Mathematics Commons^™

Five Revolutionary Ideas In The 1950s-70s Science: 90th Birthday Of Revolt Pimenov, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

This year, Revolt Pimenov, a philosophical thinker whose main ideas were in geometry of space-time, would have turned 90. In this essay, we explain how in the 1950s-70s, when he was most productive, his were two of the five natural and important revolutionary scientific ideas -- along with fuzzy logic, constructive mathematics, and scalar-tensor theory of gravitation, ideas that, in our opinions, still have potential to change the world.

How To Teach Advanced Highly Motivated Students: Teaching Strategy Of Iosif Yakovlevich Verebeichik, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The paper describes and explains the teaching strategy of Iosif Yakovlevich Verebeichik, a successful mathematics teacher at special mathematical high schools -- schools for students interested in and skilled in mathematics. The resulting strategy seems counterintuitive and contrary to all the pedagogical advice. Our explanation is not complete: it worked well for this teacher, but others who tried to follow seemingly the same strategy did not succeed. How he made it work, how can others make it work -- this is still not clear. In the words of Verebeichik himself, while mathematics itself is a science, teaching mathematics is an …

Some Quadratic Quantum P³S With A Linear One-Dimensional Line Scheme, Ian Christopher Lim

Mathematics Dissertations - Archive

It is believed that quadratic Artin-Shelter regular (AS-regular) algebras of global dimension four (sometimes called quadratic quantum P3s can be classified using a geometry similar to that developed in the 1980’s by Artin, Tate, and Van den Bergh. Their geometry involved studying a scheme (later called the point scheme) that parametrizes the point modules over a graded algebra. The notion of line scheme (which parametrizes line modules) was introduced later by Shelton and Vancliff. It is known that “generic” quadratic quantum P3s have a finite point scheme and one-dimensional line scheme. A family of algebras with these properties is presented …

Option Pricing With Investment Strategy Under Stochastic Interest Rates, Niloofar Ghorbani

Mathematics Dissertations - Archive

Equity options are the most common types of financial derivatives that give an investor the right but not the obligation to buy or sell shares of stock at a given price in the future for a premium (option price) paid at present. The Classical Black- Scholes Formula solved a longstanding mathematical problem of finding no arbitrage option price by means of stochastic Ito calculus based on Geometric Brownian Motion dynamics of the stock price and a fixed interest rate over the option time horizon. We extend the Black-Scholes Model by adding a component of investor’s buying and selling strategies for …

On Problems In Random Structures, Ryan Cushman

Dissertations

This work addresses two problems in optimizing substructures within larger random structures. In the first, we study the triangle-packing number v(G), which is the maximum size of a set of edge-disjoint triangles in a graph. In particular we study this parameter for the random graph G(n,m). We analyze a random process called the online triangle packing process in order to bound v(G). The lower bound on v(G(n,m)) that this produces allows for the verification of a conjecture of Tuza for G( …

Assessing The Impact Of Vaccination And Behavior Change On Outbreaks Of Emerging Respiratory Diseases, Mohammed Hameed Alharbi

Mathematics Dissertations - Archive

The purpose of this dissertation is to use mathematical models to evaluate the impact of characteristics of respiratory diseases like influenza and COVID-19 either alone or co-circulating and how might influenza vaccine affect this interplay. First, we assess the effects of matching and mismatching between vaccine strains and circulating strains during the Hajj. Then we evaluate the impact of the proportion of asymptomatic COVID- 19 infections on the magnitude of an epidemic under three different behavior change scenarios. Finally, we model the co-circulation of influenza and COVID-19 to investigate whether the influenza vaccine increases the combined disease burden of influenza …

Exploring Development Of Problem Solving Strategies In Emerging Mathematicians, Andrew Charles Kercher

Mathematics Dissertations - Archive

To solve an unfamiliar mathematics problem, students of the subject must know more than the appropriate prerequisite content knowledge. They must also know how best to strategically apply their knowledge, how to monitor and gauge the effectiveness of their work, and how to respond (both cognitively and emotionally) to unanticipated results. Expanding current research on the types of experiences that foster these skills is the objective of this study. In a sequence of two task-based interviews, eight graduate and four upper-division undergraduate mathematics students solved non-traditional mathematics problems, used their work on these problems as a basis to comment on …

Current Trends In Random Walks On Random Lattices, Jewgeni H. Dshalalow

Mathematics and System Engineering Faculty Publications

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset …

Mathematical Models Of Covid-19, Kate Faria

Honors Program Theses and Projects

For more than a year, the COVID-19 pandemic has been a major public health issue, affecting the lives of most people around the world. With both people’s health and the economy at great risks, governments rushed to control the spread of the virus. Containment measures were heavily enforced worldwide until a vaccine was developed and distributed. Although researchers today know more about the characteristics of the virus, a lot of work still needs to be done in order to completely remove the disease from the population. However, this is true for most of the infectious diseases in existence, including Influenza, …

Kamila Clustering For A Mixed-Type Data Analysis Of Illinois Medicare Data, Heather Elizabeth Baldacci

Undergraduate Honors Thesis Collection

The Centers for Medicare and Medicare Services (CMS) releases annual reports regarding the Market Saturation and Utilization of nationwide Medicare coverage. CMS data provide an opportunity for an in-depth analysis of Medicare usage patterns within the United States that may provide insight into socioeconomic conditions in certain regions. To discover any potential patterns, the KAMILA (KAy-means for MIxed LArge data sets) clustering algorithm has been utilized within the most recent CMS dataset from 2018. Due to the large size of the original data set, the focus of this research has been limited to Illinois Medicare data, grouped by the 102 …

Home Sweet Home: A Statistical Analysis Of Home Court Advantage In Division I College Basketball, Lauren Isabel Turnbull

Undergraduate Honors Thesis Collection

Home court advantage is a term that has been used throughout the basketball world by everyone from broadcasters and TV analysts to athletes and coaches to fans. It is a reference to the perceived benefit that the home team receives by playing in their own arena or stadium. Much of the prior research on home court advantage has centered around the reasons that it exists and how it impacts the play on the court through box score statistics. Building on this previous research, this study performs a statistical analysis of men's college basketball to determine if home court advantage is …

Secondary Teachers’ Noticing Of Students’ Mathematical Thinking As They Participate In A Professional Development Program Centered On Task-Based Student Interviews, Gurkan Kose

Theses, Dissertations and Culminating Projects

Teacher’s noticing of students’ mathematical thinking has been an important focus of research in the past two decades (e.g., Jacobs et al., 2010; Sherin et al., 2011). Noticing matters, but it is not an end in itself (Schoenfeld, 2011). It is operationalized within the context of teachers’ dispositions and knowledge which shape decisions teachers make while responding to student thinking and planning the next instructional steps. In order for teachers to adapt productive beliefs about how children learn and shift to student-centered instruction, they need to acknowledge the importance of understanding students' existing conceptions of mathematical ideas (Carpenter & Lehrer, …

2-Adic Valuations Of Square Spiral Sequences, Minh Nguyen

Honors Theses

The study of p-adic valuations is connected to the problem of factorization of integers, an essential question in number theory and computer science. Given a nonzero integer n and prime number p, the p-adic valuation of n, which is commonly denoted as νp(n), is the greatest non-negative integer ν such that p ν | n. In this paper, we analyze the properties of the 2-adic valuations of some integer sequences constructed from Ulam square spirals. Most sequences considered were diagonal sequences of the form 4n 2 + bn + c from the Ulam spiral with center value of 1. Other …

Constructing Non-Proxy Small Test Modules For The Complete Intersection Property, Benjamin Briggs, Eloísa Grifo, Josh Pollitz

Department of Mathematics: Faculty Publications

A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category Df(R), which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer-Greenlees-Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in Df(R) is proxy small. In this paper, we study a …

Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang

Electronic Theses and Dissertations

While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to …

Symmetric Presentation Of Finite Groups, And Related Topics, Marina Michelle Duchesne

Electronic Theses, Projects, and Dissertations

We have discovered original symmetric presentations for several finite groups, including 2²:^.(2⁴:(2^.S₃)), M₁₁, 3:(PSL(3,3):2), S₈, and 2^.M₁₂. We have found homomorphic images of several progenitors, including 2^*18:((6x2):6), 2^*24:(2^.S₄), 2^*105:A₇, 3^*3:_m(2³:3), 7^*8:_m(PSL(2,7):2), 3^*4:_m(4²:2²), 7^*5:(2xA₅), and 5^*6:_mS₅. We have provided the isomorphism type of …

Proper Sum Graphs, Austin Nicholas Beard

Graduate Theses/Dissertations

The Proper Sum Graph of a commutative ring with identity has the prime ideals as vertices, with two ideals adjacent if their sum is a proper ideal. This thesis expands upon the research of Dhorajia. We will cover the groundwork to understanding the basics of these graphs, and gradually narrow our efforts into the minimal prime ideals of the ring.

Math And Voting: Voting Methods, Fair Representation, And The Electoral College, Sarah Nelson

Honors Scholar Theses

Voting is in integral part of any functioning democracy, but there exist more than just one way to count votes. Some voting methods use only a voter’s top-choice candidate, while others require a ranking of all candidates, most-preferred to least-preferred, from each voter. We examine some of these ranked-choice voting methods, including the anti-plurality method, Hare’s method, and Coomb’s method.

Because of the variety of voting methods, we introduce criteria, which allow for an evaluation of the advantages and disadvantages of each method. The criteria give various definitions for what “good” or “bad” voting methods look like, depending on context. …

Frobenius And Homological Dimensions Of Complexes, Taran Funk

Department of Mathematics: Dissertations, Theses, and Student Research

Much work has been done showing how one can use a commutative Noetherian local ring R of prime characteristic, viewed as algebra over itself via the Frobenius endomorphism, as a test for flatness or projectivity of a finitely generated module M over R. Work on this dates back to the famous results of Peskine and Szpiro and also that of Kunz. Here I discuss what work has been done to push this theory into modules which are not necessarily finitely generated, and display my work done to weaken the assumptions needed to obtain these results.

Adviser: Tom Marley

From Multi-Prime To Subset Labelings Of Graphs, Bethel I. Mcgrew

Dissertations

A graph labeling is an assignment of labels (elements of some set) to the vertices or edges (or both) of a graph G. If only the vertices of G are labeled, then the resulting graph is a vertex-labeled graph. If only the edges are labeled, the resulting graph is an edge-labeled graph. The concept was first introduced in the 19th century when Arthur Cayley established Cayley’s Tree Formula, which proved that there are n^n-2 distinct labeled trees of order n. Since then, it has grown into a popular research area.

In this study, we first review several types …