Mathematics Faculty PublicationsCopyright (c) 2022 College of Saint Benedict and Saint John's University All rights reserved.
https://digitalcommons.csbsju.edu/math_pubs
Recent documents in Mathematics Faculty Publicationsen-usThu, 22 Dec 2022 17:27:45 PST3600LIGHTS OUT! on graph products over the ring of integers modulo k
https://digitalcommons.csbsju.edu/math_pubs/35
https://digitalcommons.csbsju.edu/math_pubs/35Mon, 27 Sep 2021 08:15:23 PDTLIGHTS OUT! is a game played on a finite, simple graph. The vertices of the graph are the lights, which may be on or off, and the edges of the graph determine how neighboring vertices turn on or off when a vertex is pressed. Given an initial configuration of vertices that are on, the object of the game is to turn all the lights out. The traditional game is played over Z2" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 15.26px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(85, 85, 85); font-family: "Source Sans Pro", "Helvetica Neue", Helvetica, Arial, sans-serif; position: relative;">Z2Z2, where the vertices are either lit or unlit, but the game can be generalized to Zk" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 15.26px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(85, 85, 85); font-family: "Source Sans Pro", "Helvetica Neue", Helvetica, Arial, sans-serif; position: relative;">ZkZk, where the lights have different colors. Previously, the game was investigated on Cartesian product graphs over Z2" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 15.26px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(85, 85, 85); font-family: "Source Sans Pro", "Helvetica Neue", Helvetica, Arial, sans-serif; position: relative;">Z2Z2. We extend this work to Zk" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 15.26px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(85, 85, 85); font-family: "Source Sans Pro", "Helvetica Neue", Helvetica, Arial, sans-serif; position: relative;">ZkZk and investigate two other fundamental graph products, the direct (or tensor) product and the strong product. We provide conditions for which the direct product graph and the strong product graph are solvable based on the factor graphs, and we do so using both open and closed neighborhood switching over Zk" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; font-size: 15.26px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(85, 85, 85); font-family: "Source Sans Pro", "Helvetica Neue", Helvetica, Arial, sans-serif; position: relative;">ZkZk.
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Travis Peters et al.Impartial achievement games for generating nilpotent groups
https://digitalcommons.csbsju.edu/math_pubs/34
https://digitalcommons.csbsju.edu/math_pubs/34Tue, 28 May 2019 12:00:21 PDT
We study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form T×H, where T is a 2-group and H is a group of odd order. This includes all nilpotent and hence abelian groups.
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Bret J. Benesh et al.Impartial avoidance and achievement games for generating symmetric and alternating groups
https://digitalcommons.csbsju.edu/math_pubs/32
https://digitalcommons.csbsju.edu/math_pubs/32Tue, 06 Sep 2016 13:49:27 PDT
Anderson and Harary introduced two impartial games on finite groups. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. We determine the nim-numbers, and therefore the outcomes, of these games for symmetric and alternating groups.
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Bret J. Benesh et al.Impartial avoidance games for generating finite groups
https://digitalcommons.csbsju.edu/math_pubs/31
https://digitalcommons.csbsju.edu/math_pubs/31Tue, 06 Sep 2016 13:29:34 PDT
We study an impartial avoidance game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The first player who cannot select an element without making the set of jointly-selected elements into a generating set for the group loses the game. We develop criteria on the maximal subgroups that determine the nim-numbers of these games and use our criteria to study our game for several families of groups, including nilpotent, sporadic, and symmetric groups.
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Bret J. Benesh et al.Arithmetic local constants for abelian varieties with extra endomorphisms
https://digitalcommons.csbsju.edu/math_pubs/30
https://digitalcommons.csbsju.edu/math_pubs/30Mon, 29 Aug 2016 10:14:06 PDT
This work generalizes the theory of arithmetic local constants, introduced by Mazur and Rubin, to better address abelian varieties with a larger endomorphism ring than ℤ. We then study the growth of the p^{∞}- Selmer rank of our abelian variety, and we address the problem of extending the results of Mazur and Rubin to dihedral towers k ⊂ K ⊂ F in which [F : K] is not a p-power extension.
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Sunil ChettyComparing local constants of ordinary elliptic curves in dihedral extensions
https://digitalcommons.csbsju.edu/math_pubs/29
https://digitalcommons.csbsju.edu/math_pubs/29Tue, 23 Aug 2016 08:47:44 PDT
We establish, for a substantial class of elliptic curves, that the arithmetic local constants introduced by Mazur and Rubin agree with quotients of analytic root numbers.
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Sunil ChettyOn the dimension of algebraic-geometric trace codes
https://digitalcommons.csbsju.edu/math_pubs/28
https://digitalcommons.csbsju.edu/math_pubs/28Tue, 23 Aug 2016 08:22:35 PDT
We study trace codes induced from codes defined by an algebraic curve X. We determine conditions on X which admit a formula for the dimension of such a trace code. Central to our work are several dimension reducing methods for the underlying functions spaces associated to X.
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Phong Le et al.Idempotents à la mod
https://digitalcommons.csbsju.edu/math_pubs/27
https://digitalcommons.csbsju.edu/math_pubs/27Tue, 25 Nov 2014 09:56:46 PST
An idempotent satisfies the equation x2 = x. In ordinary arithmetic, this is so easy to solve it’s boring. We delight the mathematical palette here, topping idempotents off with modular arithmetic and a series of exercises determining for which n there are more than two idempotents (mod n) and exactly how many there are.
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Thomas Q. SibleyThe Council on Undergraduate Research as a resource for mathematicians
https://digitalcommons.csbsju.edu/math_pubs/26
https://digitalcommons.csbsju.edu/math_pubs/26Tue, 25 Nov 2014 09:37:45 PSTThomas Q. SibleyMaking the most of your sabbatical
https://digitalcommons.csbsju.edu/math_pubs/25
https://digitalcommons.csbsju.edu/math_pubs/25Mon, 24 Nov 2014 14:31:18 PSTJennifer R. Galovich et al.Taking the sting out of wasp nests: a dialogue on modeling in mathematical biology
https://digitalcommons.csbsju.edu/math_pubs/24
https://digitalcommons.csbsju.edu/math_pubs/24Mon, 24 Nov 2014 14:08:30 PST
Wasps in hot climates build elongated nests, while in colder areas they tend to be circular. Mathematics cannot explain that, but there are questions about numbers of cells that can be answered.
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Jennifer C. Klein et al.The possibility of impossible pyramids
https://digitalcommons.csbsju.edu/math_pubs/23
https://digitalcommons.csbsju.edu/math_pubs/23Mon, 24 Nov 2014 13:44:45 PSTThomas Q. SibleyRhombic Penrose tilings can be 3-colored
https://digitalcommons.csbsju.edu/math_pubs/22
https://digitalcommons.csbsju.edu/math_pubs/22Mon, 24 Nov 2014 13:12:44 PSTThomas Q. Sibley et al.Sylow-like theorems in geometry and algebra
https://digitalcommons.csbsju.edu/math_pubs/21
https://digitalcommons.csbsju.edu/math_pubs/21Mon, 24 Nov 2014 08:49:36 PST
The notion of congruence provides a means to extend the Sylow theorems from group theory to a class of geometric structures called congruence spaces and to their corresponding loops. The extension of these results depends on the existence of a group acting transitively on the congruence space and preserving congruence. A partial ordering on the congruence spaces suggests a means to form all of these spaces from groups.
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Thomas Q. SibleyWhen the trivial is nontrivial
https://digitalcommons.csbsju.edu/math_pubs/20
https://digitalcommons.csbsju.edu/math_pubs/20Tue, 30 Sep 2014 09:28:23 PDTWilliam Capecchi et al.Sublimital Analysis
https://digitalcommons.csbsju.edu/math_pubs/19
https://digitalcommons.csbsju.edu/math_pubs/19Tue, 30 Sep 2014 08:51:15 PDT
The Bolzano-Weierstrass theorem asserts, under appropriate circumstances, the convergence of some subsequence of a sequence. While this famous theorem ignores the actual limit of the subsequence, it is natural to investigate such limits. This note characterizes the set of possible limits of subsequences of a given sequence.
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Thomas Q. SibleyDeconstructing bases: fair, fitting, and fast bases
https://digitalcommons.csbsju.edu/math_pubs/18
https://digitalcommons.csbsju.edu/math_pubs/18Tue, 30 Sep 2014 08:29:01 PDTThomas Q. SibleyOn classifying finite edge colored graphs with two transitive automorphism groups
https://digitalcommons.csbsju.edu/math_pubs/17
https://digitalcommons.csbsju.edu/math_pubs/17Tue, 30 Sep 2014 08:14:42 PDT
This paper classifies all finite edge colored graphs with doubly transitive automorphism groups. This result generalizes the classification of doubly transitive balanced incomplete block designs with λ=1 and doubly transitive one-factorizations of complete graphs. It also provides a classification of all doubly transitive symmetric association schemes.
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Thomas Q. SibleyPuzzling groups
https://digitalcommons.csbsju.edu/math_pubs/16
https://digitalcommons.csbsju.edu/math_pubs/16Thu, 25 Sep 2014 19:37:29 PDT
We introduce a family of puzzles that can help students understand permutation groups. In addition these puzzles provide a basis to investigate other puzzles and their groups of permutations.
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Thomas Q. SibleyColored-independence (mostly) on trees
https://digitalcommons.csbsju.edu/math_pubs/15
https://digitalcommons.csbsju.edu/math_pubs/15Wed, 23 Apr 2014 11:02:21 PDTAnne C. Sinko