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Articles 1 - 6 of 6
Full-Text Articles in Physical Sciences and Mathematics
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 …
Investigating First Returns: The Effect Of Multicolored Vectors, Shakuan Frankson, Myka Terry
Investigating First Returns: The Effect Of Multicolored Vectors, Shakuan Frankson, Myka Terry
Rose-Hulman Undergraduate Mathematics Journal
By definition, a first return is the immediate moment that a path, using vectors in the Cartesian plane, touches the x-axis after leaving it previously from a given point; the initial point is often the origin. In this case, using certain diagonal and horizontal vectors while restricting the movements to the first quadrant will cause almost every first return to end at the point (2n,0), where 2n counts the equal number of up and down steps in a path. The exception will be explained further in the sections below. Using the first returns of Catalan, Schröder, and Motzkin numbers, which …
On The Mysteries Of Interpolation Jack Polynomials, Havi Ellers
On The Mysteries Of Interpolation Jack Polynomials, Havi Ellers
HMC Senior Theses
Interpolation Jack polynomials are certain symmetric polynomials in N variables with coefficients that are rational functions in another parameter k, indexed by partitions of length at most N. Introduced first in 1996 by F. Knop and S. Sahi, and later studied extensively by Sahi, Knop-Sahi, and Okounkov-Olshanski, they have interesting connections to the representation theory of Lie algebras. Given an interpolation Jack polynomial we would like to differentiate it with respect to the variable k and write the result as a linear combination of other interpolation Jack polynomials where the coefficients are again rational functions in k. In this …
Phylogenetic Networks And Functions That Relate Them, Drew Scalzo
Phylogenetic Networks And Functions That Relate Them, Drew Scalzo
Williams Honors College, Honors Research Projects
Phylogenetic Networks are defined to be simple connected graphs with exactly n labeled nodes of degree one, called leaves, and where all other unlabeled nodes have a degree of at least three. These structures assist us with analyzing ancestral history, and its close relative - phylogenetic trees - garner the same visualization, but without the graph being forced to be connected. In this paper, we examine the various characteristics of Phylogenetic Networks and functions that take these networks as inputs, and convert them to more complex or simpler structures. Furthermore, we look at the nature of functions as they relate …
Extremal/Saturation Numbers For Guessing Numbers Of Undirected Graphs, Jo Ryder Martin
Extremal/Saturation Numbers For Guessing Numbers Of Undirected Graphs, Jo Ryder Martin
Graduate College Dissertations and Theses
Hat guessing games—logic puzzles where a group of players must try to guess the color of their own hat—have been a fun party game for decades but have become of academic interest to mathematicians and computer scientists in the past 20 years. In 2006, Søren Riis, a computer scientist, introduced a new variant of the hat guessing game as well as an associated graph invariant, the guessing number, that has applications to network coding and circuit complexity. In this thesis, to better understand the nature of the guessing number of undirected graphs we apply the concept of saturation to guessing …
Laplacian Spectra Of Kneser-Like Bipartite Graphs, Cesar Iram Vazquez
Laplacian Spectra Of Kneser-Like Bipartite Graphs, Cesar Iram Vazquez
Open Access Theses & Dissertations
Given a,b ∈N such that a > b we define a Kneser-like bipartite graph G(a,b), whose two bipartite sets of vertices represent the a-subsets and b-subsets of S = {1,...,a + b + 1}, and whose edges are pairs of vertices X and Y such that X ∩Y = ∅. We prove that the eigenvalues of the Laplacian matrix of graphs G(a,1) are all nonnegative integers. In fact, we describe these eigenvalues, and their respective multiplicities.