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David Vella, Skidmore College ABSTRACT: In this talk we look at a collection of sequences which are defined by a recursion which generalizes the famous Fibonacci sequence. We derive the generating functions of these sequences, and use this information to relate them to a pretty counting problem concerning the number of ways of expressing a positive integer n as an ordered sum of certain smaller positive integers. As a special case, which we generalize, we give a new proof of the well-known fact that the terms of the original Fibonacci sequence give the number of ways of expressing n as an ordered sum of 1`s and 2`s. Erik Wallace, Hartwick College ABSTRACT: We will present the motivation for a theorem which gives a necessary condition for the simple continued fraction of any real algebraic number. We point out that the contrapositive of the theorem can then be used as a sufficient condition for the transcendence of a number and we apply it to the number e in particular. We also consider the converse and provide a counter example, showing that it is not true in general. In conclusion we mention a problem originally posed by Jacobi, which the theorem does not come close to answering but which may be answered perhaps by applying Galois Theory to continued fractions. Gwyon Sutton, Westfield State College ABSTRACT: An overview of Hakin`s algorithm and the unsolved problem of developing a computer program to apply it: Determining if any given knot projection is the unknot. Demonstration of a program`s attempt at it, and identifying the points in Hakin`s algorithm that cannot be achieved with the program. Edward Welsh, Westfield State College ABSTRACT: Lego Bricks are very good at making rectangular structures: they have right angles built right into them. But have you ever tried to build diagonals at different angles? Pythagoras can help in a very obvious way, but we will extend classical results using something called a continued fraction. We`ll learn just what that is, and get very good at approximating irrational numbers. A warning to participants: During this talk, you will play with Lego bricks. Yangyang Liu, Dartmouth College ABSTRACT: We will present several algorithms for deciding whether or not a given hypergeometric sum is expressible in closed form. After defining “hypergeometric term” and “closed form,” we will introduce Sister Celine’s Algorithm, Gosper’s Algorithm, and the WZ method, and show applications of them. We will also discuss when these algorithms produce closed forms and what it means when an algorithm returns “no answer.” We will close by pointing out that it is very hard to construct an exhaustive hypergeometric database. Andrew Dunn, Manhattan College ABSTRACT: A C++ program has been developed to simulate both linear and star configurations of beads. A random number generator is used to select pivot points as well as the angle of rotation. A great number of randomized configurations are generated. These randomized configurations are then used to calculate the mean squared radius of gyration for each configuration. The g ratio, the ratio of the mean squared radius of gyration of the star to the mean squared radius of gyration of the linear chain, has been computed and compared to theoretical expressions. Maple has been used to visualize these configurations. Yordan D. Minev, St. Lawrence University ABSTRACT: A biometric authentication system matches physiological characteristics to a database of such characteristics. In biometric authentication, genuine users are generally those that the system should accept and imposters are those that the system should reject. One methodology for evaluating the matching performance of biometric authentication systems is the receiver operating characteristics (ROC) curve. The ROC curve graphically illustrates the relationship between type I and type II statistical classification errors when varying a threshold across a genuine and an imposter match score distributions. The performance of each biometric system can be estimated via a confidence region for a ROC curve of that system`s performance. In this project ROC confidence regions will be created using radial sweep method. Radial sweep is based on converting the type I and type II errors to polar coordinates. The technique of bootstrapping will be utilized to estimate the variability of each point on an individual ROC curve. Simulations will be performed using real biometric match score data. A radial sweep method for comparing two ROC curves will be discussed. Amelia E. Stein, SUNY Institute of Technology ABSTRACT: The Golden Ratio, F, is one of the most astonishing numbers in mathematics. It plays a major roll in mathematics, natural beauty, and society as a whole. The Golden Ratio may be seen in geometry, ancient architecture, and even the human hand. Discover the theory behind this miraculous number and learn how it has touched our lives. John Tiglias, Manhattan College ABSTRACT: An equation for the Fourier Transform in an arbitrary dimension will be derived. This equation involes Bessel functions. A C++ program has been designed to apply this equation to simulation data in one to eight dimensions. Melissa Wasson, Hartwick College ABSTRACT: This talk investigates the delay time of intersections with traffic lights by comparing the results to what the delay times would be if there was a 4-way stop at the same intersection instead. This comparison allows the effectiveness of the traffic light to be determined – if the delay time for the traffic light is greater than the delay time for the same intersection analyzed as if it had a 4-way stop then there may be a problem with having the light at the intersection. The model used gives equations for calculating three delay times: traffic light, uniform flow 4-way stop, and random flow 4-way stop. Uniform flow is when cars arrive at the intersection in fixed time intervals (ex. 1 car/5 seconds), where random flow is when cars arrive arbitrarily (more realistic). Clare Duan, Boston College ABSTRACT: Optimization models play a critical role in determining portfolio strategies for investors. The traditional Mean Variance Optimization approach has only one objective, which fails to meet the demand of investors who have multiple investment objectives. This talk is going to present a multi-objective approach to portfolio optimization problems. The proposed optimization model simultaneously optimizes portfolio risk and return for investors and produces optimal portfolio strategies for investors of any risk appetite. Detailed analysis based on convex optimization and application of the model are provided and compared to the mean variance approach. Megan Hargrave & Kristin Spampinato, Nazareth College ABSTRACT: Abstract: We will provide several ways to model the harmonic series: = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ...+ 1/n. Using calculus we will discuss various applications of the harmonic series. We will use the fact that the series diverges to infinity to demonstrate one application, namely, that it is theoretically possible, yet unrealistic, to build a bridge from America to England which involves no nails, glue, or other adhesives. Julie Muetterties, St Lawrence University ABSTRACT: A survival curve shows the proportion of a population at risk which survives up to a certain time. Such curves can be described by theoretical parametric models (such as exponential, lognormal, or Weibull) as well as nonparametric methods (such as Kaplan- Meier). We investigate methods for determining if survival curves from two populations or treatments are significantly different with applications to real data. Zach Warner, Plattsburgh State University ABSTRACT: How many ways are there to make change for a dollar using pennies, nickels, dimes, quarters and half dollars? The answer to this question can be obtained via a long, drawn-out process of listing every possible option. A much quicker and sleeker method involves using power series to create a generating function. Using this, we can solve the problem by examining specific coefficients in the function. Lauren Rudowsky, Manhattan College ABSTRACT: Suppose you randomly select real numbers between 0 and 1, and stop selecting when the sum of the chosen numbers exceeds 1. How many numbers on average do you expect to pick? This question will be answered in my presentation. Kayleigh Rose, Ithaca College ABSTRACT: We discuss the paths created by at least one creature in pursuit of another and the differential equations developed to define these curves. We investigate real world applications of these curves as well as some more imaginative cases, including the movement of creatures stationed at the vertices of regular polygons. Pam Welch, Nazareth College of Rochester ABSTRACT: The delivery of drugs can be examined using the parabolic partial differential equation. We will explore how a drug is released throughout the body through the diffusion process, focusing on time-release pills. Katie Baldiga, Williams College ABSTRACT: Is it possible to slice a polyhedron and produce only convex cross-sections? Can this be done using a slicing plane with fixed normal direction? What if we are free to rotate and translate the slicing plane? We will explore these questions using three-dimensional dualization techniques, where solutions appear in the form of paths through the dual. Alison Blank, Hamilton College ABSTRACT: Category Theory provides a foundation for mathematics in which the concept of function, as opposed to that of set membership, is the fundamental building block of mathematical structure. In this talk, I will give an introduction to the categorical way of thinking about mathematics and compare some of its basic results to our more familiar set-theoretical intuitions. Brian Simanek, Williams College ABSTRACT: Verblunsky`s Theorem establishes a one-to-one correspondence between sequences of complex numbers in the unit disk and nontrivial probability measures on the unit circle. This theorem provides a beautiful connection between measures, orthogonal polynomials, and a special class of unitary matrices called CMV matrices. I will introduce some important results pertaining to the spectrum of random CMV matrices and the zeroes of random orthogonal polynomials. Laura Mansfield, Ithaca College ABSTRACT: We investigate Gray Code and so-called Finger Games based on Gray Code. These Finger Games involve counting in Gray Code alternately on two different interacting “hands” where each “finger” represents the different bits of the Gray Code numbers. The process where a Finger Game starts and returns to a specific starting position is referred to as an orbit. We explain such orbits, from both the geometrical and numerical perspectives, based on prior investigations of Gray Code patterns. Stefana Vutova & Penyo Michev, Ithaca College ABSTRACT: Continuous parametric curves that are traced by a point mounted on a wheel, mounted on a wheel, and so on, display symmetry which is directly related to the frequencies of their complex exponentials. Frank A. Farris showed that if all frequencies are congruent to some number p mod m, where p is relatively prime to m, the curves have m-fold symmetry. We prove that if p is not prime to m, then the resultant symmetry is m / GCD(p, m)-fold. Further, we show that if all the frequencies are multiplied by some integer k, then the same curve is obtained and it retains the same symmetry as the curve produced by the original frequencies. Caitlin Owens and Alana DePoint, Ithaca College ABSTRACT: We discuss Kaprekar’s Routine regarding the exploration of a four-digit number subtraction algorithm. Furthermore, we examine its application to two and three-digit numbers. Keri Sheridan and Zack Simmons, Ithaca College ABSTRACT: We generalize Kaprekar’s Routine for three-digit numbers, and apply a subtraction algorithm to all permutations of the digits of these numbers. We discuss the fixed points and cycles of these generalized routines. Anna Foster, Marist College ABSTRACT: Cross-correlation of the long-time (73 day) average of ambient seismic noise provides an approximation of the Green`s function between station pairs for all combinations of 9 stations located in Northeast Pakistan. Causal and acausal signal-to-noise ratios of Rayleigh wave Green`s functions are compared to determine directionality of surface waves, which thus far appear to move in a general north to south pattern. Hilbert Transforms are used to analyze the group velocities for waves of different periods and to make graphical depictions of the region`s dispersion characteristics. Colin Cubinski & Darren Lim, Siena College ABSTRACT: Software Engineering (SE) is a complex field of computer science, where students must utilize their technical expertise along with communication skills. Interacting with other people (clients, other programmers, etc.) is key to the success of your software project. These aspects of software engineering have been modeled by a card game called Problems and Programmers(2003). We will demonstrate the card game as a pedagogic tool for introducing students to SE. Danielle Boucher, Westfield State College ABSTRACT: Understanding the concept of projectile motion for a skating jump can help coaches and skaters most successfully master complex figure skating jumps. In this presentation we will examine the axel jump and see how the speed and initial velocity directly affect the success of the completion of the jump. By applying Newton’s Second Law of Motion to many mathematical relations we will be able to find the skaters position in the air at any give time (t) as the skater completes their jump. This will help us see the most desirable way to achieve the best skating jump. Trina Lawton, Russell Sage College ABSTRACT: The Leibniz harmonic triangle can be produced from the familiar Pascal triangle. In this talk we will explore some properties of this and other Leibniz triangles. Nick Alena, St. Lawrence University ABSTRACT: The Mona Lisa => Leonardo DaVinci, 2.0cm Wing Span => Deadly Midge, Viagra => Spam. Is it possible to use numerical characteristics of a painting to determine its creator, or measurements of an insect to distinguish its species, or titles of emails to separate spam from legitimate messages? Linear discriminant analysis (LDA) is a statistical method used to identify group membership using a linear combination of features. The techniques extend concepts from ANOVA (analysis of variance) and regression analysis. We examine methods for choosing good discriminating variables and producing a linear combination that best distinguishes between the groups. Amanda Lowery, Russell Sage College ABSTRACT: We consider Banzhaf power distributions in weighted voting systems with four or five players. Of particular interest is the notion of hierarchy and its preservation. Sadia Choudhury, Russell Sage College ABSTRACT: Annealing is a physical process, and algorithms to simulate it can also be used to solve combinatorial optimization problems. We explore this process through an example. Daniel Bosco, Colgate University ABSTRACT: In this talk I will be introducing an old, but rarely used method for evaluating the determinant of a matrix called the Condensation Method. I will show how it is done with a few examples and then I will be giving a quiz to see if you can do it for yourself. After going over the quiz I will explain the mathematics behind this Condensation Method. Finally, I will share an interesting anecdote regarding this method`s origin. Sara Reynolds, Nazareth College of Rochester ABSTRACT: With the emergence of new strains of viruses, antibiotic resistant bacteria, and other problematic infectious diseases, the study of epidemiology has become increasingly important. Epidemiologists mathematically construct models of the spread of diseases in order to formulate and evaluate methods for their efficient prevention and treatment. The mathematics behind different models will be discussed and compared. Applications and accuracies of such models to actual epidemics or endemics will also be introduced. Andrew Burgess, SUNY Potsdam ABSTRACT: I intend to give a simple proof of the well known fact
Time allowing I’ll show some neat applications of this amazing fact. Yudishthir P. Kandel, Wesleyan University ABSTRACT: Turan`s theorem is one of the fundamental results in graph theory. It was proved in 1941 and initiated extremal graph theory. It was rediscovered many times. This talk will discuss some different proofs of the theorem. Benjamin Case and Steven Ciesla, St Lawrence University ABSTRACT: Scientific computing is requiring greater and greater processing power. Cluster computing provides a inexpensive means of creating powerful parralel processing computers. We construct a Beowulf Linux cluster for Bioinformatics work and discuss the research and construction of the cluster as well as the mechanics of how parralel processing systems are used to tackle large scale problems. Heather Eckman, Nazareth College ABSTRACT: Abstract: Stowing away in enclosed areas is no longer a safe bet. Scientists have found new technologies that detect a human heartbeat in an enclosed area. Explored is the mathematics behind this new technology, using the hyperbolic partial differential equations. This and other applications will be discussed. Gabriel Marcus, SUNY Institue of Technology ABSTRACT: Cellular automata are discrete mathematical systems characterized by a lattice of elements that evolve over time according to a set rule. While ostensibly simple, such systems can potentially lead to unpredictable complexity and variety. Because of this trait, cellular automata are used extensively as models of phenomena in the physical and social sciences. In this presentation, I intend to provide a brief theoretical introduction to CA systems and discuss applications to biology and urban dynamics. Brendan W. Sullivan, Hamilton College ABSTRACT: We examine the results of imposing restrictions on the allowed transformation sequences of a standard Iterated Function System (IFS). More specifically, we seek to extend current results on restrictions based on the previous single transformation to restrictions based on the previous two, or possibly more, transformations. This investigation is performed with the goal of identifying when a restricted IFS fractal can or cannot be produced by a standard IFS with a finite set of transformations. Leah Ziamandanis, Union College ABSTRACT: “Now it will be open to those who possess the requisite ability to examine these discoveries of mine… I judge it well to communicate them to those who are conversant with mathematics, I send them to you with proofs, which it will be open to mathematicians to examine.” These are the words of Archimedes, a great mathematician of antiquity. His groundbreaking discoveries in the world of mathematics furthered the work of his predecessors, Eudoxus and Euclid, and demonstrate insight thousands of years ahead of its time. The work of these three men produced methods and theorems foreshadowing modern-day calculus and limits that were not established for 2000 years following their existence. Although the work lacks a certain level of modern formality, it is, in essence, calculus. We can examine as mathematicians, just as Archimedes requested, a reproduction of this work. Using an ancient lens for examination, we are offered insight into the perspective of these great thinkers. The conclusion, that modern-day calculus was thought of for the first time in the years before Christ, is a way to give credit where it is deserved in the history of mathematics. Fei Yao, Colby College ABSTRACT: A young mathematician is travelling by air to the International Mathematics Olympiad Competition. Just after check-in there is a power cut and the airline computer loses data. This does not affect the passengers that much, except that none of them was issued with a boarding card. They thus sit at random in the plane. To keep her brain active, ready for the competition, the young mathematician decides to find the probability that none of the passengers are sitting in their correct seat. Kellen Myers, Colgate University ABSTRACT: In Ramsey theory, Schur\`s theorem answers a fundamental question: within any coloring of the integers, there are monochromatic solutions to x+y=z (a property called regularity). Rado\`s theorem describes regularity conditions for any linear homogeneous equation (or system). However, Rado also showed that for 2-colorings that linear homogenous equations (and systems) are 2-regular in any nontrivial circumstance. In this talk, Rado\`s result will be presented and proof of an off-diagonal generalization will be given (off-diagonal indicates that a different equation governs each of the two colors). Further discussion will present the determination of some Rado numbers (off-diagonal and regular) in some specific and some general cases. Robin Augustine Thottungal, State University of New York,New Paltz ABSTRACT: One of the major problems in scientific research today is the analysis of time series composed of experimental data. Such sequences are obtained by successively sampling over an observable quantity, which characterizes the dynamical system under investigation. When dealing with linear dynamical systems, obtaining information from the time series is relatively easy. However, when the dynamical system under consideration is nonlinear, difficulties arise. As a first step towards understanding the nonlinear behavior of dynamical systems, we transform the nonlinear time series data from the time domain to the frequency domain by using the Discrete Fourier transform. In this study, we use the logistic equation, which is a formula for approximating how populations of animals change over time. This equation measures how populations respond to predators, availability of food, land, and other changes in their environment. The logistic equation was created by the biologist Pierre Verhulst in 1845. The equation is as follows: Zulma Valcárcel, Ramapo College of New Jersey ABSTRACT: Legend has it that Josephus, a Jewish historian in the 1st century, survived the Jewish-Roman war thanks to his mathematical talent. Josephus was one of 41 Jewish rebels trapped in a cave by the Romans. Preferring suicide to capture, they decided to form a circle and every third person would commit suicide. They would proceed around the circle until no one was left. Not too fond of the idea, Josephus calculated quickly where in the circle to stand, and hence saved his life. This is the well-known Josephus problem. We consider two situations: 1) the J-2 problem, where every second remaining person is executed, and 2) the J-3 problem, where every third remaining person is executed. We first discuss the method by Knuth-Graham-Patashnik on the J-2 problem, and the difficulty to generalize it to J-3. We then approach the J-3 problem using another set of recursive relations. Based on these recursive relations, we generate algorithms and analyze the numerical outcomes. We then discuss the closed-form solution of the J-3 problem. Alexander La Point, Siena College ABSTRACT: Have you ever watched a play or movie and tried to guess what the characters will do next? By applying the mathematical techniques of game theory to the decisions characters make in the theatrical text we will try and find a mathematical structure within these decisions. Through the use of different types of game theory analysis, guessing what your favorite characters next move will be, might not be so hard. Lauren Remmes, Saint Michael`s College ABSTRACT: If you have to go somewhere in the rain, will running keep you drier? This paper will take a mathematical approach to running in the rain to determine if a person actually has less rain fall on them if they run in the rain or simply walk. The mathematics used will include vectors, volumes, and some basic physics. Several factors will be taken into consideration: velocity, wind, amount of rain falling, and the person`s size. Kevin Roode, Quinnipiac University ABSTRACT: How can math be used in solving magic tricks? Here is an approach that uses abstract thought, as well requiring a mathematical interest. Rope and card tricks will be shown and solved. Katie Berry, St. Michael`s College ABSTRACT: There exists a large class of problems for which polynomial time solutions have not yet been found which are known as NP-complete. Despite an enormous amount of work in this field, it cannot be proven that no such solutions exist. Finding a polynomial time solution for one would imply that a polynomial time solution exists for all of them. Within this so-called “gray area,” some problems are easier than others in practice. Approximation algorithms run in polynomial time and find solutions that are provably close to optimal. For some NP-complete problems, we are able to find good approximation algorithms, while for others finding a good approximation algorithm appears to be as difficult as finding a polynomial time solution. This presentation will contrast one of the easier NP-complete problems, vertex cover, with a much harder NP-complete problem, set partitioning. Sam Ferguson, Simon`s Rock of Bard College ABSTRACT: We are going to look at the historical controversy concerning Cardano’s publication of the solution to the cubic equation x3 + px2 = q, based on an algebraic method developed by Niccolo Fontana Tartaglia (we`ll also examine the mathematical reasoning behind that solution). The success of this method was first demonstrated by Tartaglia’s victory in an academic duel in 1535, but was supposed to remain a secret until Tartaglia published his findings. Instead, Gerolamo Cardano persuaded Tartaglia to tell him the formula, and with it he published Ars Magna, the first systematic work on algebra and equation theory, citing Tartaglia without permission. Andrew Rogers, Westfield State College ABSTRACT: Computers are getting small: so small that soon we will reach a road block known as quantum mechanics. The next logical step would be to research quantum computing. How far are we in the research? How powerful are these machines? How can they be used? In this talk we will see just how powerful they are and the consequences it brings with this computing power. Catalin Iordan, Williams College ABSTRACT: I will introduce the notion of unique ergodicity and explore some of its properties and connections to the concepts of minimality and periodicity of transformations through two interesting theorems. Finally, I shall give two examples of transformations which either possess the main property (unique ergodicity), yet do not exhibit one of the secondary ones (minimality) or to which the contrary applies. Andrea Adams, St. Lawrence University ABSTRACT: Bioinformatics is a field in which computer systems are used to process experimental data from biological experiments. Working with members of the Biology department, I have constructed a relational database system to aid them in processing the data from an experiment involving the results on genes expressed in neurons and mitochondria in the nemetode c.elegans when it is raised in an environment devoid of oxygen. To accomplish this, I mined data from the existing SAGE and Mitores databases to construct two relational databases: one detailing all of the genes expressed in neurons, and the other detailing the genes expressed in mitochondria. Then, once experimental data has been loaded into the database, the biologists use a web interface to access and query the databases. Andrea Austin, Saint Michael`s College ABSTRACT: The circuit partition polynomials are a relatively simple transformation of the Martin Polynomials, which encode information about families of cycles in a graph. In this talk we will explore the basics of the circuit partition polynomial. We will also discuss some of its applications and its relation to both the interlace polynomial and the Tutte polynomial. Andrea Austin, Saint Michael`s College ABSTRACT: Cardinality is the measure of the "number" of elements in a set. This talk will introduce the idea of cardinality and countable and uncountable infinite sets. The main focus will be on presenting Zorn’s lemma and some of its applications. Emma Schlatter, Smith College ABSTRACT: What happens when a grid of squares is filled with diagonal mirrors, and a beam of light sent through? What sorts of light paths emerge for different configurations of mirrors? Different-sized grids? Is there a limit to the length of path for any given grid? I will discuss the theoretical maximum path lengths for these mirrored grids, and their achievability in various cases. Julie-Anne Shaw, Westfield State College ABSTRACT: Newton’s Law of Heating and Cooling states that the rate an object heats or cools is directly proportional to the difference in temperature between the object and its environment. I investigate if and how this law can be used to model the temperature in a small reservoir. I start with air and water temperature data collected on the Westfield River, western Massachusetts, over a period of 8 months. I address the problem of “cleaning up” the data, measurement accuracy, external influences, and other factors that may affect the model’s validity. Finally, I will show a simple model and demonstrate how well it fits the actual temperatures. John Tolle, Carnegie Mellon University ABSTRACT: We all know that though the natural logarithm function grows without bound, the growth is very slow. But as "very slow" is a comparative phrase, we usually seek to understand its snail`s-pace growth relative to some other slowly-growing function, such as a root function. The relative growth rates can be established using l`Hopital`s Rule, but the result is a sort of one-dimensional view of growth without bound (GWB). But in this talk, we develop a two-dimensional perspective by looking at solutions to a simple first order differential equation. By adjusting parameters, we examine the underlying assumption concerning the growth of solutions. Then we extract the solutions. We will end up with the following classes of functions (in order of appearance): * exponential functions * "doomsday functions" * polynomial functions * linear functions * root functions * factorial "functions" * logarithmic functions By comparing growth assumptions which led to each class of functions, we can understand the relative growth rates from a perspective which l`Hopital`s Rule cannot give us. Hanh Pham, Connecticut College ABSTRACT: Given a graph G, the conditional chromatic number of G with respect to the subgraph H, X(G,H) is the minimum number of colors needed to color the vertices of G so that no color class of vertices of G contains an induced copy of H. We say H is a forbidden subgraph in each color class. In this talk we will present results on the values of X(G,H) for certain families of graphs G while forbidding complete graphs, cycles, and wheels. Gregory Loftus, University of Massachusetts Boston ABSTRACT: Bezier curves are parametric curves developed in the early 1960’s for use in CAD/CAM operations. Today Bezier curves have a wide range of applications, including graphic design and music synthesis. The speaker will give the motivations for Bezier curves through a brief history of numerical analysis and provide an overview of various applications. Joel Foisy, SUNY Potsdam ABSTRACT: This talk will be about graphs that have cycles that form a non-split 3 component link in every spatial embedding. We call such graphs intrinsically 3-linked. It remains an open question to classify the complete minor-minimal set of graphs with this property. Brendan Pankovich, University of Massachusetts Boston ABSTRACT: A regular polytope is a highly symmetric n-dimensional geometric figure. Familiar examples include the regular polygons and Platonic solids. This talk will present a quick way of computing the number of symmetries of a regular polytope from the symmetries of its faces. The proof relies on a simple application of the theory of G-Sets and orbits of a group action. Tim Morse, St. Lawrence University ABSTRACT: Anyone who has taken an introductory statistics course is familiar with the idea of assumptions. In order to make all those wonderful statistical calculations work, your professor told you to “assume the underlying population is normally distributed.” As you progressed through your statistical education the assuming undoubtedly continued. But what really happens if we leave out all the assumptions? Better yet, what if the real world refuses to cooperate with us? This talk will explore these questions and reveal what, if anything, goes wrong when the assumptions statisticians cling to no longer hold true. Dipesh Mainali, St Lawrence University ABSTRACT: Suppose that a finite set of n objects are scattered randomly in a plane. How can a shortest possible network of straight lines connect them? The Shortest Network Problem has plenty of practical applications and thus is one of the most studied Graph Theory problems. In this talk we will look at what Minimal Spanning Trees and Minimal Steiner Trees are, how they can be found, and how they can be used to find the shortest network. Shuting You, Mount Holyoke College ABSTRACT: As the demand for secure information transmission increases, it has become more crucial to find efficient and accurate primality testing and proving methods. In this talk, I will introduce the ground-breaking algorithm called the AKS primality test developed by Manindra Agrawal, Neeraj Kayal, and Nitin Saxenain in 2002. This is the first unconditional polynomial-time deterministic primality-proving algorithm. The talk will give the main idea of the algorithm with examples and discuss its complexity. Jonathan Kaptcianos, Saint Michael`s College ABSTRACT: Formed from the past idea of Sequencing by Hybridization (SBH), fragment assembly is newly explored concept of determining whether or not a reassembled strand of DNA matches the original strand. One particular way to analyze this method is by using concepts from graph theory, such as Eulerian Paths and de Bruijn graphs. By reconstructing models of data based on these ideas, it is possible to come to various conclusions about the original problem regarding reassembled strands of DNA. Specific approaches of this which will be explored include the Path Decomposition Algorithm, the Eulerian Superpath Problem, and Efficient String Reconciliation. Susan Beckhardt, Union College ABSTRACT: A graph is said to be intrinsically n-knotted if every embedding is guaranteed to have at least one knotted n-cycle. We expand techniques used by John Conway and Cameron Gordon to show that K8, the complete graph on 8 vertices, is intrinsically 8-knotted. We also discuss an alternate proof that Kn is intrinsically n-knotted for all n > 6. Jeffrey Rathbun, Canisius College ABSTRACT: Baseball is known as America`s Pastime. It is a sport played and watched by many. Even in the early days of the game, observers noticed there needed to be some kind of records kept. Without statistics, we would not be able to fully appreciate the greatness of baseball. With words like "Moneyball" and "sabrmetrics" thrown around these days, statistics are more important now than ever before. Ellen Galo, St. Lawrence University ABSTRACT: As labyrinths have become popular for meditation and entered the mainstream, the means to analyze existing labyrinths and create new esigns has also become a topic of interest to both mathematicians and non-mathematicians alike. In this talk I will survey and summarize the available literature, and indicate where labyrinth research and design connects to already-existing areas of graph theory and combinatorics. Some labyrinth history will also be mentioned as background. Zeb Engberg, Hampshire College ABSTRACT: Polynomials with coefficients in a finite field have many properties in common with the familiar integers. We have a division algorithm, we have unique factorization, we even have analogies of Fermat`s little theorem and the notion of residue classes mod some polynomial. How far can we generalize the theorems that we know and love about the integers?? Gauss called his quadratic reciprocity law the “golden theorem” of number theory and the greatest mathematician of all time never lies. In this talk, we will prove quadratic reciprocity in this fabulous new ring of polynomials over a finite field. Jasper G Burch, Saint Lawrence University ABSTRACT: Social Capital is loosely defined as resources which are embeded in a social network. There are many types of socail capital including: prestige, decision making power, knowledge and experience. In this presentation we will explore ways to measure social capital and examine how it impacts corporate change processes. We will also examine ways in which to effect the social capital available for or against a change by changing the structure of a soial network. Matt Way, St. Lawrence University ABSTRACT: We’ve all seen the tall smoke stacks releasing massive clouds of smog into the air, but were does it go? This presentation is an application of multivariable calculus and probability to predicting where the smoke will end up out of a smoke stack. We will explore the diffusion process itself, take the Gaussian Plume Model from the two-dimensional page and put it into the 3 dimensional context of a smoke stack, and even prove Laplace’s Equation using air pollution. These questions are important when installing a structure that will be emitting smoke, in relation to the type smog being emitted and what can be found in the surrounding community. Evan Cullerton, Westfield State College ABSTRACT: ABSTRACT: Have you ever thought there is a pattern to the prime numbers? This talk will include an explanation of prime numbers and how Mersenne Primes are one pattern that can be used to find prime number candidates. Mersenne Primes can be used as a guideline to lead to other patterns. The search for the highest Mersenne is in progress, however in this talk you will learn how to find prime candidates using other patterns Mersenne Primes neglect. Colleen Lanz, Canisius College ABSTRACT: The group of 2x2-matrices with integral entries and determinant 1 plays an important role in number theory. We will give a characterization of the matrices from this group that become the identity matrix when they are reduced mod N. Our methods are combinatorial and they use Farey Sequences of rational numbers. Nick Ommen, Ithaca College ABSTRACT: In this presentation, we will outline our investigation of the use of recurrence relations in symbolic integration. Specifically, we will focus on a number of general integrals that are often given in standard calculus books "as-is" and show how they can be derived through techniques that are accessible to undergraduate students. We will also demonstrate our use of Mathematica as a verification tool in conjunction with our inductive proofs of correctness. Since this project makes use of tools and techniques from mathematics as well as computer science, we will conclude by discussing some of interdisciplinary aspects of our work. Laura Beaudin, Saint Michael`s College ABSTRACT: The focus of my talk will be a mathematical model known as the Potts model. This model has applications in a variety of different areas such as economics, chemistry, physics, biology, and sociology. The model is used to study natural phenomena such by exploring interactions of internal elements of a complex substance and predicting the behavior of the object in the long run. I will begin my talk by introducing the basic structure of the model. In this section I will define its Hamiltonian, probability, and partition function. I will then focus on the partition function. It can be calculated in two ways. The first way is an exact calculation which involves the Tutte polynomial. The second is an approximation which uses certain types of Monte Carlo simulations. I will end my talk with a discussion of some real world applications of the model such as the study of tumor growth, foam flow, and human interactions. Jaclyn Siedlecki, Union College ABSTRACT: Two line segments are said to be commensurable if there exists a third line segment that measures both of the original line segments. In the sixth century BC, the ancient Pythagoreans believed that any two line segments are commensurable. This was central to much of their mathematical work as well as their philosophies about life in general. The discovery of incommensurable magnitudes created a crisis in Pythagorean mathematics and philosophy because it contradicted the underlying principles that everything could be described using whole numbers and their ratios. The crisis was resolved by Eudoxus, at the expense of much of the Pythagorean philosophy. This discovery also presented the first hint at the existence of irrational numbers. John Jaramillo, SUNYIT ABSTRACT: The Zeta – distribution, or zipf’s law, is an empirical law that states that the frequency of the usage of a word is inversely proportional to its rank on a frequency table. My project acts on the assumption that most writers find a writing style that suits them and thus generate their own frequency tables in their work that doesn’t necessarily match the one for the English language on a whole. However, the frequency tables that they do generate are unique to them and could be used to show whether a disputed piece of work was actually written by the author in question. My project attempt to demonstrate (1) that writers with different writing styles produce noticeably different zeta – distributions and (2) that such distributions could be used to identify the author of an unknown piece of writing Brian Sullivan, Canisius College ABSTRACT: Coxeter groups are groups generated by reflections. In this talk we will give a combinatorial description of the normalizer of a finite subgroup of a Coxeter group. Our method uses the actions of specific involutions on subgraphs of the Coxeter graph of the group. Stephen Huenneke, University of Massachusetts, Boston ABSTRACT: Fourier Transforms are presented as generally as a linear transform used to decompose a function. Yet many practical discrete applications of Fourier Analysis of signals exist in electronics, audio, and image processing. This talk aims to explain how a Discrete Fourier Transform can be rather simply applied and accelerated to perform blazingly fast calculations on large quantities of data, including some simple examples of signal recovery and processing. Gregory Quenell, Plattsburgh State University of New York ABSTRACT: SET (R) is a simple card game using a special deck of cards. The cards in the deck correspond to the points in a four-dimensional vector space over the field of three elements. Though the official instructions don`t put it quite this way, the object of the game is to find straight lines through this vector space. Quang Nguyen and Yin Tian, Hamilton College ABSTRACT: Some theorists have raised their doubts about the validity of predicting the future outcome of the stock market. They argue that market movements are random, thus the chance to win or lose in a certain bet is 50/50. An extreme example about a monkey throwing darts as a way of picking stocks has been a ridicule to the efforts of constructing a decent portfolio of many research analysts, economists, and investors on Wall Street. We would like to use this opportunity to present our research on the interesting topic about the randomness, if there is, of the stock market. We would like to examine whether, theoretically, one can pick stock by just throwing darts. Brenda Johnson, Union College ABSTRACT: Singular homology is an algebraic tool used to study topological spaces. In this talk, we will discuss some of the basic ideas and results in singular homology theory and show how they can be used to distinguish between spaces. This talk is also intended to provide the background material for Thomas Mazur`s talk on analyzing coverage in sensor networks. Thomas Kern, Dartmouth College ABSTRACT: A linear ordering consists of a way of placing points on a number line. I will explain how to play Ehrenfeucht-Fraisse games on linear orderings, and how these fun exercises help us discover important facts about the underlying logic of linear orderings. For instance, given only a certain quantifier depth, we may only exactly specify a finite number of linear orderings. If time permits, I will discuss my current research into what these numbers are. James Henry McConnell, Hamilton College ABSTRACT: A graph is a collection of vertices (points), some pairs of which are connected by edges (lines). An automorphism of a graph is a rearrangement of its vertices that preserves vertex connections. Thus if the vertices are unlabeled, the graph appears unchanged after an automorphism. A determining set is a set of vertices that can labeled so that we can identify the remaining vertices after an automorphism. In this talk we will formally define determining sets, give examples, and explore the minimum number size of a determining set for certain families of graphs. Sarah Taylor, SUNY Plattsburgh ABSTRACT: Two-dimensional patterns are described using symmetry groups. Different cultures are found to use several characteristic symmetries in their artwork, showing a preference in arranging design elements. I will introduce various symmetry groups and show how they appear in the artwork of diverse cultures. Charles Allaire, Westfield State College ABSTRACT: This talk will discuss the creation of a high school mathematics course for upper-classmen based upon some existing college courses at Westfield State College. This course will be a discussion course that takes a deeper look at mathematics than “just numbers in a book.” It is the hope that this course will promote mathematics in a positive way to high school students, when what they know of mathematics seem to be what is presented to them for preparation of standardized testing. Jonathan Stults, Hamilton College ABSTRACT: Erdos and Renyi proved that almost all countable random graphs are isomorphic. We will discuss the necessary random graph and probability theory to understand the heart of the proof of this remarkably counterintuitive fact. Afterwards, we will explore some unique properties of the countable random graph and its almost uniqueness. Katie Crawford, Saint Michael`s College ABSTRACT: In this talk we will describe the basic nature of functions which are continuous everywhere, but never differentiable, considering the proof that such a function exists. We will examine these functions in their historical context and highlight some of their major applications. Mario Carullo, Mary Thurston, Marist College ABSTRACT: We will show that in any similar pentagon/pentagram arrangement the ratio of the length of a side of the pentagram to the length of a side of the pentagon is the golden ratio. We will also show that cos p/5 = a/2, where a is the golden ratio. Matt Koetz, Nazareth College ABSTRACT: Coding theory is the study of methods of transmitting data efficiently across noisy channels. When data is transmitted, errors are likely to occur. Coding theory aims to reduce the number of errors, detect and correct errors, and do these things as efficiently as possible. In the search for better codes, we use results from many branches of mathematics, including linear algebra, combinatorics, graph theory, geometry, probability, and number theory. We will explore the ways in which coding theory uses each of these fields, from its basic definitions to its most beautiful results. Gary Malouf Jr., Marist College ABSTRACT: We will look at Euler`s derivation of formulas relating the roots and coefficients of a polynomial. While they are the same as those that Newton developed, we emphasize the unusual path that Euler took to reaching his desired result: specifically, how he attacked an algebraic theorem about roots and coefficients by means of logarithms, derivatives, and geometric series. Tate Shippen, St. Michael`s College ABSTRACT: The RSA cryptosystem, now 30 years old, is still used in numerous security applications today. In this talk, we will briefly consider the mathematical background of the RSA cryptosystem and the possible holes that exist in its security. We will also try to highlight some of the current applications of RSA and its effectiveness against modern attacks. Youqiong Chen, Hamilton College ABSTRACT: What makes student prefer pure math over statistics math? Is intolerance for ambiguity a factor that affects student preferences? This study collects data from Hamilton College math statistics classes to test the relationship between students' intolerance for ambiguity and student preference for pure math or statistics math. We may use Burner`s scale to test student`s intolerance for ambiguity and will conduct our own survey to test students` preference for pure math or ambiguity. We use method of covariance and also control for student`s performance in the class to test our hypothesis. David Miller, Saint Michael`s College ABSTRACT: Have you ever been hooked on the simple mathematical game called Sudoku? This game is based on the concept of Latin squares, which are nxn matrices such that the numbers 1 through n appear in each column and each row of the matrix exactly once. In this talk I will explain the mathematics behind Latin squares and look at some applications of Latin squares (e.g. eliminating the problem of varying soil quality during grain growth testing). David Miller, Saint Michael`s College ABSTRACT: When branched junction DNA molecules combine to form nanostructures, they adjoin complementary limbs. Each molecule has a particular set of these limbs and is therefore of a certain type. I am interested in increasing the efficiency of construction for these nanostructures by finding the minimal number of molecule types required for their construction. This problem can be modeled using graph theory where each nanostructure is a graph with vertices representing each branched junction molecule and edges representing adjoined complementary limbs. I intend to find the minimal number of molecule types (referred to as tile types) and edge types required for the construction of graphs under three different experimental settings. These settings are: 1) It doesn’t matter if an unintended graph can be constructed with the tile types. 2) No smaller graphs (graphs with fewer vertices) can be constructed with the tile types. 3) No smaller and no same size graphs (those with the correct number of vertices, but not isomorphic to the intended graph) can be constructed with the tile types. In this talk I will show the results for some important standard classes of graphs (and therefore the general nanostructure!). Natassia Piccolo and Jenna Schebler, Marist College ABSTRACT: In this talk we will discuss a construction of the heptadecagon, discovered by Carl Friedrich Gauss in 1796. We will briefly discuss Gauss`s contribution to this construction and the idea of Fermat Primes, which make the construction possible. We will also provide a brief demonstration of a ruler-and-compass construction of the heptadecagon. Tom Mazur, Union College ABSTRACT: We apply the ideas relating to homology that Brenda Johnson introduced in her talk to determine a criterion for verifying whether a network of detectors covers a 2-dimensional domain. In particular, we model detectors as nodes that radially cover a region of a compact domain. We assume that nodes can detect only the identities of those nodes that lie within some broadcast radius. That is, nodes can neither ascertain the distance nor direction between themselves and their neighbors. We establish a criterion via homology for verifying whether such a collection of nodes covers a domain. We ultimately modify our criterion for the case in which we generalize our assumptions. This talk is based on the work of Vin de Silva and Robert Ghrist. Sara Beth Rosenberg, Wesleyan University ABSTRACT: A bipartite graph Kn,n+1 can be decomposed into paths P2, P4,...,P2n with total number of edges 2,4,...,2n, respectively, for all odd n. The conjecture that this result is not possible for any even integer n was prompted by the observation that the predicted result is false for K2,3 since P2 and P4 cannot be achieved simultaneously. We will prove that we can actually decompose Kn,n+1 into paths P2, P4 ...P2n for all even n greater than or equal to 4. The predicted result is only false for n = 2. We will observe this result using an n x m array, Rn,m of points where the points correspond to edges in the Kn,m bipartite graph. Anna Haensch, State University of New York, New Paltz ABSTRACT: The (a,w)-derivative is an extension to the classical derivative, in that is allows one to define and study functions at points were certain functions were previously undefined. This is achieved by the introduction into the definition of the derivative of two new factors, namely a, and w. Before accepting this as a viable method for differentiation, it was necessary to prove its adherence to all the theorems surrounding classical differentiation, such as Cartheodory`s, Lagrange`s, Rolle`s, and many others. Having done so, this proves to be an acceptable method of differentation, and one that could be useful to calculus students at any level. Kimberly Elicker, Williams College ABSTRACT: The Rubik`s Cube, a major fad of the 1980`s, is a colored cubic toy in which the puzzler attempts to align all the faces with the same color. The packaging boasts "billions of possible combinations," but there are actually 43,252,003,274,489,856,000 different permutations. We will discuss the Group Theory behind a Rubik`s Cube and the mathematical search for a solution as well as some general how-to`s. We`ll also see how computer programs can find a solution to any cube, and we`ll take a look at some open problems in the field of Rubik`s Cube Group Theory. Anthony Montemayor, Simon`s Rock of Bard College ABSTRACT: Artificial neural networks have in recent years generated a lot of attention. Their skill at pattern recognition has been applied in fields as diverse as face recognition and predicting box office hits. These conceptually simple constructions yield amazingly complex behaviors. This talk will describe some of these networks and the underlying mathematics, while explaining how they might relate to biological neural networks (such as the one in your brain). Kristine Richardson, Christina Climo, Westfield State College ABSTRACT: Do you know how to solve a Sudoku puzzle? If so, you may already know the theory behind one of the most ground breaking pieces of modern medicine: the CAT scan. In this talk we will be solving a variety of puzzles from Sudoku and Challenger to transforms and CAT scans. We will take a brief look into the CAT scan procedure and how images are produced while giving the audience a chance to interpret and “solve” these puzzles together. Dustin Cidorowich, St. Lawrence University ABSTRACT: The goal of this project is to evaluate the `Draft Pick Value Chart`, which is used by most NFL teams as a way of allocating value to individual draft picks. I have used the games played and games started as a proxy for drafted players value, and will use regression methodology to to link the explanatory variables to the drafted players value. The variables used in the analysis include selection number, year, team, and position. The predicted values will be scaled and compared with the `Draft Pick Value Chart` values. Would it be in a teams best interest to trade a second and fourth round draft pick for a first round pick? Jon P. Bannon, Siena College ABSTRACT: In this talk, we will describe a transition from classical to noncommutative or "quantum" mathematics using simple examples such as the set of functions on a two point space. Such a transition is essential to 21st century mathematics since it can provide, for example, an interesting way to combine quantum mechanics and general relativity. Benjamin Purkis, Amherst College ABSTRACT: Continued fractions are useful ways of representing real numbers: but what can be learned about the properties of the representation itself? With a brief introduction to measure theory, the basic theorems of ergodic theory are explored. I will state the Individual Ergodic Theorem for probability spaces and discuss its relation to the Maximal Ergodic Theorem. I will then show that the continued fraction transformation is ergodic using Knopp’s Lemma. Several results about continued fractions now follow from the Individual Ergodic Theorem, including frequency of digits in partial quotients, and the limit of the geometric and harmonic means of partial quotients. Andrew Brouwer, SUNY Potsdam ABSTRACT: A graph can be embedded in various spaces, such as S1 (a circle) and S3 (three dimensional space). This talk will explore graphs embedded in S1 as well as various types of linking in this space, including split-links, non-split links, 3-links, and intrinsic links. Tyler Cook, SUNY Potsdam ABSTRACT: Carry Groups are formed from the cartesian product of the integers with cyclic groups. In 2006, Cook, Miller, Pollio, Riel and Spaeth noticed manipulating the carries along with their respective bases helped reveal the structure of these groups. They not only got an understanding of finitely generated carry groups, but yet countably generated carry groups also. Carry groups are interesting in and of themselves and have practical implications in Ergodic theory. Dr. Blair Madore, SUNY Potsdam ABSTRACT: J. King first introduced what he calls the Madore group in a paper from 2001. This group and a related family of groups were studied by REU students in 2001, 2002 and 2006. Using these groups we will show how analysis and algebra, two subjects that seem very far apart to undergraduates, are tightly entwined. Colin Carroll, Williams College ABSTRACT: We will present a brief introduction to Coxeter complexes and the naive, minimal and maximal compactifications. These compactifications generalize to a discrete spectrum upon weighting points in the complex, creating different tilings of n-space through the truncation of simplices. We will explore properties of these different cellulations and weightings. Kelsey Stavseth, Saint Michael`s College ABSTRACT: Chaos is a relatively new mathematical area, with real progress and development being made only recently in the early 1970’s. James Gleick said that “where chaos begins, classical science stops”. Today Chaos is a large subject and covers many fields such as ecology, economy, physiology, physics and mathematics. A basic introduction using differential equations as a foundation will be used to facilitate a meaningful discussion of basic Chaos theory: specifically Strange Attractors and the Lorenz equations, as well as a few applicable examples. Jonathan Cornfield, Colgate University ABSTRACT: The Leibniz formula for p, discovered in the late 17th century, is one of several methods of representing p as an infinite series. This talk will demonstrate how Leibniz used integral calculus and geometry to derive this formula. Further, we will look at the properties of this series and arrive at some bizarre conclusions when we try to estimate the digits of p. Jordan Volz, Bard College ABSTRACT: Let E and E` be elliptic curves defined over a finite field k with p elements. It is known that E and and E` have the same number of k-rational points, but the abelian groups they define need not be isomorphic. We consider a specific family of isogenous elliptic curves and give estimates on the densities of primes for which the two curves define isomorphic groups over k. Jordan Volz, Bard College ABSTRACT: Graphs with bounded clique-width have been shown to be of interest due to the fact that they permit polynomial time solutions to several NP-hard problems. Bipartite graphs in general do not have bounded clique-width, but we will provide a complete classification of classes of bipartite graphs defined by a single forbidden induced bipartite subgraph with respect to bounded/unbounded clique-width. Tom Garrity, Williams ABSTRACT: Recently the mathematics of statistical mechanics has been applied to the study of continued fractions. We will discuss some of the basics behind statistical mechanics and see how these basics relate to algebraic and transcendental properties of real numbers. No background in statistical mechanics will be required. The real goal is to see how some mathematics that is important in physics should remain important in non-physical realms. Sarah Hamilton, Saint Michael`s College ABSTRACT: The difficulty of VLSI chip design and layout problems sparked interest in the development of bar visibility graphs (BVGs) and their two-dimensional counterparts, rectangle visibility graphs (RVGs). However, unlike rectangular chip components which have fixed area and aspect ratios, the rectangles representing vertices in RVGs may vary in size and shape. In this talk I will introduce Unit Rectangle Visibility Graphs, whose fixed dimension restrictions provide a more accurate model for chip design applications. A graph G is an RVG if its vertices can be represented by closed rectangles in the plane with sides parallel to the axes, pairwise disjoint except possibly along their boundaries, in such a way that two vertices are adjacent if and only if there is a horizontal or vertical band of visibility joining the two rectangles. Unit Rectangle Visibility Graphs (URVGs)} have the additional restriction that the rectangles are unit squares. Our results include a characterization of what bipartite graphs are URVGs, an edge bound for trees that are URVGs, an edge bound for URVGs themselves, and which complete graphs are URVGs. Mona Merling, Bard College ABSTRACT: A bipartitioned dessin is a pair of permutations of a finite set. A dessin gives rise to (and is determined by) a graph embedded in a Riemann surface. I will talk about pairs of dessins that arise from Gassman triples of groups (G,H,H') together with pairs of elements of G. We show that the two dessins have isomorphic monodromy groups, have the same branching data and have the same number of components. Moreover, the sums of the genera of the two dessins are the same, but we give an example where the individual genera of the components of the first dessin differ from the genera of the components of the second dessin. Bryan Eckelmann, Williams College ABSTRACT: Lattice stick number sL(K) is defined to be the minimal number of sticks required to construct a polygonal representation of the knot K in the cubic lattice. This talk will demonstrate a few techniques that will help us prove that sL(3) = 12 and sL(K) ¡Ý: 14 for any other non-trivial knot K. Peter Wright, Union College ABSTRACT: Recall that the power set P(X) of a set X is the set of all subsets of X. In this talk we will discuss the relationship between union perserving functions from P(X) to P(Y) and relations from X to Y. This relationship leads us to an examination of sets on several different levels. Ultimately, the movement from elements to subsets is directed using the perspective of a field called category theory. Peter Golbus, Bard College ABSTRACT: In this talk we will present Sarkovskii’s Theorem, a characterization of discrete dynamical systems on the real number line. As an illustration, we will prove the surprising result that if a function has a periodic point of period three, then it has periodic points of all other periods. Sarah Hamilton, Saint Michael`s College ABSTRACT: A wave is defined as "any recognizeable signal that is transferred from one part of a medium to another with a recognizable velocity of propagation" (G.B. Whitham). In this talk I will introduce the notion of both traveling and dispersive waves. I will present an overview of wave motion in different settings such as traffic waves, waves of a vibrating string (the one-dimensional Wave Equation, solitons (from the Korteweg-deVries Equation), and water waves. Sarah Vallélian, Dartmouth College ABSTRACT: The Four-Color Theorem states that any planar map of countries can be colored with at most four colors so that no two countries share a border and are the same color. We will show how the infinite case of this theorem follows from the finite case by an application of the Compactness Theorem of logic. First, we will introduce the logical concepts of wffs, truth assignments, and consistency. Then we will state the Compactness Theorem and the infinite Four-Color Theorem, and close with a sketch of its proof. Jesse Welch, Marlboro College ABSTRACT: This lecture is based on an exploration of numerical solutions to the damped driven pendulum differential equations. Conducted analysis is performed via a package designed for Mathematica, specifically discussing a mapping of the parameter space. The content of the discussion will assume knowledge of advanced calculus and basic methods of differential equations. Katie Spies and Angela Upton, Marist College ABSTRACT: In 1807, Joseph Fourier completed his work on a method for solving differential equations called Fourier Transforms. We will explore Fourier`s history in order to gain a greater understanding of how and why his method was derived. Further, the influences of Fourier`s method will be discussed in order to gain perspective on his contributions to mathematics. Since Fourier transforms are helpful when solving differential equations, we will explore which types of differential equations are most applicable to these transforms. We will discuss how to apply Fourier transforms to differential equations that are not integrable in order to transform them into equations which we are able to solve. Finally, we will exhibit the use of Fourier transforms on a heat equation to illustrate the method. Christopher Jennings, Saint Michael`s College ABSTRACT: Designing computer chips looks at many different variables. Floorplanning takes a look at the general components of a chip, and tries to place them in such a way that reduces wirelength between components, meets heat requirements, and other various chip requirements. My research has centered on floorplanning algorithms, specifically one that uses spring forces (like physics). Each general component of the chip is connected to others by means of imaginary spring forces, and the algorithm attempts to find the best layout for the design of the chip. We have created both a Java and a C++ program that implements the algorithm, and have tested them using various benchmarks found online. Carrie Leonard, St. Michael`s College ABSTRACT: Mr. Monopoly is a witness to approximately 500 million players on his game board. In Monopoly the wealthiest player wins after driving opponents to bankruptcy, whether you choose to be the hat, the shoe, or my personal favorite, the racecar. Circling the 40 spaces buying, renting, and selling, we should know a winning strategy to monopolizing properties. To determine which property to buy and develop, we may ask: Is there a particular color group frequented more so than any other? If so, how much will it cost to buy and develop? The answers to these questions can be found using a stochastic process, called a Markov chain. In this presentation, we will apply a Markov chain to model the Monopoly game, which will result in a winning strategy. Jason Miller and Patricia Webster, Marist College ABSTRACT: We will be discussing the method of Lagrange multipliers, discovered by Joseph Louis Lagrange, and its derivation. Useful application to the economic industry will also be discussed with an example. Gerard D Koffi, UMASS BOSTON ABSTRACT: In this talk I will give a brief historical account of Euler`s work on infinite series and its impact on other branches of mathematics. For example, I will show that by proving that the harmonic series diverges, Euler was able to show that there are an infinitely many prime Numbers. Cameron Palmer, Middlebury College ABSTRACT: One of the most famous unsolved problems of mathematics is the Riemann hypothesis: that the non-trivial zeros of the Riemann Zeta Function are located on the line in the complex plane defined by Re(p) = ½. If this could be proved, its effects would be felt particularly in number theory, specifically in understanding the distribution of the prime numbers. However, what exactly is the Riemann Zeta Function, and why does the distribution of its zeros in the complex plane tell us anything about the real-valued primes? This talk will discuss the discovery of the Riemann Zeta Function and its extension to the entire complex plane. Applications of this function, using the distribution of the function’s zeros, will also be discussed. Robin Gane-McCalla, Hamilton College ABSTRACT: The internet can be represented as a digraph with pages as nodes and links between pages as edges. The Google Page Rank algorithm postulates which page is most important by iteratively transferring “page rank” points between connected pages. The computation of page rank by eigenvectors of a matrix or by other means will be discussed. Lastly, strategies for maximizing Page Rank will be considered. James Scott Martinez, SUNY IT ABSTRACT: The presentation will be about defining groups and finite fields. I will be discussing the use of finite Galois fields in communcation coding. Anna Blasiak, Middlebury College ABSTRACT: Given a graph, we consider the following operation, known as graph pebbling: place some number of pebbles on the vertices, and move them by removing two pebbles from one vertex and placing a new pebble on an adjacent vertex. This talk will focus on the following question: given a graph G on n vertices and any configuration of n pebbles, can we move a pebble to any vertex after some sequence of pebbling moves? Kathleen Snell, Westfield State College ABSTRACT: Deal or No Deal is a game show craze that has swept the nation, who would not want the chance to play in a win-win situation. However, are the contestants really making the most of their time on the show? This talk will discuss the mathematical strategies of taking the largest prize home from the game, how the banker calculates his mysterious offers, and how the financial theories behind the game are deeper than picking the right suitcase. Theresa Ryan, Saint Michael`s College ABSTRACT: We will take a look at the historical background of solving both the cubic and quartic equations, including mathematicians and their context. We will view a timeline in order to get a focus on where we were in the mathematical world at the time of these discoveries. Examples and relavant information on the individual mathematicians and their lives will be presented. Time permitting, we will delve into a brief focus on the insolvability of the quintic equation. Darren Lim, Siena College ABSTRACT: This talk will present Fibonacci: The Game, a card game designed to teach the principles of the Fibonacci sequence to students normally unfamiliar to recursively defined sequences. The mechanics of the game will be described, as well as the playing cards themselves. Finally, a brief demonstration of the game will conclude the talk. Tomasz Przytycki, Bard College ABSTRACT: Quandles are a non-associative abstract algebra, and in this talk, I will provide the axioms for quandles and the reidemeister moves in knot theory, and show the relationship between them. Sydnie Wells, Union College ABSTRACT: Set partitions have been used to describe Faa di Bruno`s formula for the nth derivative of the composition of two functions. We will look at how set partitions relate to this formula and can be used to prove it. Theresa A. Kelso, St. Lawrence University ABSTRACT: Many economic variables can be estimated using lags (past values of the series), which allow us to capture the effect of predictor variables that change over time. Distributed lag models can be either finite, including arithmetic and polynomial lags, or infinite, including the geometric lag. The geometric lag model is difficult to estimate as there are an infinite number of terms and the model is also nonlinear in parameters. The Koyck transformation is a very clever way around these difficulties. We investigate methods to estimate the parameters in each model and evaluate fit. Robert Rash, Westfield State College ABSTRACT: In "The Wisdom of Crowds," James Surowiecki claims that the average of all guesses is more reliable than any indivdual guess. We will see if Surowiecki is wrong by using inference testing on data gathered from the audience and as well as inference testing on data colelcted in recent statistics classes. Betzayda Rivera and Danielle Bernier, Nazareth College of Rochester ABSTRACT: Bubbles bring fun for kids yet have been the topic of mathematical research for years. In this talk we will discuss the mathematical properties associated with the Double Bubble and the Bubble Grid Plate. Geometric concepts will include angles of incidence, minimizing perimeter, maximizing area, and mean curvature. Jennifer Crounse, Fitchburg State College ABSTRACT: There are countless applications in which Pascal’s Triangle can be used to solve everyday mathematical problems. My presentation will focus on specific statistic and algebraic applications that can be integrated into secondary education. I will apply Pascal’s Triangle to statistics by exploring probability of specific outcomes of independent events and determining binomial distribution coefficients. I will also demonstrate how to find the coefficients for binomial expansion problems in algebra. Jared Schumann, Westfield State College ABSTRACT: Casual observations show that river water levels jump during October, concurrent with the seasonal fall of leaves. On the West Branch of the Westfield River (western Massachusetts), this jump is approximately 0.5 m. It is likely that this jump is a direct result of trees and other plants stopping photosynthesis. As trees become less productive (less photosynthesis) they use less water for respiration and transpiration. This increases runoff, infiltration, and hence river base flow. To verify the existence of this step jump, I used SPSS to analyze historical flow records using time series analysis and hypothesis testing. Next, I determined the amount of water used by plants for respiration and transpiration, and the resulting biomass storage associated with photosynthesis (literature review). The water increase in the river is correlated to this biomass storage resulting in an estimate of the total primary productivity for the watershed. This measure is also expressed per unit area, given the watershed area. This method / tool will be useful to foresters, biologists, watershed managers, and ecologists studying productivity of remote areas. Ambrose Sterr, Marlboro College ABSTRACT: Roman and tuscan squares are particular arrays of symbols used in experimental design. Similar to Sudoku, they are closely related to latin squares, and emphatically illustrate the cominatorial explosion - small increases in parameter size add orders of magnitude to the time of brute-force searches. This talk will explain some general constructions for roman and tuscan and report on a distributed computing search for squares with parameters for which no squares were previously known. Jacqueline Palermo, Union College ABSTRACT: Public-key encryption allows us to keep our private information secret while communicating over insecure lines. It is a major component in safeguarding against identity fraud and allows us to perform numerous tasks over the Internet, including making purchases and doing our banking, without having to worry about our information being intercepted. The RSA and El-Gamal cryptosystems, two of the most popular public-key systems, use exponentiation and modular arithmetic to encode data. We present the use of Chebyshev polynomials in place of exponentiation in both the RSA and El-Gamal cryptosystems. Furthermore, we claim that the security of these systems matches that of the classical algorithms. Max M. Levine, Skidmore College ABSTRACT: Cayley diagrams of a group can vary depending on the generators chosen. We exhibit an example of several presentations of a common group that induce distinct Cayley diagrams, leading to distinct symmetry groups of the diagrams. Matthew Bader, Nazareth College ABSTRACT: The diffusion equation can be observed throughout nature. One interesting application occurs in the generation of animal pattern structures, specifically, how a reaction-diffusion process takes place in the early stages of embryo development. This and other interesting applications in nature will be explored. Anna Boatwright, Mount Holyoke College ABSTRACT: In this talk, I will define the affine Weil zeta function of a polynomial in n variables over the finite fields with pe elements, discuss the history of the Weil zeta function, and compute the affine zeta function for several polynomials. Stephanie Tougas, Quinnipiac University ABSTRACT: This talk will look at the security of digital signatures and the mathematics behind these signatures. This will include explanations of the role that discrete logarithms and elliptic curves play in the creation of digital signatures and other cryptographic systems. Catherine Martins, Westfield State College ABSTRACT: Hands-on exploration of Brunnian links and fashion design as described in the article "Brunnian Clothes on the Runway: Not for the Bashful" by Colin Adams, Thomas Fleming and Christopher Koegel. Alex Iselin, Skidmore College ABSTRACT: A reptile is a polygon that can be partitioned into polygons congruent to each other and similar to the original. We will construct several reptiles and examine their structure and symmetry. Paul Dunkerley, Westfield State College ABSTRACT: The presentation will chronicle a class of Algebra 2 high school students who are guided along the path to discovering Euler Characteristic. What happens and how the students react to what they discover will be shared. A copy of the lesson plan and worksheets will be available to all who attend. Reid Ginoza, Bennington College ABSTRACT: Knowing the qibla, the direction of the Mecca on the horizon, is a part of daily spiritual life for a Muslim. Today this is simple enough to solve, but what about in the 9th Century, before the practice of three-dimensional drawings? This presentation focuses on a geometric method known as an analemma that Habash al-Hâ:sib created that uses only one drawn circle. Marc Pereira and Mark Tokarz, Westfield State College ABSTRACT: If you lived on a torus, and took a long journey in some direction, how would that differ from a long journey on a sphere in the same direction? Will your explorers return home? If so in how long? Every surface has characteristics that change the distance an explorer will travel. We will be examining these characteristics and imagining living in such a universe. We will also briefly discuss space-finnling curves. If you are interested in learning what kind of a world Homer Simpson would prefer to live in, tune in! Christopher Rainey, Rensselaer Polytechnic Institute ABSTRACT: Much research has gone into the investigation of how ants use chemical pheromones to communication information with one another, leading to complex, highly organized foraging trails. However, few models look at the relationship between pheromone communication and energy expenditure on both an individual and colony sized scale. I develop a stochastic model which tracks the energy reserves of individual ants, and comment on how this affects the large scale behavior of the colony. Ivana Nikolic, Ramapo College of NJ ABSTRACT: A model with a certain number of choices is observed. Given a particular position you are able to accept the alternative at the given position, or decline it. Once at the certain position one is not able to go back and accept any of the previously declined alternatives, and also not able to review any of the alternatives that would be offered after the current one. Let n be the number of the choices that we will decline before accepting one of the alternatives. First we considered a normal distribution case, and then more general case. For small n we improvise the strategies in order to get higher expectations of the distributions. Numerical methods are also applied, as well as simulations when ever closed form solutions are not available. Matt Ator and Tavish Zausner-Mannes, Skidmore College ABSTRACT: There are seven unique frieze groups in the plane. However, the standard classifications of these groups do not take into account changes of color in tiles. We will construct and classify this alternative set of frieze groups utilizing color changes in tiles. Andy Boslett & Matt Hoover, Ithaca Colege ABSTRACT: Baseball is unique in sports in that games are played in series. This provides us with an opportunity to see how travel effects home field advantage over a three game series. Does the visiting team adapt to its surroundings and do better as the series progresses or does it get fatigued from living in a hotel? Daniel Stevenson, Union College ABSTRACT: ABSTRACT: In this talk we give a basic overview of the theory of even perfect numbers. Using Euclid`s formula, we then prove that all even perfect numbers are triangular numbers, and that all even perfect numbers can be written as the sum of a progression of odd cubes. Oliver Layton and Jessica Reynolds, Skidmore College ABSTRACT: One of the many famous formulas discovered by Leonhard Euler is the summation of the reciprocals of the squares of the positive integers. In this talk, we derive the sum, which is p2/6, by using double integration. There are other approaches to obtaining this sum, but the approach via double integration can be generalized to sums of powers higher than 2, many of which are still unsolved. Regina Circosta and Elizabeth Edelheit, Skidmore College ABSTRACT: We will examine edge-to-edge isogonal tessellations using standard pattern block shapes. All 30 possible tessellations will be presented. Audience members will be invited to participate in interactive demonstrations. Jeffrey Grover, SUNY Institute of Technology ABSTRACT: In this presentation we will look at magnets and electric circuits. We will examine what happens when magnets are added to the nodes of a parallel circuit and present a simple mathematical model of this situation. Feng Lin, SUNY IT ABSTRACT: In this discussion, I`ll demonstrate how probability can make a simple problem have an unexpected solution. Matt Ator & Kellen Affleck, Skidmore College ABSTRACT: The stomachion of archimedes is an ancient 14 piece puzzle with over 500 unique solutions. We will be creating an interactive version of this puzzle using macromedia flash. Our presentation will involve the history of the puzzle and how the interactive version was made. Bill Stitson, Skidmore College ABSTRACT: There`s a new Sheriff in town, and his name is Poincaré. We`ll start with an overview of the birth of non-Euclidean geometry, and then delve into a specific hyperbolic model. We will then conclude by testing whether some (formerly) self-evident theorems will still hold under this new regime. Gerard F. Wohlrab, III, SUNY Institute of Technology ABSTRACT: With the gaming boom - mainly with poker - occurring within the last decade, more internet sites are running continuous gaming. Since most of these sites are based in jurisdictions that allow for gambling at the age of eighteen, more high school and college students are gambling on these sites. The main question arises: are these players involved in a truly random environment? We will compare several random number generating processes used by the most secure and trusted sites in existence today. Neil Bornstein and Alex Iselin, Skidmore College ABSTRACT: A reptile is a polygon that can be partitioned into polygons congruent to each other and similar to the original. We will construct several reptiles and examine their structures and symmetries. Jon Bannon, Siena College ABSTRACT: In 1905, Burnside proved that if G is a subgroup of Gl(k,C) having finite exponent n, then the order of G must be finite. The proof of this result is over 90 pages long but contains some deep ideas that we should consider. We include here at least a sketch of a short proof of this theorem with an operator-algebraic flavor. Colin Hart, State University of New York at Geneseo ABSTRACT: We use Markov Chains and Stochastic Matrices to predict outcomes and expected duration of sports matches in several sports. For instance, given the probability of a tennis player winning a rally, we predict the length and outcome of the game. This solution assumes that the probability is constant or alternatively that probability is varying and is dependant on prior performance. Additionally we also consider events with many players, such as chess and basketball tournaments. Foe example, we solve the inverse problem of calculating unknown probabilities of success of basketball teams for the current season based on the results of previous matches of all teams and then try to predict the final distribution of teams. We will illustrate our procedure with several Maple and MATLAB programs. Raymond Garzia, State University of New York at Geneseo ABSTRACT: We use stochastic processes and Kolmogorov Partial Differential Equation to predict the outcome of an investment. In our model a variable interest rate depends on trend and random fluctuations. We illustrate our procedure with several Maple and MATLAB programs. Our solution may have a practical applications related to banking, investing and gambling. Thomas Ehmann, State University of New York at Geneseo ABSTRACT: Inversion of heat propagation is an important phenomenon in physical sciences and applied mathematics. The inverse problem of determining an unknown initial temperature of an object from the knowledge the object’s temperature at time t is explored mathematically and solved numerically. We present programs written in MATLAB that accurately predict the initial temperature of the object assuming the knowledge of the temperature of the object at time t. Michael Blanding, State University of New York at Geneseo ABSTRACT: We consider the following inverse problem for the heat equations: From the knowledge of the initial temperature of an object on the surface at one end of an object and the temperature at time t at the other end we will calculate the unknown thermal diffusion coefficient. The knowledge of diffusion coefficients allows us to identify internal inconsistencies of the physical properties of the material. We solve this problem numerically and mathematically with high accuracy. Our solution could be the theoretical basis for Thermal Tomography. Our theoretical solution may have many practical applications. For instance, from measurements of temperature of both sides of a heated wing of an airplane, we can predict possible internal cracks in the wing. Laura Hutchinson, Union College ABSTRACT: When playing the game “the human knot”, where n people stand in a circle and join hands randomly across the circle, many camp counselors would be distressed to find that the knot can’t always be untangled! What type of knots can be formed, and what is the chance that the knot will be trivial? We analyze this puzzle theoretically and in a practical manner: do humans differ from string? Using stricter guidelines on the game, we can improve the chances significantly that the knot can be untied and thus the group can solve the puzzle without a fight. Megan Kiessling, Skidmore College ABSTRACT: First studied and named by Galileo in the late 16th century, the cycloid is a curve traced out by placing a point on the circumference of a circle that is rolling along a straight line (or more generally, along almost any path.) In this talk the parametric equations of the cycloid will be derived, and some of the cycloid’s fascinating properties will be explored. This talk will be followed by a talk discussing the role of the cycloid in the solution of one of the most famous problems mathematicians have encountered, the brachristochrone problem. Cassandra Allen, Skidmore College ABSTRACT: We all know that the shortest distance between point A and point B is measured along a straight line segment connecting A & B. But did you know that if we instead ask for the path which provides the least time it takes for a frictionless bead to slide from A to B (with gravity the only acting force), the answer is no longer a straight line segment but a segment of a curve known as a cycloid? Kübra Kömek and Ari Morse, Skidmore College ABSTRACT: In honor of Leonhard Euler`s 300th birthday, we will in this talk trace through Euler`s own discovery of his famous formula that:
We begin by using simple algebra and the notion of a Taylor series to derive an infinite product expansion of the sine function. From this follows not only Euler`s formula but also a famous infinite product for p due to Wallis. The line of argument we follow is not a rigorous proof, but demonstrates the incredible genius of Euler, who was born exactly 300 years and 6 days ago today. Matthew Farrelly, Siena College ABSTRACT: In 1902 William Burnside raised the question of whether a finitely generated group must be finite if each of its elements has order dividing a natural number n, called the exponent of the group. Along with this question Burnside provided cases in which it had an affirmative answer, namely for any group of exponent 2 or 3 and for all groups of exponent 4 that can be generated by 2 elements. This talk will handle the case of groups of exponent 3. Here is the exact question that I will be answering. Suppose a, b are elements in a group G and that every element can be written as a product of a, b, a-1, b-1. further assume for every element g in G, we know that g3 = e. What is the order of G? Kristin Farwell, Siena College ABSTRACT: How do you pair up couples so that they will not want to leave their partner? This problem is known as the stable marriage problem. There is an algorithm that exists that will pair up the couples so that this problem is solved. What about creating stable menage a trois? We will discuss how to create stable menage a trois and how to generalize the stable marriage algorithm to work with threesomes, foursomes, fivesomes, and n-somes?
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