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https://scholar.rose-hulman.edu
Recent documents in Rose-Hulman Scholaren-usThu, 01 Dec 2022 01:49:35 PST3600Studying Extended Sets from Young Tableaux
https://scholar.rose-hulman.edu/rhumj/vol23/iss2/5
https://scholar.rose-hulman.edu/rhumj/vol23/iss2/5Wed, 02 Nov 2022 08:13:28 PDT
Young tableaux are combinatorial objects related to the partitions of an integer and have various applications in representation theory. They are particularly useful in the study of the fibers arising from the Springer resolution. In recent work of Graham-Precup-Russell, an association has been made between a given row-strict tableau and three disjoint subsets of {1,2,...,n}. These subsets are then used in the study of extended Springer fibers, so we call them extended sets. In this project, we use combinatorial techniques to classify which of these extended sets correlate to a valid row-strict or standard tableau and give bounds on the number of extended sets for a fixed size. We are able to identify several global properties of these valid sets, and we further find an algorithm that produces a valid tableau given the extended sets in special cases.
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Eric NofzigerOn the Smallest Non-trivial Action of SAut(Fn) for Small n
https://scholar.rose-hulman.edu/rhumj/vol23/iss2/4
https://scholar.rose-hulman.edu/rhumj/vol23/iss2/4Wed, 02 Nov 2022 08:13:21 PDT
In this paper we investigate actions of SAut(F_{n}), the unique index 2 subgroup of Aut(F_{n}), on small sets, improving upon results by Baumeister--Kielak--Pierro for several small values of n. Using a computational approach for n ⩾ 5, we show that every action of SAut(F_{n}) on a set containing fewer than 20 elements is trivial.
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Reemon SpectorCompare 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 WuGeneralizations of Commutativity in Dihedral Groups
https://scholar.rose-hulman.edu/rhumj/vol23/iss2/3
https://scholar.rose-hulman.edu/rhumj/vol23/iss2/3Thu, 15 Sep 2022 13:33:49 PDT
The probability that two elements commute in a non-Abelian finite group is at most 5 8 . We prove several generalizations of this result for dihedral groups. In particular, we give specific values for the probability that a product of an arbitrary number of dihedral group elements is equal to its reverse, and also for the probability that a product of three elements is equal to a permutation of itself or to a cyclic permutation of itself. We also show that for any r and n, there exists a dihedral group such that the probability that a product of n elements is equal to its reverse is r q for some q coprime to r, extending a known result.
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Noah A. HeckenlivelyOn Cantor Sets Defined by Generalized Continued Fractions
https://scholar.rose-hulman.edu/rhumj/vol23/iss2/2
https://scholar.rose-hulman.edu/rhumj/vol23/iss2/2Tue, 13 Sep 2022 07:54:52 PDT
We study a special class of generalized continuous fractions, both in real and complex settings, and show that in many cases, the set of numbers that can be represented by a continued fraction for that class form a Cantor set. Specifically, we study generalized continued fractions with a fixed absolute value and a variable coefficient sign. We ask the same question in the complex setting, allowing the coefficient's argument to be a multiple of \pi/2. The numerical experiments we conducted showed that in these settings the set of numbers formed by such continued fractions is a Cantor set for large values of the coefficient. Using an iterated function systems construction, we prove that this is true for both real and complex cases. We also observed that in some regimes (for absolute values of the coefficient smaller than two), the set forms a peculiar fractal, and we formulate some questions and conjectures on its properties. We expect that some restrictions on the coefficients of generalized continued fractions should lead to the appearance of Cantorvals (closed bounded sets that have dense interiors but contain no isolated points or intervals) or, in the complex case, two-dimensional analogs of Cantorvals. Our projects bring together topics from number theory, dynamical systems, fractal geometry, and complex analysis. We believe it can serve as a foundation for researchers to build upon in the future.
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Danielle Hedvig et al.The Effect of Habitat Fragmentation on Plant Communities in a Spatially-Implicit Grassland Model
https://scholar.rose-hulman.edu/rhumj/vol23/iss2/1
https://scholar.rose-hulman.edu/rhumj/vol23/iss2/1Mon, 12 Sep 2022 14:10:14 PDT
The spatially implicit Tilman-Levins ODE model helps to explain why so many plant species can coexist in grassland communities. This now-classic modeling framework assumes a trade-off between colonization and competition traits and predicts that habitat destruction can lead to long transient declines called ``extinction debts.'' Despite its strengths, the Tilman-Levins model does not explicitly account for landscape scale or the spatial configuration of viable habitat, two factors that may be decisive for population viability. We propose modifications to the model that explicitly capture habitat geometry and the spatial pattern of seed dispersal. The modified model retains implicit space and is in fact mathematically equivalent to the Tilman-Levins model in the single species case. But its novel interpretation of a habitat destruction parameter better quantifies seed loss due to edge effects in fragmented habitats and results in different predictions than the Tilman-Levins model. In particular, the seed loss model predicts that species with strong dispersal traits may be most vulnerable to extinction in small habitat fragments.
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Mika T. Cooney et al.Numerical Analysis of a Model for the Growth of Microorganisms
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/9
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/9Tue, 31 May 2022 15:58:57 PDT
A system of first-order differential equations that arises in a model for the growth of microorganisms in a chemostat with Monod kinetics is studied. A new, semi-implicit numerical scheme is proposed to approximate solutions to the system. It is shown that the scheme is uniquely solvable and unconditionally stable, and further properties of the scheme are analyzed. The convergence rate of the numerical solution to the true solution of the system is given, and it is shown convergence of the numerical solutions to the true solutions is uniform over any interval [0, T ] for T > 0.
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Alexander Craig Montgomery et al.Using Differential Equations to Model a Cockatoo on a Spinning Wheel as part of the SCUDEM V Modeling Challenge
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/8
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/8Tue, 31 May 2022 12:16:15 PDT
For the SCUDEM V 2020 virtual challenge, we received an outstanding distinction for modeling a bird perched on a bicycle wheel utilizing the appropriate physical equations of rotational motion. Our model includes both theoretical calculations and numerical results from applying the Heaviside function for the swing motion of the bird. We provide a discussion on: our model and its numerical results, the overall limitations and future work of the model we constructed, and the experience we had participating in SCUDEM V 2020.
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Miles Pophal et al.Implementation of A Least Squares Method To A Navier-Stokes Solver
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/7
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/7Thu, 19 May 2022 10:30:59 PDT
The Navier-Stokes equations are used to model fluid flow. Examples include fluid structure interactions in the heart, climate and weather modeling, and flow simulations in computer gaming and entertainment. The equations date back to the 1800s, but research and development of numerical approximation algorithms continues to be an active area. To numerically solve the Navier-Stokes equations we implement a least squares finite element algorithm based on work by Roland Glowinski and colleagues. We use the deal.II academic library , the C++ language, and the Linux operating system to implement the solver. We investigate convergence rates and apply the least squares solver to the lid driven cavity problem and discuss results.
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Jada P. Lytch et al.Tiling Rectangles and 2-Deficient Rectangles with L-Pentominoes
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/6
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/6Thu, 12 May 2022 10:09:10 PDT
We investigate tiling rectangles and 2-deficient rectangles with L-pentominoes. First, we determine exactly when a rectangle can be tiled with L-pentominoes. We then determine locations for pairs of unit squares that can always be removed from an m × n rectangle to produce a tileable 2-deficient rectangle when m ≡ 1 (mod 5), n ≡ 2 (mod 5) and when m ≡ 3 (mod 5), n ≡ 4 (mod 5).
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Monica KaneA Single Criterion for Polynomial Symmetry
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/5
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/5Sat, 07 May 2022 10:11:04 PDT
We develop an understanding of the relationship between the symmetry of polynomial graphs and the calculus that underlies this symmetry. We arrive at a method to determine whether a single-variable polynomial with real coefficients has a symmetric graph. We then encode this method into a closed formula that is a necessary and sufficient condition for the polynomial to have symmetry.
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Peter A. CermakOn Isomorphic K-Rational Groups of Isogenous Elliptic Curves Over Finite Fields
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/4
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/4Sat, 07 May 2022 10:10:56 PDT
It is well known that two elliptic curves are isogenous if and only if they have same number of rational points. In fact, isogenous curves can even have isomorphic groups of rational points in certain cases. In this paper, we consolidate all the current literature on this relationship and give a extensive classification of the conditions in which this relationship arises. First we prove two ordinary isogenous elliptic curves have isomorphic groups of rational points when they have the same $j$-invariant. Then, we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$-invariant of either 0 or 1728, using Vl\u{a}du\c{t}'s characterization of the group structure of rational points.
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Ben Kuehnert et al.A New Method To Compute The Hadamard Product Of Two Rational Functions
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/3
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/3Tue, 26 Apr 2022 09:04:17 PDT
The Hadamard product (denoted by∗) of two power series A(x) =a_{0}+a_{1}x+a_{2}x^{2}+···and B(x) =b_{0}+b_{1}x+b_{2}x^{2}+··· is the power series A(x)∗B(x) =a_{0}b_{0}+a_{1}b_{1}x+a_{2}b_{2}x^{2}+···. Although it is well known that the Hadamard product of two rational functions is also rational, a closed form expression of the Hadamard product of rational functions has not been found. Since any rational power series can be expanded by partial fractions as a polynomial plus a sum of power series of the form 1/(1−ax)^{m+1}, to find the Hadamard product of any two rational power series it is sufficient to find the Hadamard product 1/(1−ax)^{m+1}∗ 1/(1−bx)^{n+1 }= (1 +ax)^{m}∗(1+bx)^{n}/(1−abx)^{m+n+1}.The Hadamard product of negative powers of quadratic polynomials have also been derived.
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Ishan KarOn the Consistency of Alternative Finite Difference Schemes for the Heat Equation
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/2
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/2Thu, 14 Apr 2022 08:53:53 PDT
While the well-researched Finite Difference Method (FDM) discretizes every independent variable into algebraic equations, Method of Lines discretizes all but one dimension, leaving an Ordinary Differential Equation (ODE) in the remaining dimension. That way, ODE's numerical methods can be applied to solve Partial Differential Equations (PDEs). In this project, Linear Multistep Methods and Method of Lines are used to numerically solve the heat equation. Specifically, the explicit Adams-Bashforth method and the implicit Backward Differentiation Formulas are implemented as Alternative Finite Difference Schemes. We also examine the consistency of these schemes.
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Tran AprilAdditional Fay Identities of the Extended Toda Hierarchy
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/1
https://scholar.rose-hulman.edu/rhumj/vol23/iss1/1Thu, 14 Apr 2022 08:53:53 PDT
The focus of this paper is the extended Toda Lattice hierarchy, an infinite system of partial differential equations arising from the Toda lattice equation. We begin by giving the definition of the extended Toda hierarchy and its explicit bilinear equation, following Takasaki’s construction. We then derive a series of new Fay identities. Finally, we discover a general formula for one type of Fay identity.
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Yu WanGraphs with Four Independent Crossings Are Five Colorable
https://scholar.rose-hulman.edu/rhumj/vol9/iss2/12
https://scholar.rose-hulman.edu/rhumj/vol9/iss2/12Sun, 20 Mar 2022 18:16:27 PDT
Albertson conjectured that if a graph can be drawn in the plane in such a way that any two crossings are independent, then the graph can be 5-colored. He proved it for up to three independent crossings. We prove this for four crossings by showing that any such graph has an independent set of size 4 with one vertex in each crossing, and give an example to show that this method fails for five independent crossings.
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Nathan HarmanAn Alternating Sum of Alternating Sign Matrices
https://scholar.rose-hulman.edu/rhumj/vol9/iss2/10
https://scholar.rose-hulman.edu/rhumj/vol9/iss2/10Sun, 20 Mar 2022 18:16:26 PDT
An alternating-sign matrix (ASM) is a square matrix with entries from {-1, 0,1}, row and column sums of 1, and in which the nonzero entries in each row and column alternate in sign. ASMs have many non-trivial parameters and symmetries that reveal their significant combinatorial structure. In this note, we will prove an identity that relates one parameter and one symmetry.
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Nathan Williams