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Relative Equilibria Of Pinwheel Point Mass Systems In A Planar Gravitational Field, Ritwik Gaur Sep 2024

Relative Equilibria Of Pinwheel Point Mass Systems In A Planar Gravitational Field, Ritwik Gaur

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we consider a planar case of the full two-body problem (F2BP) where one body is a pinwheel (four point masses connected via two perpendicular massless rods) and the other is a point mass. Relative equilibria (RE) are defined to be ordered pairs (r, θ) such that there exists a rotating reference frame under which the two bodies are in equilibrium when the distance between the point mass and the center of the pinwheel is r and the angle of the pinwheel within its orbit is θ. We prove that relative equilibria exist for …

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Graph And Group Theoretic Properties Of The Soma Cube And Somap, Kyle Asbury, Ben Glancy Aug 2024

Graph And Group Theoretic Properties Of The Soma Cube And Somap, Kyle Asbury, Ben Glancy

Mathematical Sciences Technical Reports (MSTR)

The SOMA Cube is a puzzle toy in which seven irregularly shaped blocks must be fit together to build a cube. There are 240 distinct solutions to the SOMA Cube. One rainy afternoon, Conway and Guy created a graph of all the solutions by manually building each solution. They called their graph the SOMAP. We studied how the geometric structure of the SOMA Cube pieces informs the graph theoretic properties of the SOMAP, such as subgraphs that can or cannot appear and vertex centrality. We have also used permutation group theory to decipher notation used by Knuth in previous work …

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A Machine Learning Based Approach For The Identification Of Fake Bills, Tianyang Lu, Hongyang Pang Aug 2024

A Machine Learning Based Approach For The Identification Of Fake Bills, Tianyang Lu, Hongyang Pang

Rose-Hulman Undergraduate Mathematics Journal

Fake or counterfeiting currency, which has been around as long as money has existed, is a major economic problem. Since the US dollar is the most popular form of currency globally, it is the most popular currency to counterfeit. The United States Department of Treasury estimates that between $70 million and $200 million in fake bills are in circulation. The Federal Reserve Bank uses special banknote processing systems to count each bill deposited by the bank and examine them for the possibility of counterfeits. These machines have sensors designed to detect general quality of the bills, including paper type, quality …

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Modeling The Effect Of Human Behavior On Disease Transmission, Katie Yan Jul 2024

Modeling The Effect Of Human Behavior On Disease Transmission, Katie Yan

Rose-Hulman Undergraduate Mathematics Journal

Many infectious disease models build upon the classic Susceptible-Infected-Recovered (SIR) model, a compartmental system that is used to simulate disease transmission in a population. The SIR model focuses on the transmission of disease but rarely includes behavioral or informational components that explore how disease perception influences transmission. In this paper, we propose a six-compartment behavioral SIR model that further segments the classic SIR system based on knowledge of information about the disease, and we explore how sharing information affects disease transmission. We designate two states as aware and unaware based on whether the relevant information is known by the population. …

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Uniformly Distributing Points On A Sphere, Flavio Arrigoni Jul 2024

Uniformly Distributing Points On A Sphere, Flavio Arrigoni

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we are going to present and discuss different procedures for distributing points on a sphere's surface. Furthermore, we will assess their quality with three different distribution tests. The MATHEMATICA package that we created for testing and plotting the points is publicly available.

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Modeling Virus Diffusion On Social Media Networks With The Smirq Model, Justin Browning, Arnav Mazumder, Gowri Nanda Jul 2024

Modeling Virus Diffusion On Social Media Networks With The Smirq Model, Justin Browning, Arnav Mazumder, Gowri Nanda

Rose-Hulman Undergraduate Mathematics Journal

As social networking services become more complex and widespread, users become increasingly susceptible to becoming infected with malware and risk their data being compromised. In the United States, it costs the government billions of dollars annually to handle malware attacks. Additionally, computer viruses can be spread through schools, businesses, and individuals’ personal devices and accounts. Malware affecting larger groups of people causes problems with privacy, personal files, and financial security. Thus, we developed the probabilistic SMIRQ (pSMIRQ) model that shows how a virus spreads through a generated network as a way to track and prevent future viruses. Our model is …

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Approximating Equilibria In Restricted Games, Jack W. Doyle Jul 2024

Approximating Equilibria In Restricted Games, Jack W. Doyle

Rose-Hulman Undergraduate Mathematics Journal

We consider optimal play in restricted games with linear constraints, and use ϵ-equilibria to find near-equilibrium states in these games. We present three mathematical optimization formulations -- a mixed-integer linear program (MILP), a quadratic program with linear constraints (QP), and a quadratically constrained program (QCP) -- to both approximate and identify these states. The MILP has a short runtime relative to the QP and QCP for large games (a factor 100 faster for |S|=9) and exhibits linear growth in run time, but provides only relatively weak upper bound. The QP and QCP provide a tight bound and the precise value …

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Link Element Design For A Landing-Gear Mechanism In A Statics And Mechanics Of Materials Course, Amir H. Danesh-Yazdi, Aimee Cloutier, Sean Moseley Jun 2024

Link Element Design For A Landing-Gear Mechanism In A Statics And Mechanics Of Materials Course, Amir H. Danesh-Yazdi, Aimee Cloutier, Sean Moseley

Faculty Publications - Mechanical Engineering

In this work, we describe a project involving a link element design for a landing gear mechanism as part of our Statics and Mechanics of Materials I course. During this project, students are asked to design a safe and lightweight linkage that will allow the landing gear to safely and slowly retract from a vertical position to a nearly horizontal one without breaking or stretching more than 10% of its original length. This project is introduced at the halfway point of the 10-week term, at which point students are familiar with the 2D equilibrium of rigid bodies and the concepts …

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Storage And Interaction Diagrams: Extending The Diagrammatic Framework Of Kinetic And Free-Body Diagrams To Other Conservation And Accounting Principles, Amir H. Danesh-Yazdi Jun 2024

Storage And Interaction Diagrams: Extending The Diagrammatic Framework Of Kinetic And Free-Body Diagrams To Other Conservation And Accounting Principles, Amir H. Danesh-Yazdi

Faculty Publications - Mechanical Engineering

After defining a system for analysis, a rigorous process is taught to students in their Statics and Dynamics courses on how to draw proper kinetic, free-body, and impulse-momentum diagrams. While numerous techniques and mnemonics have been mentioned in literature, any experienced instructor can tell a correct free-body diagram apart from an incorrect one. Unfortunately, this is not the case when considering scalar properties such as mass, energy, exergy, and entropy. Different fluid mechanics and thermodynamics texts have treated the diagrammatic representation of these properties either very poorly, or in the case of the latter two, not at all. In this …

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Are All Weakly Convex And Decomposable Polyhedral Surfaces Infinitesimally Rigid?, Jilly Kevo Jun 2024

Are All Weakly Convex And Decomposable Polyhedral Surfaces Infinitesimally Rigid?, Jilly Kevo

Rose-Hulman Undergraduate Mathematics Journal

It is conjectured that all decomposable (that is, interior can be triangulated without adding new vertices) polyhedra with vertices in convex position are infinitesimally rigid and only recently has it been shown that this is indeed true under an additional assumption of codecomposability (that is, the interior of the difference between the convex hull and the polyhedron itself can be triangulated without adding new vertices). One major set of tools for studying infinitesimal rigidity happens to be the (negative) Hessian MT of the discrete Hilbert-Einstein functional. Besides its theoretical importance, it provides the necessary machinery to tackle the problem …

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Operation Of A Packed Bed Reactor For Hyrdogenation, Austin Joseph Urfer Jun 2024

Operation Of A Packed Bed Reactor For Hyrdogenation, Austin Joseph Urfer

Graduate Theses - Chemical Engineering

The primary goals of this research were to develop an efficient standard operating procedure for undergraduate students and researchers performing reactions within a packed bed reactor and to determine the optimal operating conditions for phenethylamine formation from mandelonitrile hydrogenation. With the current laboratory packed bed reactor system, undergraduates can conduct gas-liquid phase reactions at a maximum temperature of 95℃ and a maximum pressure of 100 psi. The versatile system setup can be modified for unit operations laboratory instruction or future researchers analyzing heterogeneous catalysis reactions. The developed standard operating procedure provides detailed instructions on packing the reactor with a solid …

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Engineering Silicon Metalenses In Wide Field Revers-Telephoto Optical Systems For Long-Wave Infrared Thermal Imaging, Alfred Anson Moore May 2024

Engineering Silicon Metalenses In Wide Field Revers-Telephoto Optical Systems For Long-Wave Infrared Thermal Imaging, Alfred Anson Moore

Graduate Theses

A novel optical system designed to image light emitted from a CO 2 laser (10.6 μm) is presented here. This system utilizes a positive conventional lens and a negative all silicon metalens. Demonstrated here is the ability of metalenses to replace a combination of optical elements to produce more compact and light-weight optical systems. It also shows how metalenses can replace comparatively expensive custom-manufactured lenses. Several systems that utilize meta-surfaces are explored, including positive and negative metalenses, a meta-corrector, a metalens doublet imaging system that corrects for 0 th -order images overlapping 1 st -order images, and several telephoto optical …

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Modeling An Infection Outbreak With Quarantine: The Sibkr Model, Mikenna Dew, Amanda Langosch, Theadora Baker-Wallerstein Apr 2024

Modeling An Infection Outbreak With Quarantine: The Sibkr Model, Mikenna Dew, Amanda Langosch, Theadora Baker-Wallerstein

Rose-Hulman Undergraduate Mathematics Journal

Influenza is a respiratory infection that places a substantial burden in the world population each year. In this project, we study and interpret a data set from a flu outbreak in a British boarding school in 1978 with mathematical modeling. First, we propose a generalization of the SIR model based on the quarantine measure in place and establish the long-time behavior of the model. By analyzing the model mathematically, we determine the analytic formulas of the basic reproduction number, the long-time limit of solutions, and the maximum number of infection population. Moreover, we estimate the parameters of the model based …

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The Basel Problem And Summing Rational Functions Over Integers, Pranjal Jain Mar 2024

The Basel Problem And Summing Rational Functions Over Integers, Pranjal Jain

Rose-Hulman Undergraduate Mathematics Journal

We provide a general method to evaluate convergent sums of the form ∑_{k∈Z} R(k) where R is a rational function with complex coefficients. The method is entirely elementary and does not require any calculus beyond some standard limits and convergence criteria. It is inspired by a geometric solution to the famous Basel Problem given by Wästlund (2010), so we begin by demonstrating the method on the Basel Problem to serve as a pilot application. We conclude by applying our ideas to prove Euler’s factorisation for sin x which he originally used to solve the Basel Problem.

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Wang Tilings In Arbitrary Dimensions, Ian Tassin Mar 2024

Wang Tilings In Arbitrary Dimensions, Ian Tassin

Rose-Hulman Undergraduate Mathematics Journal

This paper makes a new observation about arbitrary dimensional Wang Tilings,
demonstrating that any d -dimensional tile set that can tile periodically along d − 1 axes must be able to tile periodically along all axes.
This work also summarizes work on Wang Tiles up to the present day, including
definitions for various aspects of Wang Tilings such as periodicity and the validity of a tiling. Additionally, we extend the familiar 2D definitions for Wang Tiles and associated properties into arbitrary dimensional spaces. While there has been previous discussion of arbitrary dimensional Wang Tiles in other works, it has been …

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Optimizing Buying Strategies In Dominion, Nikolas A. Koutroulakis Feb 2024

Optimizing Buying Strategies In Dominion, Nikolas A. Koutroulakis

Rose-Hulman Undergraduate Mathematics Journal

Dominion is a deck-building card game that simulates competing lords growing their kingdoms. Here we wish to optimize a strategy called Big Money by modeling the game as a Markov chain and utilizing the associated transition matrices to simulate the game. We provide additional analysis of a variation on this strategy known as Big Money Terminal Draw. Our results show that player's should prioritize buying provinces over improving their deck. Furthermore, we derive heuristics to guide a player's decision making for a Big Money Terminal Draw Deck. In particular, we show that buying a second Smithy is always more optimal …

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On The Singular Pebbling Number Of A Graph, Harmony R. Morris Jan 2024

On The Singular Pebbling Number Of A Graph, Harmony R. Morris

Rose-Hulman Undergraduate Mathematics Journal

In this paper, we define a new parameter of a connected graph as a spin-off of the pebbling number (which is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble). This new parameter is the singular pebbling number, the smallest t such that a player can be given any configuration of at least t pebbles and any target vertex and can successfully move pebbles so that exactly one pebble ends on the target vertex. We also prove that the singular pebbling number of any graph on 3 or more vertices is equal …

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Eigenvalue Algorithm For Hausdorff Dimension On Complex Kleinian Groups, Jacob Linden, Xuqing Wu Nov 2023

Eigenvalue Algorithm For Hausdorff Dimension On Complex Kleinian Groups, Jacob Linden, Xuqing Wu

Rose-Hulman Undergraduate Mathematics Journal

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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Further Generalizations Of Happy Numbers, E. Simonton Williams Oct 2023

Further Generalizations Of Happy Numbers, E. Simonton Williams

Rose-Hulman Undergraduate Mathematics Journal

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, …

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Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine Oct 2023

Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine

Rose-Hulman Undergraduate Mathematics Journal

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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Elliptic Triangles Which Are Congruent To Their Polar Triangles, Jarrad S. Epkey, Morgan Nissen, Noelle K. Kaminski, Kelsey R. Hall, Nicholas Grabill Oct 2023

Elliptic Triangles Which Are Congruent To Their Polar Triangles, Jarrad S. Epkey, Morgan Nissen, Noelle K. Kaminski, Kelsey R. Hall, Nicholas Grabill

Rose-Hulman Undergraduate Mathematics Journal

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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Structure Of A Total Independent Set, Lewis Stanton Oct 2023

Structure Of A Total Independent Set, Lewis Stanton

Rose-Hulman Undergraduate Mathematics Journal

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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A Model For The Multi-Virus Contact Process, Xu Huang Oct 2023

A Model For The Multi-Virus Contact Process, Xu Huang

Rose-Hulman Undergraduate Mathematics Journal

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 …

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K-Distinct Lattice Paths, Eric J. Yager, Marcus Engstrom Sep 2023

K-Distinct Lattice Paths, Eric J. Yager, Marcus Engstrom

Rose-Hulman Undergraduate Mathematics Journal

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(nk).

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Utilizing Graph Thickness Heuristics On The Earth-Moon Problem, Robert C. Weaver Sep 2023

Utilizing Graph Thickness Heuristics On The Earth-Moon Problem, Robert C. Weaver

Rose-Hulman Undergraduate Mathematics Journal

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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Number Of Regions Created By Random Chords In The Circle, Shi Feng Sep 2023

Number Of Regions Created By Random Chords In The Circle, Shi Feng

Rose-Hulman Undergraduate Mathematics Journal

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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The Mean Sum Of Squared Linking Numbers Of Random Piecewise-Linear Embeddings Of $K_N$, Yasmin Aguillon, Xingyu Cheng, Spencer Eddins, Pedro Morales Sep 2023

The Mean Sum Of Squared Linking Numbers Of Random Piecewise-Linear Embeddings Of $K_N$, Yasmin Aguillon, Xingyu Cheng, Spencer Eddins, Pedro Morales

Rose-Hulman Undergraduate Mathematics Journal

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 …

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Reversibility Of Stranded Cellular Automata, Allyn Loyd Sep 2023

Reversibility Of Stranded Cellular Automata, Allyn Loyd

Mathematical Sciences Technical Reports (MSTR)

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 …

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Design Of A Resonant Optical Cavity For Imaging Magneto-Optically Active Thin Film Samples, Cody Robert Brelage Aug 2023

Design Of A Resonant Optical Cavity For Imaging Magneto-Optically Active Thin Film Samples, Cody Robert Brelage

Graduate Theses - Physics and Optical Engineering

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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On Solutions Of First Order Pde With Two-Dimensional Dirac Delta Forcing Terms, Ian Robinson Jul 2023

On Solutions Of First Order Pde With Two-Dimensional Dirac Delta Forcing Terms, Ian Robinson

Rose-Hulman Undergraduate Mathematics Journal

We provide solutions of a first order, linear partial differential equation of two variables where the nonhomogeneous term is a two-dimensional Dirac delta function. Our results are achieved by applying the unilateral Laplace Transform, solving the subsequently transformed PDE, and reverting back to the original space-time domain. A discussion of existence and uniqueness of solutions, a derivation of solutions of the PDE coupled with a boundary and initial condition, as well as a few worked examples are provided.

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