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Recent documents in Rose-Hulman Scholaren-usWed, 06 Dec 2023 01:39:23 PST3600Eigenvalue Algorithm for Hausdorff Dimension on Complex Kleinian Groups
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/12
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/12Mon, 27 Nov 2023 13:55:44 PST
In this manuscript, we present computational results approximating the Hausdorff dimension for the limit sets of complex Kleinian groups. We apply McMullen's eigenvalue algorithm \cite{mcmullen} in symmetric and non-symmetric examples of complex Kleinian groups, arising in both real and complex hyperbolic space. Numerical results are compared with asymptotic estimates in each case. Python code used to obtain all results and figures can be found at \url{https://github.com/WXML-HausDim/WXML-project}, all of which took only minutes to run on a personal computer.
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Jacob Linden et al.Further Generalizations of Happy Numbers
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/11
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/11Mon, 30 Oct 2023 09:36:55 PDT
A positive integer n is defined to be happy if iteration of the function taking the sum of the squares of the digits of n eventually reaches 1. In this paper we generalize the concept of happy numbers in several ways. First we confirm known results of Grundman and Teeple and establish further results extending the known structure of happy numbers to higher powers. Then we construct a similar function expanding the definition of happy numbers to negative integers. Working with this function, we prove a range of results paralleling those already proven for traditional and generalized happy numbers. Finally, we consider a variety of special cases, in which the existence of certain fixed points and cycles of infinite families of generalized happy functions can be proven.
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E. Simonton WilliamsDivisibility Probabilities for Products of Randomly Chosen Integers
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/10
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/10Fri, 27 Oct 2023 10:03:00 PDT
We find a formula for the probability that the product of n positive integers, chosen at random, is divisible by some integer d. We do this via an inductive application of the Chinese Remainder Theorem, generating functions, and several other combinatorial arguments. Additionally, we apply this formula to find a unique, but slow, probabilistic primality test.
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Noah Y. FineElliptic triangles which are congruent to their polar triangles
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/9
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/9Fri, 27 Oct 2023 09:36:46 PDT
We prove that an elliptic triangle is congruent to its polar triangle if and only if six specific Wallace-Simson lines of the triangle are concurrent. (If a point projected onto a triangle has the three feet of its projections collinear, that line is called a Wallace-Simson line.) These six lines would be concurrent at the orthocenter. The six lines come from projecting a vertex of either triangle onto the given triangle. We describe how to construct such triangles and a dozen Wallace-Simson lines.
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Jarrad S. Epkey et al.Structure of a Total Independent Set
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/8
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/8Thu, 26 Oct 2023 08:21:05 PDT
Let $G$ be a simple, connected and finite graph with order $n$. Denote the independence number, edge independence number and total independence number by $\alpha(G), \alpha'(G)$ and $\alpha''(G)$ respectively. This paper establishes an upper bound for $\alpha''(G)$ in terms of $\alpha(G)$, $\alpha'(G)$ and $n$. We also describe the possible structures for a total independent set containing a given number of elements.
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Lewis StantonA Model for the Multi-Virus Contact Process
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/7
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/7Fri, 20 Oct 2023 15:20:57 PDT
We study one specific version of the contact process on a graph. Here, we allow multiple infections carried by the nodes and include a probability of removing nodes in a graph. The removal probability is purely determined by the number of infections the node carries at the moment when it gets another infection. In this paper, we show that on any finite graph, any positive value of infection rate $\lambda$ will result in the death of the process almost surely. In the case of $d$-regular infinite trees, We also give a lower bound on the infection rate in order for the process to survive, and an upper bound for the process to die out.
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Xu HuangApplying Hallgren’s algorithm for solving Pell’s equation to finding the irrational slope of the launch of a billiard ball
https://scholar.rose-hulman.edu/math_mstr/184
https://scholar.rose-hulman.edu/math_mstr/184Mon, 02 Oct 2023 12:51:22 PDT
This thesis is an exploration of Quantum Computing applied to Pell’s equation in an attempt to find solutions to the Billiard Ball Problem. Pell’s equation is a Diophantine equation in the form of x^{2} − ny^{2} = 1, where n is a given positive nonsquare integer, and integer solutions are sought for x and y. We will be applying Hallgren’s algorithm for finding irrational periods in functions, in the context of billiard balls and their movement on a friction-less unit square billiard table. Our central research question has been the following: Given the cutting sequence of the billiard ball’s movement, can you find the irrational slope value in which the billiard ball was put in motion?
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Sangheon ChoiHuman and Technical Factors in the Adoption of Quantum Cryptographic Algorithms
https://scholar.rose-hulman.edu/math_mstr/183
https://scholar.rose-hulman.edu/math_mstr/183Mon, 02 Oct 2023 12:51:18 PDT
The purpose of this research is to understand what factors would cause users to choose quantum key distribution (QKD) over other methods of cryptography. An Advanced Encryption Standard (AES) key can be exchanged through communication using the Rivest, Shamir, Adleman (RSA) cryptographic algorithm, QKD, or post-quantum cryptography (PQC). QKD relies on quantum physics where RSA and PQC use complex mathematics to encrypt data. The BB84 quantum cryptographic protocol involves communication over a quantum channel and a public channel. The quantum channel can be technically attacked by beamsplitting or intercept/resend. QKD, like other forms of cryptography, is vulnerable to social attacks such as industrial espionage. QKD products can transmit over maximum distances ranging from 40 km up to 150 km with key rates as low as 1.4 kb/s up to at least 300 kb/s. A survey and focus group discussion with a defense contracting company revealed that while nobody fully trusts current security systems, they are more concerned about social engineering attacks before attacks on cryptography. The company is not interested in implementing QKD unless the range capabilities are improved or there is regulation requiring them to use it.
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Alyssa Pinkstonk-Distinct Lattice Paths
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/6
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/6Fri, 29 Sep 2023 09:01:26 PDT
Lattice paths can be used to model scheduling and routing problems, and, therefore, identifying maximum sets of k-distinct paths is of general interest. We extend the work previously done by Gillman et. al. to determine the order of a maximum set of k-distinct lattice paths. In particular, we disprove a conjecture by Gillman that a greedy algorithm gives this maximum order and also refine an upper bound given by Brewer et. al. We illustrate that brute force is an inefficient method to determine the maximum order, as it has time complexity O(n^{k}).
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Eric J. Yager et al.Utilizing graph thickness heuristics on the Earth-moon Problem
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/5
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/5Thu, 28 Sep 2023 10:31:20 PDT
This paper utilizes heuristic algorithms for determining graph thickness in order to attempt to find a 10-chromatic thickness-2 graph. Doing so would eliminate 9 colors as a potential solution to the Earth-moon Problem. An empirical analysis of the algorithms made by the author are provided. Additionally, the paper lists various graphs that may or nearly have a thickness of 2, which may be solutions if one can find two planar subgraphs that partition all of the graph’s edges.
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Robert C. WeaverNumber of Regions Created by Random Chords in the Circle
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/4
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/4Thu, 07 Sep 2023 09:56:43 PDT
In this paper we discuss the number of regions in a unit circle after drawing n i.i.d. random chords in the circle according to a particular family of distribution. We find that as n goes to infinity, the distribution of the number of regions, properly shifted and scaled, converges to the standard normal distribution and the error can be bounded by Stein's method for proving Central Limit Theorem.
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Shi FengThe Mean Sum of Squared Linking Numbers of Random Piecewise-Linear Embeddings of $K_n$
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/3
https://scholar.rose-hulman.edu/rhumj/vol24/iss2/3Thu, 07 Sep 2023 08:35:48 PDT
DNA and other polymer chains in confined spaces behave like closed loops. Arsuaga et al. \cite{AB} introduced the uniform random polygon model in order to better understand such loops in confined spaces using probabilistic and knot theoretical techniques, giving some classification on the mean squared linking number of such loops. Flapan and Kozai \cite{flapan2016linking} extended these techniques to find the mean sum of squared linking numbers for random linear embeddings of complete graphs $K_n$ and found it to have order $\Theta(n(n!))$. We further these ideas by inspecting random piecewise-linear embeddings of complete graphs and give introductory-level summaries of the ideas throughout. In particular, we give a model of random piecewise-linear embeddings of complete graphs where the number of line segments between vertices is given by a random variable. We find further that in our model of the random piecewise-linear embeddings, the order of the expected sum of squared linking numbers is still $\Theta(n (n!))$.
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Yasmin Aguillon et al.Reversibility of Stranded Cellular Automata
https://scholar.rose-hulman.edu/math_mstr/182
https://scholar.rose-hulman.edu/math_mstr/182Tue, 05 Sep 2023 09:00:13 PDT
Cellular automata, such as the Stranded Cellular Automaton (SCA) model created by Joshua and Lana Holden, can be used to model weaving patterns. Similar models can be constructed to model macrame patterns, where strands are knotted together. If a rule is injective, then it is reversible. If a rule is surjective, then every configuration has at least one predecessor. In this paper, we will discuss the injectivity and surjectivity of several new SCA models in order to find reversible rules. We will also analyze the number of configurations with no predecessors and the number of configurations that map to the same successor.
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Allyn LoydDesign of a Resonant Optical Cavity for Imaging Magneto-Optically Active Thin Film Samples
https://scholar.rose-hulman.edu/optics_grad_theses/31
https://scholar.rose-hulman.edu/optics_grad_theses/31Thu, 31 Aug 2023 07:01:10 PDT
This document describes the design and fabrication of an optical resonator system to investigate magneto-optic properties of thin film samples. This system uses an open-air optical resonator to enable photons to make multiple passes through each thin film and thus increase the magnitude of the Faraday rotation that each sample imposes onto the light that exits the system. This system promises many future experiments to study the magneto-optic properties of thin film and nano-particle samples. Using an optical resonator to enhance Faraday rotation should enable an improved signal-to-noise ratio in taking measurements and images with a photodetector.
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Cody Robert BrelageSodium Alginate Toughening of Gelatin Hydrogels
https://scholar.rose-hulman.edu/chemical_engineering_fac/199
https://scholar.rose-hulman.edu/chemical_engineering_fac/199Wed, 23 Aug 2023 05:30:47 PDTAdam Nolte et al.Wrinkling-To-Delamination Transition In Thin Polymer Films On Compliant Substrates
https://scholar.rose-hulman.edu/chemical_engineering_fac/198
https://scholar.rose-hulman.edu/chemical_engineering_fac/198Fri, 18 Aug 2023 11:37:36 PDTAdam Nolte et al.Workshop: Taking It To The Next Level... Game-Based Learning In Engineering Education
https://scholar.rose-hulman.edu/chemical_engineering_fac/197
https://scholar.rose-hulman.edu/chemical_engineering_fac/197Fri, 18 Aug 2023 11:37:35 PDTDaniel D. Anastasio et al.Work in Progress: Developing a Multi-dimensional Method for Student Assessment in Chemical Engineering Laboratory Courses
https://scholar.rose-hulman.edu/chemical_engineering_fac/196
https://scholar.rose-hulman.edu/chemical_engineering_fac/196Fri, 18 Aug 2023 11:37:34 PDTDaniel Anastasio et al.Work In Progress: Content Validation For An Engineering Process Safety Decision-Making Instrument (EPSRI)
https://scholar.rose-hulman.edu/chemical_engineering_fac/195
https://scholar.rose-hulman.edu/chemical_engineering_fac/195Fri, 18 Aug 2023 11:37:33 PDTDaniel Anastasio et al.Work In Progress: Career Ready... Or Not? A Career-Readiness Activity For Senior Chemical Engineering Students
https://scholar.rose-hulman.edu/chemical_engineering_fac/194
https://scholar.rose-hulman.edu/chemical_engineering_fac/194Fri, 18 Aug 2023 11:37:32 PDTD. D. Anastasio et al.