Kayla Barker and Garrison Koch, Stockton University
A Bipartite Graph Reduction Game
A game-labeling of a bipartite graph is one in which the vertices have non-negative labels and the label sums of the partite sets are equal. A reduction across an edge is one in which the endpoints are both reduced by the same integral amount. We introduce a single player game on a game-labeled bipartite graph G where each move is a reduction across an edge and no move produces a negative label. The goal of the game is to reduce all labels in G to 0, and if the player succeeds in doing so, the player wins the game. In our research, we present a necessary and sufficient sum condition for detecting the solvability of the game. We also demonstrate the connection between this game and Double Choco Puzzles.
Haoyuan Chen, Franklin & Marshall College
Semigroups of non-negative integer-valued matrices
Factorization-theoretic aspects of semigroups of matrices have received much attention over the past decade. Much of the focus has been on the multiplicative semigroups of nonzero divisors in rings of matrices; that is, factorization in rings of matrices. More recently, factorizations of upper triangular matrices over the nonnegative integers and over more general semirings have been considered. Here, we continue the study of the semigroup of upper-triangular matrices over the nonnegative integers as well as the larger semigroup of all square n × n matrices over the semiring of nonnegative integers. We extend the notion of divisor-closed semigroups to a noncommutative setting. After giving a characterization of irreducible elements in these matrix semigroups, we use the almost divisor-closed result along with precise computations, to determine arithmetical invariants that measure the degree to which factorization in these semigroups is nonunique. The result of this study has been published on Communications in Algebra with the same title.
Sarah Hartman, Ryan McAllister, and Griffin McVay, Messiah University
Nonparametric Curve Fitting: Techniques and Applications
Regression is a common tool for analyzing relationships between two variables which requires assumptions about form and variability. We will provide an introduction to nonparametric curve fitting, which attempts to remove the need for these assumptions. Three basic approaches of nonparametric curve fitting will be discussed and demonstrated using simulated and real world examples, including COVID case data and data gathered from our own campus.
Emma Miller, Moravian University
Considering Quantum Games
Quantum Tic-Tac-Toe is a commonly used teaching metaphor in quantum mechanics that aids students in visualizing abstract concepts such as superposition. In his papers, Alan Goff outlines 4 rules to change the classical rules of Tic-Tac-Toe to a quantum version, however, these rules keep the game combinatorial. Because of this, we are able to use mathematical techniques of combinatorial game theory to study the outcome classes of these games as well as optimal strategies. Furthermore, we extend Goff’s rules to other position combinatorial games such as Hex that rely on a single game token in a single cell. We compare the known classical outcome classes and strategies of Tic-Tac-Toe and Hex to quantum versions of these games.
Maria Nicos Alain Pasaylo, Anne Arundel Community College
Make your own "Message Protector"
During the 20th century, advances in mathematics and technology prompted the proliferation of many new encryption methods. Among these, the Hill Cipher, a polygraphic substitution cipher introduced in 1929, pioneered the use of modular arithmetic and linear algebra in an encryption algorithm. We will discuss its mathematical framework, method of plaintext attack, and Python implementation.
Isaac Reiter, Kutztown University of Pennsylvania
They are Perfectly Identical: Perfect Matching Transitivity of Circulant Graphs
A perfect matching M of a graph G is a subset of E(G) such that every vertex is incident to exactly one edge in M. G is perfect matching transitive if for any two perfect matchings M1 and M2, there exists an automorphism on V(G) that maps M1 to M2. In this talk, I will discuss the PM-transitivity of circulant graphs.
