Alon Liberman

Senior Independent Study Theses

This project uses computer vision, natural language processing and audio analysis to automatize the highlights generation task for tennis matches. Computer vision techniques such as camera shot detection, hough transform and neural networks are used to extract the time intervals of the points. To detect the best points, three approaches are used. Point length suggests which points correspond to rallies and aces. The audio waves are analyzed to search for the highest audio peaks, which indicate the moments where the crowd cheers the most. Sentiment analysis, a natural language processing technique, is used to look for points where the commentators …

Counting The Moduli Space Of Pentagons On Finite Projective Planes, Maxwell Hosler

Senior Independent Study Theses

Finite projective planes are finite incidence structures which generalize the concept of the real projective plane. In this paper, we consider structures of points embedded in these planes. In particular, we investigate pentagons in general position, meaning no three vertices are colinear. We are interested in properties of these pentagons that are preserved by collineation of the plane, and so can be conceived as properties of the equivalence class of polygons up to collineation as a whole. Amongst these are the symmetries of a pentagon and the periodicity of the pentagon under the pentagram map, and a generalization of …

Wildfire Simulation Using Agent Based Modeling: Expanding Controlled Burn Season, Morgan C. Kromer

Senior Independent Study Theses

The United States is home to many different and unique forests. Prior to the 21st century, the United States Forests Service assumed that the best way to protect these forests was to put all efforts to keeping them alive. An enemy to these efforts were wildfires, thus the US adopted a complete fire suppression approach. At the turn of the century, the US realized that wildfires are a necessary part of a forest ecosystem, as they help return nutrients to the soil and reduce ground fuels. However, after suppressing all fires for over 100 years, the forests evolved into a …

Wittgenstein On Miscalculation And The Foundations Of Mathematics, Samuel J. Wheeler

Philosophy Faculty Publications

In Remarks on the Foundations of Mathematics, Wittgenstein notes that he has 'not yet made the role of miscalculating clear' and that 'the role of the proposition: "I must have miscalculated"...is really the key to an understanding of the "foundations" of mathematics.' In this paper, I hope to get clear on how this is the case. First, I will explain Wittgenstein's understanding of a 'foundation' for mathematics. Then, by showing how the proposition 'I must have miscalculated' differentiates mathematics from the physical sciences, we will see how this proposition is the key to understanding the foundations of mathematics.

Multicolor Ramsey And List Ramsey Numbers For Double Stars, Jake Ruotolo

Honors Undergraduate Theses

The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of order. For a graph H, the k-color Ramsey number r(H; k) of H is the smallest integer n such that every k-edge-coloring of K_n contains a monochromatic copy of H. Despite active research for decades, very little is known about Ramsey numbers of graphs. This is especially true for r(H; k) when k is at least 3, also known as the multicolor Ramsey number of …

Local Finiteness And Automorphism Groups Of Low Complexity Subshifts, Ronnie Pavlov, Scott Schmieding

Mathematics: Faculty Scholarship

We prove that for any transitive subshift X with word complexity function c_n(X), if lim inf(log(c_n(X)/n)/(log log log n)) = 0, then the quotient group Aut(X, σ)/〈 σ〉 of the automorphism group of X by the subgroup generated by the shift σ is locally finite. We prove that significantly weaker upper bounds on c_n(X) imply the same conclusion if the gap conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of X of range n in terms of word complexity, which may be …

Measure-Theoretically Mixing Subshifts With Low Complexity, Darren Creutz, Ronnie Pavlov, Shaun Rodock

Mathematics: Faculty Scholarship

We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any f : N → N with f (n)/n increasing and ∑ 1/f (n) < ∞, that there exists an extremely elevated staircase with word complexity p(n) = o(f (n)). This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.

Math Department 2021-2022 Newsletter, Mathematics Department

Mathematics Newsletter

Welcome to our annual Mathematics Department Newsletter. As we are slowly coming out of the pandemic, we were able to resume many of our normal events.

Mathematical Reasoning: Writing And Proof (Pretext Edition), Ted Sundstrom

Open Textbooks

Mathematical Reasoning: Writing and Proof is a text for the first college mathematics course that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics.

Mathematical Reasoning: Writing and Proof (PreTeXt Edition) was developed as part of the Accelerating Open Educational Resources Initiative at Grand Valley State University, with support from the University Libraries and the President's Innovation Fund.

Mathematical Reasoning: Writing and Proof was written by Ted Sundstrom, Professor Emeritus of Mathematics at Grand Valley State University. This textbook was converted into PreTeXt by Ian Curtis, Editorial Assistant for the …

Grade K-5 Teachers’ Perceptions Of Professional Development That Supports Mathematics Instruction, Shannon Annette Manley

Walden Dissertations and Doctoral Studies

Many Grade K-5 teachers in the United States do not receive the mathematics support they need from the professional development (PD) activities offered by their school districts. The purpose of this qualitative research was to explore the perceptions of Grade K-5 teachers on the PD activities they received from their school district to support mathematics instruction. The conceptual framework that supported this study was andragogy, an adult learning theory that takes the learner’s needs into account and values the connection to real-world situations. The research question addressed how Grade K-5 teachers perceive the PD that they were offered by their …

A Brief Treatise On Bayesian Inverse Regression., Debashis Chatterjee Dr.

Doctoral Theses

Inverse problems, where in a broad sense the task is to learn from the noisy response about some unknown function, usually represented as the argument of some known functional form, has received wide attention in the general scientific disciplines. However, apart from the class of traditional inverse problems, there exists another class of inverse problems, which qualify as more authentic class of inverse problems, but unfortunately did not receive as much attention.In a nutshell, the other class of inverse problems can be described as the problem of predicting the covariates corresponding to given responses and the rest of the data. …

Secret Sharing And Its Variants, Matroids,Combinatorics., Shion Samadder Chaudhury Dr.

Doctoral Theses

The main focus of this thesis is secret sharing. Secret Sharing is a very basic and fundamental cryptographic primitive. It is a method to share a secret by a dealer among different parties in such a way that only certain predetermined subsets of parties can together reconstruct the secret while some of the remaining subsets of parties can have no information about the secret. Secret sharing was introduced independently by Shamir [139] and Blakely [20]. What they introduced is called a threshold secret sharing scheme. In such a secret sharing scheme the subsets of parties that can reconstruct a secret …

Under Pressure: A Case Study Of The Effects Of External Pressure On Mlb Players Using Twitter Sentiment Analysis, Jonathan Huntley

Honors Projects in Mathematics

Performance under pressure and psychological momentum are well-documented topics in sports psychology, but most research focuses on “in-game” pressure. This study views pressure more broadly to examine how the external pressure of fans, quantified using the sentiment of tweets mentioning the players, can affect how MLB players perform. Although external pressure is intangible, it can impact a player’s psyche and performance. This investigation focuses on players Chris Sale and David Price. A new process was developed leveraging the Vader package in Python that can generate tweet sentiment to compare to several performance metrics from Baseball Reference. Results proved to be …

Participatory Action Research: Undergraduate Researchers Engaging Secondary Students In Social Justice Mathematics, Isabelle Miller, Alexis Grimes, Camryn Adkison

The Journal of Purdue Undergraduate Research

No abstract provided.

Some Contributions To Free Probability And Random Matrices., Sukrit Chakraborty Dr.

Doctoral Theses

No abstract provided.

Excursions In Summation, Brock Erwin

Fall Showcase for Research and Creative Inquiry

Using polynomials from series representation of functions to approximate other functions on the closed interval from [-1,1].

Non-Local Approximation Properties, Kira Pierce

Fall Showcase for Research and Creative Inquiry

This project concerns the approximation properties of a given set where X is a scattered sequence and Ï•(x) = 1/x* ln(1 + x^2 ). Similar approximation sets are commonly used in interpolation problems and are especially helpful due to their Fourier representation. For our work, we will work to prove the following theorem.

Some Topics In Leavitt Path Algebras And Their Generalizations., Mohan R. Dr.

Doctoral Theses

The purpose of this section is to motivate the historical development of Leavitt algebras, Leavitt path algebras and their various generalizations and thus provide a context for this thesis. There are two historical threads which resulted in the definition of Leavitt path algebras. The first one is about the realization problem for von Neumann regular rings and the second one is about studying algebraic analogs of graph C ∗ -algebras. In what follows we briefly survey these threads and also introduce important concepts and terminology which will recur throughout.

Scaling Limits Of Some Random Interface Models., Biltu Dan Dr.

Doctoral Theses

In this thesis, we study some probabilistic models of random interfaces. Interfaces between different phases have been topic of considerable interest in statistical physics. These interfaces are described by a family of random variables, indexed by the ddimensional integer lattice, which are considered as a height configuration, namely they indicate the height of the interface above a reference hyperplane. The models are defined in terms of an energy function (Hamiltonian), which defines a Gibbs measure on the set of height configurations. More formally, letÏ• = {Ï•x}x∈Z dbe a collection of real numbers indexed by the d-dimensional integer lattice Z d. …

On The Inertia Conjecture And Its Generalizations., Soumyadip Das Dr.

Doctoral Theses

This thesis concerns problems related to the ramification behaviour of the branched Galois covers of smooth projective connected curves defined over an algebraically closed field of positive characteristic. Our first main problem is the Inertia Conjecture proposed by Abhyankar in 2001. We will show several new evidence for this conjecture. We also formulate a certain generalization of it which is our second problem, and we provide evidence for it. We give a brief overview of these problems in this introduction and reserve the details for Chapter 4.Let k be an algebraically closed field, and U be a smooth connected affine …

Perron Theorem For Delay Differential Equations, William Barker

Theses and Dissertations

This note discusses results furthering the study of bounded solutions of non-homogeneous second-order differential equations with delay. The equation of interest has already been shown to possess unique bounded solutions in accordance to Perron's Theorem. We will determine the negativity of the solutions given a positive forcing term by establishing a uniform bound regarding small time delay.

Expansion Of The Department Of Mathematics At Princeton University And The Founding Of The School Of Mathematics At The Institute For Advanced Study: 1900-1950, Andrew Beman-Cavallaro

Great Plains Graduate Review

From 1900 to 1950 Princeton, New Jersey, hosted two of the most prestigious institutions and provided a location for the expansive mathematical investigations taking place just before, during, and immediately after the Second World War. Princeton University and the Institute for Advanced Study housed, fed, and provided workspaces for an array of Mathematicians uncovering new research methodologies (resulting in the defeat of both the Nazi Party and the Empire of Nippon), the foundation for modern experimental Mathematics, and expansions of Theoretical and Applied Physics.

Commuting Isometries And Invariant Subspaces In Several Variables., Sankar T. R. Dr.

Doctoral Theses

A very general and fundamental problem in the theory of bounded linear operators on Hilbert spaces is to find invariants and representations of commuting families of isometries.In the case of single isometries this question has a complete and explicit answer: If V is an isometry on a Hilbert space â„‹, then there exists a Hilbert space Hu and a unitary operator U on â„‹u such that V on â„‹u and[ S ⊗ IW 0 0 U] ∈ B((l 2 (ℤ+) ⊗ W) ⊕ â„‹u),are unitarily equivalent, whereW = ker V∗ ,is the wandering subspace for V and S is the …

College Algebra Through Problem Solving (2021 Edition), Danielle Cifone, Karan Puri, Debra Maslanko, Ewa Stelmach

Open Educational Resources

This is a self-contained, open educational resource (OER) textbook for college algebra. Students can use the book to learn concepts and work in the book themselves. Instructors can adapt the book for use in any college algebra course to facilitate active learning through problem solving. Additional resources such as classroom assessments and online/printable homework is available from the authors.

For The Love Of Mathematical Research: A Conversation With Undergraduate Research Students, Mike Janssen

Faculty Work Comprehensive List

"One of my passions as a professor is creating opportunities for students to ask questions about mathematics."

Posting about students' perspectives on mathematics research from In All Things - an online journal for critical reflection on faith, culture, art, and every ordinary-yet-graced square inch of God’s creation.

https://inallthings.org/for-the-love-of-mathematical-research-a-conversation-with-undergraduate-research-students/