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The idea behind bipolar plots is to extend the traditional polar coordinate system. Complexities arise from simple functions and simplicities arise from "complex" functions. This is a very accessible talk; familiarity with polar coordinates and a little trigonometry is helpful. Graphs are essential in this talk.
I will present a talk on space groups. My object has been to identify certain space groups by investigating the matrix group GL(n,Z) and its finite subgroups. These subgroups were generated with the aid of the computer algebra system Magma. Really, folks, it is fairly interesting!
This research was funded by the Howard Hughes Medical Institute as part of a summer research program at Benedictine University under the direction of Dr. Donald Taylor and Dr. Lisa Townsley.
For my senior paper, I investigated a particular binary operation on strings (blocks) of numbers. The operation is called the block product. In the introduction I provide the audience with background information on the subject of block products. In particular, I make reference to two types of blocks, respectively, circular and cyclic. In the main portion of the presentation, I introduce the audience to the definition of semi-cyclic blocks. I then show that the block product of two circular blocks is circular and the block product of two cyclic blocks is semi-cyclic
This past May marked the 1000th anniversary of the death of Gerbert, the remarkable French mathematician and churchman who became Pope. I will report on the results of a summer research project, in which I tried to determine when and where Gerbert learned about Arabic mathematics.
Discrete dynamical systems known as cellular automata will be presented. We study the long-term behavior of these structures when generated over group and monoid alphabets. This research relates algebraic properties of these generating alphabets, such as those of automorphisms and subgroups, to the long-term dynamics of cellular automata.
Do you remember making cubes from paper in kindergarten? There is an analogous construction for a hypercube from eight cubes, but what are the rules for "folding" them together? Observations from our kindergarten problem can help answer this question.
Busemann points are the points in the metric boundary of a metric space that are limits of geodesic rays. We answer a question posed by Marc Rieffel by giving an if and only if condition for the existence of non-Busemann points of graph metrics.
The Isoperimetric Problem is one that has been pondered since the days of the Greeks. The problem is to find the shape that covers the largest area with any simple closed curve of a given perimeter. Several proofs will be discussed, including a technique called Steiner symmetrization.
An overview of a classic example of a chaotic dynamical system, the Henon map, with special emphasis on creating a nearly complete bifurcation diagram for this map. Computer -generated images will be used to demonstrate the dynamics of this map for various parameters.
Like infinite series, infinite products have some very interesting properties, but the topic is seldom taught in a calculus class. We will investigate some of these properties.
This presentation is about work done for the Youngstown State University physics department. It includes results on cosmic rays, angular correlation of the decay of cobalt, and the mathematics of Poisson parameter estimation.
The random filling of space with parked cars can be applied in the understanding of several other problems. To understand these problems better, I have examined the spatial distribution in the car parking problem. This has been done with serial and parallel methods.
In this talk we will begin with the definition of "freeness" as it pertains to group theory and other objects in abstract algebra. From there, we will investigate other times mathematical freedom occurs, such as in category theory and topology.
A question I'm sure many of you have wondered about before is: How many numbers can I choose from 1 to n such that no pair of them differs by a square number? Let's say we can choose K(n) numbers in that range. If you're anything like me, you certainly hoped that
limn K(n)/n = 0
Luckily this turns out to be true! As if that wasn't enough, a bound on the order of K(n) can (and will!) be established.
In this talk I will discuss the seventeen crystallographic groups and show some representative patterns that I created which have fractal boundaries and tile the plane. Fractals were generated through an iterative process using Visual Basic.
This research was conducted while I was enrolled in the McNair scholarship program at Akron University.
In this talk we present a classical theorem due to Pick. Pick's Theorem tells us how to calculate the area of a polygon based on the number of elementary triangles within the polygon. The proof uses Euler's formula.
Jello, seat cushions, and helium balloons are all very normal objects, which can be difficult to simulate in a computer program. This talk will discuss the mathematics behind physical modeling of 3D deformable objects.
Many math problems have multiple solutions, some better than others. The alternating group A4 provides a counterexample to the converse of Lagrange's Theorem because it has no subgroup of order six. I will explain several proofs which show there is no possible way to obtain a subgroup of A4 of order six.
For what values of n is it possible to divide the integer set {1, 2,...,n} into k subsets of equal cardinality such that the sum of the elements in each subset is a constant? This presentation will explore this question with different specifications of n and k.
It is known that C4 C4 C4 contains an (8, 8, 8, 1) relative difference set. Results of an exhaustive search for other such sets will be shared, along with an overview of additional topics dealing with relative difference sets.
This research was conducted in collaboration with Nick Bauer as part of an REU program at Central Michigan University under the direction of Dr. Ken Smith.
A topological proof of the fact that there are at most five regular polyhedra is given. The proof uses Euler's formula for a sphere:
v + f - e = 2
where v, f and e are the numbers of vertices, faces, and edges, respectively.
Disappearing faces and pennies appearing from nowhere may be magic tricks, but they are also Mathematics! These are just two of the amazing properties of hexaflexagons. This presentation will look at the theory behind the hexaflexagon's unique properties. It will also feature instructions for making these delightful "flexing" puzzles.
While certain molecules have self-symmetries, other molecules do not have this property, but attain symmetries within a crystal structure. We will briefly introduce molecular symmetries, and then discuss crystallographic symmetries in both planar and three-dimensional settings. We will include a discussion of the symmetry groups of different molecular compounds.
A great deal is known about the arithmetic mean, harmonic mean, and geometric mean of real numbers. In this talk we will examine the arithmetic mean, harmonic mean and geometric mean of complex numbers. In particular we will examine properties that we expect such means to possess.
This project provides a methodology for obtaining sets of polynomial coefficients, refinement coefficients or integral displacements for finite polynomial refinement given the other two sets. Transforming the refinement equation of an nth degree polynomial into vectors in Rn utilizing a common basis yields a solvable system of matrices and vectors. Patterns that emerge in this system provide a foundation for many proofs involving refinement of polynomials.
Knots in a cubic lattice are placed in a 3D grid. I illustrate theorems concerning the composition of these knots, finding better upper bounds for the stick numbers of composed knots in relation to those of the original knots. Other theorems discuss composition of trefoil knots and figure-eight knots.
The complements of hyperbolic knots often contain surfaces with interesting geometric properties. However, identifying knots with these totally geodesic surfaces has proven difficult. We will explore extensions of manifolds, termed orbifolds, with unique geometric structures that can generate these special knots.
A look at an argument verifying the statement: If Euclidean geometry is consistent then hyperbolic geometry is consistent. The purpose will be to demonstrate techniques of the proof as well as express the meaning and interest of such a statement and its implications.
I will present several formulas that we found for computing the eccentricity, radius, and center of n-fold tensor products of graphs.
In closely contested elections, what are the chances that any one vote cast will affect the outcome of an election? We will extend the question to consider effects of various sized blocks of voters.
Take a triangle and draw segments from the vertices to determined points on the opposite sides. These segments form a triangle inside the original triangle. Taking inspiration from a 2001 Putnam Exam problem, we will investigate how the sides and area of this inner triangle are related to those of the original.
A classic calculus problem involves determining where to sit on a level floor in order to maximize the viewing angle for a picture on a wall. We will discuss interesting generalizations using calculus and geometry.
