Many people go through life without realizing that mathematics is all around them in nature. Leonardo Fibonacci was not one of these people. Fibonacci saw the connection between mathematics and nature in the simple 'multiplication' of rabbits. His insight helped him to develop the Fibonacci numbers. Even after hundreds of years mathematicians still find ways to apply the Fibonacci numbers to their own work. In this presentation, I will discuss the Fibonacci numbers, Fibonacci’s golden numbers, the manifestation of these numbers in nature, and some of the ways other mathematicians have applied these numbers.

Rene Descartes’ trust in human senses superseded his trust in European theology and spurred his contributions to the scientific revolution of the Renaissance, the historical transition from the Medieval Era to the Modern Age. In this presentation, Descartes’ philosophy and some of his accomplishments in mathematics and physics will be presented in order to illustrate the progress of science in Europe during the 17th Century. The basis of Descartes’ analytical geometry of a cone to produce graphs of an ellipse, circle, parabola, and hyperbola will be used to examine applications to optics and light diffraction with lenses.

The early 1700’s saw the dawn of the Catholic Enlightenment and with it the birth of an intelligent and caring woman in Maria Gaetana Agnesi. The eldest of twenty-one children, Agnesi’s genius was evidenced by her early command of modern languages, her gifted public speaking, and her appreciation for the study of mathematics. In this presentation, I will examine Agnesi’s dilemma: to apply her knowledge to help educate others or to pursue her other passion, caring for the infirm and elderly by joining a convent.

"Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does lightning travel in a straight line." Benoit Mandlebrot. This is obvious, but often ignored, and some still continue to try to squeeze life into the over-simplistic models of Euclidean geometry. Fortunately, fractals have revolutionized many diverse fields. From physics, chemistry, and biology to art, economics, and topology, fractals have changed our view of the world around us. This new way of thinking is due to the 'father of fractal geometry', Benoit Mandlebrot, whose meandering and self-labeled chaotic life led to his search for order in chaos and a new description of so-called chaotic structures in nature. In this presentation, I will discuss Mandlebrot’s progression toward an understanding of chaos and explore the fruition of his work: the Mandlebrot set.

Undergraduate applied mathematics students usually spend the majority of their studies in a one-dimensional world when learning about diffusion and its approximation. The typical reasons for not delving into two-dimensional diffusion studies are complexity and time constraints. But the approximation of two-dimensional diffusion using finite differences presents an excellent example of the concepts of numerical stability and dissipation. As a result, they provide a good introduction to Von Neumann analysis. This presentation will provide a brief explanation of stability and dissipation for the two-dimensional diffusion equation using Von Neumann analysis.

We will give several interesting examples of spirals, explain how they can be described, and present a discrete method for creating them.

We define two 0-1 sequences to be compatible if it is possible to delete certain 0’s from both sequences in order to make the two sequences complementary. The general conjecture is that the sequences will be compatible with positive probability when the occurrence of 1’s within the sequences is some probability less than 0.5. We will consider small finite examples of compatible sequences.

Currently, a person diagnosed with glioblastoma (GBM), a highly invasive cancer of the structural cells in the brain, has an extremely low chance of long-term survival. One of the obstacles in treating GBM is the inability of current medical imaging technology to observe growth at an extremely small scale. Our task, in cooperation with IBM Corporation and researchers at Harvard University, is to develop a continuum model that accounts for both the proliferation and migration of tumor cells. In formulating this model, we will use a system of partial differential equations to describe the dynamics of the tumor and its effects on the surrounding brain tissue. In addition, we will employ finite difference methods to approximate the solution to the system. Our goal is to utilize the model in understanding patterns in the initial stages of tumor growth.

We will describe two new computer programs. The first can generate all the possible vertex magic total labelings (VMTL) for K3, 3. We will show how this program is not computationally feasible for looking for vertex magic labelings of larger complete bipartite graphs. The second program searches for a special VMTL. This program is also useful for large graphs. We will describe the labelings that this program will find, and why they play a central role in the theory.

A general classification problem may be described as follows: Given a multivariate observation, which is known to belong to one of several populations, determine which population is most likely. Traditional methods for dealing with this problem often lack flexibility. Observations are often a ssumed to be normally distributed, for instance. Traditional methods cannot work with categorical data or incomplete data in a natural way. CART (Classification and Regression Trees) techniques are being applied to data from a small private undergraduate institution. Current processing and results will be discussed.

Abstract genetic algorithms (GA’s) are an incredibly useful way of finding good solutions to hard optimization problems in a time-and-space-efficient manner, but they are poorly understood from a theoretical standpoint. One effort to better understand GA’s has emerged through the notion of convergence time. We will define and examine this idea, and compare the convergence times in GA’s that use both ranking and proportional selection. We will see why the idea of convergence time is useful and at the same time limited in analyzing GA performance.

Minimizing long-run average cost per stage in stochastic network problems requires solving the countably infinite system of nonlinear equations generated by Bellman’s equation. One approximation method transforms Bellman’s equation into a linear program, the system of linear constraints that yields the exact solution. We will explore a method of further simplifying this linear program by fitting functions to Bellman’s equation. Our result is a small linear program that is computationally simple to solve for a reasonably accurate average cost.

Placing a magnet on a pendulum and an array of magnets beneath it, the understood behavior of the pendulum gives way to unpredictable orbits and end states. Without knowledge of the exact initial state, the end state cannot be predicted with any degree of accuracy. This is because the pendulum creates fractals along the boundaries of its basins of attraction. I have written software that has allowed me to show that the boundary is a fractal.

For those looking to shave a few strokes off their golf game or master the intricacies of billiards, a good understanding of vectors, scalars, and motion could go a long way. One of the most famous equations, Newton’s Second Law, simultaneously demonstrates the importance of vectors, scalars, and motion. In this presentation we will provide an overview of these notions, discuss the history of the development and application of vectors, and emphasize the relationship to three-dimensional motion.

The ratio of successive entries of the Fibonacci sequence of integers converges to a number which can be used to construct a two-dimensional 'golden spiral'. The 'Tribonacci' sequence of integers converges to another ratio which can be used to construct a similar spiral in three dimensions. I will discuss some of the properties of this spiral and the number on which it is based, which is approximately 1.839286755.

Due to the popularity of Texas Hold’em poker, many mathematical techniques have been implemented to help determine an optimal winning strategy. One important statistic is a hand’s probability of winning (the hand’s winning percentage). This presentation illustrates a new technique that calculates the exact winning percentage of a hand. These values were tested via Monte Carlo simulations. While not equal, the calculated and simulated results are similar enough to validate the approach.

Using a classic differential equation for the deflection curve of an elastic beam, we derive a general mathematical model for a beam subject to a non-symmetric system of loads. The governing differential equation is them solved by a standard calculus method. Four arbitrary constants are determined from natural constraints on the deflection function and its derivative. Examples with specific material and geometric parameters for the beam are given and the absolute maximum of the deflection function is found. If time permits, possible applications in design problems will be discussed.

In most US elections every voter may vote for one candidate and the candidate receiving the most votes wins. This voting system has a number of flaws, most notably the 'spoiler effect'; two similar candidates can lose to a third even though their common policies are supported by a majority. A number of voting systems solve this problem, including approval voting, Condorcet methods, and single transferable vote. Various mathematical criteria are used to analyze these voting systems. The importance of strategy in the design of voting systems will be discussed.

Given its iterative nature, it is natural to study Newton's method as a discrete dynamical system. Of particular interest are the various open sets of initial seeds that fail to converge to a root. We shall examine a certain family of 'bad' polynomials that contain extraneous, attracting periodic cycles. The family relies on only one parameter, and we have developed and implemented computer programs to locate values of this parameter for which Newton's method fails on a relatively large set of initial conditions. In doing so, we have discovered some rather surprising dynamical figures including Mandlebrot-like sets, tricorns, and swallowtails. We have also uncovered analytic and numerical evidence that aids in explaining the existence of such figures.

The Tower of Hanoi problem, in which a pile of different sized disks must move from one pole to another, is solved using a recursive process. In this variation of the Tower of Hanoi problem, two piles of disks must trade places. How can you do this? Does the recursion still work? What is the least number of moves possible? Learn all this and more!