Jacquiline Beckvermit, Sara Linville, Caleb Haag and Travis Morton, Fort Lewis College
Title: Not Your Normal Norm

Abstract: Most mathematics, mainly calculus, is based solely on the Euclidian norm. We will look at what it would be like if, rather than the Euclidian-normed plane, we instead used the “Taxicab” norm. We will give certain definitions and unique criteria to make this norm a genuine norm. We will show how this norm works, how it differs from the Euclidian norm, and if one can embed and rotate a shape, such as a triangle, in the “Taxicab” norm.

Tyler Branyan, Andrew Kreeger, and Melissa Wiederrecht, University of Wyoming

Title: Primality Testing: Comparison of the implementation of AKS and Miller-Rabin primality tests

Our presentation will consist of an introduction to both AKS primality testing and Miller-Rabin primality testing. We will present how we implemented each of these methods and present results based on these implementations. This project was conducted while learning about distributed computational number theory and its implications on cryptography.

Jessica Coyle, Colorado College

Title: Simulation of the Effect of Fire Regimes on Ponderosa Pine Forest Structure

Abstract: Ponderosa pine (Pinus ponderosa) is a tree of considerable ecological and economic interest found throughout western North America.  Ecologists believe that a fire regime of low-intensity ground fires was a primary force that maintained pre-settlement forest structure.  I use an agent-based Java program to investigate the mechanisms by which fire changes forest structure and to simulate the long-term affects of natural fire regimes in Ponderosa Pine forests.  I use statistical methods to analyze the spatial point patterns of the results to determine whether different fire-regimes have different effects.  Results are in progress and will be presented at the conference.

Andreea Erciulescu , Colorado State University

Title: The Root of a Complex Number: An Algebraic Method  and its Applications in Fermat's Last Theorem

Abstract: I have created a method for finding the roots of a complex number in a new algebraic way, different from the familiar graphical method. I  then  apply this to something very special... what else but the great Fermat's Last Theorem?  This project makes use of the polar form a complex number, de-Moivre's Formula, close fields, Fermat's Last Theorem, unique factorization domains and Euclidean rings.

Adam Forland, Western State College

Title: Determining Matrix Products from Vector States

Abstract: A very powerful technique in mathematics is to study an object by studying mappings on that object. In this talk we consider a set of functions mapping 2x2 matrices to the complex numbers. These functions, known as vector states, are constructed from the unit vectors in C². By carefully studying this space of functions, information about the product structure of the group of 2x2 unitary matrices can be deduced. Using complex numbers, linear algebra, and calculus this presentation will outline the background and proof of a construction that recovers the product structure.

Brian Geisinger, Fort Lewis College

Title: The Art of Pursuit and Evasion

Abstract:  Have you ever found yourself in a situation that called for drastic measures?  Have you ever had to make a decision by filtering through an almost seemingly infinite amount of choices?   In this talk, we will cover the fundamentals of differential game theory by looking at basic definitions, understanding how to find an optimal path, and how they relate to a simple motion game.

Peter Lemke, Blackhills State University

Title:  Schwarzschild-like Solution of Einstein Manifold

Abstract: The object of this article is to obtain a Schwarzschild like exact solution to the Einstein Manifold in vacuum with nonzero cosmological constant. That is, where is the Ricci tensor,  is the metric tensor and is the nonzero cosmological constant. We shall discuss some mathematical preliminaries and properties regarding the Riemann curvature tensor, Ricci tensor and the metric tensor, that are useful in the construction and solution of the Einstein Manifold. Finally, we shall present the main theorem and discuss the solution in the context of the classical Schwarzschild solution.

Chris MacLellan and Daniel Peterson, University of Wyoming

Title: P-1 and P+1: The Search for Stronger Pseudoprimes

We discuss the attempt to construct a composite likely to be passed as prime by statistical primality tests.   In particular, we attempt to construct a squarefree composite with p-1|n-1 for all p|n, and p+1|n+1 for all p|n.  This construction is found by sieving for primes conducive toward the desired product.  The sieve conditions and sieve implementation are focused on in detail.

Cole McGee, Colorado State University Pueblo

Title:  Three-Body Dynamics in the Earth-Moon System

Abstract: Background will be provided on the nature of three body problems and their relative value in terms of controlled movement in particular situations. Some of the software developed to simulate and optimize trajectories in the circular-restricted earth-moon system will be explained and sample results demonstrated. Tool development will be summarized, and different problems tackled in the project thus far will be presented. A few sample trajectories going past, or even into orbit, around Lagrange points will be provided. The optimal-control problem will be reviewed briefly, as it is still in development. Finally target goals for the end of the semester will be offered.

Mike Moore, Emporia State University

Title:  To Mock a Mockingbird:  An Overview

Abstract: Ever thought about using birds to explain mathematical logic principals?   In his book “To Mock a Mockingbird”, Smullyan uses birds as combinators in an effort to explain some of the advanced topics found in the field of combinatory logic.  This discussion will focus on an introduction to Smullyan’s birds, their derivations, properties, and logic principles.  This will then lead to the discussion of what it means to have a basis for all birds.  From fond birds to egocentric birds, we will enter a “forest” of discovery.

Note: This talk will be followed by a second related talk by Zheng Yang : Deeper into the Forest of Talking Birds

Jette Petersen, Colorado College

Title: Frobenius Numbers and the Frobenius Level Problem

Abstract:  The Frobenius number of a set of relatively prime integers is the greatest possible integer that cannot be represented as a nonnegative linear combination of elements is the set.  For A = (a, b) and gcd(a,b) = 1, the Frobenius number is given by: ab – (a+b).  In 1996, Ramirez-Alfonsin proved that finding the Frobenius number for sets of three or more elements is NP-hard.  We show how to generate infinite families of G-sets (G(k)) and corresponding S-sets of numbers (S(k)) that cannot be expressed as a positive linear combination of elements from the corresponding G(k).  Given a natural number n, we give a method for solving the Frobenius level problem, i.e. finding the smallest k such that n first appears in S(k).

Christine Roether, Emporia State University

Title:  When Was the Last Time YOU Were at the Zoo?

Abstract: In the fall of 2007, the Emporia State math modeling class was presented with a unique opportunity.  The director of the local zoological park requested assistance in determining the typical number of zoo visitors.  With no prior statistical analysis or records present the students collected information, ran analysis, and developed conclusions all based on this data.  This presentation addresses the data in relation to the numerical analysis and final results.

James Spotts, Western State College

Title:  Infinite Factorization Using Weierstrass’ Factorization

Abstract: The fundamental theorem of algebra tells us that a polynomial equation of degree n has n zeros.  The questions we will address in this talk are; can we factor functions with infinitely many zeros?  If so, then what kind of representation will this require?  Weierstrass’s factorization theorem addresses these questions.

Katie Sowards, Adams State College

Title:  Finite Groups as Matrices

Abstract: The collection S_X of all permutations forms a group under composition of functions, and the collection P_n of all permutation matrices of size nxn forms a group under matrix multiplication.  Cayley’s Theorem uses group actions to show that every finite group is isomorphic to a subgroup of a symmetric group S_n.  We have shown that every symmetric group is isomorphic to a finite collection of permutation matrices.  Together, we have that every finite group can be represented by a group of matrices.  Sometimes these representations can be large and we may want to find if they are reducible so that we have a smaller representation with which to work.  An interesting way to find all of the irreducible representations of a group is to use group algebras, idempotents, and character tables.

Kami Wilson, University of Colorado at Boulder

Title:  Fragmentation of Bacterial Aggregates

Abstract: Flocculation, whereby particles aggregate into clusters, or flocs, is a widespread phenomenon exhibited by many planktonic lifeforms in suspension. While many microbial communities flocculate, such as brewers yeast, marine algae, and sewage treatment plants, not much theoretical work has addressed the behavior of bacterial communities. A moment’s reflection reveals the significantly greater challenges associated with the analysis of this fundamental state of multicellular existence; their polydisperse, diaphanous, and fractal nature defies straightforward experimental quantification and modeling. In our project, we are using geometry, physics, and Principal components analysis (PCA) to model the yield stresses as well as identify the plane along which fragmentation is most likely to occur.

Zheng Yang, Emporia State University

Title:  Deeper into the Forest of Talking Birds

Abstract: Based on Smullyan’s famous book “To Mock a Mockingbird,” we will discuss several important aspects of birds in Smullyan’s forest that stand for different kinds of combinators. We will not only use them to derive more kinds of birds, but also find the smallest basis of birds possible. With this analogy in hand, we can easily explore topics like logical and arithmetical birds that will lead to Gödel’s Incompleteness Theorem.

Note: This is a follow-up talk to Mike Moore's: To Mock a Mockingbird