Consecutive and Parity-Consecutive Complete Lucas Sequences when the Period Equals the Modulus The Lucas sequence of the first kind (LSFK) is denoted (Un(p,q))n≥0, where Un is its nth term. The LSFK is defined recursively by Un = pUn−1−qUn−2 with initial terms U0 = 0 and U1 = 1, for integers p and q. The period length of (Un(p,q))n≥0 (mod m), denoted π(m), is the least positive integer r such that Un+r ≡ Un (mod m) for all n ≥ 0. Although the distribution of sequences has been studied, none have classified the conditions when each residue occurs exactly once. This motivates the following definitions. A sequence is complete when π(m) = m and the m repeating terms of (Un(p,q) (mod m))π(m)−1 n=0 are some permutation of the values 0,1,2,...,m − 1. Investigation of complete sequences reveals unique patterns. There exist complete consecutive sequences where Un ≡ n (mod m) for all 0 ≤ n ≤ m−1. Furthermore, parity-consecutive complete sequences exist, in which (Un(p,q))n≥0 (mod m) decomposes into the disjoint subsequences (U2n(p,q))n≥0 and (U2n+1(p,q))n≥0 modulo m containing all even and odd terms, respectively. Our research determines the values of p and q that yield the various forms of completeness under certain moduli m
Scheduling classes at the departmental level is a challenging and time-consuming task. The mathematical technique of linear programming has the potential to simplify this challenge by building a model of linear constraints to find the most optimal solution that satisfies all the constraints. In this project, we are implementing a linear programming model using the DOCplex library in Python. The objective function represents instructor satisfaction with different courses, and the constraints represent limitations, such as the fact that one instructor cannot teach two courses at the same time. We will present example schedules produced by using the ConflictRefiner function to relax low-priority constraints when it is impossible to satisfy all constraints simultaneously. We will also present a combination of constraints and preferences that improve the distribution of courses throughout the day.
The Communities in a network are detected by investigating the corresponding graph and finding dense clusters of vertices. The decoding algorithm SASH determines the initial codeword of communities that would most likely result in specific clusters. SASH checks various candidate codewords at clustering types until the codeword with the smallest discrepancy from the observed dataset is located. Using the dataset Zachary’s Karate Club, errors within the algorithm that lead to a significantly lowered accuracy from expectations will be highlighted, as well as potential ways that could amend the issues to optimize performance.
Homotopical combinatorics uses tools from combinatorics to explore and understand structures in equivariant homotopy theory. One object of study in this field is a G-transfer system. We will define transfer systems and present the differences between abelian and non-abelian transfer systems. We then explore structural characteristics of the lattice of transfer systems for some non-abelian groups, such as D₉ and F₅. For these groups, we classify the (co)saturation of the transfer systems. We also present a proof that width, the number of arrows needed to generate a complete transfer system, can be determined by the prime factorization of the order for a dihedral group.
This research project investigates the learning competencies prioritized in Numbers and Operations courses designed for preservice elementary teachers across other midwestern universities. The use of learning competencies is to strengthen the preparation for elementary teachers and in this study, we examine how different universities view which pieces are more essential than others. We developed a Qualtrics-implemented questionnaire survey where participants are asked which topics are taught and within the topics, what competencies are prioritized. Our findings reveal that Operations (addition/subtraction, multiplication/division, operation) and Type of Number (decimals, fractions, whole numbers) were listed frequently and discussed further upon. Whereas, Number Theory topics and Proportional Reasoning were barely discussed within the survey. Recognizing these disparities and examining them further, provides an opportunity to narrow in on what competencies are more crucial than others in order to have the most effective preparation for preservice elementary teachers.
Some people call it a new “miracle drug.” Others are far more skeptical. However, one thing is certain: the surge in popularity of weight-loss drugs around the country is not going unnoticed. With obesity rates in the United States reaching over 40%, people are eager to find new ways to develop healthier lifestyles. Also known as Anti-Obesity Medications (AOMs), these drugs are still relatively new to the market and carry high, ongoing costs. Our research seeks to create an actuarial model to evaluate the financial impact of covering AOMs within employer-sponsored health plans. Current research on this subject is limited and studies that are available reflect conflicting results. We seek to create an accessible (Excel-based), functioning ROI model for AOM coverage that accounts for drug costs, expected utilization, weight-related risk reduction, and future medical cost offsets. We believe that the successful production of a dependable actuarial model could help employers make a more informed choice when it comes to the coverage of AOMs for their employees.
The utilization of analytics in women’s professional tennis, the Women’s Tennis Association (WTA), has occurred much more recently when compared to other professional sports. Despite unique difficulties in predicting match outcomes, there have been a spate of recent articles overviewing probabilistic and data-mining methods to do so. Our research work: (1) performed a literature review of data-mining methods for men’s professional tennis (ATP) and women’s professional tennis (WTA); (2) integrated knowledge about predictive statistics learned from prior years’ data-exploration; and (3) adds generalizations, efficiencies, documentation, and new features to functions used to summarize and organize data for predictive models.Data for player statistics and results of WTA tournaments was obtained from a GitHub repository under a Creative Commons license. We computed and analyzed a set of related summary statistics for use in comparison of data-mining methods. We discuss the data-analytic methods and compare predictive abilities. We edited original functions in R with the focus of wrangling the data across an appropriate time window, court surface, and player rank and of summarizing statistics for individual players and opponents. The process and final forms of the functions, along with the dissemination process via an R package, are described.
I have been teaching at UW - Eau Claire since 2006, covering courses in undergraduate statistics (introductory and upper-level) and Master’s-level data mining and programming. My research is in data-mining techniques, with a focus on penalized regression. My recent (last ~ 6 years... Read More →
This project investigates how instructional pedagogy and physical classroom design can encourage students to justify their reasoning and develop conceptual understanding in mathematics. Traditional classroom structures, where students passively receive information while teachers deliver instruction from the front of the room, have remained largely unchanged since the early days of public education. This study asks how both spatial design and teaching strategies can be reimagined to better support students’ sense-making and reasoning.To explore this question, I conducted a comprehensive review of existing literature on mathematics pedagogy, classroom discourse, and learning environments. Drawing from this research, I am constructing a diorama that synthesizes research-based practices into a visual and spatial model.The primary outcome of this project is a three-dimensional model of an ideal mathematics classroom that supports justification, collaboration, and conceptual understanding. This model illustrates how thoughtfully designed tasks, strategic teacher moves, and an intentional classroom culture can work together within a supportive physical space to help students move from memorization and empirical reasoning toward deeper mathematical meaning.
