Rose-Hulman Undergraduate Mathematics JournalCopyright (c) 2021 Rose-Hulman Institute of Technology All rights reserved.
https://scholar.rose-hulman.edu/rhumj
Recent documents in Rose-Hulman Undergraduate Mathematics Journalen-usThu, 30 Dec 2021 01:37:10 PST3600Decomposable Model Spaces and a Topological Approach to Curvature
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/8
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/8Tue, 28 Dec 2021 13:29:47 PST
This research investigates a model space invariant known as k-plane constant vector curvature, traditionally studied when k=2, and introduces a new invariant, (m,k)-plane constant vector curvature. We prove that the sets of k-plane and (m,k)-plane constant vector curvature values are connected, compact subsets of the real numbers and establish several relationships between the curvature values of a decomposable model space and its component spaces. We also prove that every decomposable model space with a positive-definite inner product has k-plane constant vector curvature for some integer k>1. In two examples, we provide the first instance of a model space with (m,k)-plane constant vector curvature and leverage our theorems to efficiently calculate the k-plane constant vector curvature values of a decomposable model space. This research further characterizes model spaces by assigning new basis-independent values to its various subspaces and allows us to easily construct model spaces with prescribed curvature values.
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Kevin M. TullyWinning Strategy For Multiplayer And Multialliance Geometric Game
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/7
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/7Tue, 28 Dec 2021 12:39:59 PST
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 calculate the upper bound and lower bound of the length of Geometric Game. We are also able to prove some interesting results in terms of the winning strategy of 2-player games, and some special cases of multiplayer games and multialliance games.
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Jingkai YeHurwitz Actions on Reflection Factorizations in Complex Reflection Group G6
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/6
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/6Tue, 28 Dec 2021 12:03:40 PST
We show that in the complex reflection group G_{6}, reflection factorizations of a Coxeter element that have the same length and multiset of conjugacy classes are in the same Hurwitz orbit. This confirms one case of a conjecture of Lewis and Reiner.
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Gaurav Gawankar et al.Lie-Derivations of Three-Dimensional Non-Lie Leibniz algebras
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/5
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/5Tue, 28 Dec 2021 11:40:18 PST
The concept of Lie-derivation was recently introduced as a generalization of the notion of derivations for non-Lie Leibniz algebras. In this project, we determine the Lie algebras of Lie-derivations of all three-dimensional non-Lie Leibniz algebras. As a result of our calculations, we make conjectures on the basis of the Lie algebra of derivations of Lie-solvable non-Lie Leibniz algebras.
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Emily H. BelangerThe Optimal Double Bubble for Density $r^p$
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/4
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/4Tue, 28 Dec 2021 09:40:05 PST
In 2008 Reichardt proved that the optimal Euclidean double bubble---the least-perimeter way to enclose and separate two given volumes---is three spherical caps meeting along a sphere at 120 degrees. We consider R^{n}with densityr^{p}, joining the surge of research on manifolds with density after their appearance in Perelman's 2006 proof of the Poincaré Conjecture. Boyer et al. proved that the best single bubble is a sphere through the origin. We conjecture that the best double bubble is the Euclidean solution with the singular sphere passing through the origin, for which we have verified equilibrium (first variation or ``first derivative'' zero). To prove the exterior of the minimizer connected, it would suffice to show that least perimeter is increasing as a function of the prescribed areas. We give the first direct proof of such monotonicity in the Euclidean plane. Such arguments were important in the 2002 Annals proof of the double bubble in Euclidean 3-space.
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Jack Hirsch et al.A Proof of a Generalization of Niven's Theorem Using Algebraic Number Theory
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/3
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/3Fri, 24 Dec 2021 06:53:29 PST
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 then show for which rational multiples of π the squares of the sine, cosine, and tangent functions are rational, providing a generalized form of Niven’s theorem. We end with a discussion of a few related combinatorial identities.
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Caroline NunnA Mathematical Model Regarding Change in Preferences of Refugee Settlements
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/2
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/2Fri, 17 Dec 2021 15:33:36 PST
Where cultures meet, there is bound to be conflict to some extent. This especially applies in the case of refugees grouped together when seeking asylum, with different styles of life, socialization, and conflict resolution meeting in one place. This paper focuses specially on three types of conflict resolution(negotiation, mediation, and arbitration) and constructs a differential equation model to study how the interactions between populations cause the number of people following each resolution method to shift. It was found that when there is no existing outside authority or environmental bias towards a resolution method, the method with the greatest number of followers will also be the one to take over the final population. However, in the presence of an outside force promoting or discouraging certain methods, although some groups will be given advantages over others, the final outcome is also still partially under the influence of the initial population. Outside of stable equilibria representing situations where one method ends up taking over the entire population, we also found certain unstable equilibria that carry key information about the basins of attraction of the stable equilibria.
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Raaghav Malik et al.Convergence Properties of Solutions of a Length-Structured Density-Dependent Model for Fish
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/1
https://scholar.rose-hulman.edu/rhumj/vol22/iss2/1Fri, 17 Dec 2021 15:20:05 PST
We numerically study solutions to a length-structured matrix model for fish populations in which the probability that a fish grows into the next length class is a decreasing nonlinear function of the total biomass of the population. We make conjectures about the convergence properties of solutions to this equation, and give numerical simulations which support these conjectures. We also study the distribution of biomass in the different age classes as a function of the total biomass.
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Geigh ZollicofferIrreducibility and Galois Groups of Random Polynomials
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/10
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/10Tue, 06 Jul 2021 15:49:43 PDT
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.
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Hanson Hao et al.The Cost of a Positive integer
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/9
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/9Tue, 06 Jul 2021 15:49:35 PDT
The cost C_{S} of a positive integer m relative to a set S of binary operations is defined to be the lesser of m and the minimum of C_{S}(a) + C_{S}(b) where a and b are positive integers and m = a ◦ b for some binary operation ◦ ∈ S. The cost of a positive integer measures the complexity of expressing m using the operations in S, and is intended to be a simplification of Kolmogrov compelexity. We show that, unlike Kolmolgorov complexity, C_{S} is computable for any finite set S of computable binary operations. We then study C_{S} for various choices of S. If S = {∗} and m > 1 then C_{S}(m) is the sum of the prime factors of m (with repetition). A positive integer m is defined to be completely multiplicative if C_{{+,∗}} (m) = C_{{∗}} (m). We show that if m is completely multiplicative then m is of the form 2a ∗ 3b ∗ 5c. The converse is an open question. If S = {+, ∗} we prove logarithmic upper and lower bounds implying that C_{S}(m) ∈ O(log m). We use the notations +1 and −1 to denote the binary operations a + b and a − b restricted to the cases where b = 1, and we conjecture that C_{{+,∗,−}} = C_{{+1,∗,−1}} . However, we give an example to show that C_{{+,∧}}≠ C{+1,∧} where ∧ represents exponentiation. Several interesting open questions are also discussed.
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Maxwell NorfolkLebesgue Measure Preserving Thompson Monoid and Its Properties of Decomposition and Generators
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/8
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/8Tue, 06 Jul 2021 15:49:28 PDT
This paper defines the Lebesgue measure preserving Thompson monoid, denoted by G, which is modeled on the Thompson group F except that the elements of G preserve the Lebesgue measure and can be non-invertible. The paper shows that any element of the monoid G is the composition of a finite number of basic elements of the monoid G and the generators of the Thompson group F. However, unlike the Thompson group F, the monoid G is not finitely generated. The paper then defines equivalence classes of the monoid G, use them to construct a monoid H that is finitely generated, and shows that the union of the elements of the monoid H is a set of equivalence classes, the union of which is G.
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William LiRepeat Length Of Patterns On Weaving Products
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/7
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/7Tue, 06 Jul 2021 15:49:20 PDT
On weaving products such as fabrics and silk, people use interlacing strands to create artistic patterns. Repeated patterns form aesthetically pleasing products. This research is a mathematical modeling of weaving products in the real world by using cellular automata. The research is conducted by observing the evolution of the model to better understand products in the real world. Specifically, this research focuses on the repeat length of a weaving pattern given the rule of generating it and the configuration of the starting row. Previous studies have shown the range of the repeat length in specific situations. This paper will generalize the precise repeat length in one of those situations using mathematical proofs. In the future, the goal is to further generalize the findings to more situations.
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Zhuochen LiuDisjointness of Linear Fractional Actions on Serre Trees
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/6
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/6Tue, 06 Jul 2021 15:49:13 PDT
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 in specific special cases.
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Henry W. TalbottNew Results on Subtractive Magic Graphs
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/5
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/5Tue, 06 Jul 2021 15:49:06 PDT
For any edge xy in a directed graph, the subtractive edge-weight is the sum of the label of xy and the label of y minus the label of x. Similarly, for any vertex z in a directed graph, the subtractive vertex-weight of z is the sum of the label of z and all edges directed into z and all the labels of edges that are directed away from z. A subtractive magic graph has every subtractive edge and vertex weight equal to some constant k. In this paper, we will discuss variations of subtractive magic labelings on directed graphs.
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Matthew J. Ko et al.Directed Graphs of the Finite Tropical Semiring
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/4
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/4Tue, 06 Jul 2021 15:48:58 PDT
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.
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Caden G. ZonnefeldThe Degeneration of the Hilbert Metric on Ideal Pants and its Application to Entropy
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/3
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/3Tue, 06 Jul 2021 15:48:49 PDT
Entropy is a single value that captures the complexity of a group action on a metric space. We are interested in the entropies of a family of ideal pants groups $\Gamma_T$, represented by projective reflection matrices depending on a real parameter $T > 0$. These groups act on convex sets $\Omega_{\Gamma_T}$ which form a metric space with the Hilbert metric. It is known that entropy of $\Gamma_T$ takes values in the interval $\left(\frac{1}{2},1\right]$; however, it has not been proven whether $\frac{1}{2}$ is the sharp lower bound. Using Python programming, we generate approximations of tilings of the convex set in the projective plane and estimate the entropies of these groups with respect to the Hilbert metric. We prove a theorem that, along with the images and data produced by our code, suggests that the lower bound is indeed sharp. This theorem regards the degeneration of the Hilbert metric on the convex set $\Omega_{\Gamma_T}.
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Marianne DeBrito et al.An Introduction to Fractal Analysis
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/2
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/2Tue, 06 Jul 2021 15:48:41 PDT
Classical analysis is not able to treat functions whose domain is fractal. We present an introduction to analysis on a particular class of fractals known as post-critically finite (PCF) self-similar sets that is suitable for the undergraduate reader. We develop discrete approximations of PCF self-similar sets, and construct discrete Dirichlet forms and corresponding discrete Laplacians that both preserve self-similarity and are compatible with a notion of harmonic functions that is analogous to a classical setting. By taking the limit of these discrete Laplacians, we construct continuous Laplacians on PCF self-similar sets. With respect to this continuous Laplacian, we also construct a Green's function that can be used to find solutions to the Dirichlet problem for Poisson's equation.
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Lucas YongIrrational Philosophy? Kronecker's Constructive Philosophy and Finding the Real Roots of a Polynomial
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/1
https://scholar.rose-hulman.edu/rhumj/vol22/iss1/1Tue, 06 Jul 2021 15:48:33 PDT
The prominent mathematician Leopold Kronecker (1823 – 1891) is often relegated to footnotes and mainly remembered for his strict philosophical position on the foundation of mathematics. He held that only the natural numbers are intuitive, thus the only basis for all mathematical objects. In fact, Kronecker developed a complete school of thought on mathematical foundations and wrote many significant algebraic works, but his enigmatic writing style led to his historical marginalization. In 1887, Kronecker published an extended version of his paper, “On the Concept of Number,” translated into English in 2010 for the first time by Edward T. Dean, who confirms that Kronecker is “notoriously difficult to read.” In his paper, Kronecker proves that a so-called “algebraic number”, meaning any root of a polynomial with integer coefficients, can be isolated from the other roots of that polynomial, as Dean says, “using solely talk of natural numbers.” To ease the reader’s comprehension of Kronecker’s prose, here we explicate in detail the argument contained in that paper.
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Richard B. SchneiderNumerical Integration Through Concavity Analysis
https://scholar.rose-hulman.edu/rhumj/vol21/iss2/7
https://scholar.rose-hulman.edu/rhumj/vol21/iss2/7Thu, 07 Jan 2021 14:56:27 PST
We introduce a relationship between the concavity of a C2 func- tion and the area bounded by its graph and secant line. We utilize this relationship to develop a method of numerical integration. We then bound the error of the approximation, and compare to known methods, finding an improvement in error bound over methods of comparable computational complexity.
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Daniel J. PietzExponents of Jacobians of Graphs and Regular Matroids
https://scholar.rose-hulman.edu/rhumj/vol21/iss2/6
https://scholar.rose-hulman.edu/rhumj/vol21/iss2/6Thu, 07 Jan 2021 14:56:13 PST
Let G be a finite undirected multigraph with no self-loops. The Jacobian Jac (G) is a finite abelian group associated with G whose cardinality is equal to the number of spanning trees of G. There are only a finite number of biconnected graphs G such that the exponent of Jac (G) equals 2 or 3. The definition of a Jacobian can also be extended to regular matroids as a generalization of graphs. We prove that there are finitely many connected regular matroids M such that Jac (M) has exponent 2 and characterize all such matroids.
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Hahn Lheem et al.