Digital Commons Network^™

The Spectrum Of Nim-Values For Achievement Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben

Mathematics Faculty Publications

We study an impartial achievement game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The game ends when the jointly selected elements generate the group. The last player able to make a move is the winner of the game. We prove that the spectrum of nim-values of these games is {0, 1, 2, 3, 4}. This positively answers two conjectures from a previous paper by the last two authors.

How Effective Is The Efficiency Gap?, Thomas Q. Sibley

Mathematics Faculty Publications

Gerrymandering has affected U. S. politics since at least 1812. A political cartoon that year decried this tactic by then Massachusetts Governor Elbridge Gerry. (Gerrymandering is manipulating the boundaries of districts to benefit a group unfairly.)

While we may feel we know a gerrymander when we see one, finding a meaningful metric has proven challenging. This article uses elementary mathematics to investigate the efficiency gap, a recent model proposed to measure gerrymandering.

Fantasy On A Baseball Seam, Genevieve Ahlstrom, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.

Impartial Achievement Games For Generating Nilpotent Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben

Mathematics Faculty Publications

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.

Impartial Avoidance And Achievement Games For Generating Symmetric And Alternating Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben

Mathematics Faculty Publications

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.

Comparing Local Constants Of Ordinary Elliptic Curves In Dihedral Extensions, Sunil Chetty

Mathematics Faculty Publications

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.

On The Dimension Of Algebraic-Geometric Trace Codes, Phong Le, Sunil Chetty

Mathematics Faculty Publications

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.

Impartial Avoidance Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben

Mathematics Faculty Publications

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.

Arithmetic Local Constants For Abelian Varieties With Extra Endomorphisms, Sunil Chetty

Mathematics Faculty Publications

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.

Computing Local Constants For Cm Elliptic Curves, Sunil Chetty, Lung Li

Mathematics Faculty Publications

Let E/k be an elliptic curve with CM by O. We determine a formula for (a generalization of) the arithmetic local constant of Mazur-Rubin at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to describe extensions F/Q over which the O-rank of E grows.

When The Trivial Is Nontrivial, William Capecchi, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.

Groups Of Graphs Of Groups, David P. Byrne, Matthew J. Donner, Thomas Q. Sibley

Mathematics Faculty Publications

We classify all groups of color preserving automorphisms (isometries) of edge colored complete graphs derived from finite groups.

Undergraduate Students' Self-Reported Use Of Mathematics Textbooks, Aaron Weinberg, Emilie Wiesner, Bret Benesh, Timothy Boester

Mathematics Faculty Publications

Textbooks play an important role in undergraduate mathematics courses and have the potential to impact student learning. However, there have been few studies that describe students' textbook use in detail. In this study, 1156 undergraduate students in introductory mathematics classes were surveyed, and asked to describe how they used their textbook. The results indicate that students tend to use examples, instead of the expository text, to build their mathematical understanding, which instructors may view as problematic. This way of using the textbook may be the result of the textbook structure itself, as well as students' beliefs about reading and the …

The Graph Distance Game, Wayne Goddard, Anne Sinko, Peter J. Slater, Honghai Xu

Mathematics Faculty Publications

In the graph distance game, two players alternate in constructing a maximal path. The objective function is the distance between the two endpoints of the path, which one player tries to maximize and the other tries to minimize. In this note, we examine the distance game for various graphs, and provide general bounds, exact results for special graphs, and an algorithm for trees. Computer calculations suggest interesting conjectures for grids.

The Probabilistic Zeta Function, Bret Benesh

Mathematics Faculty Publications

This paper is a summary of results on the P_G(s) function, which is the reciprocal of the probabilistic zeta function for finite groups. This function gives the probability that s randomly chosen elements generate a group G, and information about the structure of the group G is embedded in it.

Sublimital Analysis, Thomas Q. Sibley

Mathematics Faculty Publications

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.

Queen's Domination Using Border Squares And (A,B)-Restricted Domination, Anne Sinko, Peter J. Slater

Mathematics Faculty Publications

In this paper we introduce a variant on the long studied, highly entertaining, and very difficult problem of determining the domination number of the queen's chessboard graph, that is, determining how few queens are needed to protect all of the squares of a k by k chessboard. Motivated by the problem of minimum redundance domination, we consider the problem of determining how few queens restricted to squares on the border can be used to protect the entire chessboard. We give exact values of "border-queens" required for the k by k chessboard when 1≤k≤13. For the general case, we …

A Classification Of Certain Maximal Subgroups Of Alternating Groups, Bret Benesh

Mathematics Faculty Publications

This paper addresses and extension of Problem 12.82 of the Kourovka notebook, which asks for all ordered pairs (n,m) such that the symmetric groups S_n embeds in S_m as a maximal subgroup. Problem 12.82 was answered in a previous paper by the author and Benjamin Newton. In this paper, we will consider the extension problem where we allow either or both of the groups from the ordered pair to be an alternating group.

Efficient Domination In Knights Graphs, Anne Sinko, Peter J. Slater

Mathematics Faculty Publications

The influence of a vertex set S ⊆V(G) is I(S) = Σ_v∈S(1 + deg(v)) = Σ_v∈S |N[v]|, which is the total amount of domination done by the vertices in S. The efficient domination number F(G) of a graph G is equal to the maximum influence of a packing, that is, F(G) is the maximum number of vertices one can dominate under the restriction that no vertex gets dominated more than once.

In …

A Classification Of Certain Maximal Subgroups Of Symmetric Groups, Benjamin Newton, Bret Benesh

Mathematics Faculty Publications

Problem 12.82 of the Kourovka Notebook asks for all ordered pairs (n,m) such that the symmetric group Sn embeds in Sm as a maximal subgroup. One family of such pairs is obtained when m=n+1. Kalužnin and Klin [L.A. Kalužnin, M.H. Klin, Certain maximal subgroups of symmetric and alternating groups, Math. Sb. 87 (1972) 91–121] and Halberstadt [E. Halberstadt, On certain maximal subgroups of symmetric or alternating groups, Math. Z. 151 (1976) 117–125] provided an additional infinite family. This paper answers the Kourovka question by producing a third infinite family of ordered …

Undergraduates' Use Of Mathematics Textbooks, Bret Benesh, Tim Boester, Aaron Weinberg, Eimilie Wiesner

Mathematics Faculty Publications

No abstract provided.

On Classifying Finite Edge Colored Graphs With Two Transitive Automorphism Groups, Thomas Q. Sibley

Mathematics Faculty Publications

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.

Deconstructing Bases: Fair, Fitting, And Fast Bases, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.

Taking The Sting Out Of Wasp Nests: A Dialogue On Modeling In Mathematical Biology, Jennifer C. Klein, Thomas Q. Sibley

Mathematics Faculty Publications

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.

The Possibility Of Impossible Pyramids, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.

Rhombic Penrose Tilings Can Be 3-Colored, Thomas Q. Sibley, Stan Wagon

Mathematics Faculty Publications

No abstract provided.

Beauty Bare, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.

Changing Modes Of Thought: Non-Euclidean Geometry And The Liberal Arts, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.

Homomorphisms For Equidistance Relations, Thomas Q. Sibley

Mathematics Faculty Publications

This paper presents necessary and sufficient conditions for the existence of homomorphisms for equidistance relations in terms of the closed subsystems (the Fundamental Theorem of Homomorphisms). Further, it shows that every closed subsystem of a 1-point homogenous equidistance system is a coset of a unique homomorphism. Affine spaces and other incidence geometries can be seen as examples of equidistance systems.

Equidistance Relations: A New Bridge Between Geometric And Algebraic Structures, Thomas Q. Sibley

Mathematics Faculty Publications

This paper investigates the transformations of certain geometric structures into algebraic ones and conversely. The algebraic notion of absolute value corresponds with the geometric one of equidistance. Further, latin squares with absolute values correspond to regular equidistance relations, "near groups" yield 1 point homogenous equidistance relations, and "near commutative groups" yield 2-point homogenous equidistance relations.