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Papers by Marion Scheepers

In the spring 2020 semester, one group assignment in the course Communication in the Mathematical... more In the spring 2020 semester, one group assignment in the course Communication in the Mathematical Sciences at Boise State University, taught by Dr. Marion Scheepers, instructed students to write a paper which explores Brahmagupta N-triples. A triple (x,y,k) of integers is a Brahmagupta N-triple if the equation x2-Ny2=k holds for them. Two groups found a previously unknown sequence of Brahmagupta N-triples involving record primes. This finding led to new research questions and the current project. This project investigates whether there are for each k patterns related to the N, x, or y for new record values of x or y coefficients at corresponding N, relations among these record coefficients, possible growth patterns, and modular patterns for record N\u27s, and whether for these record values the number N is required to be a prime number

Many practical problems have the goal of identifying, with limited resources, a small number of o... more Many practical problems have the goal of identifying, with limited resources, a small number of objects from a large collection - be it a faulty circuit in a complex device, an infected individual in a population, a cryptographic key in a cyber attack, or a person of interest in a series of crimes. Although some such search problems are believed to require exhaustive search in general, many practical instances have yielded to carefully designed efficient search strategies. Our research focuses on the design and analysis of such efficient search techniques using combinatorial structures called splitting systems. The smaller a splitting system is, the more efficiently large-scale searches based on it can be executed. We hope to identify techniques for creating very small splitting systems in an attempt to speed up these processes. The methods used in this investigation stem from the fields of discrete mathematics and combinatorics

Finite groups are mathematical platforms for modern cryptography. Security protocols are often vu... more Finite groups are mathematical platforms for modern cryptography. Security protocols are often vulnerable to subtle exploits. A well-chosen group can be used to foil these exploits. To identify suitable groups, attack scenarios are modeled by two-player games. This research focuses on two classes of such games. For one class of games we give a complete analysis over finite Abelian groups. We report partial results for non-Abelian groups and for the other class of games. Game Theory

Consider a square matrix M in Z2, equal to its transpose, with a northwest-southeast diagonal of ... more Consider a square matrix M in Z2, equal to its transpose, with a northwest-southeast diagonal of zeros. The mcds operation, when applied repeatedly to such a matrix M, terminates either in the zero matrix or else in several matrices, each with at most six ones located in specific positions within the matrix. The variability in outcomes for the results of this operation suggests a basis for a finite combinatorial game. In this project we explore winning strategies for the game in question and examine the possible ending configurations of the process upon which it is based

arXiv (Cornell University), Nov 8, 2010

arXiv (Cornell University), May 19, 2014

arXiv (Cornell University), Mar 19, 2016

arXiv (Cornell University), Jul 24, 1992

arXiv (Cornell University), Nov 28, 2018

Topology and its Applications, May 1, 2019

$Research paper thumbnail of A sequential property of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="sans-serif">C</mi><mi>p</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathsf {C}_p(X)</annotation></semantics></math>Cp(X) and a covering property of Hurewicz$

Proceedings of the American Mathematical Society, 1997

Proceedings of the American Mathematical Society, Nov 1, 1995

arXiv (Cornell University), Apr 5, 2019

arXiv: Combinatorics, 2019

The study of sorting permutations by block interchanges has recently been stimulated by a phenome... more The study of sorting permutations by block interchanges has recently been stimulated by a phenomenon observed in the genome maintenance of certain ciliate species. The result was the identification of a block interchange operation that applies only under certain constraints. Interestingly, this constrained block interchange operation can be generalized naturally to simple graphs and to an operation on square matrices. This more general context provides numerous techniques applicable to the original context. In this paper we consider the more general context, and obtain an enumeration, in closed form, of all simple graphs on n vertices that are ``sortable" by the graph analogue of the constrained version of block interchanges. We also obtain asymptotic results on the proportion of graphs on n vertices that are so sortable.