2010-2011 Seminar Series
One Sweet Orchard Problem
Consider standing in the center of a circular orchard with trees planted neatly in a rectangular grid. If the tree trunks are too big we will not be able to see out of the orchard. In this talk, we will find out just how big the trunks need to be to completely obscure our view out of the orchard.
Monday, May 2
Mathematical Methods of Rotation
Rotations are one of the most intuitive operations we have, yet formalizing a verbal definition can be extremely difficult. For this we will turn to the language of mathematics. We will look at two different methods of modeling rotations in the plane, 2x2 rotation matrices, and complex multiplication. Generalizing to 3-Space we will find that the complexity of rotation matrices goes up significantly, and will wonder if a higher dimensional analog to the structure of complex numbers can be found. It turns out there is, it is the structure known as quaternions. By looking at quaternions we will see how 3D rotational modeling can be understood in an intuitive fashion and arrive at some interesting results.
Friday, April 29
Final Digit Strings from Cubes
There are many properties associated with the integers Z. Integers are referred to as primes, some are perfect, others are squares, and some are cubes. Examples of cubes are 1, 8, 27,…, they represent the volume of a cube with integer side length. Unfortunately not all integers are cubes, however more numbers than we think have cubes involved with their digits. For example, let s be the string of decimal digits 3141596535 … 5275045519 formed by the first billion digits of pi. Is there an integer n such that n3 ends in s? We will reveal the answer to this question and more, but be sure to bring your calculator so you can help find the solution.
Friday, April 22
Oscillation within Nerve Cells of the Heart
The human heart beating can be described as an oscillation. As the heart beats, charge builds up within the cells of the heart and is then released causing the muscle to contract. Once the charge has been released, charge begins building one again in the heart and the release of that charge is triggered by the Sinoatrial (SA) node. The SA node is the body's built in pacemaker that keeps the heart in normal rhythm. In the event of an unhealthy heart, an artificial pacemaker is implanted in the heart to maintain a normal rhythm.
I will be studying the oscillation within nerve cells of the heart. This will require understanding simple harmonic oscillators, forced harmonic oscillators, and relaxation oscillators. Possible models to be considered include the Hodgkin-Huxley model, the Van der Pol oscillator, and the Fitzhugh-Nagumo model.
Wednesday, April 20
The Game of SET
SET is a game in which a player creates a “set” from cards with geometric images. The game begins with twelve cards placed on a table, where the goal is to select three cards that together form a “set”. A set is formed by three cards who either share an attribute on the cards or they all differ in an attribute on the cards. With this in mind, how many cards can we lay down to be played without ever containing a set? Using a four dimensional vector field over three element, we can model SET and build an argument for the number of cards not containing a set.
Wednesday, April 13
Generalizations of Cantor’s Uncountability Proof
In the late 1800’s, famous Mathematician Georg Cantor proved the real numbers are uncountable with his now famous diagonalization argument. Although this proof is the most commonly know version, it was not Cantors first. Almost a decade and a half before his diagonalization argument Cantor published a different paper in which he proved that the real numbers are uncountable. In this presentation I will discuss Cantor’s first proof and extend it with the use of perfect sets to prove R^k is uncountable.
Monday, April 11
Algebraic Coding Theory
When information is transmitted, it is very possible that the received information will have errors. Algebraic Coding Theory is a branch of mathematics that improves the accuracy of such transmissions. Whether it is the simple scanning of a product at the grocery store, or the transmitting of an image from a satellite, we find algebraic codes in almost every aspect of modern communication. This important field of mathematics implements ingenious strategies to detect, and sometimes correct errors in messages. In this presentation, I will be exploring several of theses strategies, including the Hamming Code.
Wednesday, April 6
Sickness is something that seems to always be traveling through the WSC student netwok. We studied network properties and network immunization concepts. In our research we explored two different measures of centrality or importance to the network. Our goal was to determine which of these measures would give us the appropriate number of persons and which persons to immunize to stop a disease from spreading though the WSC student netwok.
Monday, April 4
A Mathematical Fry Cook
The carnival food business is all about efficiency. Profits depend on quickly delivering as much freshly prepared food as possible to hungry customers. In the case of fried dough, or ``elephant ears," more ears can be produced by efficiently packing them in the fryer. In an attempt to fit more elephant ears in the fryer than conventional methods allow, we explore packing circles and irregular objects, as well as the packings' relationship to tiling.
Friday, April 1
Modeling Disease Spread in Complex Networks
Networks can be used to describe many real-world systems in areas such as biology, technology, and sociology. Several models in mathematical epidemiology have been extended to complex networks. I will use the susceptible-infected-susceptible (SIS) model to examine the spread of infectious diseases in various types of networks. I will derive the critical spreading rates required to produce endemics in these networks. I will also discuss the results of simulations we performed using real networks of Western State College students.
Wednesday, March 2