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Articles 1 - 13 of 13
Full-Text Articles in Algebra
Generalizations Of Commutativity In Dihedral Groups, Noah A. Heckenlively
Generalizations Of Commutativity In Dihedral Groups, Noah A. Heckenlively
Rose-Hulman Undergraduate Mathematics Journal
The probability that two elements commute in a non-Abelian finite group is at most 5 8 . We prove several generalizations of this result for dihedral groups. In particular, we give specific values for the probability that a product of an arbitrary number of dihedral group elements is equal to its reverse, and also for the probability that a product of three elements is equal to a permutation of itself or to a cyclic permutation of itself. We also show that for any r and n, there exists a dihedral group such that the probability that a product of n …
On Isomorphic K-Rational Groups Of Isogenous Elliptic Curves Over Finite Fields, Ben Kuehnert, Geneva Schlafly, Zecheng Yi
On Isomorphic K-Rational Groups Of Isogenous Elliptic Curves Over Finite Fields, Ben Kuehnert, Geneva Schlafly, Zecheng Yi
Rose-Hulman Undergraduate Mathematics Journal
It is well known that two elliptic curves are isogenous if and only if they have same number of rational points. In fact, isogenous curves can even have isomorphic groups of rational points in certain cases. In this paper, we consolidate all the current literature on this relationship and give a extensive classification of the conditions in which this relationship arises. First we prove two ordinary isogenous elliptic curves have isomorphic groups of rational points when they have the same $j$-invariant. Then, we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$-invariant of either 0 …
Winning Strategy For Multiplayer And Multialliance Geometric Game, Jingkai Ye
Winning Strategy For Multiplayer And Multialliance Geometric Game, Jingkai Ye
Rose-Hulman Undergraduate Mathematics Journal
The Geometric Sequence with common ratio 2 is one of the most well-known geometric sequences. Every term is a nonnegative power of 2. Using this popular sequence, we can create a Geometric Game which contains combining moves (combining two copies of the same terms into the one copy of next term) and splitting moves (splitting three copies of the same term into two copies of previous terms and one copy of the next term). For this Geometric Game, we are able to prove that the game is finite and the final game state is unique. Furthermore, we are able to …
A Proof Of A Generalization Of Niven's Theorem Using Algebraic Number Theory, Caroline Nunn
A Proof Of A Generalization Of Niven's Theorem Using Algebraic Number Theory, Caroline Nunn
Rose-Hulman Undergraduate Mathematics Journal
Niven’s theorem states that the sine, cosine, and tangent functions are rational for only a few rational multiples of π. Specifically, for angles θ that are rational multiples of π, the only rational values of sin(θ) and cos(θ) are 0, ±½, and ±1. For tangent, the only rational values are 0 and ±1. We present a proof of this fact, along with a generalization, using the structure of ideals in imaginary quadratic rings. We first show that the theorem holds for the tangent function using elementary properties of Gaussian integers, before extending the approach to other imaginary quadratic rings. We …
Irreducibility And Galois Groups Of Random Polynomials, Hanson Hao, Eli Navarro, Henri Stern
Irreducibility And Galois Groups Of Random Polynomials, Hanson Hao, Eli Navarro, Henri Stern
Rose-Hulman Undergraduate Mathematics Journal
In 2015, I. Rivin introduced an effective method to bound the number of irreducible integral polynomials with fixed degree d and height at most N. In this paper, we give a brief summary of this result and discuss the precision of Rivin's arguments for special classes of polynomials. We also give elementary proofs of classic results on Galois groups of cubic trinomials.
Disjointness Of Linear Fractional Actions On Serre Trees, Henry W. Talbott
Disjointness Of Linear Fractional Actions On Serre Trees, Henry W. Talbott
Rose-Hulman Undergraduate Mathematics Journal
Serre showed that, for a discrete valuation field, the group of linear fractional transformations acts on an infinite regular tree with vertex degree determined by the residue degree of the field. Since the p-adics and the polynomials over the finite field of order p act on isomorphic trees, we may ask whether pairs of actions from these two groups are ever conjugate as tree automorphisms. We analyze permutations induced on finite vertex sets, and show a permutation classification result for actions by these linear fractional transformation groups. We prove that actions by specific subgroups of these groups are conjugate only …
Directed Graphs Of The Finite Tropical Semiring, Caden G. Zonnefeld
Directed Graphs Of The Finite Tropical Semiring, Caden G. Zonnefeld
Rose-Hulman Undergraduate Mathematics Journal
The focus of this paper lies at the intersection of the fields of tropical algebra and graph theory. In particular the interaction between tropical semirings and directed graphs is investigated. Originally studied by Lipvoski, the directed graph of a ring is useful in identifying properties within the algebraic structure of a ring. This work builds off research completed by Beyer and Fields, Hausken and Skinner, and Ang and Shulte in constructing directed graphs from rings. However, we will investigate the relationship (x, y)→(min(x, y), x+y) as defined by the operations of tropical algebra and applied to tropical semirings.
Mathematical Magic: A Study Of Number Puzzles, Nicasio M. Velez
Mathematical Magic: A Study Of Number Puzzles, Nicasio M. Velez
Rose-Hulman Undergraduate Mathematics Journal
Within this paper, we will briefly review the history of a collection of number puzzles which take the shape of squares, polygons, and polyhedra in both modular and nonmodular arithmetic. Among other results, we develop construction techniques for solutions of both Modulo and regular Magic Squares. For other polygons in nonmodular arithmetic, specifically of order 3, we present a proof of why there are only four Magic Triangles using linear algebra, disprove the existence of the Magic Tetrahedron in two ways, and utilizing the infamous 3-SUM combinatorics problem we disprove the existence of the Magic Octahedron.
The Name Tag Problem, Christian Carley
The Name Tag Problem, Christian Carley
Rose-Hulman Undergraduate Mathematics Journal
The Name Tag Problem is a thought experiment that, when formalized, serves as an introduction to the concept of an orthomorphism of $\Zn$. Orthomorphisms are a type of group permutation and their graphs are used to construct mutually orthogonal Latin squares, affine planes and other objects. This paper walks through the formalization of the Name Tag Problem and its linear solutions, which center around modular arithmetic. The characterization of which linear mappings give rise to these solutions developed in this paper can be used to calculate the exact number of linear orthomorphisms for any additive group Z/nZ, which is demonstrated …
Colorings And Sudoku Puzzles, Katelyn D. May
Colorings And Sudoku Puzzles, Katelyn D. May
Rose-Hulman Undergraduate Mathematics Journal
Map colorings refer to assigning colors to different regions of a map. In particular, a typical application is to assign colors so that no two adjacent regions are the same color. Map colorings are easily converted to graph coloring problems: regions correspond to vertices and edges between two vertices exist for adjacent regions. We extend these notions to Shidoku, 4x4 Sudoku puzzles, and standard 9x9 Sudoku puzzles by demanding unique entries in rows, columns, and regions. Motivated by our study of ring and field theory, we expand upon the standard division algorithm to study Gr\"obner bases in multivariate polynomial rings. …
New Theorems For The Digraphs Of Commutative Rings, Morgan Bounds
New Theorems For The Digraphs Of Commutative Rings, Morgan Bounds
Rose-Hulman Undergraduate Mathematics Journal
The digraphs of commutative rings under modular arithmetic reveal intriguing cycle patterns, many of which have yet to be explained. To help illuminate these patterns, we establish a set of new theorems. Rings with relatively prime moduli a and b are used to predict cycles in the digraph of the ring with modulus ab. Rings that use Pythagorean primes as their modulus are shown to always have a cycle in common. Rings with perfect square moduli have cycles that relate to their square root.
Monoidal Supercategories And Superadjunction, Dene Lepine
Monoidal Supercategories And Superadjunction, Dene Lepine
Rose-Hulman Undergraduate Mathematics Journal
We define the notion of superadjunction in the context of supercategories. In particular, we give definitions in terms of counit-unit superadjunctions and hom-space superadjunctions, and prove that these two definitions are equivalent. These results generalize well-known statements in the non-super setting. In the super setting, they formalize some notions that have recently appeared in the literature. We conclude with a brief discussion of superadjunction in the language of string diagrams.
Strengthening Relationships Between Neural Ideals And Receptive Fields, Angelique Morvant
Strengthening Relationships Between Neural Ideals And Receptive Fields, Angelique Morvant
Rose-Hulman Undergraduate Mathematics Journal
Neural codes are collections of binary vectors that represent the firing patterns of neurons. The information given by a neural code C can be represented by its neural ideal JC. In turn, the polynomials in JC can be used to determine the relationships among the receptive fields of the neurons. In a paper by Curto et al., three such relationships, known as the Type 1-3 relations, were linked to the neural ideal by three if-and-only-if statements. Later, Garcia et al. discovered the Type 4-6 relations. These new relations differed from the first three in that they were …