Manhattan College, Riverdale, NY
ABSTRACTS
Session IA
1
Using Statistics: A Study of Popular Breakfast Cereals
Jonathan Zicari (Rishiparna R. Pate), St. John's University
Statistics isn't just the dull task of crunching a horde of irrelevant numbers into their mean, median, and mode, as is the common misconception. There are many real-world applications to even its most basic principles. Many statisticians have an agenda when studying something, but, regardless of the outcome of their research, even if it performed exactly as they had preconceived it, there are always new ideas and new questions to be answered. In a personal study of popular breakfast cereal's, I was able to collect data, analyze it, and form interesting conclusions about a product essential in many people's morning routine. Level: 1
2
An Investigation into the Patterns of Uses and Effects of
Self-Medication in Caribbean Immigrant Communities
Julius Adenihun (Dr. Nkechi Madonna Agwu),
Borough of Manhattan Community College, City University of New York
The aim of this interdisciplinary project was to determine the pattern and extent of self-medication in Caribbean immigrant communities. A survey was conducted on 339 self-identified non-minor persons of Caribbean heritage living/working in New York City about their self-medication practices. The results of this survey were used to select the common alternative medications for chemical analysis. The chemical analysis indicated that some medications had toxicities and contra-indications that might directly lead to adverse health risks. Specifically, the presentation will focus on the statistical methodology used for the design, planning and implementation of the survey, on the statistical techniques used for analyzing the data, on the involvement of the student presenter as a research intern throughout the study and the benefits of this internship, on the teaching of statistics within the framework of community-based nursing to BMCC nursing 112 students through their involvement and training as survey interviewers and on the development of project-based curriculum materials from this study for teaching concepts in MAT 150 � Introduction to Statistics. Level: 1
3
Analysis of the Cumulative Homerun Ratio
Jason Limb and Sophia Obamije (Father Gabriel Costa), United States Military Academy
The cumulative homerun ratio(CHR) is a sabermetrics statistic that describes the number of home runs a player hits per at bat(HR/AB) over a time period. In our presentation we will present a list of all 500+ HR hitters with their CHR graphs and provide the quadratic polynomial which "best fits" the data for each individual. We determine the area under each curve and then compare this number to the Contemporary CHR (CCHR) for each hitter. From the CCHR we will determine the linear polynomial which "best fits" the data for each CCHR and then calculate
the net-area between the corresponding "pairs" of regression functions. We attempt to give an interpretation as to what these metrics signify? Finally, we discuss possible future trends of other players, such as Alex Rodriguez, and other potential 500+ home run hitters." Level: 2
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Session IB
1The Quadrature of Curvilinear Figures
Dena Veselica (Rehana Patel), St. John's University
The ancient Greeks were interested in taking a given shape, and using only a compass and straightedge, constructing a square with the same area as the original shape (a process known as quadrature). While it was found that this is possible for all rectilinear figures, the interesting question was whether or not curvilinear figures are quadrable as well. Hippocrates of Chios, ca. 440 B.C. succeeded in the quadrature of a crescent shaped figure called the lune. I will discuss the history of quadrature and explain how it took two millennia to determine whether or not the circle is quadrable. Level: 1
2 A Simple Construction to Approximate Any Arc Length
Walter Jacob (Thomas Osler), Rowan University
Given any angle, it is impossible, using straight edge and compass techniques, to construct a straight line whose length is the radian measure of that angle. We give a simple construction method which gives us approximations to the angle of arbitrary accuracy. This construction is a generalization of a method for constructing pi/2 to arbitrary accuracy which relates to Vieta's product of nested radicals. Level: 1
3 Unusual Methods for Constructing a Hyperbola
David Grochowski (Dr. Thomas Osler), Rowan University
This presentation details three methods with which one can geometrically construct a hyperbola. In the first method, a hyperbola is constructed using its asymptote and a family of ellipses. In the second method, a hyperbola is constructed using a family of circles instead of using a family of ellipses. In the last method, a hyperbola is constructed using just the asymptote and a line parallel to the asymptote. Level: 1
4 Toricelli Solids
Anna Haensch and Joan Kim (Diego Dominici), SUNY New Paltz
Torricelli's Trumpet (also known as Gabriel's 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. Level: 2
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Session IC
1 A Statistical Analysis of the Game Left, Right, Center
Kaitlyn Berletic and Lauren Rudowsky (Rosemary Farley), Manhattan College
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? Level: 1
2 Knight's Tour on a Pillow
Matthew Dalzell (Dr. Kathryn Weld), Manhattan College
A knight's tour is a list of squares that a knight lands on in a sequence of traditional knight moves. In the Knight's 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 Knight's Tour problem on a pillow chessboard. Level: 1
3 Statistical Analysis of "Komi" Handicapping in Go
Benjamin J. Harshfield (Matthew A Tom), Emmanuel College
In the game of Go, players take turns placing stones on a 19x19 grid, aiming to surround the largest amount of territory. Empirical evidence has shown that the player who moves first, "black", has a distinct advantage. To balance the game, white receives "komi,� compensation usually between 3 and 7 points. However, debates still rage about how many points it takes for black and white to compete on equal footing. By analyzing over 30,000 professional games, we will ascertain the optimal level of compensation and any dependence on the skill levels of the players. Level: 1
4 "Slave Market's" optical illusion - Bust of Voltaire
Justin Morle (Professor Rehana Patel), St. John's University
In Salvador Dali's painting "Slave Market," an optical illusion is present. As an art major I wanted to understand the mathematical dynamics of a painting I love. I made an analysis as to
why the optical illusion can be seen by everyone: The geometric placement of objects combined with the distribution of color forces the eye along a spiral path that allows the viewer to see the illusion. This spiral defies the conventional perspective of classical paintings. I will present my analysis of the measurements and angles of this spiral and discuss why it makes the painting so effective. Level: 1
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Session ID
1 Triangular Numbers and Pascal's Triangle
Professor Marty Lewinter, Purchase College
The n-th triangular number is the sum 1 + 2 + 3 + �+ n, which equals n(n+1)/2. The first few triangular numbers are 1, 3, 6, 10, and 15. The first few rows of Pascal's triangle are:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
Each entry is the sum of the number directly above it and the latter's preceding entry. We establish an interesting identity relating triangular numbers and Pascal's triangle. Level: 2
2 Primitive Roots
Dr. John McCabe, Manhattan College
Let p be a prime number. An integer g is a primitive root for p if the powers of g,
g, g^2, g^3, ... 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 does have a primitive root. The proof is essentially non-constructive. We will discuss the problem of determining a primitive root for a given prime p. We will discuss applications to public key cryptography and to the generation of pseudorandom numbers. And we will discuss Artin's conjecture. Level: 2
3 Random Number Generation
Daniel Kendris (John McCabe), Manhattan College
One way to construct a random number generator is to generate the sequence g, g^2, g^3, ... (mod p) where p is a prime number and g is a primitive root mod p. We will design 3 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 and the Kolmogrov-Smirnov tests were used to test these generators. Level: 1
4 Date to Day-of-week Conversion
Zachary Chao Lin (Dr. Kardos), The College of New Jersey
My presentation would be about how to convert any date after year 1752 into day of the week. For instance, I know May 8, 2006 is a Monday immediately. The algorithm is simple. I add the number for the month (zero for May) and the date of month to get a sum. The sum is 8 and it is divided by 7 (number of days in a week). The remainder is 1, so it is a Monday. Now why is the month number for May zero and what about other months and years? Answer: math patterns observation! Level: 1
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Session IE
1 The Development of Calculus Notation
Brendan Kennedy (Marvin Littman), Westchester Community College
Calculus was born as early as the 17th century. Soon after, people were seeing its ancient writing on the wall. Newton and Leibniz, co-discoverers of the calculus, each had their own notation for this new mathematical study. I will talk about the evolution of calculus notation, from its first published forms to the modern forms used today. Level: 1
2 Nested Derivatives: A Simple Method for Computing
Series Expansions of Inverse Functions
Diego Dominici, State University of New York at New Paltz
We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first N terms of the series. We show several examples of its application in calculating the inverses of some special functions. Level: 1
3 Arithmetic Mean � Geometric Mean Inequality
Behailu Mammo, Hofstra University
In this talk, after outlining two proofs of the Arithmetic Mean � Geometric Mean Inequality, we will apply this inequality to solve some optimization problems without using calculus. If time permits, we will also discuss some other applications of this inequality. Level: 2
4 The Math in Chocolate
Raquel Estevez (Rehana Patel), St. John's University
There is math in chocolate? That is probably the question that comes to mind when you read this title. The truth of the matter there is more math in chocolate than ever one would have thought. There is math from the time chocolate is liquid being turned to solid, to the moment it is in your mouth. Ever-wonder why chocolate melts in your mouth and not in your hands? Well this property involves several mathematical concepts such as size, angels, time, and even patterns. How about how chocolate flows; did you ever think that the velocity at which chocolate flows can be represented by a graph; how about viscosity? Level: 1
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Session IF
1 A Polynomial Multiplication Complexity Analysis
Roland Varriale (Dr. R. Gregory Taylor), Manhattan College
Various algorithms for polynomial multiplication have been devised to improve on the time efficiency of traditional polynomial multiplication (O(n^2 )). 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 2^15. 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. Level: 2
2 An application of extended set theory (XST)
Gregory Brown (Thurmon Whitley), University of New Haven
A talk about the (potential) practical applications of Extended Set Theory in computer applications. This covers the mathematical background of XST's set operations and their roots in traditional set theory. If possible, some real world examples will be demonstrated.
Programming examples will be covered, but focus will be placed on the abstract and mathematical aspects of XST.
In particular, common operations such as restrictions and scope transformations will be discussed. Reference: http://www.xprogramming.com/xpmag/index.htm Level: 1
3 Genetic Algorithm for the Minimum Tollbooth Problem
Christopher Kollmann (Dr. John Loase)
Corby Harwood, and Matthew Stamps (Dr. Lihui Ba), Concordia College
This presentation considers the minimum tollbooth problem (MINTB) for determining a tolling strategy in a transportation network that requires the least number of toll locations, and simultaneously causes the most efficient use of the network. Current nonlinear program solvers require unreasonable amounts of time. Thus, a more efficient heuristic method has been investigated. This presentation describes the genetic algorithm used to solve MINTB, and reports numerical results on small networks. Level: 2
PM
Session IIA
1 Convenient Matrix Formalism for Power Series Solution
of Ordinary Differential Equations with Real Analytic Terms
Amy Orianna (Lydia Novozhilova), Western Connecticut State University
A matrix form of conventional power series solution of nonlinear ODE is presented. The main component is the development of a universal matrix for handling any real analytic terms of a nonlinear equation. Maple is used for examples, convergence analysis, and graphical comparisons with exact solutions of model equations. An application of the developed matrix form to time-dependent dynamical systems is also presented. The matrix form is used to construct a phase portrait of a specific system. Level: 1
2 Standard Map on a Torus
Frank Wang, LaGuardia Community College
Kovalevskaya's study of the rigid body motion and Poincare's research on the three-body problem prove that few dynamical systems are integrable. Because differential equations that model a physical situation rarely admit exact solutions, routine presentation of algorithms for solving special classes of differential equations commonly adopted in an undergraduate differential equations course is inadequate in preparing students for the modern approach. We use the standard map, which describes a pendulum with periodic kicks and has characteristics of a large class of nonintegrable systems, to demonstrate how a student can use numerical solutions to visualize bifurcation and geometrical properties of ODE. Level: 2
3 A Friendly Introduction to Chaos
Joseph Manthey, Saint Joseph College
The article, �Simple mathematical models with very complicated dynamics�, published in Nature by May in 1976 set off a firestorm of research on chaotic dynamical systems in mathematics and many related scientific disciplines. In this session we will explore the discrete logistic equation and consider some of its wild behaviors including fixed points, bifurcations, two-cycles, period-doubling, chaos and sensitive dependence on initial conditions. The mathematical background required to understand these concepts is minimal as the discrete logistic equation is simple quadratic iteration rule. Level: 1
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Session IIB
1 Applied Mathematics in Neuroscience
Felix Apfaltrer, BMCC / CUNY
An expository presentation of some of the mathematical research done in computational neuroscience will be given. In particular, some mathematical models of neurons will be briefly described and some of their properties will be mentioned (Hodgkin-Huxley, Fitzhugh-Nagumo and integrate-and-fire neurons). These model neurons are used in neural network simulations, which usually use the simpler models because of the (computational) complexity of the resulting networks. Network models will briefly described and their implementations will be briefly described and some results in visual neuroscience will be shown. Level: 1
2 The Error Function
Lucio Prado, Borough of Manhattan Community College - The City University of New York
This talk will be about a fundamental function that appears in different mathematical contexts, namely, the error function. We will illustrate with facts in:
a) Analysis/Calculus (Liouville's theory);
b) Algebra (Differential Algebra);
c) Statistics (Probability Theory/Normal Distribution);
d) Numerical Analysis (Numerical Integrations). Level: 1
3 Chaplain's Office Model for Operational Resource Tracking (COMFORT)
Adam Maciuba & Michael Mingler (Major Howard McInvale), United States Military Academy
In support of the Global War on Terror, the Office of the Chief of Chaplains requested assistance from Operations Researchers at West Point to identify and implement a means of effectively assigning Army chaplains to deploying/deployed units to ensure appropriate coverage of services and reduce the workload required to develop the assignment plan. With the current rate of deployment of active, reserve, and guard forces the demand for chaplains exceeds the chaplains available. The current, by hand, process requires constant updating and has difficulty forecasting requirements and resources. We will determine a more efficient system for the Chief of Chaplains. Level: 1
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Session IIC
1 4,000 Years of Newton's Method
Larry Blaine, Plymouth State University
Newton's root-finding algorithm is simple, powerful, and elegant, and perhaps deserves more attention than it usually gets in introductory calculus courses. What may be surprising is that some special cases of the algorithm, derived by purely geometric means, were known in ancient Babylon. In this talk, we will explore this ancient history, then move forward several millennia to the present day. A few interesting and computationally sophisticated applications of Newton-like methods will be presented. Level: 1
2 Cubic Equations
Malgorzata Szostek (Richard Churchill), Hunter College
While the method of solving quadratic equations had been known since Babylonian times, the equations of third degree defied the mathematicians until the XVI century. At that time, dal Ferro and Tartaglia analyzed special cases of cubic equations, but it was Girolamo Cardan who first published the results in 1545. A few decades later, Franciscus Vieta introduced an alternate method of solving cubic equations using a trigonometric identity. I will present Vieta's technique, which allows us to avoid the use of Cardan's lengthy formula. Level: 1
3 A New Method of Finding Pi
Charlie Siggins (Xiaoou Jiang), College of St. Joseph in Vermont
Over the past two years, out of curiosity over how mathematicians knew what the value of Pi was, I developed my own way. It is based on geometry and trigonometry. A few days ago I finally put the last touches on the formula itself, not merely the process. The basic concept is similar to the old Archimedes method of inscribing and circumscribing regular polygons, only I actually turn the circle into a regular polygon, and I do not use the area of the polygon but rather the perimeter and the distance across the polygon. Level: 1>
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Session IID
1 Introduction to Partial Words
Joel Dodge (Ada Peluso), Hunter College
I will define a partial word as a generalization of a word over an alphabet and introduce some of the terminology used to discuss partial words. I will then present some basic results on partial words and discuss an open problem or two. Level: 1
2 Unfoldings of the Cube
Robert Suzzi Valli (Richard Goldstone), Manhattan College
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. Level: 2
3 A Brief History of Galois Theory
Jennifer Sieviec (Melkana Brakalova-Trevithick), Fordham University
The quest of discovering whether any particular equation can be solved by radicals has been an age old pursuit of numerous individuals dating as far back as 1600 B.C. The work of the young mathematician Evariste Galois (1811-1832) shed an important light on this area and was instrumental in discovering the connection between group theory and solutions of polynomial equations. In this talk I will discuss the life of Evariste Galois and some of his proofs and findings, and I will provide a method for determining the irreducibility of a polynomial over
a group. Level: 2
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Session IIE
1 An Invariant Separating Rational Tangles from Irrational Tangles
Alexis Morley-Lyons (Matthew Horak), Trinity College
A mathematical knot is a closed 1-dimensional string, which is embedded into 3-dimensional space. A tangle is a smaller part of a knot which can be surrounded by a circle and has loose ends crossing the circle at four points. The most accessible class of tangles are rational tangles, which relate to how DNA recombines during replication. We define a property that helps to classify any given tangle as rational or irrational. For a checkerboard shaded diagram, we associate a related graph and prove that if the graph contains no separating edge, then the tangle is irrational. Level: 2
2 Nimber Sequences with No Preperiods for Three-Element Subtraction Sets
Brittany C. Shelton (Michael A. Jones), Montclair State University
Nimber sequences describe optimal behavior in the two-player combinatorial game, single-pile Nim. Because the nimber sequences for all 2-element subtraction sets are known [Berlekamp, Conway, and Guy, 1982], we focus our attention on 3-element subtraction sets and classify six types of these sets that generate nimber sequences without preperiods by explicitly giving the associated nimber sequence. For a fixed period length, we count the number of nimber sequences generated by each of the types of subtraction sets. Numerical calculations support our conjecture that these are the only subtractions sets that generate nimber sequences with these properties. Level :1
3 Undecidability of the Rationals
Jodi Black (Carol Wood), Wesleyan University
By 1920 �Hilbert's program� had developed around the view that all mathematics follows from some set of axioms and that there is some such axiom system which is provably consistent. Just over a decade later, Godel showed that Hilbert's program failed for the natural numbers. Taking any set of true statements about the natural numbers as axioms, the list of other true statements that can be derived is incomplete. In 1949, Julia Robinson would pick out the integers in the rational numbers thus demonstrating the undecidability of the rationals. I will give a proof of Robinson's result and discuss applications. Level: 2
