» Vol. 10, Issue 1, 2009 «

Title: Fixed Points of Number Derivatives Modulo n Author: Franque Bains, Californaia State University, Los Angeles Author Bio     Abstract: A number derivative is a function that satisfies the Product Rule. In this paper, we find all solutions to the equation f (x)=x, where f is a number derivative on the ring of integers modulo an integer n. Thinking of number derivatives as analogues of the ordinary derivative from Calculus, we can think of this equation as a "differential equation" of sorts; solutions to it will then be rough analogues of exponential functions. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Computing the Arrow Polynomial Author: Kumud Bhandari, McKendree University Author Bio     Abstract: Determining if two knots are not equivalent in an efficient manner is important in the study of knots. The arrow polynomial, which is calculated from a virtual knot diagram and is invariant under the Reidemeister moves, can be used to determine if two knots are not equivalent and determine a lower bound on the virtual crossing number. In this paper, we present the necessary data structures and algorithms to represent a link diagram on a computer and calculate the arrow polynomial. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Differential Equations and the Method of Upper and Lower Solutions Authors: Jacob Chapman, University of Alabama at Birmingham Author Bio     Abstract: In this paper, some background material regarding differential equations and initial value problems is presented. The method of upper and lower solutions, which is used for determining existence of periodic solutions to periodic differential equations, is then discussed. Theorems regarding periodicity and the first-order case of upper and lower solutions are proven. The method is applied to some examples from pure mathematics along with the logistic equation, and corresponding graphs generated in MATLAB illustrate the periodic behavior and stability of solutions. The second-order case of upper and lower solutions is then introduced, and an example is taken from pure mathematics in addition to one regarding a simple undamped pendulum subject to periodic forcing. In conclusion, it is noted that the method of upper and lower solutions is used for existence purposes only and should not be used if analytical solutions can be obtained; the method somewhat resembles the intermediate value theorem and squeeze theorem; and it is useful mainly for nonlinear periodic differential equations when analytical solutions do not exist. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: On the Order of a Group Containing Nontrivial Gassmann Equivalent Subgroups Authors: Michael DiPasquale, Wheaton College Author Bio     Abstract: Using a result of de Smit and Lenstra, we prove that the order of a group containing nontrivial Gassmann equivalent subgroups must be divisible by at least five primes, not necessarily distinct. We then investigate the existence of Gassmann equivalent subgroups in groups with order divisible by exactly five primes. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Numerical Solutions for Intermediate Angles for the Laplace-Young Capillary Equations Authors: Genevieve Dupuis, University of Notre Dame Jessica Flores, University of Puerto Rico Author Bio     Author Bio     Abstract: Capillarity is the phenomena of fluid rise against a solid vertical wall. For our research, we consider bounded cases of intermediate corner angles ( p/2< a + g < p/2+2 g ), where g is the angle of contact and 2a is the wedge angle. The Laplace-Young Capillary equations are used to determine the rise of the fluid, especially at corners. While there exist asymptotic expansions for the height rise occurring at the corner of an intermediate angle, not all coefficients are known analytically. Therefore, numerical solutions are necessary, even though only a few numerical methods have been published. We explain our least-squares finite element method used in determining solutions to the Laplace-Young Capillary equations, and then give our numerical results. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Proof of Solvability for the Generalized Oval Track Puzzle (revised 04/18/2010) Author: Sam Kaufmann, Carnegie Mellon University Andreas Kavountzis, Carnegie Mellon University Author Bio     Author Bio     Abstract: The oval track puzzle (also known as Top Spin) is a game consisting of 20 numbered tiles in an oval shaped track. Also, there is a fixed window (the swapping window) of 4 tiles that reverses the order of the tiles within the window, leaving the other 16 tiles fixed. The object of the puzzle is to reorder the tiles into counting order using the mechanisms of the puzzle. Our paper presents conditions for both solvability and non-solvability for the general oval track puzzle with n total tiles and k tiles in the swapping window. This paper answers questions left over from the work done by Eric Wilbur in his paper entitled Topspin: Solvability of Sliding Number Game from Volume 2, Issue 2 of the RHIT Mathematics Journal. Using his notation and terminology as a reference, we reproved some cases as well as proved open problems from his paper. Article: Downloadable PDF     Additional Downloads: Picture of Oval Track     » BACK TO TOP Title: A Beckman-Quarles Type Theorem for Laguerre Transformations in the Dual Plane Authors: Timothy Ferdinands, Calvin College Landon Kavlie, Calvin College Author Bio     Author Bio     Abstract: In 1953, Beckman and Quarles proved a well-known result in Euclidean Geometry that any transformation preserving a distance r must be a rigid motion. In 1991, June Lester published an analogous result for circle-preserving transformations in the complex plane. In our paper, we introduce the notion of dual numbers and the geometry of the dual plan. We forcus on the set of vertical parabolas and non-vertical linear P with a distance between pairs of parabolas defined to be the difference of slopes at their point(s) of intersection. We then prove that any bijective transformation from P to itself which preserves our distance 1 induces a fractional linear or Laguerre transformation of the dual plane. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Analyzing Human Papillomavirus Vaccine Stockpiles Author: Jamie D. Lloyd, Virginia Commonwealth University Author Bio     Abstract: The development of a vaccine to prevent the contraction of the high-risk strands of human papillomavirus (HPV) 6, 11, 16 and 18 has the potential to prevent 70% of all cervical cancers. The Center for Disease Control and Prevention (CDC) currently recommends that girls aged 11-12 receive the HPV vaccine. At present, eighteen states have already decided or are considering to make HPV vaccination mandatory for adolescent girls. As the HPV vaccine becomes mandatory, the demand for the vaccine is expected to dramatically rise. This increase in demand could make our nation vulnerable to interruptions in HPV vaccine production. If an interruption occurs, many adolescent girls and women could be at an unnecessary risk of acquiring HPV if they were to miss routine HPV immunizations. One major factor in the prevention of HPV vaccine shortages is the creation of vaccine stockpiles by the CDC. In this paper, mathematical models are used to determine and analyze stockpile levels sufficient to minimize the effects of a production interruption for the HPV vaccine. The results indicate that the stockpile level is highly sensitive to the vaccine coverage rate and the duration of the production interruption. To protect against a six month interruption in vaccine production, a stockpile of at least 3M is recommended. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: The Period and the Distribution of the Fibonacci-like Sequence Under Various Moduli Author: Hiroshi Matsui , Kwansei Gakuin High School, Nishinomiya City JAPAN Masakazu Naito , Kwansei Gakuin High School, Nishinomiya City JAPAN Naoyuki Totani , Kwansei Gakuin High School, Nishinomiya City JAPAN Author Bio     Author Bio     Author Bio     Abstract: We reduce the Fibonacci sequence mod m for a natural number m, and denote it by F (mod m ). We are going to introduce the properties of the period and distribution of F (mod m). That is, how frequently each residue is expected to appear within a single period. These are well known themes of the research of the Fibonacci sequence, and many remarkable facts have been discovered. After that we are going to study the properties of period and distribution of a Fibonacci-like sequence that the authors introduced in article in the previous issue of Undergraduate Math Journal. This Fibonacci-like sequence also has many interesting properties, and the authors could prove an interesting theorem in this article. Some of properties are very difficult to prove, and hence we are going to present some predictions and calculations by computers. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Josephus Problem Under Various Moduli Author: Toshiyuki Yamauchi, Kwansei Gakuin High School, Nishinomiya City JAPAN Takahumi Inoue , Kwansei Gakuin High School, Nishinomiya City JAPAN Soh Tatsumi, Kwansei Gakuin High School, Nishinomiya City JAPAN Author Bio     Author Bio     Author Bio     Abstract: We are going to study the Josephus Problem and its variants under various moduli in this article. Let n be a natural number. We put n numbers in a circle, and we are going to remove every second number. Let J(n) be the last number that remains. This is the traditional Josephus Problem. The list { J(n) , n = 1,2,...,20 } is {1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 9, 11, 13, 15, 1, 3, 5, 7, 9 }. When this sequence is reduced mod 4 , then we have {1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1 }. Next we are going to study a variant of the Josephus Problem in which two numbers are to be eliminated at the same time, and let J2(n) be the last number that remains. If the sequence { J2(2n) , n = 1, 2, ...63 } is reduced mod 2 , then we have {1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 }. The pattern that exists in the sequence is obvious if you look at the sequence carefully. In this way we get interesting patterns of sequences for the Josephus Problem and its variants under various moduli. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Bounds on Biased and Unbiased Random Walks Author: Daniel Parry, New York University Author Bio     Abstract: The pre-asymptotic convergence of Markov chains is a relatively new field of study only two or three decades old and is still an active area of research. One example of a pre-asymptotic behavior is the cutoff phenomenon explored by Diaconis and his collaborators. A Markov chain has a cutoff if it remains far from stationary for a long period, after which it converges within a small number of iterations. As his most famous example, Diaconis showed that seven shuffles is enough to randomize the order of a deck of cards, but after six shuffles the card order is still far from uniformly randomized. Fully understanding the phenomenon would help improve the efficiency of calculating Markov chains in their "long run" states. Though many examples have been analyzed, in general the cutoff phenomenon is still not well understood [1]. Our goal in this paper is to explore the cutoff phenomena for some random walks on one-dimensional lattices. After reviewing some facts about discrete Markov chains in general, we describe spectral and probablistic bounds that describe their convergence. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: L(d,j,s) Minimal and Surjective Graph Labeling Author: Michelle Lingscheit, Simpson College Kiersten Ruff, Simpson College Jeremy Ward, Simpson College Author Bio     Author Bio     Author Bio     Abstract: Interference between radio signals can be modeled using distance labeling where the vertices on the graph represent the radio towers and the edges represent the interference between the towers. The distance between vertices affects the labeling of the vertices to account for the strength of interference. In this paper we consider three levels of interference between signals on a given graph, G. Define D(x,y) to represent the distance between vertex x and vertex y. An L(d,j,s) labeling of graph G is a function f from the vertex set of a graph to the set of positive integers, where |f(x)-f(y)| ³ d if D(x,y)=1, |f(x)-f(y)|³ j if D(x,y)=2, and |f(x)-f(y)|³ s if D(x,y)=3 for positive integers m and d where d>j>s. In this paper we will examine surjective and minimal labeling of different families of graphs including paths, cycles, caterpillars, complete graphs, and complete bipartite graphs. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP