Open Access. Powered by Scholars. Published by Universities.®

Mathematics Commons™

Open Access. Powered by Scholars. Published by Universities.®

2016

Discipline
Institution
Keyword
Publication
Publication Type
File Type

Articles 391 - 420 of 1329

Full-Text Articles in Mathematics

Quaternion Algebras And Hyperbolic 3-Manifolds, Joseph Quinn Jun 2016

Quaternion Algebras And Hyperbolic 3-Manifolds, Joseph Quinn

Dissertations, Theses, and Capstone Projects

I use a classical idea of Macfarlane to obtain a complex quaternion model for hyperbolic 3-space and its group of orientation-preserving isometries, analogous to Hamilton’s famous result on Euclidean rotations. I generalize this to quaternion models over number fields for the action of Kleinian groups on hyperbolic 3-space, using arithmetic invariants of the corresponding hyperbolic 3-manifolds. The class of manifolds to which this technique applies includes all cusped arithmetic manifolds and infinitely many commensurability classes of cusped non-arithmetic, compact arithmetic, and compact non-arithmetic manifolds. I obtain analogous results for actions of Fuchsian groups on the hyperbolic plane. I develop new …

Go to article

The Remedy That's Killing: Cuny, Laguardia, And The Fight For Better Math Policy, Rachel A. Oppenheimer Jun 2016

The Remedy That's Killing: Cuny, Laguardia, And The Fight For Better Math Policy, Rachel A. Oppenheimer

Dissertations, Theses, and Capstone Projects

Nationwide, there is a crisis in math learning and math achievement at all levels of education. Upwards of 80% of students who enter the City University of New York’s community colleges from New York City’s Department of Education high schools fail to meet college level math proficiencies and as a result, are funneled into the system’s remedial math system. Once placed into pre-college remedial arithmetic, pre-algebra, and elementary algebra courses, students fail at alarming rates and research indicates that students’ failure in remedial math has negative ripple effects on their persistence and degree completion. CUNY is not alone in facing …

Go to article

Method Of Lines Transpose: High Order L-Stable {O}(N) Schemes For Parabolic Equations Using Successive Convolution, Matthew F. Causley, Hana Cho, Andrew J. Christlieb, David C. Seal Jun 2016

Method Of Lines Transpose: High Order L-Stable {O}(N) Schemes For Parabolic Equations Using Successive Convolution, Matthew F. Causley, Hana Cho, Andrew J. Christlieb, David C. Seal

Mathematics Publications

We present a new solver for nonlinear parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a one-dimensional heat equation solver that uses fast $\mathcal O(N)$ convolution. This fundamental solver has arbitrary order of accuracy in space and is based on the use of the Green's function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution (or resolvent expansions). These resolvent expansions facilitate our proofs of stability and convergence, and permit us to construct …

Go to article

The Collatz Conjecture And Integers Of The Form 2KB−M And 3KB−1, Patrick Wiltrout, Eric Landquist Jun 2016

The Collatz Conjecture And Integers Of The Form 2KB−M And 3KB−1, Patrick Wiltrout, Eric Landquist

Furman University Electronic Journal of Undergraduate Mathematics

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 2kb−m and 3kb−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 …

Go to article

The Search For One As A Prime Number: From Ancient Greece To Modern Times, Angela Reddick, Yeng Xiong Jun 2016

The Search For One As A Prime Number: From Ancient Greece To Modern Times, Angela Reddick, Yeng Xiong

Furman University Electronic Journal of Undergraduate Mathematics

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 …

Go to article

On The Dead End Depth Of Thompson's Group F, Justin Halverson Jun 2016

On The Dead End Depth Of Thompson's Group F, Justin Halverson

Furman University Electronic Journal of Undergraduate Mathematics

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 …

Go to article

Relative Goldbach Partitions And Goldbach's Conjecture, Houston Hutchinson Jun 2016

Relative Goldbach Partitions And Goldbach's Conjecture, Houston Hutchinson

Furman University Electronic Journal of Undergraduate Mathematics

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.

Go to article

Integers Of The Form A2±B2, Robert Zeman Jun 2016

Integers Of The Form A2±B2, Robert Zeman

Furman University Electronic Journal of Undergraduate Mathematics

This paper explores which integers can be expressed in the form a2±2b2 by using rings of the form Z[√d], particularly when d = 2 and d = −2.

Go to article

Paths And Circuits In G-Graphs Of Certain Non-Abelian Groups, A. Dewitt, A. Rodriguez, Jennifer Daniel Jun 2016

Paths And Circuits In G-Graphs Of Certain Non-Abelian Groups, A. Dewitt, A. Rodriguez, Jennifer Daniel

Furman University Electronic Journal of Undergraduate Mathematics

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 Dn. 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.

Go to article

The Relationships Between Cg, Bfgs, And Two Limited-Memory Algorithms, Zhiwei (Tony) Qin Jun 2016

The Relationships Between Cg, Bfgs, And Two Limited-Memory Algorithms, Zhiwei (Tony) Qin

Furman University Electronic Journal of Undergraduate Mathematics

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.

Go to article

Finding Prime Numbers: Miller Rabin And Beyond, Christina Mcintosh Jun 2016

Finding Prime Numbers: Miller Rabin And Beyond, Christina Mcintosh

Furman University Electronic Journal of Undergraduate Mathematics

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.

Go to article

Carolinas Mathematics Undergraduate Research Conference Abstracts, John Harris Jun 2016

Carolinas Mathematics Undergraduate Research Conference Abstracts, John Harris

Furman University Electronic Journal of Undergraduate Mathematics

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.

Go to article

Notes On Gabriel's Horn, Joseph Krenicky, Jan Rychtář Jun 2016

Notes On Gabriel's Horn, Joseph Krenicky, Jan Rychtář

Furman University Electronic Journal of Undergraduate Mathematics

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.

Go to article

Derham Cohomology Of The Rectangular Torus, Eric M. Katerman Jun 2016

Derham Cohomology Of The Rectangular Torus, Eric M. Katerman

Furman University Electronic Journal of Undergraduate Mathematics

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.

Go to article

The University Of North Carolina At Greensboro Regional Undergraduate Mathematics Conference Abstracts, Jan Rychtář Jun 2016

The University Of North Carolina At Greensboro Regional Undergraduate Mathematics Conference Abstracts, Jan Rychtář

Furman University Electronic Journal of Undergraduate Mathematics

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 …

Go to article

On The Nonexistence Of Singular Equilibria In The Four-Vortex Problem, Marshall Hampton, Andrea Peterson, Heather Stoller, Albert Wang Jun 2016

On The Nonexistence Of Singular Equilibria In The Four-Vortex Problem, Marshall Hampton, Andrea Peterson, Heather Stoller, Albert Wang

Furman University Electronic Journal of Undergraduate Mathematics

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.

Go to article

Properties Of The Iterates Of The Weierstrass-℘ Function, Walter H. Chen, Michael S. Willis Jun 2016

Properties Of The Iterates Of The Weierstrass-℘ Function, Walter H. Chen, Michael S. Willis

Furman University Electronic Journal of Undergraduate Mathematics

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 ℘(z1 + z2), 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 z0 and adding it to itself, both in the fundamental parallelogram C/Γ and …

Go to article

A Dynamical Programming Solution For Shortest Path Itineraries In Robotics, Martin Talbot Jun 2016

A Dynamical Programming Solution For Shortest Path Itineraries In Robotics, Martin Talbot

Furman University Electronic Journal of Undergraduate Mathematics

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 …

Go to article

Vertex Magic, Daisy Cunningham Jun 2016

Vertex Magic, Daisy Cunningham

Furman University Electronic Journal of Undergraduate Mathematics

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 …

Go to article

Some Geometry Of H(RN), Christopher Frayer Jun 2016

Some Geometry Of H(RN), Christopher Frayer

Furman University Electronic Journal of Undergraduate Mathematics

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(Rn). Specifically, we concentrate on understanding lines in H(R). In particular, we show that for any two points A, B, ∈ H(Rn), there exist infinitely many points on the line joining A and B. We characterize some points on the lines formed using closed …

Go to article

Tiling By (K, N)-Crosses, Joanne Charlebois Jun 2016

Tiling By (K, N)-Crosses, Joanne Charlebois

Furman University Electronic Journal of Undergraduate Mathematics

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].

Go to article

Fourier And Wavelet Representations Of Functions, Nicholas G. Roland Jun 2016

Fourier And Wavelet Representations Of Functions, Nicholas G. Roland

Furman University Electronic Journal of Undergraduate Mathematics

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 …

Go to article

A Lie Algebra Of Integrals For Keplerian Motion Restricted To The Plane, Jason Osborne Jun 2016

A Lie Algebra Of Integrals For Keplerian Motion Restricted To The Plane, Jason Osborne

Furman University Electronic Journal of Undergraduate Mathematics

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.

Go to article

Excellent Rings With Singleton Formal Fibers, Dan Lee, Leanne Leer, Shara A. Pilch, Yu Yasufuku Jun 2016

Excellent Rings With Singleton Formal Fibers, Dan Lee, Leanne Leer, Shara A. Pilch, Yu Yasufuku

Furman University Electronic Journal of Undergraduate Mathematics

In this paper we construct a non complete excellent local ring A such that the natural map Spec  → Spec A is bijective.

Go to article

Planar Double Bubbles On Flat Walls, G. Christopher Hruska, Dmitriy Leykekhman, Daniel Pinzon, Brian Shay Jun 2016

Planar Double Bubbles On Flat Walls, G. Christopher Hruska, Dmitriy Leykekhman, Daniel Pinzon, Brian Shay

Furman University Electronic Journal of Undergraduate Mathematics

In this paper, we investigate some properties of planar soap bubbles on a straight wall with a single corner. We show that when a wall has a single corner and a bubble consists of two connected regions, the perimeter minimizing bubble must be one of the types, two concentric circular arcs or a truncated standard double bubble, depending on the angle of the corner and areas of the regions.

Go to article

The Strong Shadowing Property On The Unit Interval, James O. Warren, Joseph H. Brown Jun 2016

The Strong Shadowing Property On The Unit Interval, James O. Warren, Joseph H. Brown

Furman University Electronic Journal of Undergraduate Mathematics

We study continuous maps of the unit interval into itself. We determine necessary and suffcient conditions so that all pseudo-orbits can be approximated by orbits with the same initial point.

Go to article

Quantifying Chaos In Dynamical Systems With Lyapunov Exponents, Michael Van Opstall Jun 2016

Quantifying Chaos In Dynamical Systems With Lyapunov Exponents, Michael Van Opstall

Furman University Electronic Journal of Undergraduate Mathematics

In this paper we analyze the dynamics of a four dimensional mechanical system which exhibits sensitive dependence on initial conditions. The aim of the paper is to introduce the basic ideas of chaos theory while assuming only a course in ordinary differential equations as a prerequisite.

Go to article

Divisability Tests, Apoorva Khare Jun 2016

Divisability Tests, Apoorva Khare

Furman University Electronic Journal of Undergraduate Mathematics

In this paper, we give a new method to test the divisibility of any positive integer by another. First, we outline the usual test, in which one proceeds from right to left, i.e. the direction opposite the one taken while carrying out long division. After pointing out some of the problems with this method, we give another method, which forms the core of this paper. This latter method works along the same lines as the earlier one, but here one proceeds from left to right. Examples of some well known divisibility tests for special divisors are given along with some …

Go to article

Maximum Matchings In Complete Multipartite Graphs, David Sitton Jun 2016

Maximum Matchings In Complete Multipartite Graphs, David Sitton

Furman University Electronic Journal of Undergraduate Mathematics

How many edges can there be in a maximum matching in a complete multipartite graph? Several cases where the answer is known are discussed, and then a new formula is given which answers this question.

Go to article

Compactifications Of Topological Spaces, Jay Blankespoor, John Krueger Jun 2016

Compactifications Of Topological Spaces, Jay Blankespoor, John Krueger

Furman University Electronic Journal of Undergraduate Mathematics

Given a locally compact, Hausdorff space, χ, there are several ways to compactify it. We examine two of these compactifications – the one-point compactification, X̌, and the Stone-Čech compactification, β(χ) – and give conditions which guarantee that X̌ ≠ β(χ). We also give new examples of topological spaces for which X̌ = β(χ).

Go to article

First Prev Next Last