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Steven J. Miller, Brown University ABSTRACT: Let RS (resp., RA) denote the average number of runs scored (resp., runs allowed) in a baseball game by a team. It was numerically observed years ago that a good predictor of a team`s won-loss percentage is RS2 / (RS2 + RA2), though no one knew WHY the formula worked. We review elementary concepts of probability and statistics and discuss how one can build and solve a model for this problem. In the course of investigating this problem we discuss how one attacks problems like this in general (what are the features of a good model, how to solve it, and so on). The only pre-requisite is simple calculus. Hilary Spring, Mount Holyoke College ABSTRACT: The ability to study the interior of an object without destroying it is an important industrial tool. We present a method for using the steady state heat equation to access the interior of a two-dimensional region of known material using only the boundary information. Robert Paul Chase, UMass Amherst ABSTRACT: Systems of Partial Differential equations have solutions that exhibit wavelike phenomena like recognizable features traveling at constant speed and interactions between waves. Glimm`s scheme, proven in the 1960s proves that given initial data, hyperbolic systems of partial differential equations can be solved locally. Actually implementing this scheme is the topic of my senior thesis, and I hope to share this with the conference. Xing Ni Chen, Union College ABSTRACT: Although there is extensive ongoing research into two-sided matching problems, there even the simplest marriage matching problem remains interesting. In this paper, we deal with the model wherein a marriage market consists of n men and n women. Each of the 2n people has a linear preference among those of the opposite sex, and n couples will be formed through this matching. A matching is said to be stable if we cannot find a man in one pair and a woman in another pair each of whom prefers the other to their present partner. We show that the set of all stable matchings can be represented by a distributive lattice, in which each stable matching corresponds to a point in the lattice. Furthermore, we discuss Charles Blairs proof that for every distributive lattice, we can find a marriage matching game whose set of stable matchings is isomorphic to this lattice. There exists a men-optimal stable matching and a women-optimal stable matching. The men-optimal stable matching is found at the very top of the corresponding lattice, and the women-optimal stable matching will be found at the very bottom. Alternatively, if we put the women-optimal matching on the top, then the men-optimal matching will have to be at the bottom. More generally, the lattice order reflects opposing collective preference of the men and the women. Maksim Sipos, Ithaca College ABSTRACT: In this talk we discuss properties of convex figures of constant width in the Euclidean plane. In particular, we show how to continuously constuct the family of Reuleaux polygons starting from a circle while preserving the constant width in the intermediate steps. Bill Dunbar, Simon's Rock College of Bard ABSTRACT: The fact that the infinite sum in the title converges to a finite number is proven in most calculus courses. The exact value of the sum is often not revealed, though. Euler found it first, and I'll use a proof which is similar to the one he gave. It is quite elementary, barely using calculus. Jennifer Richinick, Keene State College ABSTRACT: A triple of positive integers (a, b, c) is a Pythagorean Triple if and only if (a2) + (b2) = (c2). Integers a and b will also be the lengths of the sides of a right triangle and integer c will be the length of the hypotenuse. Let d equal the length of the segment that is perpendicular to the hypotenuse and that passes through the vertex of the right angle. It can then be proved that 1/(a2)+ 1/(b2) = 1/(d2). The presenter will call the triple (a, b, d) an Upside Down Pythagorean Triple. The objective is to determine the Upside Down Pythagorean Triples that are integers and discuss some of their properties. Todd Shayler, Williams College ABSTRACT: Given a knot K, what is the minimum number of rigid sticks necessary to construct K? This invariant is known as the stick number of K. We will focus specifically on torus knots, discussing some known results and presenting new research. Matthew Danziger, Hamilton College ABSTRACT: Do you remember the card game War? Two-player war is a game where both players get half the deck and turn over their top card to determine who has the higher card. It is a game of pure luck, but what would happen if we introduced an element of cheating? We change the game by viewing our bottom card after seeing our opponents card. We could then make a more informed decision on whether to use our unknown top card or known bottom card. Does this form of cheating give us a mathematical edge? My talk will show the mathematical modeling involved in determining strategies that give us the greatest advantage. Gregory Quenell, SUNY Plattsburgh ABSTRACT: Draw line segments connecting (0,9) with (1,0), (0,8) with (2,0), (0,7) with (3,0), and so on. The upper right edge of the resulting pattern suggests a curve, called the envelope of this family of line segments. We discuss an elementary way to find an equation for such a curve, and explore some of the places where envelopes turn up, such as game theory, sliding-ladder problems, and arts and crafts. Jasper G. Burch, Saint Lawrence University ABSTRACT: A Norm is defined by || * ||p = ( |e1|p |e2|p |e3|p ... |en|p )1/p . In this presentation, the norm will be examined as a function of p. It will be shown to be a monotone decreasing function and || * ||? will be found. Cases when 0 < p < 1 will then be examined. Properties of the norm as p approaches 0 will be explored. Ryan Decker, Siena College ABSTRACT: The integral _{a}^{2a}(1/x)dx=ln|2| takes the same value for any choice of positive a. It is natural to ask which other functions have this property, namely for which functions f(x) so that _{a}^{2a}f(x)dx=ln|2| for every positive a? We prove that the answer to this question depends on the smoothness of f. If f is analytic then the solution is essentially unique, but for any finite degree of smoothness we can construct many such functions by using the fact that such a function satisfies the functional equation 2f(2a)=f(a). (Note: If you would like a .PDF copy of the abstract, e-mail my supervisor Jon Bannon at jbannon@siena.edu) Krista Sueltenfuss, Westfield State College ABSTRACT: Everyone has played tic-tac-toe at one time or another, whether it was on a placemat at a restaurant, in sidewalk on the playground, or even in the middle of class on the side of your notes. The object is obviously to get three of your X or O in a row first. But have you ever considered what would happen if the surface was not flat? What if you played it on something like a torus or a Klein bottle? We will not only discuss how to play, but also how strategies and rules change. Julie-Anne Shaw, Westfield State College ABSTRACT: Regression Analysis will be used to explore music melody sequences. Consider a given melody sequence that consists of n consecutive notes. For each note in the sequence (x = 1, 2,..., n) we can measure: y = total distance traveled in the melody sequence after x notes. The total distance after x notes is the cumulative sum of the distances between (x 1) pairs of adjacent notes. The distance between each pair of notes is measured in (musical) half steps. This model (y as a function of x) is a monotone function which can be approximated and studied using a linear model. In this project, we will focus on melodies from the music of The Beatles. After deciding on a set of songs to examine, we will choose two musical phrases from each song, and transform this musical data into numerical (x, y) data. For each musical phrase, we will construct the graph of y vs. x, and will use linear regression to compute the slope (which in this model will be a measure of variability in the melody sequence), yintercept, and correlation coefficient. We will explore these results, looking for (and interpreting) patterns both within songs and between songs. Stephen C. Sawyer, Westfield State College ABSTRACT: The recent popularity of the Sudoku puzzle has created a lot of interests in our department on not just own to solve the puzzle, but how to create one. With this in mind, I have done statistical research that will help to detect different properties of the puzzles such as difficulty, how to better solve the puzzles, and ultimately try and discover how to create and rank a puzzle on my own. Jessica Jones Scannell, Simmons College ABSTRACT: Using transition matrices, this project looks at the projections of wild Atlantic salmon (Salmo salar L.) populations under current conditions as well as under several different types of perturbations. Howie Austin and Samuel Patt, Skidmore College ABSTRACT: We will discuss the history and development of check digits by examining ISBN, UPC, Luhn's Algorithm, and CRC's. We will describe the algorithms used and how and what types of errors they can detect and correct. Tom Kern, Dartmouth College ABSTRACT: Mathematical statements and the ways they are logically related form an important foundation for all of mathematics. Knowing the structure of the entirety of logic would be useful, so I will draw a picture of it, and examine its shape. Jordan Volz, Bard College ABSTRACT: The figure equation is a geometric process for determining the characteristic polynomials of a graph. Combining the figure equation with the algebra of graphs we derive many interesting formulas for the characteristic polynomials of new graphs. Cory Jemison, Hamilton College ABSTRACT: Ever wonder what happened to all those people who built the giant statues on Easter Island? I will discuss the natural and social conditions on Easter Island and show through mathematical modeling how these conditions eventually brought about the demise of the Easter Island society. John Adams, Union College ABSTRACT: A graph defined by a set of vertices and edges can be represented by a unique polynomial, called the Tutte polynomial, from which many attributes of the graph can be derived. For example, one recent result relates the expected length of minimal spanning trees of random graphs to an integral of a function of its Tutte polynomial. In this talk, I will present my algorithm for the computation of the Tutte polynomial using the idea of linear dependence of edges in a graph. Jane Gimian, Williams College ABSTRACT: Infinity is an ancient concept. Yet, until 150 years ago, it was mistakenly believed that all infinite things are equal in size. For, how can their be anything bigger than that which goes on forever? Today, however, thanks to the work of Georg Cantor in the late 19th Century, we not only know that there are different sized infinities, but we can even create a continuum of infinities in order by size. This talk will explore the definition of aleph numbers that are used to designate the size, or cardinality, of infinite sets. We will also briefly look at the continuum hypothesis and the difficulties of finding the exact placement for the cardinality of the real numbers among the cardinalities of other infinite sets. Jacob Mitchell, Westfield High School ABSTRACT: The Traveling Salesman Problem is a popular problem among mathematicians and computer scientists because it is probably the simplest example of an NP-hard problem. Rather than finding the shortest route through a brute force search, it is significantly less time consuming to use an approximation algorithm. This lecture will involve an introduction to TSP, the difficulties in solving it, and discussion of an original approximation algorithm that utilizes ellipses. Don Kaupelis, SUNY Plattsburgh ABSTRACT: High school mathematics is likely to be one of the least favorite subjects in America`s public schools. Many students suffer math anxiety and are often bored in their math classes. What if this wasn`t so? What if, for example, in Geometry quadratic equations were presented as Omar Khayyam saw them in 1100 AD? This talk will be looking at creative ways to integrate history of mathematics into a high school classroom, in order to create intrest in mathematical topics and reduce math anxiety. Kristen MacMurray, St. Lawrence University ABSTRACT: Assume every person in a group of people has a unique tidbit of gossip to share. How many conversations must occur before everyone in the group knows all the gossip? It depends on what we assume about the conversations. The gossip number assumes that conversations occur between two people who tell each other everything they know. The email gossip number assumes that one person shares all the gossip that he or she knows with all his or her friends in a mass mailing. We discuss some interesting results about the gossip number and the email gossip number of a graph. Jeff Cluckey, St. Lawrence University ABSTRACT: The optimal assignment problem discusses how to assign workers to jobs in the most effective way, given a measure of how effective each worker is at each job. We discuss a solution to this problem and some applications. Mallori Morrison, Westfield State College ABSTRACT: How do you teach students in your class if they are terrified of Mathematics and the material being covered in your course? How do you teach students who are dealing with anxieties all semester long? Come learn how we alleviate these pressures in our classrooms. Take part in a non-traditional geometric learning activity that will give you an inside look at insights we give prospective teachers that they will later be able to use in their own classrooms. Eric Frazer Lock, Hamilton College ABSTRACT: A summary of research conducted on 3D fractal trees as part of the Research Experience for Undergraduates program at Ithaca College. First, we outline a method for constructing and classifying fractal trees in three dimensions. We go on to give conditions required for self-contact in symmetric trees and prove that the canopy of a self-contacting tree forms a connected surface. In addition, we provide an example of a tree that fills three dimensional space, and show that a tree constructed similarly in n-dimensions will fill n-dimensional space. Kaitlyn Berletic and Lauren Rudowsky, Manhattan College ABSTRACT: The game of Left, Right, Center involves three dice and at least three players. Each player starts out with three tokens. Within each turn, the dice determine what the player does with his or her tokens. Each die has a face with the letter L, the letter R, and the letter C. The remaining faces contain a dot. Left, Right, Center is a game of probability. We have written a program using the computer algebra system Maple that simulates this game. Using the simulation we are able to analyze statistically Left, Right, Center. Our investigations will include questions such as: If you want to win the game, does it matter where you sit? Jonathan Prigoff, Williams College ABSTRACT: Every day, millions of people use RSA encryption to send and receive all kinds of private data, including credit card numbers, bank information, and trade secrets. But what exactly does that little lock icon at the bottom of your computer screen mean? This talk will detail how RSA works and why it is considered secure, but also touch on what possible holes exist in the system that many of us take for granted every day. Will Bastian, Tim Gildea, Skidmore College ABSTRACT: We will adress the Lempel-Ziv-Welch compression algorithm from a coding theory standpoint. That is, we will address the process under which the dictionary is created and furthermore examine the entropy of the compressed string. Katie Lerch and Megan Schoellhamer, Skidmore College ABSTRACT: We will be discussing the life and work of Claude Shannon, specifically his discoveries about the mathematical theory of communication and solving the problem of most efficiently transmitting information. We will discuss in depth the notion of entropy and his source encoder channel decoder destination model. Eric Callahan & Bryan Reynolds, Ithaca College ABSTRACT: The home field advantage in baseball is unique in professional sports. We look to analyze this home field advantage by comparing baseball to other professional sports. Additionally, we will compare home field advantage between American League and National Leaque, and the distribution of of the winning run. Billy Jackson, Westfield State College ABSTRACT: If I were to tell you that I am a liar, would you believe me? I cannot be a liar because then Id be telling a true statement and not lying. But if I were not a liar, then saying so would be a lie. In this talk, I will explain, try to make light of, or show loopholes in some well-known paradoxes such as Zenos Paradox and Aristotles Wheel Paradox. Josh Bolton, Williams College ABSTRACT: Every continous function on the interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. I will look at several different applications involving this astonishing theorem. Robert Suzzi Valli, Manhattan College ABSTRACT: The matrix-tree theorem can be used to show that there are 384 ways to cut open a cubical surface and unfold it into a planar "net." The number of distinct planar shapes obtained is bounded by the number of orbits the cut patterns fall into under the action of the isometry group of the cube. Each cut pattern is a spanning tree of the vertex-edge graph, so we obtain the number of orbits from Burnside`s lemma after finding the number of spanning trees fixed by each isometry. We then describe a combinatorial process on the unfolded shapes themselves that generates enough unfolded shapes to attain the upper bound. Catherine Sheard, Hugh C. Williams High School and St. Lawrence University ABSTRACT: Sudoku, a highly addictive game that has swept the nation, involves numbers but is it math? Placing the numbers 1 through 9 in little boxes so that each digit appears only once per row, column, and sub-square isnt math, is it? Yes, it is! This talk will explore the mathematics behind generating and solving Sudokus, and will also present the history behind these puzzles. From its humble beginnings as Eulers Latin Square to its recent explosion throughout Europe and America, Sudoku has sent puzzle enthusiasts scrambling for erasers and computer programmers scratching their heads over the complexity of these seemingly simple grids. Nick Gallucci, Williams College ABSTRACT: In 1976 two mathematicians solved a century old conjecture dealing with the coloring of maps. For the first time an important proof was solved that relied heavily on computers for evidence and computation. While their results were interesting and groundbreaking, it was their methods that really ushered in a new age of mathematical possibility. Come and see what they found and how far weve come. Kristin Retenski and Hillary Price, Skidmore college ABSTRACT: R.W. Hamming`s work in coding theory, including error detecting, error correcting, and his method of decoding. Isaac Gerber, Williams College ABSTRACT: Infinity. You use it. I use it. But what is it? Where did it come from? Who invented it? Does it really exist? To try and answer some of these questions, we will go on a trip back to the origins of our modern concept of infinity. Beginning with the ancient Greeks, our quest of this most elusive of quantities will have us travel through centuries of mathematical thought. During the course of our voyage we will visit such thinkers as Aristotle, Galileo, and George Cantor to see just what they had to say about infinity. Julie Shumway, Marlboro College ABSTRACT: A cwatset is an algebraic structure related to groups and vector spaces. The fact that the sum of two cwatsets may or may not be a cwatset gives rise to the idea of "prime" cwatsets. We prove some sufficient conditions for a cwatset to be prime, discovering an infinite class of prime cwatsets along the way, and finish with a theorem on the sum of two cwatsets. Joseph Gangestad, Williams College ABSTRACT: Rare eclipses of the Sun and stars can be seen on Earth only along a narrow path, but to make the most of these events the paths must be known to exacting detail years in advance. We will demonstrate how to compute such a path. Kate Harkey, Williams College ABSTRACT: The Fibonacci sequence adds the previous two numbers to get the next: {1,1,2,3,5,8,13,21,34,....}. There are many interesting applications. In music, it is used to determine tunings, and in visual art to dtermine the length or size of content or formal elements. My Nguyen, Williams College ABSTRACT: The powerful theorem, which states that "every bounded sequence in R has a convergent subsequent," was first proven by Bolzano, a Czech mathematician, and developed independently and published years later by Karl Weierstrass, a German mathematician. We will prove the theorem by proving two lemmas, which state that " a bounded monotone sequence converges," and that "every sequence has a monotonic subsequence." Nathan Cook, Williams College ABSTRACT: The concept of the infinite has captivated and befuddled us for ages. What is infinity? How many infinities are there? Are there infinities beyond infinities? We will discuss the nature of infinity through the analysis of sets and cardinalities and unravel some of these mysteries. Matt Furmaga, Williams College ABSTRACT: Group theory is a fundamental branch of abstract algebra and the backbone of early encryption schemes. I`ll discuss zip codes, bar codes, credit card schematics, and the Diffie-Hellman public key exchange. Pierre Bordeaux, Williams College ABSTRACT: Seemingly abstract mathematical ideas can be found in some of the strangest places. We can use fractal geometry to describe patterns in nature and the volatility of the stock market. Phi seems to be inexplicably linked to aesthetic beauty. But, perhaps one of the More surprising appearances of mathematics in our lives is its presence in one of the biggest blockbuster trilogies of the past decade. The underlying principles of The Matrix storyline rely heavily on one of the most profound mathematical theorems of the past century, Gdels incompleteness theorem. What was the Architect rambling about? What does being the One really mean? What in the world happened at the end of the last movie? All these questions and more will be answered by examining the incompleteness theorem and applying our new knowledge to understanding the mathematics behind The Matrix. A basic understanding of The Matrix Trilogy plot structure is assumed. Emily Luidens, Hamilton College ABSTRACT: What is 2? It is the number of my hands: it is the amount, in ounces, of water left in my glass: it is the number of people that compose a couple. While the concepts of hands, water, and people are indubitable and not difficult to define, the number 2 is, in Bertrand Russells words, a metaphysical entity about which we can never feel sure that it exists or that we have tracked it down. Russell defines 2 to be the class of couples the label for a class that contains all sets of 2 objects. Similarly, Gottlob Frege claims that numbers are independent objects that can be defined in terms of sets. Paul Benacerraf, on the contrary, after concluding that numbers are not sets, claims that numbers do not exist at all! In my talk, I will explore and compare Freges and Benacerrafs opposing views, the former of which I believe to be more convincing. Dane M. Johnson, Hamilton College ABSTRACT: As an exercise in real analysis, we were asked to prove that every bounded sequence of real numbers provides a linear transformation from the vector space of all absolutely summable real sequences into R. That is, the space of bounded sequences is a subspace of the so-called algebraic dual space of the space of absolutely summable sequences. A question that interested as well as frustrated me was whether or not there were other such linear transformations on this vector space. I will demonstrate that there are infinitely many of them, although an explicit representation of them is elusive. Jennifer Haghpanah, Quinnipiac University ABSTRACT: The first recorded Chinese Remainder problem was stated by Sun-Tsu in Suang-Ching. This problem has evolved over the past 1500 years into a modern theorem, and is useful in solving a system of congruences. In this talk, I will discuss the history of the Chinese Remainder Theorem and demonstrate a solution to Sun-Tsu`s problem. Darlene M. Olsen, Saint Michael's College ABSTRACT: Did you ever ask the question of how many possible ways there are to tie a necktie? Furthermore, what factors determine an aesthetic tie knot? This problem can be answered using mathematics. We will discover the mathematical ways for describing how to tie necktie knots. We will also classify knots according to their size and shape. You will be provided with a list of all 10 aesthetic knots as determined by Thomas Fink and Yong Mao Meghan Proudman, Westfield State College ABSTRACT: What can go wrong in a Calculus classroom? Students may rely too much on the instructor and not have any independent work, or students may fail to communicate with each other during group work. These are only some of the problems I will be discussing along with different solutions to these issues. Conor Quinn, Williams College ABSTRACT: A regular tiling is one in which any flag (the union of an edge, vertex, and tile) can be mapped onto another tile by a symmetry of the tiling. It has been shown that the only regular tilings are those of the square, equilateral triangle, and regular hexagon. We will explore this definition and analyze some of the properties of the symmetry groups of simple tilings. Matthew Rogala, Ithaca College ABSTRACT: Using an operational technique, we will find the Laplace Transform of the Bessel Functions. Then, we will explore possible connections between the Bessel Functions, the Fibonacci Polynomials, and other Orthogonal Polynomials. Jolie Baumann and Peta-Gay McCarthy, Ithaca College ABSTRACT: The primary focus of our research is Chebyshev Polynomials. We used the recursive definition to derive an explicit formula for these polynomials. After this preliminary work, we continued to search for relationships between Fibonacci Polynomials, Chebyshev Polynomials of different types, Trigonometric functions, and Bessel functions. Mike Califano, College of Saint Rose ABSTRACT: This talk will include the uprise of mathematics in the early western civilazations such as in Babylonia, Egypt, Greece, and early Europe. Topics being discussed will include the uprise of number systems, arithmetic, and geometry as well as some algebra, logic, and number theory. Dustin Adams and Justin Ryan, Ithaca College ABSTRACT: The Gambler`s Ruin problem involves two players who simultaneously bet capital and play a game with the ultimate goal of winning all capital. For illustrative purposes, we use a two-sided coin in our game. We ultimately derive a solution to the difference equation explaining the probabiltiy of a player winning a game with N total dollars beginning with i dollars. The same procedure is followed to solve for the average number of plays a particular game will last. All of this information is then used to predict probable outcomes of the game. We conclude with a discussion of different stopping strategies. Katherine Dieber, Williams College ABSTRACT: In 1991, Kenneth and Andreas Hsu found Bachs Invention #1 seemed to keep its self-similarity after several iterations of thinning, that is, removing every other note. I will show that a random sequence of notes has the same behavior. Jillian Cavanna and Brett Hotchkiss, Ithaca College ABSTRACT: We examined the geometric implications of culturally influenced patterns, with emphasis on Islamic star patterns. We applied the ideas of group theory, including dihedral groups and frieze patterns to determine the mathematical characteristics of these patterns. The educational applications for our research include methods for reproducing such patterns in the program Geometers Sketch Pad- examining complicated mathematical principles via basic geometric functions. Variations of these ideas were also applied to questions of tiling the plane. Michael Fiske, Westfield State College ABSTRACT: What else can go wrong in a calculus classroom? Students may express mathphobia and anxiety, or plagerize their assignments. This will be an extension on the previous TA talk. Amelia E. Stein, SUNY Institute of Technology ABSTRACT: Proprietary information stored in databases is vulnerable to attack when left unprotected. We are interested in evaluating the cryptographic products and protocols to surmise the best results. We will discuss the procedures used to test selected encryption products for security and performance capabilities. We look at the performance, security, usability, and scalability of the selected products. Based on analysis of the given results and research, the recommendations are made for the best cryptographic protocol. Diane Lunman, Nazareth College ABSTRACT: We`ve seen elliptic curves used in such areas applications as Andrew Wiles proof of Fermat`s Last Theorem, the factorization of integers and primality testing. In this talk we will explore the uses of elliptic curves in cryptology, their benefits and faults as well as some interesting characteristics. Nela Vukmirovic, Williams College ABSTRACT: Firstly, this talk will explore edge-to-edge tilings of regular polygons. If we require that all the vertices are of the same type, there are exactly 11 distinct tilings, including the three regular ones. Secondly, if we consider monohedral tilings whose vertices are regular, again there are only 11 possible combinations. Finally, we consider the correspondence between the two and the principle of duality. Elizabeth Schwartzman, Williams College ABSTRACT: Factorials are easy, right? Well, the larger the numbers get, it can get nearly impossible to compute them quickly. Thanks to a derivation initiated by DeMoivre and finalized by Stirling, we can approximate n! for large values of n, and even calculate the error using methods faster than the conventional multiplication formula. Joy Kogut, Simmons College ABSTRACT: The population of the United States of America is constantly growing. In long term planning offices such as the Social Security Administration, the predicted population for the future is vital information to determine the course of the system. To evaluate whether or not the Social Security system will be able to support the retired workers in future years, we must predict the population. A support proportion of retired to working people helps estimate the ability of the current system. If this proportion is small, the system can support the retired. In order to predict the population accurately, the 2000 census provided the necessary data needed to model the population growth. Assumptions about birth rates and mortality rates within age groups were created in order to build the model. Modeling the years from 2000 until 2040 showed the predicted growth in the population increased approximately 70 million people. The overall population increase, however, does not show context. In the year 2000, the support proportion is roughly 0.24 while in 2040 the proportion increases roughly to 0.45. We examined different assumptions for this model to see how sensitive the model is to various estimates for birth rates, death rates, etc in the model. The goal of creating a mathematical model is to create realistic and accurate assumptions for the model and use those assumptions to make accurate predictions. Reed Tinsley, Wheaton College ABSTRACT: ABSTRACT: Let F(n) denote the n-th term in the Fibonacci sequence F(0) = 0, F(1) = 1, F(2) = 1, 2, 3, 5, 8, 13, 21 . . . It is easy to see that 2 divides the third Fibonacci number F(3) = 2, that 3 divides F(4 )= 3, that 5 divides F(5) = 5, and that 7 divides F(8) = 21. In this talk, we will see that if p is any prime > 5, then p divides either the (p-1)th Fibonacci number F(p-1) or the (p+1)th Fibonacci number F(p+1), but not both. Jennifer Weiser, Wheaton College ABSTRACT: The Four Color Theorem is well known in Graph Theory, which allows us to color any map with at most four colors. But what happens when a country owns more then one area in a map? This brings forth the idea of empires, since each area belonging to a country requires the same color. This talk will show how empires have an application to testing printed circuit boards. Christie Mallet, Westfield State College ABSTRACT: Have you ever wondered in how many ways a polygon can be cut into a triangle? Or how about the number of ways that parenthesis can be placed in a sequence of numbers to be multiplied? The sequence 1,1,2,5,14,42,132,.....known as Catalan numbers will give you the answer. These numbers first appeared in 1838 when Eugene Catalan showed that there are Cn ways of parenthesizing a nonassociative product of n+1 factors. According to Richard P. Stanley there are at least 66 combinatorial interpretations of Catalan numbers. We will look at a few sets and show why they describe Catalan numbers. Ashley Goyette, Westfield State College ABSTRACT: Catalan numbers, founded by Eugene Charles Catalan, is the solution of the problem of dividing polygons into triangles with diaginals that do not intersect. They are shown by the explicit formula Cn = ((1/(n+1))(2n)!/((n+1)!n!). In my talk I would like to show how Catalan numbers and the Pascal triangle relate using the different patterns within the columns formed by the numbers in the triangle. Andrew Canaday, Marist College ABSTRACT: In this talk, we discuss computational methods for solving large linear systems of the form Ax = b. Householder transformations can be employed to factor the coefficient matrix into the product QR, where Q is an orthogonal matrix and R is an upper triangular matrix. This type of factorization leads to improvements in stability and computational efficiency over classical Gaussian elimination algorithms. We will describe Householder transforms and discuss the general factorization algorithm. Seth Gemme, Westfield State College ABSTRACT: The Josephus problem involves a group of m people arranged in a circle. Starting at the first position, every nth person is eliminated until only one person J(m,n) remains. I will discuss several interesting patterns that I found in the raw data, graphical data, and cases with specific values of m and n. Also, I will briefly discuss my approaches to finding a generalized solution. Michael P. Higgins, University of Connecticut ABSTRACT: In recent years mathematics has become a major theme in Hollywood motion pictures and in works of fine literature. This talk will explore how mathematics and mathematicians are portrayed in these works, and the stereotypes that some of these movies and novels exploit. We will attempt to answer the questions of whether or not mathematical society benefits from this new found attention. Matthew Handelman, Hamilton College ABSTRACT: The first twenty years of the 20th century produced two of the greatest names in Mathematics and Literature, Franz Kafka and Kurt Gdel. We will argue that Gdel`s Incompleteness Theorem provides an interesting and poignant way to read and interpret The Trial. In particular, we will show how the world of The Law in The Trial is, in both a mathematical sense and a literal sense, incomplete. William S. Swindlehurst, Westfield State College ABSTRACT: An optimization problem involving the purchase of lumber that comes in different fixed lengths for use as baseboard trim is introduced. Analysis leads to linear Diophantine Equations in two variables and several possible equations containing an objective quantity that must be optimized. Students are encouraged to solve the posed problem using a "brute force" method, Microsoft Excel, and Geometer`s Scetchpad software. The Diopahntine Equation can have infinite solutions, but only several make practical sense, and one or two solutions are optimal depending on the student`s perception of what optimal is. Many extensions to the posed problem are possible, with varying degrees of rigor. Elizabeth Darrow, SUNY Geneseo ABSTRACT: Even though computers can do much of the computational work for us, some problems are still computationally complex. One practical way to reduce the complexity of a problem is to use an approximation method. Approximation methods can be important if the simulation of a complex system is desired. In this talk, we discuss the fitting of output of one approximation method with complexity O (n2) to the output of a more complex (O(n3)) and more accurate approximation method that predicts the thermodynamic stability of DNA fragments. The goal is to then extrapolate the less complex method in the hopes of practically improving its accuracy without making it more computational complex. Darren Lim, Siena College ABSTRACT: One of the many popular Next Generation board games, "Ticket to Ride" combines the gameplay of cards with aspects of Graph Theory. In a programming course at Siena College, we will use Ticket to Ride to teach students various concepts, including the difference between adjacency matrices and adjacency lists, edge weights and attributes, and Dijstra`s shortest path algorithm. Walter Hannah, Ithaca College ABSTRACT: We will investigate internal rays of the Mandelbrot set by describing mappings from the unit disc to each bulb. We will also see what these rays can tell us about how the bulbs are attached to one another. Antoaneta Kraeva, Williams College ABSTRACT: Experiments maintain an ideal population size of bacteria by adding or removing some organisms at certain moments. To describe this process we develop a new mathematical model using impulsive differential equations. Darren Lim, Siena College ABSTRACT: The Protein Folding problem has been of keen interest to scientists for the past 3 decades. Recently, computer technology and techniques have been used to find a protein`s functional structure in silico using only the protein`s primary structure (amino acid sequence) information. In this talk, I will describe an algorithm for solving the HP model of the folding problem in two and three dimensional spaces. Kate Belin, Bard College ABSTRACT: Suppose we are flipping a fair coin until either the sequence HHT or the sequence THH occurs. Does one sequence have a greater chance of occurring before the other? Chip-firing games can be used to determine the probability of one sequence beating the other. Elizabeth Adams, Williams College ABSTRACT: A number field is an extension of the rational numbers. We can associate a positive integer, called the class number, to each cubic field. Class numbers give us information about prime factorization in the ring of integers of a number field, and while factorization into primes is unique in the regular integers, it is not always unique in the ring of integers of a number field. Despite the fact that we can determine class number for any given field with certainty, it is much harder to find classes of fields with a given class number. I will discuss my current senior thesis research on cubic fields with even class number, which is interestingly related to the units of the cubic field. Abstract algebra is assumed. Joseph DeFilippo and Darren Lim, Siena College ABSTRACT: We will present our study of various computer platforms running short term processes. The intractability of the HP Protein Folding problem forces the researchers to investigate various methods for speeding up their algorithm. One such method is configuration of the computer system (Processor, Memory, Operating System). We will present the results of a sample simulation performed on various machine setups to see how well our algorithm performs when ported. Christopher Higgins, Saint Michael's College ABSTRACT: Over the past few years, ESPN`s coverage of the World Series of Poker has caused a poker frenzy around the world. Although poker is a game of chance and imperfect information, it favors the mathematically inclined. This talk will explain how simple odds and mathematical expectation can be applied to poker in order to implement a winning strategy. Applications of game theory will also be addressed in order to demonstrate the importance of implementing "mixed strategies" and bluffing. Anthony Marcuccio, Williams College ABSTRACT: Consider a group of ten pirates dividing 100 gold pieces according to a rigid, hierarchal method. The fiercest pirate proposes a distribution of the gold, and all ten pirates vote. If he gets half the vote, they divide the treasure as agreed. Otherwise, he is thrown overboard, and it is up to the second fiercest pirate to propose a distribution. It isn`t too hard to determine what the smartest plan would be for the fiercest of ten pirates, but if that number grows to 200 or even 500 pirates, the puzzle has some unexpected solutions. Christian Damberg and Darren Lim, Siena College ABSTRACT: We will present our study of software enhancements on an algorithm for solving the HP Protein Folding problem. Efficiently solving an NP-Complete problem requires a combination of savvy programming, optimized heuristics, and a fast machine. We will take a look at the evolution of one such algorithm, from its initial design phase, through the numerous improvements for speed. Suzanne S. Switzer, Smith College ABSTRACT: Previous surveys of Original Articles published in The New England Journal of Medicine in 1979 and 1989 revealed increasing use of statistical methods over time. In a recent survey of articles published in 2004-2005 we found that there was a continued trend toward increased use of newer and more sophisticated statistical methods not typically included in an introductory statistics course. In this talk I will describe this study and discuss how this increasing sophistication of statistical methods has potential implications regarding clinicians` comprehension of new research as well as statistical education more generally. Andrea Austin, Saint Michael's College ABSTRACT: The first digit in many financial records tend to follow the expected frequencies given by Benford`s Law. We will present a history of Benfords Law: how it was developed and how it can be applied to identify possible fraud. Illustrations of the application of Benford`s Law to data will be discussed. Gordon Phillips, Williams College ABSTRACT: Can Ford circles help approximate the national debt? My bank balance? My GPA? No. Ford circles are a geometric method of displaying complexity of rational numbers on the upper-half plane. Here we will describe these notions, discuss the Farey sequence, and offer a bit of insight into an area of number theory known as Diophantine approximation--an area in which we search for excellent rational approximations to irrational real numbers. Jake Randall and Rohan Mehra, Williams College ABSTRACT: We will show how Euler`s Equation and the calculus of variations can be applied to determine the optimal strategy for consumption over the course of one`s life. Douglas R. Hammond, Williams College ABSTRACT: What`s the best strategy for gambling and investments? We will discuss the Kelly rule for even-money betting and volatility-pumping of investments. Jessica M Belanger, Fitchburg State College ABSTRACT: Edwin A. Abbotts book, Flatland, reaches far beyond the obvious two-dimensional, geometric perspective of A Square. Abbott creates a metaphorical society that not only criticizes the two-dimensional, political and social 19th century English society of his time, but also attempts to explain higher dimensions. By introducing concrete mathematical concepts, my presentation will explore a fifth dimensional world that parallels todays 21st century society. The imaginary world is literally turned inside out: klein bottles populate the planet, a mobius strip that is wrapped around a wormhole. This bizarre perspective parallels Abbotts original Flatland and portrays societys progression through time. Andrey Taran, Siena College ABSTRACT: In this presentation, we investigate the dynamics of simple maps. From these basic ideas, we will introduce the fundamental ideas of bifurcation and chaos. Julie Jaenicke, Colby College ABSTRACT: Piero della Francesca had two passions - art and geometry. The Renaissance artist was a master of painting and also of linear perspective. Piero wrote what he understood of the math behind perspective in the book A Treatise on Perspective which influenced many Renaissance artists, including Leonardo da Vinci. Discussed is the integration of geometry and art within the works of Piero. Specific paintings are analyzed including his most famous painting, The Flagellation of Christ. Phillip Monin, Canisius College ABSTRACT: I will open with the definition of the Ihara zeta function on finite graphs and state some of its number-theoretic and spectral properties. Then I will provide an extension of the Ihara zeta function to infinite graphs, which are limits of finite graphs, and present some explicit calculations. Michael Knight, Bard College ABSTRACT: aspects of prime numbers and some weird functions and theorems Emily Sheldon, St. Lawrence University ABSTRACT: In this talk we attempt to derive an equation from the Beta-binomial distribution that can be used to apply sequential probability ratio testing to biometric devices. We first examine sequential analysis testing methods and then apply them to examples of multiple independent bernoulli trials. We use these examples to illustrate the decision of when to stop testing. Lastly we examine the Beta-binomial distribution and derive an equation that can be used in sequential analysis methodology. Jane Leary, Colby College ABSTRACT: Hypatia is considered by many to be the original woman mathematician. She was raised in the world of education by her father Theon, the leading scholar of Alexandria, Egypt. She is considered one of the most intelligent mathematicians, scientists and philosophers of all time. Her influence on the world of mathematics and specifically the role of women in this field continues today. Discussed are her mathematics and the studies that have developed as a result of her influence. Lisa Rogers, Rensselaer Polytechnic Institute ABSTRACT: Armed with computational prowess and the mathematical ability to attenuate dynamic systems through analysis, it is possible to simulate even the most complex reactions involved in the inner workings of the human biological clock. The goal of this talk is to delineate the process of creating a revised human sleep wake cycle model based on limit cycle oscillations. Aspects of circadian biology theory will be utilized, with an emphasis on the buildup of restorative REM/NREM oscillations to demonstrate the unfailing circadian regulation process. The model draws from dynamic systems theory and Michaelis-Menton kinetics. Jaehong Cho, Williams College ABSTRACT: The ladder game is a simple and easy game played in Korean elementary schools to randomly group people into teams. A series of vertical lines are drawn on a paper, players randomly place horizontal lines in between, and each player goes down the ladder. Why does it work? What if there are infinitely many players? are the questions we are going to attempt to solve. Raluca Dragusanu, St. Lawrence University ABSTRACT: Traditional time-series models such as Autoregressive (AR) and Moving Average (MA) models are based on the homoskedasticity assumption, which translates into a constant variance for the errors of a model. This assumption has been shown to be inappropriate when dealing with some economic and financial market data. A new class of models - conditional heteroskedastic models was developed to deal with data that does not exhibit constant variance of the errors. The most well- known models in this class are the Autoregressive Conditional Heteroskedastic model (ARCH) and its generalized version (GARCH). Stock market volatility, the square root of the variance of stock returns presents a very good application of this type of model. In finance, volatility is the expression of risk. Since we must take risks to achieve rewards, finding appropriate methods to forecast volatility is necessary in order to optimize our behavior and, in particular, our portfolio. I will present the general properties of the ARCH and GARCH models and use both Monte Carlo simulations and known financial time series data to test their performance. Stephanie Hildebrand, College of St. Rose ABSTRACT: For most people not associated in the mathematical field, math is boring and/or too complicated to understand. However, by examing the influence of Math in Art and Architecture, Math is no longer inaccessible to the general public while providing easy examples. Aleksey Panasyuk, SUNY Institute of Technology Utica/ Rome ABSTRACT: Problem: investigating the effect of an earthquake on a building, some of the simplistic assumptions guiding the research will be that each floor has a fixed mass, and the floors are coupled like springs that obey Hookes Law, with the focus upon the horizontal displacement of the floors. The model is than that of a system of coupled springs: with a two story building having two masses, a three story having three and so on up until n stories. I want to investigate the behavior of the floors in the model for floors = 1 to n where n could be 100 stories or higher depending on the feasibility of the calculations. I will use matrices and the help of MATLAB in computing results for large values of n. If I am successful, I will go further and consider a model that is not so simplistic by adding other factors to it. Dana Camp, Southern Connecticut State University ABSTRACT: This project examines the high tech classroom and accompanying teaching style. Because the concept of technology in the classroom is so broad, for the purpose of this experiment it has been defined as the use of an LCD projector and a computer primarily by the teacher, that is students were not be supplied with any computers, certain calculators, etc. It is hypothesized that the proper use of LCD projectors and limited PCs in the classroom as intuitional tools can help a student to learn more material. The experimental design takes two similar cohorts of students and approaches them with the two differing styles after a flush-out period. Students were tested on the lesson material before and after each lesson to see how much of an understanding of the material was gained by the high tech or low tech lesson. Michael Gillmor, Williams College ABSTRACT: The heat equation models the dissipation of heat through a substance over time. The equation can be solved using Fourier series. Applications of this solution to other areas of physics such as particle diffusion will then be explored. Caitlin E. Scott, Mount Holyoke College ABSTRACT: Due to van der Waals and electrostatic interactions, as well as the rough surface of the protein, the dynamics of water molecules near the protein surface is different than that of the bulk phase. First, we model the interaction of a simple system consisting of two particles using the Lennard-Jones potential. Then, we extend our studies to more complex systems, such as the water and ion interactions. In these cases, the potential of mean force calculated from a molecular dynamics simulation is used as the external field. The Brownian dynamics of the solvent in the different external field of the solute is calculated in order to understand how the solute affects the diffusivity. We access how well the diffusivity can be described with this simplified dynamics. Gregory M. Vlahos, Fitchburg State College ABSTRACT: Einsteins theory of special relativity brought about many changes in Physics. One very popular characteristic of special relativity is known as time dilation. Although the concept is hard to accept, there is an equation that can be easily derived to explain this effect. The derivation uses both mathematical and physical equations such as the Pythagorean Theorem and the relationships between position, velocity and time. The end result gives interesting results about the speed of light and how fast matter can really travel, plus more. Sixuan Chen, Mount Holyoke College ABSTRACT: Nash equilibrium, named after Nobel Laureate John Nash, is a widely known solution concept for zero-sum games. Another solution proposed by Nash, a somewhat less famous albeit no less important one, is the Nash bargaining solution for non-zero-sum games. In this talk, we will explain and analyze this solution by evaluating his assumptions and also looking into how it is applied to some examples of bargaining games. Amit Desai and Abhishek Maity, Hamilton College ABSTRACT: We graphed the solution curve to an innocuous initial value problem, y' = (4 - t*y)/(1 + y2), y(1) = 0, for Differential Equations class, and the computer lied to us. We used the program Maple to calculate the solution curve and, when pushed, Maple gave us numerical nonsense, graphical nonsense, and graphics that didn't match the numerics. In our talk we will exhibit the lies our computer told us. We also hope to explain the lies, but about that we make no promises. ______________ Caitlin O'Connell, Williams College ABSTRACT: The movements of an airplane are commonly described in terms of yaw, pitch, and roll, and more specifically described in terms of linear transformations from a plane`s initial axis. We will find matrix representations of the movements of an airplane using linear algebra. Thomas Heacock and Kyle Psaty, Hamilton College ABSTRACT: The Avian Flu, more commonly known as the bird flu, has been the topic of discussion recently not only in the United States but throughout the world. Although human infection has been rather isolated to date, history`s past has proven that the avian flu can be extremely deadly to humans as well as bird populations across the globe. Using a Kermack-McKendrick system of differential equations and published research of the flu, we have established a model showing the potential consequences of an outbreak. To assist the model, we will provide a brief background of the virus and the major historical outbreaks which have occured. Alexandra Constantin, Williams College ABSTRACT: The expectation-maximization (EM) algorithm is an algorithm for finding maximum likelihood estimates of parameters in probabilistic models, where the model depends on unobserved latent variables. EM alternates between performing an expectation (E) step, which computes the expected value of the latent variables, and a maximization (M) step, which computes the maximum likelihood estimates of the parameters given the data and setting the latent variables to their expectation. In this talk, I will describe how the EM algorithm can be used to estimate the means of a mixture of distinct Gaussian distributions. I will provide an example of how this technique can be used for text classification. Susan Reid, Williams College ABSTRACT: Also known as the golden section, the golden mean, and the divine proportion, the golden ratio manifests itself everywhere in nature. We will explore the history and character of this most irrational number, from its discovery by the ancient Greeks, to its use as the standard for the perfect proportion of the human body in Renaissance art, to the prevalence of the spiral of Archimedes in nature. We will also look at some of its mathematical oddities, such as the interesting qualities of its inverse and its square, as well as its relationship to the Fibonacci numbers. Nicholas Linse, Unites States Military Academy ABSTRACT: In the modern baseball era, managers use starting pitchers for fewer innings and relief pitchers in situational roles more than in the past. Are these strategies paying off? The question this study will seek to answer is if managers are getting more or less out of their pitching staffs in the modern baseball era compared with the past 40 years. Michelle Donnelly, Williams College ABSTRACT: Though the conclusion of the intermediate value theorem may seem simple, it is an essential component of many proofs in calculus. Its uses range from helping to prove the fundamental theorem of calculus to proving the existence of the square root of 2. Alex Zaliznyak, Williams College ABSTRACT: Pontryagin`s principle is used to find optimal solutions to problems in control theory. I will outline how it works by analogy to Lagrange multipliers, and then use it to find solutions to a few problems usually solved with the calculus of variations, including minimum area surfaces and optimal economic strategies. Christopher Hardin, Smith College ABSTRACT: Suppose you think of a function f from the real numbers into some set as a function of time, and that you want to predict f(t) based on the previous values of f (that is, f(s) for s<:t). If f is continuous, you can predict f(t) by taking a limit. But what if f is arbitrary? In that case, predicting the value of f(t) based on previous values seems hopeless. Nevertheless, the axiom of choice lets us exhibit a strategy for guessing f(t) based on previous values that, for any f, is correct for almost every t. The strategy also almost always predicts the values of f on an interval of the future. Wesley Tjosvold, Williams College ABSTRACT: Want to make money? The Black-Scholes Differential Equation describes how the price of a specific kind of financial security should move. This talk will present this equation, prove it, and discuss some of its uses and limitations. First derived in 1973, the equation finally won its surviving discoverers the Nobel Price in Economics in 1997. Ashley Jankowski, Wheaton College ABSTRACT: Amartya K. Sens Theorem concludes that individual rights can not exist in society in which there are more than three alternatives. The proof has been recognized as accurate but the implications of the theorem have been open to interpretation. In this talk, we will see that the conclusion of Sens Theorem is not as disturbing as commonly accepted. Geoff P. O'Donoghue, Williams College ABSTRACT: Molecular vibrations of symmetrical molecules can be predicted using group theory methods. The secular determinant will be used to make a quantitative prediction of normal frequencies. Degeneracies of the normal modes of vibration are also treated. A qualitative study of the vibrational spectrum can be made. Bobby Cole, Hamilton College ABSTRACT: Life is full of questions: If a tree falls in the forest and nobody is around to here it, does it make a sound? What if the hokey pokey really is what it`s all about?Should a player roll to get out of jail or simply pay the fine? And the most important question of all, is Baltic Ave worth the $60? My talk will seek to answer these age old questions (the last two anyway) using a healthy mix of economics and probability theory. Some basic familiarity with ANOVA testing and game theory will make it easier to understand, but everybody is sure to take something useful away. Krenar Komoni, Norwich University ABSTRACT: The mathematics senior project at Norwich University involves the understanding and implementation of the wavelet analysis functions. We have looked at wavelets and tried to understand the importance of them in the field of signal processing. We also have looked at different application of wavelets in the industry and government such as fingerprint image compression, de-noising of music, filtering of frequency sounds, etc. There are many types of wavelets that are used, and each works better for a certain application. The wavelet types that we have looked and analyzed the most are the Haar wavelets. We also look at the vector spaces, basis functions, and the Orthogonal Decomposition theorem for Haar wavelets. Haar wavelets are very common and are very easy to grasp for students who start and are beginners in wavelet theory. Michael Chmutov, The Ohio State University ABSTRACT: The word `categorification` has been introduced recently, but the concept itself is not a new one. It is a philosophical idea that instead of certain integers that describe properties of objects we want to study, we may consider vector spaces that have those integers as dimensions. This provides a new structure to these numbers and allows to use powerful methods from linear algebra to search for relationships between them. The most classical example is the homology theory of manifolds, which may be viewed as a categorification of the Euler characteristic. More recently, within the last 10 years, M. Khovanov very successfully applied this idea to knot theory. Now Khovanov homology theory is one of the most active development areas of knot theory. In this talk, I will treat these examples in more detail and try to give a feeling for what a categorification is. If time permits, I will describe a homology theory for graphs that is based on Khovanov`s construction. Peter G Nunns, Williams College ABSTRACT: "When there`s no more room in hell, the dead shall walk the earth," promises the zombie drama Dawn of the Dead. But where shall they walk, and how fast? Zombies, reanimated corpses whose sole purpose is to consume the flesh of the living, clearly pose a desperate threat to humanity (as chillingly argued in The Zombie Survival Guide). But many of the key questions posed by the living dead have yet to be answered. An accurate mathematical model of the spread and survival rate of an outbreak has yet to be proposed. This talk represents a mathematical attempt to understand the progress of a zombie outbreak, using data documented in movies such as Night of the Living Dead. Understanding the progress of a zombie outbreak, with the aid of mathematics, will of course assist the efforts of those responding to the zombie menace as well as the actions of the individual citizen during such a crisis. It will, of course, also greatly increase your appreciation of "horror" movies. Anthony Francis, Norwich University ABSTRACT: Generating functions are a way of encoding sequences of numbers as coefficients of power series. We will take a brief look at some of the applications of generating functions, including solving recurrence relations, counting the number of ways that a department chair could schedule courses given several restrictions, and finding the number of ways we can build fountains out of coins. Kevin Hahm, Williams College ABSTRACT: Euler's equation is a differential equation that you solve to find a function y = f(x) that minimizes some integral. We will derive Euler`s equation and illustrate how it can be applied to find optimal economic strategies. Shelby Kimmel and Matthew Earle, Williams College ABSTRACT: In 1931, Kurt Godel proved that in most formal systems (including arithmetic, number theory, and real analysis), there are statements that can`t be proven true or false using the axioms of the system. We will discuss the proof and its important implications for mathematics and for the philosophy of intelligence. Laura Brill, Dartmouth College ABSTRACT: A partition of a nonnegative integer n is a weakly decreasing sequence of integers that sum to n. We discuss symmetric functions (meaning that you get the same function if you permute the variables) -- and give a definition of Schur functions, which are symmetric and can be indexed by partitions of n. We then discuss products of Schur functions, and end with statement of a theorem that makes computation of the (Kronecker) product of Schur functions much easier. Jeanie Oudin, Williams College ABSTRACT: Have you ever wondered how many primes exist less than some x? The Prime Number Theorem can help to answer this question by describing the asymptotic distribution of the prime numbers. In this talk I will discuss the history of the Prime Number Theorem, its applications in number theory, and its implications in the world beyond math. Diana Davis, Williams College ABSTRACT: The cheapest way to enclose area in the Euclidean plane is by a circle, but what if the plane has varying density? What if we only consider a pie-shaped sector of the plane with varying density? I`ll show how to eliminate shapes (such as the circle) that we now know cannot be minimizing, and give conjectures and evidence for the best shape. Mark Gaudette, Wentworth Institute of Technology ABSTRACT: Taking an engineering approach to teaching algebra to the non-mathematical-seeming student. This talk will cover the importance of conveying to students, early on in their college careers, how to do many of the standard basic algebra problems in an engineering-like way. As a student majoring in electromechanical engineering and also serving as a college peer math tutor, I have noticed the many mathematical deficiencies that some students have on my campus. I will present some of the basic algebraic tricks that have proven to work well for me, as well as for the students who I tutor. These techniques include, dimensions concepts, graphical concepts for equations with examples. Math is the universal language so why not have everybody understand its building blocks! Yordan D. Minev, St. Lawrence University ABSTRACT: One methodology for evaluating the matching performance of biometric authentication systems is the receiver operating characteristics (ROC) curve. A biometric authentication system matches physiological characteristics to a database of such characteristics. The ROC curve graphically illustrates the relationship between type I and type II statistical errors when varying a threshold across a genuine and an imposter match distributions. In biometric authentication, genuine users are generally those that the system should accept and imposters are those that the system should reject. In this project ROC confidence regions are created using radial sweep methods. Radial sweep is based on converting the type I and type II errors to polar coordinates. The goal of the project is to estimate the performance of each biometric system via a confidence region and to identify the most effective method for computing such a confidence region for a ROC curve of that systems performance. Skyler Wengreen, Westfield State College ABSTRACT: The objective of this problem is to find the minimum number of queens (chess piece) which can be placed on a n x n chessboard to which every square is protected or occuppied by a queen but where a queen does not protect another queen. I intend to demonstrate how you can do this on a 5 x 5, 8 x 8, 11 x 11, as well as demostrating how to solve some of the finite solutions with mathematical equations. Patrick Karas, Dartmouth College ABSTRACT: How is visual information from our eyes processed into information that our brain can interpret? The most basic processing of visual data is done by simple cells in the primary visual cortex. I will first present the current theory on how simple cells transform information from the visual field into nerve impulses in the brain, covering very basic anatomy and neuron function. I will then present Gabors family of linear filters and generalize them to 2-dimensional space in preparation for their application to simple cell function. Finally, I will show how 2D Gabor filters maximally model the processing of data by simple cells, leading to a better understanding of how the brain interprets visual information. Matt Ollis, Marlboro College ABSTRACT: The Oberwolfach Problem, posed by Ringel in the 1960s, involves seating arrangements for meals at a math conference. In graph theoretic language, it asks for a decomposition of the complete graph into 2-factors, all of which are isomorphic to a given 2-factor. We describe a recent approach to the problem that uses graceful labellings of paths. We also offer two seemingly vulnerable conjectures that, if true, give solutions to the problem for many currently unsolved cases. (All of the graph theoretic terminology will be defined in the talk: familiarity with modular arithmetic is the only math that will be assumed.) Rachel R. Roe-Dale, Skidmore College ABSTRACT: Several experimental and clinical studies have documented that the order in which chemotherapy drugs are administered affects the outcome of cancer treatment. Mathematical models can be used to explain this drug sequencing phenomenon. As a first approximation, the exponential model is used to describe cell growth. Treatments that affect cells based on cell cycle state and resistance level are also simulated. These models can be used to verify the preferred regimens for the experimental and clinical findings of Bonadonna, a notable breast cancer researcher. Vanessa Mahoney, Worcester State College ABSTRACT: I will be using the measurements of a dumpster to determine it`s volume. Then I will calculate, using partial derivatives, which measurements of another dumpster, with the same volume, will minimize construction costs. Cheryl Areson, Wheaton College ABSTRACT: In professional sports drafts, individual team strategizing may not necessarily yield an optimal allocation of players. Given four basic assumptions, well see that strategic behavior may yield an outcome that is less desirable to every team than an outcome yielded from sincere behavior (where teams choose their first choice in each round). Tracy Zaino, Hann Yang, & Andrew Lyons, Hamilton College ABSTRACT: Imagine youre Bart Simpson standing on an overpass. Youve had a rough day, so you decide the only thing to do is spit at the on-coming traffic. But at what angle should you spit to hit the farthest possible car? These and other questions raised by simulating projectile motion have led to several conjectures, including: a formula for finding the angle that maximizes distance, the maximum angle another angle will land at the same point on the downward slope at the same speed and the maximum angle, and the landing angle are complementary. We have proven some of the conjectures, but not all. This process illustrates a valuable use of modeling that is often overlooked that of simulation in helping to create conjectures. Joshua Heyman, Westfield State College ABSTRACT: I will address the enumeration of all possible Sudoku grids and the minimum number of clues required to solve a "well-posed" Sudoku grid. Joseph Carvalho, Westfield State College ABSTRACT: My talk will include a brief introduction to BIBDs (balanced incomplete block designs) and Planar Near-Rings. I will then go on to show how Planar Near-Rings can be used to construct BIBDs, creating a finite Integral Planar Near-Ring. Martha Rogers, Williams College ABSTRACT: The generalized Stokes Theroem illustrates how a simply stated math theorem can yield more complicated looking calculus theorems. We will start with the generalized Stokes Theorem and derive Gauss` Divergence Theorem, Green`s Theorem, Stokes Theorem, and the Fundamental Theorem of Calculus. Donna Beers, Simmons College ABSTRACT: Alexander McCall Smiths widely acclaimed novels, the No. 1 Ladies Detective Agency series, have beautiful book covers that are decorated with frieze and wallpaper patterns. The heroine of the series is a smart and savvy proprietor who loves her native Botswana. This talk will show how an instructor can use Botswana and its culture as a rich source of examples to illustrate and analyze symmetry and planar patterns. Michael Abdow, Westfield State College ABSTRACT: Bach`s Harmonization of the Lutheran hymns, also known as the Chorales, is the pinnacle of musical ingenuity and beauty. Music students are often given the main melody (called the Cantus Firmus) and asked to Harmonize them in four voices (SATB), only to be humbled by later comparing their attempt with Bach`s. There has been recent interest amongst computer scientists who are trying to "automate" such harmonizations. What is it about these Harmonizations by Bach that make them so "special". Is there an underlying pattern that eludes us when we try to understand them from the traditional point of view? James D. Hall, St. Lawrence University ABSTRACT: The statistical procedure known as bootstrapping is used to approximate a sampling distribution for any statistic by resampling from an original sample with replacement in order to draw conclusions about the shape, center and variability of the sample statistic. These methods avoid traditional assumptions such as assuming a certain population is normally distributed. We give a brief description of bootstrapping techniques and demonstrate via computer simulation (using the statistical software packages R and Fathom) the effectiveness, in terms of coverage and average width, of bookstrap confidence intervals compared to traditional confidence intervals in standard situations and in cases where standard assumptions fail. Kristen Johnson, Westfield State College ABSTRACT: Wilhelm Lenz proposed a model to his student Ernest Ising who solved the one dimensional case in 1925 as an attempt at describing a mathematical model for phase transitions, for example just when liquid turns to gas. The Ising model can be used to discuss changes in state of a wide variety of instances: protein folding, flocking birds, and beating heart cells. However in the 80 years since the model was introduced it has never, except in special cases, been solved. So what is the story behind this model that is so important but cant even be solved? This is what will be discussed in this presentation. Charles Vick, Hamilton College ABSTRACT: All mathematicians know a great proof when they see one. Great proofs are brief but convincing: their results are mathematically significant at all levels but can require little mathematical background to understand. I will focus on the aesthetics of mathematical proofs, discussing the qualities that make a great proof and why it feels so good to write one. We will also explore the possibility that aesthetically pleasing proofs are more than just pretty faces but can, in fact, provide insight into the connection between the human mind and the mathematical realm. Max Yazhbin, Westfield High School ABSTRACT: How does the water flow along the surface z2 = c2 - 16x2 - 9y2? We will solve this problem using well understood partial derivative methods which generate gradient curves. The results yield surprising outcomes. Applications include using these techniques on surfaces such as river beds and mountains. Sarah Walker, Saint Michael's College ABSTRACT: DNA nanotechnology is an emerging scientific field in which biologists use strands of DNA to construct two and three-dimensional nano-scale structures. When biologists seek to construct a specific structure, e.g. a graph, one of the most basic questions is: how many strands of DNA must they design? We can use topological graph theory to relate the minimum and maximum genus of a graph to the minimum and maximum number of DNA strands required to construct the corresponding nanostructure, and to specify when a graph can be made from just one strand of DNA. Significant strides have been made in constructions where the edges of the graph are traced out by two strands of DNA, one in each direction. Our research involves adapting the results from double stranded constructions to structures with four strands on each edge. Nick Yates, Williams College ABSTRACT: Are you tired of embroidering geometry all day, and knitting number theory all night, with ne`er a stitch crossed between them? Are you searching for a precious metal to spice up your math life? Look no further! Here we introduce an explicitly describable smooth curve in the plane that spirals in to the golden ratio phi while simultaneously capturing the number theoretic structure of its best rational approximates F_{n+1}/F_{n}. Further, by investigating lines tangent to our spiral, we find a surprising connection to the much-loved golden rectangle of geometric glory. By weaving together these strands from number theory, analysis, and geometry, we increase our understanding of what (mathematically) is gold, in a world where all that glitters is not. Janet Langley, Wentworth Inst of Tech ABSTRACT: I will be presenting on the topic of modeling of the motion of a object when a force varied at its frequcny of the strongest resonance frequency. I will also be calculating the resonant frequencies/ disambiguation of different objects string, particle and elongated mass due to their density and volume. David J. Covert, Canisius College ABSTRACT: This talk will be an introduction to Non-Commutative Functional Analysis based on a course given at Canisius College by Dr. David Blecher (University of Houston) in the Fall of 2004. I will start with some key definitions in functional analysis and work my way to the non-commutative world. This talk will conclude with Ruan`s Theorem and how this (fairly) newly-developed field helped solve the Halmos-Similarity Problem. Lily Davidoff and Brittany Bannish, Mount Holyoke College ABSTRACT: With the threat of bio-terrorism, the recent cases of avian influenza, the outbreak of SARS, and the thousands of cases of influenza every winter, it is clear that understanding how certain diseases spread temporally and geographically, and how to best contain the spread, is of utmost importance. We construct a spatial, multi-patch SEIRS (susceptible, exposed, infected, recovered) differential equations model for disease dynamics, which incorporates the travel/movement of individuals between distinct geographic locations. A theoretical analysis provides a threshold condition for whether a disease will die out or persist. Numerical simulations demonstrate the impact of travel rates on this threshold and on the size of an outbreak. In addition, simulations illustrate that quarantine via travel restriction can actually be counterproductive under certain conditions, leading to an increase in the number of infections. Sarah Hamilton, Saint Michael's College ABSTRACT: Systems of differential equations are being used to model the effect of HIV on the immune system. This talk will give an introduction to such models. In particular it will focus on models which compare the benefits of different treatment regimes such as Structured Treatment Interruptions. Sean P Bradley, Manhattan College ABSTRACT: All of our professors would tell us that the time limits of the semester force them to leave out some interesting topics. One such topic is Gershgorin`s Theorem: this theorem allows us to estimate the location of the eigenvalues of a square complex matrix. I will present the theorem, its short proof, and some applications. One semester of Linear Algebra will be sufficient background for this talk. Michelle A. Lastrina, Mount Holyoke College ABSTRACT: Does the spectrum of a graph determine its shape? Seidel switching is a technique for generating pairs of graphs that are cospectral but not necessarily isomorphic, thus telling us that the shape of a graph cannot be determined from its spectrum. Gregory Quenell explored Seidel switching in The Combinatorics of Seidel Switching in order to answer this very question. We will introduce how to construct a pair of graphs via the Seidel technique and illustrate the construction by example. The construction can also be restricted in order to generate cospectral pairs of regular graphs. This is of interest because cospectral pairs of regular graphs are rarer than cospectral pairs of non-regular graphs. Kimberly Elicker, Williams College ABSTRACT: The Fibonacci sequence is everywhere, even if it doesn`t appear to be! This talk will use simple generating functions to uncover the hidden, "fibbing" Fibonacci sequence in common fractions. Roland Varriale, Manhattan College ABSTRACT: Various algorithms for polynomial multiplication have been devised to improve on the time efficiency of traditional polynomial multiplication( O(n2 )). In this talk I will discuss and compare time efficiencies of the Karatsuba algorithm, the Fast Fourier Transform, and traditional polynomial multiplication as applied to polynomials up to the order 215. The timing of these algorithms will then be compared to the computational time used by the Maple mathematics software. The results will be plotted using graphical software for further examination. Brian Simanek, Williams College ABSTRACT: I will discuss some recent results in Commutative Algebra. Specifically, I will answer the following question: "Given a complete local ring T and a finite set C of nonmaximal prime ideals in T, when does there exist an integral domain A such that the completion of A is T and A contains a principle prime ideal with semi-local formal fiber whose maximal ideals are exactly the elements of C?" Gregory Knop, Canisius College ABSTRACT: This talk is based on an unpublished paper written by Per Enflo, Robert Heath, and Angela Spalsbury. It discusses the following: In the 1970s, new regulations regarding phosphorus loading in Lake Erie were put into place. As a result, certain species of algae that had all but vanished in the 1950s and 1960s returned. In the 1980s, the zebra mussel was introduced to Lake Erie. This resulted in a reduction in the total amount of algae in the lake at first, but certain species of algae returned to much higher levels years later. This talk will discuss a mathematical model based on modified predator-prey relations to study the above scenario. Leigh-Anne Cioffredi, Wheaton College ABSTRACT: The Bubonic Plague ripped through Europe in the Middle Ages. Could the Black Death return? This talk will examine this question using Mathematical Modeling. David Dunham, United States Military Academy ABSTRACT: The US Air Force currently employs Systems Effectiveness Analysis Simulation (SEAS), an agent-based, stochastic simulation of the battlefield. One simulated scenario involves a loose nuke incident, with terrorists attempting to escape Iran before US forces can find them. The US forces in the scenario have well-developed logic, however the terrorists behavior is rather ill-defined. The development of refined object-oriented algorithms for the terrorist side could develop into a new class for the SEAS library, with inheritance for behaviors extending to different subclasses of terrorists for future use. Jennifer King and Lindsey Krok, Westfield State College ABSTRACT: Many times professors present mathematics and do not always tell students where it came from, and occasionally are unsure of the history themselves: it is time we stop taking these accomplishments for granted. Its time we give recognition to these mysterious mathematicians for the time and effort they put in to change our field of study. In particular, female mathematicians have been short changed in the recognition of their efforts. We will also give one possible solution on how to improve future mathematicians knowledge of their history. Travis Gingras, St. Lawrence University ABSTRACT: Statistics and sports have been related for many years, and recently the art of using statistics to observe players tendencies has become more and more common among coaches. This project looks to investigate patterns of shots taken by the St. Lawrence Mens hockey team using a geographic information system. ArcGIS is a mapping program generally designed for geographical data, but in this project we have defined a database to store information about individual shots in multiple hockey games while placing them on a map of the offensive zone of a hockey rink. We can then study patterns and look for the trends that might benefit individual players or the team as a whole. Theresa Ryan, St. Michael's College ABSTRACT: Fractals are geometric patterns that by zooming in on a part of the pattern yield a similar picture. The Mandelbrot Set is an example of a fractal that has been called on of the most intricate and beautiful objects in mathematics. We will discuss the process of generating the Mandelbrot set which is based on a simple equation involving complex numbers. John R. Burke, Marist College ABSTRACT: We will discuss classes of closed geodesics on some familiar surfaces in space. We will graph representatives of these classes by numerically solving the differential equations that determine geodesics on parameterized surfaces. Colleen Dalton, Lindsay Jardin, Marlee Berg, Westfield State College ABSTRACT: Life is either daring adventure or nothing. - Helen Keller This is just one of the numerous quotations and famous mottos that we will consider in our talk. You will also learn many useful ways to employ these quotations and mottos in many different learning environments. They can be used to motivate students, inspire learners, capture the interest of readers, provoke an audience, and challenge prevailing stereotypes of mathematics. As both current students and future teachers we will illustrate the impact of quotations in mathematics classes through both the eyes of students and teachers. We will also share many references and resources so you have access to an ample library of mathematical and educational quotations for your own use. William S Zwicker, Union College ABSTRACT: Here are three general methods, each of which can be used to create a variety of specific voting systems: Scoring rules: Each voter assigns points to each candidate. The winner is the candidate with the most points. Mean proximity voting: Each voter is identified with a point in space. Each candidate is identified with a point in space. The mean position g of the voters is calculated. The candidate closest to g wins. Distance-to-unanimity voting: The outcome of any election is clear when the vote is unanimous. When a vote is not unanimous, the winner corresponds to whichever unanimous choice is closest to the actual vote. In what ways are these three general types of voting different? We use a theorem of Huygens to prove that the three classes are exactly the same. Daniel Kendris, Manhattan College ABSTRACT: One way to construct a random number generator is to generate the sequence: g, g2, g3, ... (mod p) where p is a prime number and g is a primitive root mod p. We will design three such generators, one for a 16-bit processor, one for a 32-bit processor, and one for a 64-bit processor. Sophie Germaine primes will play an important role in the design. The chi-square test and the Kolmogorov-Smirnov test were used to test these generators. Ross B. Gingrich, Southern Connecticut State University ABSTRACT: While the quadratic formula for solving quadratic equations is well known, many students have not seen Girolamo Cardanos method for solving the general cubic equation x3 + bx2 + cx + d = 0. Cardano published his solution in his "Ars Magna or The Rules of Algebra" in 1545 CE. In this talk, we will look at the history of his solution, the method that he used, and its modern formulation. John McCabe, Manhattan College ABSTRACT: Let p be a prime number. An integer g is a primitive root for p if the powers of g: g, g2, g3, ..., when reduced mod p, give all the residue classes of integers mod p. A proof will be presented of the theorem that any prime number p does have a primitive root. If the prime number p is very large, determining a primitive root for p is believed to be in general a computationally hard problem. We will discuss some computational stategies. Primitive roots have applications to public key cryptography, and to the generation of pseudo-random numbers. If time permits we will discuss this, and also, if time permits, Artin`s conjecture. Some acquaintance with abstract algebra will be needed. Anna Haensch and Joan Kim, SUNY New Paltz ABSTRACT: Torricellis Trumpet (also known as Gabriels Horn), the solid of revolution achieved by rotating the graph of the function, f(x) = 1/x around the x-axis, is known to have a finite volume but an infinite surface area. Our goal is to search for other such objects with the same properties, which we call Torricelli solids, by analyzing the solids of revolution generated by functions of the form f(x) = x-p, f(x) = [ln(x)]-p, and other special functions. By doing this, we will expose trends amongst these functions, and make predictions about their behavior. In addition, we will consider the possibility of the existence of isolated solids in three-dimensional space with the same properties of finite volume and infinite surface area. Catalin Iordan, Williams College ABSTRACT: Groups are a very powerful mathematical concept and they apply to many more areas than just number theory. A very interesting connection can be found between Group Theory and elliptic curves which opens a wide range of practical applications. I will define a binary operation on the points of an arbitrary elliptic curve and prove that it generates a group structure. Also, I will explore some of the implications of this fact in the realm of cryptography. Jeffrey Connor, Westfield State College ABSTRACT: How Many Sudoku Puzzles are THERE??? ABSTRACT: Sudoku puzzles typically consist of a nine-by-nine grid of squares. Some of the squares contain numbers: most of the squares are blank. The goal is to fill in the blank squares with digits from 1 to 9 so that each row, each column, and each of the nine three-by-three blocks making up the grid contains just one of each of the nine digits. The additional constraint of the three-by-three blocks reduces the enormous number of possible nine-by-nine Latin squares to "a smaller but still-humungous number: 6,670,903,752,021,072,936,960". Further refinements can be made to arrive at a more manageable 3546146300288. This presentation will offer a brief examinination of a method of counting the number of Sudoku grids, relying on Bertram Felgenhauer`s "Enumerating Possible Sudoku Grids". Time permitting, other counting methods will be outlined. Alan D. Taylor, Union College ABSTRACT: We consider the following kind of hat problem. Several people (perhaps finitely many, perhaps infinitely many) are to have hats of various colors placed on their heads. Each will be able to see at least some of the other hats, but not his or her own. No communication is allowed after the hats have been placed and each will be asked to guess (simultaneously and independently) the color of his or her own hat. But before the hats are placed, the people are allowed to get together to plan a strategy as to how they might guess. A remarkable observation of Yuval Gabay and Michael OConnor is that if the number of people is infinite and each can see all but finitely many of the other hats, then there is a strategy ensuring that only finitely many guess incorrectly. We will present a general framework for this kind of question and a number of results, in both the finite and infinite case, obtained by Chris Hardin, myself, and others. Andrew Crossett, Canisius College ABSTRACT: Technical advances in the field of computational neuroscience have given rise to many new and complex situations dealing with neuronal data. Because of the relatively small number of samples that can be taken at any given time, it is important to apply both intuitive and non-intuitive statistical methodology. We will begin by showing that an elementary framework for estimating the firing rate of neurons can be inefficient. Therefore, we will turn our attention to three main points of more non-intuitive methods: the use of maximum likelihood and Bayesian methods to estimate parameters, applying modern nonparametric methods to neuronal data and finally showing that many analyses that are based on Poisson assumptions can be applied to non-Poisson data. Jamey T. Lewis, St. Michael's College ABSTRACT: Force-directed graphing involves applying a physical analogy to graphs of nodes and edges, allowing forces of attraction and repulsion to push and pull the system into a state of equilibrium. This method has been used to aid in computer chip design, simulating the structure of gates and wires in the chip as nodes and edges. We have modified the basic model in order to apply it to the floorplanning stage of chip design, in which the set of gates in the chip is partitioned into logical clusters or blocks which are then placed in the chip space as a pre-processing step for the actual placement and routing. In this presentation I will discuss our modifications of the "spring-embedder" force-directed algorithm, such as the use of nodes with width and height, effective repelling perimeter and pressure equalization equations. I will also present a brief summary of the experimental results that demonstrate the feasibility of this approach. This work responds to an industry problem presented by Cadence Design Systems, a company that develops chip layout tools. Jamey T. Lewis, St. Michael's College ABSTRACT: Wavelet analysis involves representing a signal or any mathematical function in terms of a basic waveform called a Wavelet, in order to extract information about how the signal changes over the domain. Similar to Fourier analysis, which represents signals as the sum of sinusoids, Wavelets in their current form have been around only since the 1980s (though with connections to Haar`s work in the early 20th century). In this presentation I will discuss the basics of Wavelets, and then provide an example from one of the many interesting fields to which Wavelet analysis is applied. Emily Fertig, Williams College ABSTRACT: Scientists and mathematicians are using data from the Wilkinson Microwave Anisotropy Probe (WMAP) to try to determine the shape of the three-dimensional spatial universe. WMAP measures cosmic background radiation, the relics of radiation emitted soon after the Big Bang. Mathematician Jeffery Weeks has interpreted the data as consistent with a finite dodecahedral-shaped universe with spherical geometry, in which traveling far enough in any direction will bring you back where you started. Others claim that the data is more consistent with a Euclidean universe. Nicole Mercier, Westfield State College ABSTRACT: For over two millennia mathematicians tried to determine whether it was possible to double a cube, trisect an angle, or square a circle using an unmarked straightedge and a compass. Other geometric constructions are possible using the same instruments, like bisecting an angle or doubling the length of a segment, so it`s natural to wonder about these "Three Greek Problems of Antiquity". In the early nineteenth century, the young mathematicians Niels Henrik Abel and variste Galois invented tools which innitiated what we now know as group theory. This allowed mathematicians to solve not only the three Greek problems but other fundamental problems like the soluability of the quintic equation as well. This talk will not only discuss the Three Greek Problems and the fundamental discoveries of Abel and Galois, but the poignant history surrounding their short lives as well. Daniel Dadush, Brown University ABSTRACT: Ever wanted to know how to best plan your workday? Or how to teach a computer to play roulette? Then you`ll want to learn about Markov Decision Processes and the tools we have to solve them. This talk will give a basic introduction to what Markov Decision Problems are and their applications. Daniel Marcus, SUNY Geneseo ABSTRACT: A description of the spread of influenza through a small world network which has been expanded to handle multiple cities. Each city is in itself a small world network of people, and the whole model is a small world network of cities. The model changes the dynamics of the epidemic curve of a single small world network by allowing the transition of influenza from one city to another. Jonathan Huang, Dartmouth College ABSTRACT: ABSTRACT: Minesweeper is an addictive computer game that comes with Microsoft Windows operating systems. It turns out that there is a link between this time-draining distraction and the $1,000,000 Millinium math problem, P=NP. We will introduce the P=NP problem, the concept of NP-completeness, and the SAT problem. Then we shall show how SAT reduces to Minesweeper. (1) "Minesweeper is NP-complete", Mathematical Intelligencer Vol 22, No. 4, 2000, p 9-15. Nicholas LaDuc Campany, Hamilton College ABSTRACT: Anyone familiar with the sport knows that baseball is a game of numbers. Professional baseball is simply littered with statistics. This particular study will look at the stats for players in the year immediately preceding their status as a free agent as well as the year following, in an attempt to determine whether one should expect the same sort of productivity in both years. Will the player perform better in the season after the trade (equity theory) as a result of being satisfied with their new team, or will the player perform better in the year preceding the trade (expectancy theory) in an attempt to increase their market value? This analysis provides significant convincing evidence in support the latter hypothesis, perhaps pointing to a greed-based motive for performance. Gwyon Thomas Sutton, Westfield State College ABSTRACT: I`ll be starting with a brief origin of the mobius strip and the birth of topology. I`ll briefly intoduce its origin from Gauss`s ideas (never formally) onto his `students` Johann B. Listing (who actually first used the Mobius strip) and of course the independent August Ferdinand Mbius as well. The core of the presentation will involve modern applications of the Strip and other topological concepts, from the 1960`s and up. The mobius strip (and more importantly, the mathematical concepts of geometric transformations) were used to develop electronic resistors, and in a more modern example, graphical transformations and rendering (transformation of 3-D objects and projecting 3-D images on a 2-D surface/screen) The heaviest concentrations will be the modern graphics algorithms to approximate these transformations that have risen from the more exact formulas used in the math world: I will go into detail about these algorithms` origins (the mathematical formula they derive from) as well as demonstrating (graphically and in the form of equations/numeric values) the results of `exact` transformations using pure mathematics, and then the "approximate" transformations by the algorithms. Matthew Brewer, Keene State College ABSTRACT: A nomogram is a chart with three or more sides and/or curves such that the relationship between variables can be displayed along a straight line. Special nomograms are presented which will, for a particular date after the institution of the Gregorian Calendar(post 1582), give the day of the week which that date occured. Rosica Dineva, Mount Holyoke College ABSTRACT: This talk is based on the article Topological Abelian Groups, by Fred Wright (1957). If G is a topological abelian group (a TAG), and A is a nonempty subset of G, the set s(A)={x є: G: x+a: A } is a semigroup. In particular, these semigroups generate interesting subgroups of G. We will highlight some of the properties of s(A) and briefly discuss how these relate to the structure of the topological abelian group they are contained in. Vladimir Chernov, Dartmouth College ABSTRACT: Arnold observed that one can associate a knot to a wave front propagating on a surface according to the Huygens principle. Low used similar ideas to relate cosmological causality to linking. We show that generalized linking numbers (constructed in a joint work with Yu. Rudyak) often allow one to conclude that two events are causally related without the knowledge of the front propagation history. This conclusion can be made from the current shapes of the fronts of the events. Julian F. Fleron, Westfield State College ABSTRACT: The Mbius band has been a favorite in the categories of mathematical magic and recreational mathematics for over a century. It has also had important applications and deep connections to art throughout its long history. Here we recycle this object once again, showing how it can be used as a tool to create elebratory artwork, including: garland, hearts, and stars. The audience will be invited to try their hand at building their own examples of this artwork, including the symbol of the Holocaust remembrance documentary Paperclips. These examples set the stage for a powerful metaphor for mathematics, the Mbius metaphor, which will be explored to help conclude this participatory talk. Geoffrey Scott, Dartmouth College ABSTRACT: The hexaflexagon is a mysterious geometrical toy that exhibits equally mysterious mathematical properties. We explore methods that mathematicians use to build and analyze hexaflexagons, and discover some surprising traits of these seemingly simple objects. Wayne Younghans, St. Michael ABSTRACT: The study of Number Theory during our undergraduate years introduces us to many intersting types of numbers that have many applications. In this presentation I will describe an interesting group of numbers called the Stirling Numbers, and display some of the applications of these numbers in the world around us. I will be primarily focus on the Stirling Numbers of the second kind and their interesting characteristics. David Quinzi and Garrett Jones, SUNY Geneseo ABSTRACT: The increasing amount of research being done in silico has made providing researchers access to computational research tools more important than ever. We will present our methods and examples of providing broad simplified access to in silico computational research. These methods can be used to make use of idle processor time on servers as well as provide a simple user interface for research applications run on distributed systems. Christina Callear, SUNY Geneseo ABSTRACT: Group testing, or pooling, is a widely used general procedure applicable whenever a large group of objects is to be subjected to the same test. In many cases, a number of objects or complexes of the objects to be tested will produce an undesirable result. Group testing offers an advantage over individual testing when the number of these target objects or complexes is small in comparison to the size of the group as a whole. Group testing has proved useful in a range of fields including medical diagnostics, industry and most recently in working with DNA. This work addresses the development, optimization and expansion of a probabilistic group testing method that leads to the identification of portions of cohorts or complexes that collectively produce an observable result. Pam Welch, Nazareth College of Rochester ABSTRACT: In 1640, Philip Naude wrote to Euler asking how many ways there were to add seven different positive integers to get 50. A brief history behind integer partitions will be given along with the fascinating way that Euler came up with the solution. Jacqueline Dresch, SUNY Geneseo ABSTRACT: Circulant graphs have recently been used as starting points in constructions of small-world networks used to study disease dynamics. We take random induced subgraphs of circulant-like graphs, representing the susceptible population after random vaccination. Then, by computing the expected size of the largest component in these subgraphs we have a bound on the number of individuals who can contract the disease being studied after one infection. Ashish Dixit, St. Lawrence University ABSTRACT: My research involves development of an Automated Theorem Proving Assistant using Java and exploration of the fundamental concepts of first-order logic through the automation process. The goal is an enhanced understanding of the techniques of first-order logic through analysis of the process of development. The talk will focus on the software development process in this project and the problems encountered. It will also include a report on the results of the project. Tomasz Przytycki & Wui-Ming Gan, Bard College ABSTRACT: The talk will begin with an overview of the pigeon-hole principle. Then we will prove, using the pigeon-hole principle, that for any sequence of mn+1 distinct real numbers, there exists an increasing subseuence of length m+1 OR a decreasing subsequence of length n+1 (or both). Klarenc Hoxha, SUNY IT ABSTRACT: From a Mathematicans point of view a Mathematical Model is a set of expressions and equations which describe a physical pehnomenon. For this presentation I will illustrate the physical system of a fallen rock with an Initial Velocity V1 and a initial Time T1. With the given data I will focus my presentation in mostly two major points: a) Building the Mathematical Model of the Physical Phenomenon and b) Describing how the model fits in our everyday life. Also to Ilustrate this experiment I will demonstrate the procedure with a users imput data on a C program. Mona Merling, Bard College ABSTRACT: Besides his remarkable theories in physics, Huygens is also the author of some very useful mathematical theorems. I will give a proof of the famous inequality that was named after him. I will use mathematical induction in an unusual and clever manner. Stefan A Elrington, Williams College ABSTRACT: We provide a simple algorithm for partially predicting cubes of integers. Matthew Dalzell, Manhattan College ABSTRACT: A knights tour is a list of squares that a knight lands on in a sequence of traditional knight moves. In the Knights Tour problem, a knight must land on every square of a chessboard once. It has been explored on traditional boards with varying number of squares, in which case the existence of a tour depends on the dimensions of the board. What happens when the chessboard is printed on a sphere, viewed as a pillow with the chessboard printed on both sides? We will explore the solutions to the Knights Tour problem on a pillow chessboard. Andrija Perunicic, Bard College ABSTRACT: Suppose that n people throw their hats in the air. After walking around, everybody picks up a hat from the floor. What is the chance that no one gets his or her hat back? As the crowd participating gets really large (as n tends to infinity), the probability gets closer to 1/e. I present a combinatorial proof residing on the Inclusion-Exclusion Principle. Daniel FitzGerald, SUNY Geneseo |
