Papers by Marion Scheepers
The internet connected modern world raises many security and privacy concerns. Cryptography is on... more The internet connected modern world raises many security and privacy concerns. Cryptography is one of the main tools in addressing these concerns. Classical cryptosystems relied on internet connected devices having significant computational and memory resources. Advances in technology have led to smaller, less resource rich devices, having internet connectivity. Examples include implantable medical devices such as insulin pumps, pacemakers, deep brain stimulators, etc. New cryptosystems that provide high security and privacy despite limited computational and memory resources are in critical demand. Such cryptosystems are called Lightweight cryptosystems. In this project we develop a lightweight cryptosystem which is based on a different mathematical platform than that of current candidates for being a recommended standard by the National Institute of Standards and Technology
In 2003, Prescott et al. hypothesized a special sorting operation active during ciliate genome ma... more In 2003, Prescott et al. hypothesized a special sorting operation active during ciliate genome maintenance. This operation, called cds, involves block interchanges of “permutations”, or unsorted lists. Christie (1996) discovered that for permutations sortable by cds, cds sorts them using the fewest possible block interchanges of any kind. Adamyk et al. (2013) discovered an efficient way of quantifying the non-cds-sortability of a permutation called the strategic pile. We investigate permutations with maximal strategic pile, aiming to determine when such a permutation has a given number of available cds moves. We complete this characterization when the number of available moves is close to maximal and when the number of available moves is minimal. We discover a collection of symmetries on these permutations that preserves the number of available cds moves and the maximality of the strategic pile. We then study permutations that are “symmetric”, count the number of permutations with maximal strategic pile and a given number of symmetries, and rediscover a classical theorem of Wilson about prime numbers. Adamyk et al. (2013) discovered a natural two player game using permutations that are not cds-sortable. We discover new sufficient conditions for player ONE to have a winning strategy in this game
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
We show that (1) Rothberger bounded subgroups of σ-compact groups are characterized by Ramseyan p... more We show that (1) Rothberger bounded subgroups of σ-compact groups are characterized by Ramseyan partition relations. (Corollary 4) (2) For each uncountable cardinal κ there is a T 0 topological group of cardinality κ such that ONE has a winning strategy in the point-open game on the group and the group is not a subspace of any σ-compact space. (Theorem 8) (3) For each uncountable cardinal κ there is a T 0 topological group of cardinality κ such that ONE has a winning strategy in the point-open game on the group and the group is σ-compact.
arXiv (Cornell University), May 19, 2014
We present a unified approach, based on dominating families in binary relations, for the study of... more We present a unified approach, based on dominating families in binary relations, for the study of topological properties defined in terms of selection principles and the games associated to them.
arXiv (Cornell University), Mar 19, 2016
Prior studies of the efficiency of the block interchange (swap) and the reversal sorting operatio... more Prior studies of the efficiency of the block interchange (swap) and the reversal sorting operations on (signed) permutations identified specialized versions of the these operations. These specialized operations are here called context directed reversal, abbreviated cdr, and context directed swap, abbreviated cds. Prior works have also characterized which (signed) permutations are sortable by cdr or by cds. It is now known that when a permutation π is cds sortable in n steps, then any n consecutive applicable cds operations will sort π. Examples show that this is not the case for cdr. This phenomenon is the focus of this paper. It is proven that if a signed permutation is cdr sortable, then any cdr fixed point of it is cds sortable (the cds Rescue Theorem). The cds Rescue Theorem is discussed in the context of a mathematical model for ciliate micronuclear decryption. It is also known that if applications of cds to a permutation π reaches a cds fixed point in n steps, then any n consecutive applicable cds operations will terminate in a cds fixed point of π. This is not the case for cdr: It is proven that though for a given signed permutation the number of cdr operations leading to different cdr fixed points may be different from each other, the parity of the number of operations is the same (the cdr Parity Theorem). This result provides a solution to two previously formulated decision problems regarding certain combinatorial games.
arXiv (Cornell University), Jul 24, 1992
Given a free ideal J of subsets of a set X, we consider games where player ONE plays an increasin... more Given a free ideal J of subsets of a set X, we consider games where player ONE plays an increasing sequence of elements of the σ-completion of J, and player TWO tries to cover the union of this sequence by playing one set at a time from J. We describe various conditions under which player TWO has a winning strategy that uses only information about the most recent k moves of ONE, and apply some of these results to the Banach-Mazur game.
arXiv (Cornell University), Nov 28, 2018
The special purpose sorting operation, context directed swap (CDS), is an example of the block in... more The special purpose sorting operation, context directed swap (CDS), is an example of the block interchange sorting operation studied in prior work on permutation sorting. CDS has been postulated to model certain molecular sorting events that occur in the genome maintenance program of some species of ciliates. We investigate the mathematical structure of permutations not sortable by the CDS sorting operation. In particular, we present substantial progress towards quantifying permutations with a given strategic pile size, which can be understood as a measure of CDS non-sortability. Our main results include formulas for the number of permutations in Sn with maximum size strategic pile. More generally, we derive a formula for the number of permutations in Sn with strategic pile size k, in addition to an algorithm for computing certain coefficients of this formula, which we call merge numbers.
Topology and its Applications, May 1, 2019
We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, t... more We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in [6] holds while the classical Borel Conjecture fails.
Proceedings of the American Mathematical Society, 1997
Cp(X) has the monotonic sequence selection property if there is for each f , and for every sequen... more Cp(X) has the monotonic sequence selection property if there is for each f , and for every sequence (σn : n < ω) where for each n σn is a sequence converging pointwise monotonically to f , a sequence (fn : n < ω) such that for each n fn is a term of σn, and (fn : n < ω) converges pointwise to f. We prove a theorem which implies for metric spaces X that Cp(X) has the monotonic sequence selection property if, and only if, X has a covering property of Hurewicz.
Proceedings of the American Mathematical Society, Nov 1, 1995
We give a direct proof of the fact that if player TWO of a certain infinite game on a metric spac... more We give a direct proof of the fact that if player TWO of a certain infinite game on a metric space has a winning strategy, then the space is a union of countably many of its compact subsets.
arXiv (Cornell University), Apr 5, 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.
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.
arXiv (Cornell University), Aug 21, 2018
We show that a statement concerning the existence of winning strategies of limited memory in an i... more We show that a statement concerning the existence of winning strategies of limited memory in an infinite two-person topological game is equivalent to a weak version of the Singular Cardinals Hypothesis.
arXiv (Cornell University), Aug 22, 2018
We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, t... more We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in [8] holds while the classical Borel Conjecture fails.
Uploads
Papers by Marion Scheepers