Furman University Electronic Journal of Undergraduate MathematicsCopyright (c) 2019 Furman University All rights reserved.
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Recent documents in Furman University Electronic Journal of Undergraduate Mathematicsen-usMon, 04 Feb 2019 20:08:13 PST3600The Collatz Conjecture and Integers of the Form <em>2<sup>k</sup>b−m</em> and <em>3<sup>k</sup>b−1</em>
https://scholarexchange.furman.edu/fuejum/vol17/iss1/1
https://scholarexchange.furman.edu/fuejum/vol17/iss1/1Wed, 01 Jun 2016 12:10:57 PDT
One of the more well-known unsolved problems in number theory is the Collatz (3n + 1) Conjecture. The conjecture states that iterating the map that takes even n ∈ N to n/2 and odd n to (3n+1)/2 will eventually yield 1. This paper is an exploration of this conjecture on positive integers of the form 2^{k}b−m and 3^{k}b−1, and stems from the work of the first author's Senior Seminar research. We take an elementary approach to prove interesting relationships and patterns in the number of iterations, called the total stopping time, required for integers of the aforementioned forms to reach 1, so that our results and proofs would be accessible to an undergraduate. Our results, then, provide a degree of insight into the Collatz Conjecture.
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Patrick Wiltrout et al.The Search for One as a Prime Number: From Ancient Greece To Modern Times
https://scholarexchange.furman.edu/fuejum/vol16/iss1/1
https://scholarexchange.furman.edu/fuejum/vol16/iss1/1Wed, 01 Jun 2016 12:10:43 PDT
It has often been asked if one is a prime number, or if there was a time when most mathematicians thought one was prime. Whether or not the number one is prime is simply a matter of definition, but definitions are often decided by the use of mathematics. In this paper we will survey the history of the definition of prime as applied to the number one, from the ancient Greeks to the modern times. For the Greeks the numbers (αριθμος) were multiples of the unit, and for this reason one did not fall into the category of primes (a subdivision of the numbers). This view held with few exceptions until Stevin (c. 1585) argued successfully that one was a number, at which point it finally made sense to ask if one is prime. This was followed by a period of confusion which began to dissipate with Gauss' Disquisitiones Arithmeticae. Our survey will show that for most of history, one was not considered a prime, and there was no point in time where a clear majority of mathematicians viewed one as prime.
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Angela Reddick et al.On the Dead End Depth of Thompson's Group <em>F</em>
https://scholarexchange.furman.edu/fuejum/vol15/iss1/2
https://scholarexchange.furman.edu/fuejum/vol15/iss1/2Wed, 01 Jun 2016 12:10:33 PDT
Thompson’s group F was introduced by Richard Thompson in the 1960’s and has since found applications in many areas of mathematics including algebra, logic and topology. We focus on the dead end depth of F, which is the minimal integer N such that for any group element, g, there is guaranteed to exist a path of length at most N in the Cayley graph of F leading from g to a point farther from the identity than g is. By viewing F as a diagram group, we improve the greatest known lower bound for the dead end depth of F with respect to the standard consecutive generating sets.
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Justin HalversonRelative Goldbach Partitions and Goldbach's Conjecture
https://scholarexchange.furman.edu/fuejum/vol15/iss1/1
https://scholarexchange.furman.edu/fuejum/vol15/iss1/1Wed, 01 Jun 2016 12:10:28 PDT
In this note, we utilize techniques from discrete mathematics to develop first an inequality, and then second a counting formula that is connected to Goldbach's conjecture. In order to do this, we introduce the notion of a Relative Goldbach Partition.
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Houston HutchinsonIntegers of the Form a<sup>2</sup>±b<sup>2</sup>
https://scholarexchange.furman.edu/fuejum/vol14/iss1/1
https://scholarexchange.furman.edu/fuejum/vol14/iss1/1Wed, 01 Jun 2016 12:10:18 PDT
This paper explores which integers can be expressed in the form a^{2}±2b^{2} by using rings of the form Z[√d], particularly when d = 2 and d = −2.
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Robert ZemanPaths and Circuits in G-Graphs of Certain Non-abelian Groups
https://scholarexchange.furman.edu/fuejum/vol13/iss1/1
https://scholarexchange.furman.edu/fuejum/vol13/iss1/1Wed, 01 Jun 2016 12:10:08 PDT
In [BJRTD08], necessary and suffcient conditions were given for the existence of Eulerian and Hamiltonian paths and circuits in the G-graph of the dihedral group D_{n}. In this paper, we consider the G-graphs of the quasihedral, modular, and generalized quaternion group. These groups are of rank 2 and we consider only the graphs Γ(G, S) where |S|= 2.
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A. Dewitt et al.The Relationships Between CG, BFGS, and Two Limited-memory Algorithms
https://scholarexchange.furman.edu/fuejum/vol12/iss1/2
https://scholarexchange.furman.edu/fuejum/vol12/iss1/2Wed, 01 Jun 2016 12:09:57 PDT
For the solution of linear systems, the conjugate gradient (CG) and BFGS are among the most popular and successful algorithms with their respective advantages. The limited-memory methods have been developed to combine the best of the two. We describe and examine CG, BFGS, and two limited-memory methods (L-BFGS and VSCG) in the context of linear systems. We focus on the relationships between each of the four algorithms, and we present numerical results to illustrate those relationships.
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Zhiwei (Tony) QinFinding Prime Numbers: Miller Rabin and Beyond
https://scholarexchange.furman.edu/fuejum/vol12/iss1/1
https://scholarexchange.furman.edu/fuejum/vol12/iss1/1Wed, 01 Jun 2016 12:09:52 PDT
This expository paper motivates and explains the Miller Rabin test and gives some generalizations of it. The Miller Rabin test is a standard probabilistic test used to find large prime numbers quickly.
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Christina McIntoshCarolinas Mathematics Undergraduate Research Conference Abstracts
https://scholarexchange.furman.edu/fuejum/vol11/iss1/3
https://scholarexchange.furman.edu/fuejum/vol11/iss1/3Wed, 01 Jun 2016 12:09:41 PDT
On Friday, March 24, 2006, Furman University hosted the Carolinas Mathematics Undergraduate Research Conference. The conference was supported by the Mathematical Association of America (NSF Grant DMS-0241090). These are the abstracts for the eight undergraduate talks given on that day.
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John HarrisNotes on Gabriel's Horn
https://scholarexchange.furman.edu/fuejum/vol11/iss1/2
https://scholarexchange.furman.edu/fuejum/vol11/iss1/2Wed, 01 Jun 2016 12:09:35 PDT
A smooth bounded solid of finite volume and infinite surface is constructed. It is a variant of the classical Gabriel’s horn that is often taught in Calculus classes.
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Joseph Krenicky et al.DeRham Cohomology of the Rectangular Torus
https://scholarexchange.furman.edu/fuejum/vol11/iss1/1
https://scholarexchange.furman.edu/fuejum/vol11/iss1/1Wed, 01 Jun 2016 12:09:29 PDT
For the special case of a rectangular at torus, we present and prove DeRham's Theorem, which says that cohomology is given by closed differential forms modulo exact forms.
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Eric M. KatermanThe University of North Carolina at Greensboro Regional Undergraduate Mathematics Conference Abstracts
https://scholarexchange.furman.edu/fuejum/vol10/iss1/2
https://scholarexchange.furman.edu/fuejum/vol10/iss1/2Wed, 01 Jun 2016 12:09:17 PDT
It was a very chaotic day, says Kathryn Sikes. Indeed, mutants spread everywhere, according to Brian Stadler. Bacterial wars raged all over the place, adds Dan MacMartin. Everybody was stealing, reported Christian Sykes. There were no limits to it, witnessed by Samuel Grundman. Only the fittest survived and got out of the prison, noted by Joseph Krenicky. The group was set free by Steven Piantadosi. We almost got lost in cyclic paths, said Heather Allmond. At least, our weight was a perfect number, smiles Michael Shiver, because we were not oversized thanks to Martha Shott. Finally, a picture was taken by Gavin Taylor and you could win a date if you listened to David Dombrowski. Math can be a lot of fun.
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Jan RychtářOn the Nonexistence of Singular Equilibria in the Four-vortex Problem
https://scholarexchange.furman.edu/fuejum/vol10/iss1/1
https://scholarexchange.furman.edu/fuejum/vol10/iss1/1Wed, 01 Jun 2016 12:09:05 PDT
In this paper we provide a partial answer to a question recently posed by Hassan Aref et. al. in their article Vortex Crystals, namely whether there are certain singular equilibria of point vortices. We prove that there are no such equilibria in the four-vortex case.
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Marshall Hampton et al.Properties of the Iterates of the Weierstrass-℘ Function
https://scholarexchange.furman.edu/fuejum/vol9/iss1/3
https://scholarexchange.furman.edu/fuejum/vol9/iss1/3Wed, 01 Jun 2016 12:08:53 PDT
This paper discusses several properties of the Weierstrass-℘ function, as defined on the fundamental parallelogram C/Γ, where C is the complex plane and Γ is the lattice generated by ω1 and ω2. Using the addition formula for ℘(z_{1} + z_{2}), we develop a reccurence relation for ℘(nz) in terms of ℘(z). We then examine the degree of this expression, some coefficients, and patterns concerning the poles of this function. We also consider the geometric interpretation of taking an arbitrary z_{0} and adding it to itself, both in the fundamental parallelogram C/Γ and the elliptic curve generated by ℘(z) and ℘′(z).
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Walter H. Chen et al.A Dynamical Programming Solution for Shortest Path Itineraries in Robotics
https://scholarexchange.furman.edu/fuejum/vol9/iss1/2
https://scholarexchange.furman.edu/fuejum/vol9/iss1/2Wed, 01 Jun 2016 12:08:47 PDT
In robotics, more precisely Autonomous Mobile Robotics (AMR), robots, much like human beings, are confronted regularly with the problem of finding the best path to take from a source location to a destination location. This is an optimization concern, since the robot wants to minimize its cost in time or in energy while achieving its goal. Different algorithms exist for shortest path computation; the famous Dijkstra’s Shortest Path Algorithm will solve single-source shortest path problems in near linear time (O(mn log n)). However, for certain complex optimization path-planning problems, this algorithm alone is insufficient. We will study a dynamical formulation using Integer Programming (IP) to solve complex path-planning problems in robotics.
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Martin TalbotVertex Magic
https://scholarexchange.furman.edu/fuejum/vol9/iss1/1
https://scholarexchange.furman.edu/fuejum/vol9/iss1/1Wed, 01 Jun 2016 12:08:42 PDT
This paper addresses labeling graphs in such a way that the sum of the vertex labels and incident edge labels are the same for every vertex. Bounds on this so-called magic number are found for cycle graphs. If a graph has an odd number of vertices, algorithms can be found to produce different magic-vertex graphs with the maximum and minimum magic number. Also, every cycle graph with an odd number of vertices can be made into a vertexmagic graph if the odd numbers or even numbers are placed on the vertices. Some interesting problems arise when one begins to look at cycle graphs with an even number of vertices. Bounds for the magic number change, and it becomes harder to make these graphs vertex-magic. We have shown some algorithms for finding vertex-magic cycle graphs with a magic number that lies within the bounds.
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Daisy CunninghamSome Geometry of <em>H</em>(R<sup>n</sup>)
https://scholarexchange.furman.edu/fuejum/vol8/iss1/1
https://scholarexchange.furman.edu/fuejum/vol8/iss1/1Wed, 01 Jun 2016 12:08:29 PDT
If X is a complete metric space, the collection of all non-empty compact subsets of X forms a complete metric space (H(X), h), where h is the Hausdorff metric. In this paper we explore some of the geometry of the space H(R^{n}). Specifically, we concentrate on understanding lines in H(R). In particular, we show that for any two points A, B, ∈ H(R^{n}), there exist infinitely many points on the line joining A and B. We characterize some points on the lines formed using closed and bounded intervals of R and show that two distinct lines in H(R) can intersect in infinitely many points.
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Christopher FrayerTiling by (<em>k</em>, <em>n</em>)-Crosses
https://scholarexchange.furman.edu/fuejum/vol7/iss1/1
https://scholarexchange.furman.edu/fuejum/vol7/iss1/1Wed, 01 Jun 2016 12:08:19 PDT
We investigate lattice tilings of n-space by (k, n)-crosses, establishing necessary and sufficient conditions for tilings with certain small values of k. We give a necessary condition for tilings corresponding to nonsingular splittings with general values of k. We also prove one case of a conjecture made by Stein and Szabó in [4].
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Joanne CharleboisFourier and Wavelet Representations of Functions
https://scholarexchange.furman.edu/fuejum/vol6/iss1/1
https://scholarexchange.furman.edu/fuejum/vol6/iss1/1Wed, 01 Jun 2016 12:08:09 PDT
Representations of functions are compared using the traditional technique of Fourier series with a more modern technique using wavelets. Under certain conditions, a function can be represented with a sum of sine and cosine functions. Such a representation is called a Fourier series. This classical method is used in applications such as storage of sound waves and visual images on a computer. One problem with this sum is that it is infinite. In use, only a finite number of terms can be used. More accuracy requires more terms in the series, but more terms require more time to compute and more space to store. A new type of sum called a wavelet series was first introduced in the 1980’s. With these new series the same accuracy often takes fewer terms. Since wavelet representations can be more accurate and take less computer time, they are often more useful.
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Nicholas G. RolandA Lie Algebra of Integrals for Keplerian Motion Restricted to the Plane
https://scholarexchange.furman.edu/fuejum/vol5/iss1/2
https://scholarexchange.furman.edu/fuejum/vol5/iss1/2Wed, 01 Jun 2016 12:07:58 PDT
In this paper we consider the slightly simpler problem of Keplerian motion restricted to the plane rather than Keplerian motion in three dimensions as done in [3, page 11]. We parallel the three-dimensional problem in that we use the same actions to find invariants (integrals) but rather than working in a six-dimensional phase space to find six independent integrals we restrict ourselves to a four-dimensional phase space. In doing this, we find that we have three independent integrals and thus we have a three-dimensional Lie algebra.
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Jason Osborne