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Full-Text Articles in Discrete Mathematics and Combinatorics
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.
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 …