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Analysis Of Don't Break The Ice, Amy Hung, Austin Uden
Analysis Of Don't Break The Ice, Amy Hung, Austin Uden
Rose-Hulman Undergraduate Mathematics Journal
The game Don't Break the Ice is a classic children's game that involves players taking turns hitting ice blocks out of a grid until a block containing a bear falls. We present Don't Break the Ice as a combinatorial game, and analyze various versions with an eye towards both normal and misere play. We present different winning strategies, some applying to specific games and some generalized for all versions of the game.
The Secret Santa Problem Continues, Daniel Crane, Tanner Dye
The Secret Santa Problem Continues, Daniel Crane, Tanner Dye
Rose-Hulman Undergraduate Mathematics Journal
We explore the Secret Santa gift exchange problem. A group of n people draws names at random, giving a gift to the person drawn. First, we examine the probabilities of gift exchanges under various scenarios when everyone draws names at once, similar to Montmort's matching problem. We then consider the probabilities of certain gift exchanges when people take turns drawing names and develop a strategy to maximize the likelihood of receiving a gift from the most generous participant.
An Appreciation Of Euler's Formula, Caleb Larson
An Appreciation Of Euler's Formula, Caleb Larson
Rose-Hulman Undergraduate Mathematics Journal
For many mathematicians, a certain characteristic about an area of mathematics will lure him/her to study that area further. That characteristic might be an interesting conclusion, an intricate implication, or an appreciation of the impact that the area has upon mathematics. The particular area that we will be exploring is Euler's Formula, $e^{ix}=\cos{x}+i\sin{x}$, and as a result, Euler's Identity, $e^{i\pi}+1=0$. Throughout this paper, we will develop an appreciation for Euler's Formula as it combines the seemingly unrelated exponential functions, imaginary numbers, and trigonometric functions into a single formula. To appreciate and further understand Euler's Formula, we will give attention to …
Super-Walk Formulae For Even And Odd Laplacians In Finite Graphs, Chengzheng Yu
Super-Walk Formulae For Even And Odd Laplacians In Finite Graphs, Chengzheng Yu
Rose-Hulman Undergraduate Mathematics Journal
The number of walks from one vertex to another in a finite graph can be counted by the adjacency matrix. In this paper, we prove two theorems that connect the graph Laplacian with two types of walks in a graph. By defining two types of walks and giving orientation to a finite graph, one can easily count the number of the total signs of each kind of walk from one element to another of a fixed length.
On The Long-Repetition-Free 2-Colorability Of Trees, Joseph Antonides, Claire Kiers, Nicole Yamzon
On The Long-Repetition-Free 2-Colorability Of Trees, Joseph Antonides, Claire Kiers, Nicole Yamzon
Rose-Hulman Undergraduate Mathematics Journal
A word w =uu is called a long square if u is of length at least 3; a word w is called long-square-free if w contains no sub-word that is a long square. If there exists a k-coloring of the vertices of a graph G such that, for any path P in G, the word generated by the coloring of P is long-square-free, then G is called long-repetition-free} k-colorable. We show that every rooted tree of radius r <= 7 is long-repetition-free 2-colorable. We also show that there exists a class of trees which are not long-repetition-free 2-colorable.
Scott Sentences In Uncountable Structures, Brian Tyrrell
Scott Sentences In Uncountable Structures, Brian Tyrrell
Rose-Hulman Undergraduate Mathematics Journal
Using elementary first order logic we can prove many things about models and theories, however more can be gleamed if we consider sentences with countably many conjunctions and disjunctions, yet still have the restriction of using only finitely many quantifiers. A logic with this feature is L_{\omega_1 , \omega}. In 1965 Scott proved by construction the existence of an L_{\omega_1 , \omega} sentence that could describe a countable model up to isomorphism. This type of infinitary sentence is now known as a Scott sentence. Given an infinitary cardinal \kappa, we wish to find a set of conditions such that if …
A Mathematical Model Of Cardiovascular And Respiratory Dynamics In Humans With Transposition Of The Great Arteries, Corey Riley
A Mathematical Model Of Cardiovascular And Respiratory Dynamics In Humans With Transposition Of The Great Arteries, Corey Riley
Rose-Hulman Undergraduate Mathematics Journal
Transposition of the Great Arteries (TGA) is a congenital heart defect in humans in which the pulmonary artery and the aorta are transposed, causing oxygen-poor blood to bypass the lungs and be recirculated throughout the body. In many cases, an atrial and/or ventricular septal defect also forms to allow the oxygen-rich and oxygen-poor blood to mix in the heart, temporarily sustaining the patient's life. In this paper, we create a model of cardiovascular and respiratory dynamics for a human patient with TGA by extending a current model of normal heart function. The goal of this research is to predict blood-oxygen …
Generalization Of Pascal's Rule And Leibniz's Rule For Differentiation, Rajeshwari Majumdar
Generalization Of Pascal's Rule And Leibniz's Rule For Differentiation, Rajeshwari Majumdar
Rose-Hulman Undergraduate Mathematics Journal
We generalize the combinatorial identity for binomial coefficients underlying the construction of Pascal's Triangle to multinomial coefficients underlying the construction of Pascal's Simplex. Using this identity, we present a new proof of the formula for calculating the nth derivative of the product of k functions, a generalization of Leibniz's Rule for differentiation.
The Unimodular Determinant Spectrum Problem, Wilson Lough
The Unimodular Determinant Spectrum Problem, Wilson Lough
Rose-Hulman Undergraduate Mathematics Journal
We present results related to the determinant spectrum of matrices with entries restricted to quartic roots of unity. We completely characterize determinant spectra for small orders and present conjectures on the elements and structures of higher-order spectra.
Existence And Uniqueness Of Solutions Of An Einstein-Maxwell Pde System, Toby Aldape
Existence And Uniqueness Of Solutions Of An Einstein-Maxwell Pde System, Toby Aldape
Rose-Hulman Undergraduate Mathematics Journal
We consider a nonlinear coupled system of partial differential equations with asymptotic boundary conditions which is relevant in the field of general relativity. Specifically, the PDE system relates the factors of a conformally flat spatial metric obeying the laws of gravity and electromagnetism to its charge and mass distributions. The solution to the system is shown to be existent, smooth, and unique. While the discussion of the PDE assumes knowledge of physics and differential geometry, the proof uses only the PDE theory of flat space.
The Log Convex Density Conjecture In Hyperbolic Space, Leonardo Di Giosia, Jahangir Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu
The Log Convex Density Conjecture In Hyperbolic Space, Leonardo Di Giosia, Jahangir Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu
Rose-Hulman Undergraduate Mathematics Journal
The Euclidean log convex density theorem, proved by Gregory Chambers in 2015, asserts that in Euclidean space with a log convex density spheres about the origin are isoperimetric. We provide a partial extension to hyperbolic space in which volume and perimeter densities are related but different.
Classification Of Four-Component Rotationally Symmetric Rose Links, Julia Creager, Nirja Patel
Classification Of Four-Component Rotationally Symmetric Rose Links, Julia Creager, Nirja Patel
Rose-Hulman Undergraduate Mathematics Journal
A rose link is a disjoint union of a finite number of unknots. Each unknot is considered a component of the link. We study rotationally symmetric rose links, those that can be rotated in a way that does not change their appearance or true form. Brown used link invariants to classify 3-component rose links; we categorize 4-component rose links using the HOMFLY polynomial.
Variations On The Euclidean Steiner Tree Problem And Algorithms, Jack Holby
Variations On The Euclidean Steiner Tree Problem And Algorithms, Jack Holby
Rose-Hulman Undergraduate Mathematics Journal
The Euclidean Steiner Tree Problem (ESTP) involves creating a minimal spanning network of a set of points by allowing the introduction of new points, called Steiner points. This paper discusses a variation on this classic problem by introducing a single Steiner line‚ whose weight is not counted in the resulting network, in addition to the Steiner points. For small sets, we arrive at a complete geometric solution. We discuss heuristic algorithms for solving this variation on larger sets. We believe that, in general, this problem is NP-hard.
Anallagmatic Curves And Inversion About The Unit Hyperbola, Stephanie Neas
Anallagmatic Curves And Inversion About The Unit Hyperbola, Stephanie Neas
Rose-Hulman Undergraduate Mathematics Journal
In this paper we investigate inversion about the unit circle from a complex perspective. Using complex rational functions we develop methods to construct curves which are self-inverse (anallagmatic). These methods are then translated to the split-complex numbers to investigate the theory of inversion about the unit hyperbola. The analog of the complex analytic techniques allow for the construction and study of anallagmatic curves about the unit hyperbola.
Giuga's Primality Conjecture For Number Fields, Jamaris Burns, Katherine Casey, Duncan Gichimu, Kerrek Stinson
Giuga's Primality Conjecture For Number Fields, Jamaris Burns, Katherine Casey, Duncan Gichimu, Kerrek Stinson
Rose-Hulman Undergraduate Mathematics Journal
Giuseppe Giuga conjectured in 1950 that a natural number n is prime if and only if it satisfies the congruence 1n-1+2n-1+ ... + (n-1)n-1 = -1 mod n. Progress in validating or disproving the conjecture has been minimal, with the most significant advance being the knowledge that a counter-example would need at least 19,907 digits. To gain new insights into Giuga's conjecture, we explore it in the broader context of number fields. We present a generalized version of the conjecture and prove generalizations of many of the major results …
Bounds On The Number Of Irreducible Semigroups Of Fixed Frobenius Number, Clarisse Bonnand, Reid Booth, Carina Kaainoa, Ethan Rooke
Bounds On The Number Of Irreducible Semigroups Of Fixed Frobenius Number, Clarisse Bonnand, Reid Booth, Carina Kaainoa, Ethan Rooke
Rose-Hulman Undergraduate Mathematics Journal
In 2011, Blanco and Rosales gave an algorithm for constructing a directed tree graph whose vertices are the irreducible numerical semigroups with a fixed Frobenius number. Laird and Martinez in 2013 studied the levels of these trees and conjectured what their heights might be. In this paper, we give an exposition on irreducible numerical semigroups. We also present some data supporting the conjecture of Laird and Martinez, and give a lower and upper bound on the number of irreducible numerical semigroups with fixed Frobenius number.
Complex Symmetry Of Truncated Composition Operators, Ruth Jansen, Rebecca K. Rousseau
Complex Symmetry Of Truncated Composition Operators, Ruth Jansen, Rebecca K. Rousseau
Rose-Hulman Undergraduate Mathematics Journal
We define a truncated composition operator on the spaces P_n of n-degree polynomials with complex coefficients. After doing so, we concern ourselves with the complex symmetry of such operators, that is, whether there is an orthonormal basis that gives them a symmetric matrix representation.
The Calculus Of Proportional 𝛼-Derivatives, Laura A. Legare, Grace K. Bryan
The Calculus Of Proportional 𝛼-Derivatives, Laura A. Legare, Grace K. Bryan
Rose-Hulman Undergraduate Mathematics Journal
We introduce a new proportional alpha-derivative with parameter alpha in [0,1], explore its calculus properties, and give several examples of our results. We begin with an introduction to our proportional alpha-derivative and some of its basic calculus properties. We next investigate the system of alpha-lines which make up our curved yet Euclidean geometry, as well as address traditional calculus concepts such as Rolle's Theorem and the Mean Value Theorem in terms of our alpha-derivative. We also introduce a new alpha-integral to be paired with our alpha-derivative, which leads to proofs of the Fundamental Theorem of Calculus Parts I and II, …
Computing The Autocorrelation Function For The Autoregressive Process, Omar Talib, Souleimane Cheikh Sidi Mohamed
Computing The Autocorrelation Function For The Autoregressive Process, Omar Talib, Souleimane Cheikh Sidi Mohamed
Rose-Hulman Undergraduate Mathematics Journal
In this document, we explain how complex integration theory can be used to compute the autocorrelation function for the autoregressive process. In particular, we use the deformation invariance theorem, and Cauchy’s residue theorem to reduce the problem of computing the autocorrelation function to the problem of computing residues of a particular function. The purpose of this paper is not only to illustrate a method by which one can derive the autocorrelation function of the autoregressive process, but also to demonstrate the applicability of complex analysis in statistical theory through simple examples.
Filling In The Gaps: Using Multiple Imputation To Improve Statistical Accuracy, Ashley Peterson, Emily Martin
Filling In The Gaps: Using Multiple Imputation To Improve Statistical Accuracy, Ashley Peterson, Emily Martin
Rose-Hulman Undergraduate Mathematics Journal
Missing data is a problem that many researchers face, particularly when using large surveys. Information is lost when analyzing a dataset with missing data, leading to less precise estimates. Multiple imputation (MI) using chained equations is a way to handle the missing value while using all available information given in the dataset to predict the missing values. In this study, we used data from the Survey of Midlife Development in the United States (MIDUS), a large national study of health and well-being that contains missing data. We created a complete dataset using MI. Following that we performed multiple regression analyses …
The Conformable Ratio Derivative, Evan Camrud
The Conformable Ratio Derivative, Evan Camrud
Rose-Hulman Undergraduate Mathematics Journal
This paper proposes a new definition for a conformable derivative. The strengths of the new derivative arise in its simplicity and similarity to fractional derivatives. An inverse derivative (integral) exists showing similar properties to fractional integrals. The derivative is scalable, and exhibits particular product and chain rules. When looked at as a function with a parameter, the ratio derivative K&alpha [f] of a function f converges pointwise to f as &alpha &rarr 0, and to the ordinary derivative as &alpha &rarr 1. The conformable derivative is nonlinear in nature, but a related operator behaves linearly within a power series …
Frog Model Wakeup Time On The Complete Graph, Nikki Carter, Brittany Dygert, Stephen Lacina, Collin Litterell, Austin Stromme, Andrew You
Frog Model Wakeup Time On The Complete Graph, Nikki Carter, Brittany Dygert, Stephen Lacina, Collin Litterell, Austin Stromme, Andrew You
Rose-Hulman Undergraduate Mathematics Journal
The frog model is a system of random walks where active particles set sleeping particles in motion. On the complete graph with n vertices it is equivalent to a well-understood rumor spreading model. We given an alternate and elementary proof that the wakeup time, that is, the expected time for every particle to be activated, is &Theta(log n). Additionally, we give an explicit distributional equation for the wakeup time as a mixture of geometric random variables.
Klein Link Multiplicity And Recursion, David Freund, Sarah Smith-Polderman
Klein Link Multiplicity And Recursion, David Freund, Sarah Smith-Polderman
Rose-Hulman Undergraduate Mathematics Journal
The (m,n)-Klein links are formed by altering the rectangular representation of an (m,n)-torus link. Using the braid representation of a (m,n)-Klein link, we generalize a previous braid word result and show that the (m, 2m)-Klein link can be expressed recursively. Applying braid permutations, we determine a formula for the number of components for an (m,n)-Klein link and classify the Klein links that are equivalent to knots.
New Upper Bounds On The Distance Domination Numbers Of Grids, Michael Farina, Armando Grez
New Upper Bounds On The Distance Domination Numbers Of Grids, Michael Farina, Armando Grez
Rose-Hulman Undergraduate Mathematics Journal
In his 1992 Ph.D. thesis Chang identified an efficient way to dominate m-by-n grid graphs and conjectured that his construction gives the most efficient dominating sets for relatively large grids. In 2011 Goncalves, Pinlou, Rao, and Thomasse proved Chang's conjecture, establishing a closed formula for the domination number of a grid. In March 2013, Fata, Smith and Sundaram established upper bounds for the k-distance domination numbers of grid graphs by generalizing Chang's construction of dominating sets to k-distance dominating sets. In this paper we use algebraic and geometric arguments to improve the upper bounds established by …
Perfect Zero-Divisor Graphs Of 𝕫ₙ, Bennett Smith
Perfect Zero-Divisor Graphs Of 𝕫ₙ, Bennett Smith
Rose-Hulman Undergraduate Mathematics Journal
The zero-divisor graph of a ring R, denoted &Gamma(R), is the graph whose vertex set is the collection of zero-divisors in R, with edges between two distinct vertices u and v if and only if uv=0. In this paper, we restrict our attention to &Gamma(Zn), the zero-divisor graph of the ring of integers modulo n. Specifically, we determine all values of n for which &Gamma(Zn) is perfect. Our classification depends on the prime factorization of n, with relatively simple prime factorizations corresponding to perfect graphs. In fact, …
An Algebraic Characterization Of Highly Connected 2𝑛-Manifolds, Shamuel Auyeung, Joshua Ruiter, Daiwei Zhang
An Algebraic Characterization Of Highly Connected 2𝑛-Manifolds, Shamuel Auyeung, Joshua Ruiter, Daiwei Zhang
Rose-Hulman Undergraduate Mathematics Journal
All surfaces, up to homeomorphism, can be formed by gluing the edges of a polygon. This process is generalized into the idea of a (n,2n)-cell complex: forming a space by attaching a (2n-1)-sphere into a wedge sum of n-spheres. In this paper, we classify oriented, (n-1)-connected, compact and closed 2n-manifolds up to homotopy by treating them as (n,2n)-cell complexes. To simplify the calculation, we create a basis called the Hilton basis for the homotopy class of the attaching map of the (n,2n …
Recursion In Topological Invariants Of Twist And Rational Knot Exteriors, Christian Gorski
Recursion In Topological Invariants Of Twist And Rational Knot Exteriors, Christian Gorski
Rose-Hulman Undergraduate Mathematics Journal
No abstract provided.
Symmetries Of Cairo-Prismatic Tilings, John Berry, Matthew Dannenberg, Jason Liang, Yingyi Zeng
Symmetries Of Cairo-Prismatic Tilings, John Berry, Matthew Dannenberg, Jason Liang, Yingyi Zeng
Rose-Hulman Undergraduate Mathematics Journal
We study and catalog isoperimetric, planar tilings by unit-area Cairo and Prismatic pentagons. In particular, in counterpoint to the five wallpaper symmetry groups known to occur in Cairo-Prismatic tilings, we show that the five with order three rotational symmetry cannot occur.
Sums Of Reciprocals Of Irreducible Polynomials Over Finite Fields, Spencer Nelson
Sums Of Reciprocals Of Irreducible Polynomials Over Finite Fields, Spencer Nelson
Rose-Hulman Undergraduate Mathematics Journal
We will revisit a theorem first proved by L. Carlitz in 1935 in which he provided an intriguing formula for sums involving the reciprocals of all monic polynomials of a given degree over a finite field of a specified order. Expanding on this result, we will consider the equally curious case where instead of adding reciprocals all monic polynomials of a given degree, we only consider adding reciprocals of those that are irreducible.
A Practical Study Of Longitudinal Reference Based Compressed Sensing For Mri, Samuel Birns, Bohyun Kim, Stephanie Ku, Kevin Stangl
A Practical Study Of Longitudinal Reference Based Compressed Sensing For Mri, Samuel Birns, Bohyun Kim, Stephanie Ku, Kevin Stangl
Rose-Hulman Undergraduate Mathematics Journal
Compressed sensing (CS) is a new signal acquisition paradigm that enables the reconstruction of signals and images from a low number of samples. A particularly exciting application of CS is Magnetic Resonance Imaging (MRI), where CS significantly speeds up scan time by requiring far fewer measurements than standard MRI techniques. Such a reduction in sampling time leads to less power consumption, less need for patient sedation, and more accurate images. This accuracy increase is especially pronounced in pediatric MRI where patients have trouble being still for long scan periods. Although such gains are already significant, even further improvements can be …