Mathematical Sciences Technical Reports (MSTR)Copyright (c) 2023 Rose-Hulman Institute of Technology All rights reserved.
https://scholar.rose-hulman.edu/math_mstr
Recent documents in Mathematical Sciences Technical Reports (MSTR)en-usFri, 03 Nov 2023 06:02:38 PDT3600Applying 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 PinkstonReversibility 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 LoydCompare and Contrast Maximum Likelihood Method and Inverse Probability Weighting Method in Missing Data Analysis
https://scholar.rose-hulman.edu/math_mstr/181
https://scholar.rose-hulman.edu/math_mstr/181Wed, 19 Oct 2022 13:28:29 PDT
Data can be lost for different reasons, but sometimes the missingness is a part of the data collection process. Unbiased and efficient estimation of the parameters governing the response mean model requires the missing data to be appropriately addressed. This paper compares and contrasts the Maximum Likelihood and Inverse Probability Weighting estimators in an Outcome-Dependendent Sampling design that deliberately generates incomplete observations. WE demonstrate the comparison through numerical simulations under varied conditions: different coefficient of determination, and whether or not the mean model is misspecified.
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Scott SunAnalysis of a Quantum Attack on the Blum-Micali Pseudorandom Number Generator
https://scholar.rose-hulman.edu/math_mstr/180
https://scholar.rose-hulman.edu/math_mstr/180Thu, 29 Sep 2022 16:16:19 PDT
In 2012, Guedes, Assis, and Lula proposed a quantum attack on a pseudorandom number generator named the Blum-Micali Pseudorandom number generator. They claimed that the quantum attack can outperform classical attacks super-polynomially. However, this paper shows that the quantum attack cannot get the correct seed and provides another corrected algorithm that is in exponential time but still faster than the classical attack. Since the original classical attacks are in exponential time, the Blum-Micali pseudorandom number generator would be still quantum resistant.
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Tingfei FengStructure of Number Theoretic Graphs
https://scholar.rose-hulman.edu/math_mstr/179
https://scholar.rose-hulman.edu/math_mstr/179Thu, 29 Sep 2022 13:59:42 PDT
The tools of graph theory can be used to investigate the structure imposed on the integers by various relations. Here we investigate two kinds of graphs. The first, a square product graph, takes for its vertices the integers 1 through n, and draws edges between numbers whose product is a square. The second, a square product graph, has the same vertex set, and draws edges between numbers whose sum is a square. We investigate the structure of these graphs. For square product graphs, we provide a rather complete characterization of their structure as a union of disjoint complete graphs. For square sum graphs, we investigate some properties such as degrees of vertices, connectedness, hamiltonicity, and planarity.
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Lee TrentThe Primitive Root Problem: a Problem in BQP
https://scholar.rose-hulman.edu/math_mstr/178
https://scholar.rose-hulman.edu/math_mstr/178Thu, 29 Sep 2022 13:48:09 PDT
Shor’s algorithm proves that the discrete logarithm problem is in BQP. Based on his algorithm, we prove that the primitive root problem, a problem that verifies if some integer g is a primitive root modulo p where p is the largest prime number smaller than 2n for a given n, which is assumed to be harder than the discrete logarithm problem, is in BQP by using an oracle quantum Turing machine.
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Shixin WuComputer Program Simulation of a Quantum Turing Machine with Circuit Model
https://scholar.rose-hulman.edu/math_mstr/177
https://scholar.rose-hulman.edu/math_mstr/177Tue, 07 Dec 2021 16:30:53 PST
Molina and Watrous present a variation of the method to simulate a quantum Turing machine employed in Yao’s 1995 publication “Quantum Circuit Complexity”. We use a computer program to implement their method with linear algebra and an additional unitary operator defined to complete the details. Their method is verified to be correct on a quantum Turing machine.
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Shixin WuProbability Distributions for Elliptic Curves in the CGL Hash Function
https://scholar.rose-hulman.edu/math_mstr/176
https://scholar.rose-hulman.edu/math_mstr/176Sun, 08 Aug 2021 16:59:08 PDT
Hash functions map data of arbitrary length to data of predetermined length. Good hash functions are hard to predict, making them useful in cryptography. We are interested in the elliptic curve CGL hash function, which maps a bitstring to an elliptic curve by traversing an inputdetermined path through an isogeny graph. The nodes of an isogeny graph are elliptic curves, and the edges are special maps betwixt elliptic curves called isogenies. Knowing which hash values are most likely informs us of potential security weaknesses in the hash function. We use stochastic matrices to compute the expected probability distributions of the hash values. We generalize our experimental data into a theorem that completely describes all possible probability distributions of the CGL hash function. We use this theorem to evaluate the collision resistance of the CGL hash function and compare this to the collision resistance of an “ideal” hash function.
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Dhruv Bhatia et al.Topological and H^q Equivalence of Cyclic n-gonal Actions on Riemann Surfaces - Part II
https://scholar.rose-hulman.edu/math_mstr/175
https://scholar.rose-hulman.edu/math_mstr/175Wed, 30 Sep 2020 16:36:32 PDT
We consider conformal actions of the finite group G on a closed Riemann surface S, as well as algebraic actions of G on smooth, complete, algebraic curves over an arbitrary, algebraically closed field. There are several notions of equivalence of actions, the most studied of which is topological equivalence, because of its close relationship to the branch locus of moduli space. A second important equivalence relation is that induced by representation of G on spaces of holomorphic q-differentials. The notion of topological equivalence does not work well in positive characteristic. We shall discuss an alternative to topological equivalence, which we dub equisymmetry, that may be applied in all characteristics. The relation is induced by families of curves with G-action, and it works well with rotation constants and q-differentials, which are also defined in positive characteristic. After giving an overview of the various equivalence relations (conformal/algebraic, topological, q-differentials, rotation constants, equisymmetry) we focus on the interconnections among rotation constants, q-differentials, and equisymmetry.
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Sean A. BroughtonModeling Braids with Space-Varying and Time-Varying Stranded Cellular Automata
https://scholar.rose-hulman.edu/math_mstr/174
https://scholar.rose-hulman.edu/math_mstr/174Mon, 10 Aug 2020 14:11:38 PDT
Braids in a traditional sense and braids in a mathematical sense are wildly different outlooks on the same concept. Using cellular automata to represent and analyze braids is a way to bridge the gap between them. Joshua and Lana Holden and Hao Yang have previously worked on developing and expanding upon a Stranded Cellular Automata (SCA) model capable of representing many different braids and weaves. Continuing their work, we were able to devise a more user-friendly method for interacting with the model such that even those without a mathematical background can construct and analyze braids of their own. This paper will also discuss the addition of space-varying and time-varying rulesets to expand upon the types of braids and weaves the SCA model is able to represent.
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Brian ChanThe Game of Life on the Hyperbolic Plane
https://scholar.rose-hulman.edu/math_mstr/173
https://scholar.rose-hulman.edu/math_mstr/173Sun, 24 May 2020 11:51:48 PDT
In this paper, we work on the Game of Life on the hyperbolic plane. We are interested in different tessellations on the hyperbolic plane and different Game of Life rules. First, we show the exponential growth of polygons on the pentagon tessellation. Moreover, we find that the Group of 3 can keep the boundary of a set not getting smaller. We generalize the existence of still lifes by computer simulations. Also, we will prove some propositions of still lifes and cycles. There exists a still life under rules B1, B2, and S3.
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Yuncong GuRepeat Length Of Patterns On Weaving Products
https://scholar.rose-hulman.edu/math_mstr/172
https://scholar.rose-hulman.edu/math_mstr/172Sun, 03 Nov 2019 14:22:09 PST
Interlacing strands have been used to create artistic weaving 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 LiuForward Selection via Distance Correlation
https://scholar.rose-hulman.edu/math_mstr/170
https://scholar.rose-hulman.edu/math_mstr/170Wed, 22 May 2019 13:41:21 PDTTy AdamsPeriodicity and Invertibility of Lattice Gas Cellular Automata
https://scholar.rose-hulman.edu/math_mstr/171
https://scholar.rose-hulman.edu/math_mstr/171Wed, 22 May 2019 11:05:42 PDT
A cellular automaton is a type of mathematical system that models the behavior of a set of cells with discrete values in progressing time steps. The often complicated behaviors of cellular automata are studied in computer science, mathematics, biology, and other science related fields. Lattice gas cellular automata are used to simulate the movements of particles. This thesis aims to discuss the properties of lattice gas models, including periodicity and invertibility, and to examine their accuracy in reflecting the physics of particles in real life. Analysis of elementary cellular automata is presented to introduce the concept of cellular automata and construct foundations for the analysis of properties of lattice gas.
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Jiawen WangEffects of Framing in Exams on Student Performance
https://scholar.rose-hulman.edu/math_mstr/169
https://scholar.rose-hulman.edu/math_mstr/169Thu, 16 May 2019 20:19:45 PDTMariana Lane et al.Stranded Cellular Automaton and Weaving Products
https://scholar.rose-hulman.edu/math_mstr/168
https://scholar.rose-hulman.edu/math_mstr/168Sun, 02 Sep 2018 16:16:21 PDT
In order to analyze weaving products mathematically and find out valid weaving products, it is natural to relate them to Cellular Automaton. They are both generated based on specific rules and some initial conditions. Holden and Holden have created a Stranded Cellular Automaton that can represent common weaving and braiding products. Based on their previous findings, we were able to construct a Java program and analyze various aspects of the automaton they created. This paper will discuss the complexity of the Stranded Cellular Automaton, how to determine whether a weaving product holds together or not based on the automaton and the Java program we used for investigation.
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Hao YangBranching matrices for the automorphism group lattice of a Riemann surface
https://scholar.rose-hulman.edu/math_mstr/167
https://scholar.rose-hulman.edu/math_mstr/167Thu, 22 Mar 2018 16:04:38 PDT
Let S be a Riemann surface and G a large subgroup of Aut(S) (Aut(S) may be unknown). We are particularly interested in regular n-gonal surfaces, i.e., the quotient surface S/G (and hence S/Aut(S)) has genus zero. For various H the ramification information of the branched coverings S/K -> S/H may be captured in a matrix. The ramification information, in particular strong branching, may be then be used in analyzing the structure of Aut(S). The ramification information is conjugation invariant so the matrix's rows and columns may be indexed by conjugacy classes of subgroups. The only required information is a generating vector for the action of G, and the subgroup structure. The latter may be computed using Magma or GAP. The signatures and generating vectors of the subgroups are not required.
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Sean A. BroughtonStudy of Extinction Time
https://scholar.rose-hulman.edu/math_mstr/165
https://scholar.rose-hulman.edu/math_mstr/165Fri, 25 Aug 2017 09:36:21 PDT
In this thesis we study an epidemic spreading through a finite population where each individual can be susceptible (S), infective (I), and return to being susceptible (S), and we track the number of individuals in each state as time progresses. In contrast to the deterministic case which is modeled by systems of ODEs, we consider infection and recovery to be stochastic (random) events. Interest is in the (random) time T at which the epidemic dies out. For a large number of initial infectives, the time for extinction is governed by the ratio of the infection and recovery rates. For a small number of initial infectives, the epidemic may die out quickly due to random effects even if the infection and recovery rates would predict otherwise.
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Yilin WangVariance of Clusterings on Graphs
https://scholar.rose-hulman.edu/math_mstr/164
https://scholar.rose-hulman.edu/math_mstr/164Wed, 23 Aug 2017 13:43:25 PDT
Graphs that represent data often have structures or characteristics that can represent some relationships in the data. One of these structures is clusters or community structures. Most clustering algorithms for graphs are deterministic, which means they will output the same clustering each time. We investigated a few stochastic algorithms, and look into the consistency of their clusterings.
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Thomas Vlado Mulc