Keynote address:
Peter Winkler, Dartmouth
What Can We Learn from Mathematical Puzzles?
Mathematical puzzles (as opposed to problems) are designed for
entertainment, more than enlightenment,
but can nonetheless be invaluable challenges to our intuition and
creativity. We'll present some of our
favorite examples, then talk a bit about their histories and how they make
people think differently.
Morning Sessions:
I A) Differential Equations
Matt Buchta (Dr. Lydia Novozhilova)
Western Connecticut State University
Connection Between Logistic Model with Periodic Harvesting and Mathieu Equation
We found that a classic logistic model with periodic harvesting can be transformed into a Mathieu equation. Relevant facts from rich theory of this classic linear differential equation will be discussed. Then we present solutions for the model based on this relationship with Mathieu equation. CAS Maple is extensively used for computations and visualization of the results. Level: 1
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Orianna DeMasi (Dr. Lydia Novozhilova)
Western Connecticut State University
The Struggle for Existence: Analytical Study of a General Model in Population Dynamics
Qualitative analysis of population dynamics models for competing species is traditionally taught in modern ODE courses. In addition to its own benefit, qualitative analysis is frequently used because analytical solutions to such models are uncommon. The mathematical physics community has recently developed a simple and formal method of finding exact solutions to certain polynomial systems of nonlinear equations. We give a brief description of this method and then apply it to a problem from population dynamics. Level: 1
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Brian Juskiewicz
Institution: Norwich University
Analysis of RLC Circuits
using Differential Equations
RLC Circuits are built of resisters, inductors and capacitors. The behavior of RLC circuits at arbitrary times can be studied using differential equations. The solutions to these equations yield the properties used to describe the circuits effects including dampening. More heuristic approaches for larger scale circuits will be briefly explained. Level: 2
I B) Number Theory
Kevin LeClair (Darlene Olsen)
Norwich University
Perplex Numbers
To reiterate the properties and different uses for Imaginary Numbers. Introduce a newer number system, called Perplex numbers. The Perplex Numbers, like the imaginary numbers, are simply an extension to the real number system. The perplex number is introduced by adding a number h, such that the absolute value of h equals negative One. I will then compare and contrast the Imaginary versus the Perplex Number system. Then I will proceed to explain different uses for the perplex number system. Level: 1
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Diego Dominici
SUNY New Paltz
Working with 2s and 3s
We establish an equivalent condition to the validity of Collatz's conjecture, using elementary methods. We derive some conclusions and show several examples. We also offer a variety of exercises, problems and conjectures. Level: 1
Dr. John McCabe
Manhattan College
Fermat and Infinite Descent
AFermat showed that the equation
(1) x^4 + y^4 = z^4
has no solutions in positive integers. He did this by looking instead at the
(more general) equation
(2) x^4 + y^4 = z^2,
which he showed has no solutions in positive integers, by applying his
method of infinite descent.
At first sight this is puzzling. Why could he not have applied his method
to equation (1)? It will be seen that geometry provides the answer. Level: 2
Jane Akhuetie
College of Mount Saint Vincent
Integer Points on an Elliptic Curve
Abstract: Consider the elliptic Curve
E: y^2 = x^3 - x.
This curve has 3 obvious integer points, namely:
(0,0), (1,0), and (-1,0).
It will be shown that these are in fact the only integer points on the curve
E. Level: 1
IC) Applications and Modeling
Andrew Dunn (Marvin Bishop)
Manhattan College
Computer Modeling of Linear and Star Polymers
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. Level: 1
Christopher M. Frenz
College of Mount Saint Vincent
Protein Stability Engineering Using Neural Network Hybrid Systems
Computer scientists are increasingly faced with the challenge of working with complex systems in which the relationships between the variables that comprise the system are highly interdependent and not clearly defined. One such system is the prediction of mutation-induced changes in protein stability. A neural network based methodology of predicting mutation-induced changes of protein stability has been developed that has an accuracy of 93%. This neural network is utilized as the basis of two approaches to stability engineering. One, in which it serves as a knowledge base for an AI-neural network hybrid system that is capable of engineering a protein sequence of the correct stability, and a second approach in which it is utilized as the fitness function of a genetic algorithm that is capable of producing a range of protein sequences of the desired stability. Level: 2
John Tiglias (Marvin Bishop)
Manhattan College
The Fourier Transform in Arbitrary Dimensions
An equation for the Fourier Transform in arbitrary dimensions will be derived. This equation involves Bessel Functions. A C++ program has been designed to apply this equation to simulation data in one to eight dimensions.
level: 1
Joseph Landry
Norwich University
Finding Patterns in Sporadic Physical Phenomena (Space Weather)
In an event list of seemly sporadic physical phenomena, some events
occurrences may not be random. I demonstrate a strategy for the extraction
of the recurrences of these patterns from the event list, and apply this
strategy to a fabricated event list for purposes of demonstration. Then I
apply this strategy to thirty-six years of galactic cosmic radiation
measured from earth to understand the solar control of space weather and the
depletion of cosmic rays. Level: 2
I D) Probability/Statistics
Lauren Rudowsky (Dr. Rosemary Farley)
Manhattan College
A Surprising Expected Value
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.
level: 1
Desislava Slavova
Norwich University
Applying Benford's law
Abstract: In 1938, the physicist Frank Benford noticed that in some
numerical data sets the probability of the first significant digit of the
number is not distributed uniformly, instead there is a set probability rate
expressed by a logarithmic function. His discovery, known as Benford's law,
has many applications in accounting and economics. This presentation will
show how data collected from the stock market relates to Benford's law.
Level: 1
Kevin Garland and Chris Johnson (Bjorn Schellenberg) CMSV
An Urban Legend About Expected Values
Abstract: An urban legend holds that a teacher of probability asked his students to roll a die 200 times and write down all numbers. Next day he took a look at each sheet and told the dumbfounded students exactly who had cheated. The legend holds that he simply looked for a repeat of the same number 5 or more times.
We will calculate the probability of a repeat string of length 5 or more, using only basic concepts. Then we will verify our calculations by Monte Carlo simulation. Level: 1
Darlene Olsen
Norwich University
A Mathematician Working in a Nonacademic Position
Abstract: Are you near completion of a degree in mathematics and looking for
a career outside of education? Work for the Department of Labor and help
provide your state with current labor market information. Come learn about
the exciting and rewarding job opportunities with the New York State
Department of Labor. Level: 1
I E) Knot Theory and Dynamical Systems
Peter Golbus (Bob McGrail)
Bard College
Discrete Dynamical Systems on the Real Number Line
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. Level: 1
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Robin Augustine Thottungal (Diego Dominici)
State University of New York, New Paltz
Logistic Equations: Behavior in Frequency Domain
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:
Xn+1 = a*Xn*(1 - Xn) where a is a constant representing the growth rate. The talk will be about the results that we have found by studying the logistic equation in the frequency domain. Level: 1
Alexandra Phelan (K. Weld)
Manhattan College
An Introduction to Knot Coloring
Take a string, tangle it up and glue the two
ends together. Now try to untangle it
without cutting the string. This is one of the basic concepts of knot
theory. Over a
century old, knot theory is today one of the most active areas of modern
mathematics. In
this talk, we will explore the foundation concepts of knot theory, focusing
on mod p
labeling invariants of knots.
Level: 1 (Prequel to Matt Dalzell�s talk.)
Matt Dalzell (K. Weld) Manhattan College
The Expanded Kauffman-Harary Conjecture
Abstract: The Kauffman-Harary Conjecture says for a an alternating knot and
a prime
determinant, each non-trivial p-labeling assigns different labels to
different arcs.
Kauffman believed the conjecture could be expanded to include more types of
knots. In
this talk, we will explore the expanding of the Kauffman-Harary Conjecture
to all
alternating knots. Level: 1, if you listen to the preceding talk.
ABSTRACTS FOR THE AFTERNOON SESSIONS
II A) Differential Equations
Justin Alperin and Michael Shoushani (Lydia Novozhilova)
Western Connecticut State University
Qualitative and quantitative analysis of a model of infectious diseases
We present the phase analysis of a classic SIR (S=susceptible, I=infectibles, R=removables) model in epidemiology. We also perform a Painleve test of integrability of the SIR system of ODEs in the spirit of Sohya Kovalevskaya work on the motion of a rigid body with a fixed point (now called Kovalevskaya top). Numerical solutions to the system governing the spread of infection disease are implemented using CAS Maple. Level: 2
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Robert Decker
University of Hartford
Periodic Solutions to the Single Neuron Differential Equations
The single neuron differential equation y'=1/(1+exp(-K*(y-f(t)))) can be used to model the behavior of a neuron in the brain. When the input f(t) is constant there are three possible states: two stable fixed points and one unstable fixed point between the stable ones. If the input f(t) is made to be periodic with a small amplitude, the stable fixed points turn into stable periodic solutions, and the unstable fixed point turns into an unstable periodic solution. Thus it was believed that the periodically driven neuron may have at most three periodic solutions; this turns out not to be the case. Level: 1
Kristal Atkinson (Robert Decker)
University of Hartford
Multiple Periodic Solutions to the Single Neuron Equation
The presenter will show how it is possible to construct a periodic input f(t) to the single neuron differential equation
y'=1/(1+exp(-K*(y-f(t)))), for which the differential equation has multiple periodic solutions. One can use an input signal f(t) of the form exp(-b*(t-a)2)*sin(k*pi*t) to create a single "finger" at t=a, which lifts the solution curves enough to create a new periodic solution. A sum of such inputs can be used to create multiple periodic solutions. To make the input periodic, the independent variable t is replaced with a Fourier approximation to the identity g(t)=t. This result could have implications for neural scientists.
level: 2
II B) Algebra/Number Theory
John Massey (Brian Hopkins)
St. Peters College
Not Quite Pythagorean Triples
The Pythagorean Theorem has been helping students correctly identify the missing length of a right triangle for over 2000 years. For integers a, b, and c with a^2 + b^2 = c^2, it has been proven that when using three-part partitions that the number of three-part partitions of and the three-part partitions of b sum to the total number of three-part partitions of c, corresponding with the Pythagorean Theorem. What we investigated were the cases when numbers summed to a value slightly different from a perfect square yet the three-part partition concept still held true.
Rachel Brudnicki (Susan Diesel)
Norwich University
Understanding Sudoku Puzzles
Sudoku puzzles are very popular right now. While the puzzles themselves do not require any mathematical calculations to solve, generating the puzzles is another matter entirely. Using Burnside�s Lemma, Group Theory and Latin Squares this discussion will show the ways in which all possible 9x9 grids can be reduced into valid Sudoku puzzles. The main focus will be on reductions using permutations and the multiple symmetries of a square. There will also be a short comparison between Sudoku puzzles, Latin Squares and Magic Squares. Some knowledge of Linear Algebra required. Level: 2
Vanessa Abrisqueta, Mesfin Fekadu, (Brian Hopkins)
St. Peters College
Ramsey Theory of Finite Groups
Given a finite group with non-identity elements x, y, z, not necessarily distinct, how many colors are necessary to avoid a "monochromatic solution" to xy = z? This question was answered in the 1970s for groups of 16 elements when researchers were working on Ramsey theory of graphs. Last summer, we were part of an REU that worked on this for other groups. Our results include a complete characterization of which groups require 3 colors.
II C) Analysis
Walter Jacob (Dr. Thomas Osler)
Rowan University
Results from Euler's E46
In this talk we will look at Leonhard Euler's METHODUS UNIVERSALIS SERIERUM CONVERGENTIUM SUMMAS QUAM PROXIME and discuss his method for approximating sums of series using integrals, his discussion of approximating integrals using both rectilinear and curvilinear triangles, and his evaluations of the first ten terms of ζ(2), the first million terms of the harmonic series, and the exact value of the infinite sum of the reciprocals of squares. Level: 1
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Anna Haensch (Diego Dominici
SUNY New Paltz
An extension of the classical derivative
We extend the usual definition of the derivative in a way that Calculus I students can easily comprehend and which allows calculations at branch points. We prove new versions of the classical theorems
Rolle, Lagrange, Cauchy, L�Hopital, Taylor, etc.) and consider possible extensions to complex variables. Level: 1
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Luthfia Syarbaini
St. John's University
Accelerating Into A Derivative
Taking a freshman calculus applied to physics class, I was frightened about
my term paper. I was presented with the problem of Newton's Second Law and
how it conflicted with Einstein's Theory of Relativity. When Newton
discovered the Second Law of Motion, it worked well for everyday objects.
The law worked and satisfied the world for
several of hundreds of years. When Einstein found out that the equation does
not satisfy objects that have small masses, the equation had to be
questioned. I was told that the equations of the two theories are equal.
However, I had to find out what had to be ignored in order to simplify
Einstein's theory to Newton's Second Law. By differentiating the function
and simplifying with basic algebra, I was able to see that the Second Law of
Motion coincides with the theory of Relativity. Level: 1
II D) General Interest
Robin Schwartz and Anusha Mehar
College of Mount Saint Vincent
The Fairly Odd Parent Functions (and Even too!!): Using "Parent Functions" as an Organizing Principle throughout High School and Beyond
Studying parent functions (such as linear, quadratic, trig, reciprocal that would be used to run a regression) builds sharper graphing, algebraic, and critical thinking skills that naturally lead to an excellent discussion of zero, infinity, undefined and limits. These precalculus modeling concepts were highlighted in a college freshman core curriculum course (survey class) incorporating graphs from the newspaper (business, economy, health, etc.) and TI-84 graphing technology. These topics allow for clarity of common misconceptions (i.e. dividing by zero, exponent �rules�, fractions) leading to improvement of student quantitative literacy, comprehension and confidence. Level: 1
Victoria Wilson
Norwich University
Reality vs. Relativity
Reality is the quality or state of being actual or true. Relativity begins to
conflict the human perception of reality in time and space. Einstein's
Relativity theory tells us that a clock moving at a relativistic speed will be
seen to be running slow and the length of any object in a moving frame will
appear shortened in the direction of motion. We will derive the equations used
to calculate time dilation and length contraction from the Galilean and Lorentz
Transformation equations. Level: 1
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Shreshth Jain
St. John's University
Ticking your way into a good G.P.A.
Every college student can sympathize with time management issues. Trying to
handle a full course load and still having a social life is perhaps the
biggest challenge for a student. I formulated a model for optimizing my
G.P.A. within the constraints of a day, taking into account the different
difficulty levels of courses, differences in
credit hours, nutritional requirements, stress and the necessity of
�?�breaks.�?� My presentation will demonstrate how simple mathematics can be
used to solve common problems and will display the usefulness of mathematics
for students from all majors. Level: 1
II E) Graph Theory and Platonic Solids/Knots
Michael Zalewski
Norwich University
Dijsktra's Algorithm and the Shortest Route Problem
Have you ever wondered just how internet routing sites like Mapquest and Google Maps develop the shortest route between your origin and destination? It is believed that these proprietary algorithms are based on Dijsktra's Algorithm, which calculates the lowest cost route for a directed graph. This presentation will explore Dijsktra�s Algorithm and its application to the shortest route problem as well as comparisons to similar techniques employed in solving this type of problem. Level: 1
Tom Foley
Saint Joseph's University (Phila., PA)
The Physics of TWO of the "Five Platonic Circuits"
Is it possible to construct a regular tetrahedron using straws and paper clips? Yes, of course. Is it possible to construct a regular hexahedron: a) using straws and paper clips, or b) using just business cards without glue or sticky tape? Yes, again -- in both cases. However, the part a) construction of the hexahedron is NOT rigid. Why? Details will be discussed and several demonstrations will be presented. Physics plays an important role in the two circuits.
level: 1
Title: Wandering about Hamiltonians
Institution: Saint Peter's College
Name: Kristofer Gryte
Advisor: Dr. Brian Hopkins
Abstract:
To date, means of classifying Hamiltonian circles on Platonic solids has proven a rather formidable task. As the number of vertices increase, establishing an efficient method of differentiating between unique Hamiltonian cycles and those which are the same upon rotation and reflection is paramount. Subsequently, building upon prior work on the dodecahedron, we attempt to create novel techniques for 1) classifying unique Hamiltonian cycles, 2) establishing sub-classification topologies inherent in a given cycle, and 3) writing proofs particular to the Icosahedral Platonic solid. And as an additional aside, if time permits, we shall discuss the relevance of Hamiltonian circles on surfaces in concrete application.
Level 2
