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Spring 2015 MAA Seaway Section Meeting, Colgate University
Student Talk Abstracts
Katharine Ahrens, Ithaca College
“In an Ocean of Ashes”: Order and Chaos in Math and Literature
Abstract: We explore the connections between math, as seen in the interplay of chaos and fractals,
and literature, as seen in the form and structure of Dante’s Divine Comedy and Stoppard’s play
Arcadia. We argue that Dante seems to intuitively grasp the order in the chaos that Benoit
Mandelbrot and those that followed after him would formalize within fractal theory and dynamical
systems theory centuries later. Stoppard is a separate case, both showing that order and chaos are
human themes, but also bringing some of the mathematics directly into his play. Stoppard’s
intentional and unintentional connections serve as a bridge between the mathematics of fractals
and the literature of Dante, and show us that these two disparate disciplines connect on a
fundamental and human level.
Kathleen Allen, Hamilton College
Analyzing Diet and Exercise Perceptions and Habits of College Students
Abstract: This project uses statistical analysis to examine the best types of information to present
students with in order to encourage them to alter their dietary and exercise habits. Information
provided was from one of six categories: nutritional value, physical appearance, mental health,
physical health, mortality and general. This analysis looks specifically at whether the level of
influence of a given category is gender specific. Data was acquired from a survey administered to
Hamilton students. Female Hamilton students were more likely to change their dietary habits,
perhaps due to their lower weight satisfaction. Based upon the study, statements on how nutrition
and exercise can improve mental health (e.g., depression and stress levels) would be the best types
of information to present college students with to encourage them to alter their habits. In light of
the obesity epidemic and the rise of Western diseases, determining the best ways in which to
increase positive dietary and exercise habits is particularly important.
Danielle Armaniaco, SUNY Oneonta
The Linearity Number
Abstract: The geometry of the triangle and its associated points, lines, and circles has often been
viewed as inexhaustible. In particular, the computation of the linearity number and the application
of the Theorems of Ceva and Menelaus helped in the discovery of important points of concurrency
as well as in providing a di↵erent perspective on Simson’s Line. In this presentation, we will
illustrate some of these concepts.
Toryn Avery, Ithaca College
Erin Nannen, Ithaca College
Kaprekar’s Constant
Abstract: Kaprekar’s algorithm takes any four-digit number containing at least two di↵erent digits,
arranges the digits in ascending and then in descending order to get two four-digit numbers, then
subtracts the smaller number from the larger number. We repeat this process with each new
output, and the algorithm will eventually terminate with the number 6174. The constant, 6174, is
a fixed point of this process, independent of the initial number used. We explore a variety of
results that come from adapting Kaprekar’s Algorithm to numbers with a di↵erent number of
digits in base 10 and to other base systems with varying digit length. We will also look at patterns
in the resulting constants, as well as the number of steps it takes to reach the constant or cycle
resulting from this algorithm.
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Spring 2015 MAA Seaway Section Meeting, Colgate University
Rose Berns-Zieve, Hamilton College
What Are Numbers?
Abstract: Understanding the concept of equality lies at the foundation of mathematics. How do we
determine, though, when two things are equal? This is the question that Barry Mazur addresses in
his piece “When is One Thing Equal to Some Other Thing?”
The initial focus of understanding equality begins with the natural numbers. Mazur gives three
di↵erent definitions of the natural numbers (all of which di↵er from Peano’s axiom definition.) He
then uses Peano’s axioms along with category theory to create the Peano category definition.
Ryan Bianconi, Ithaca College
Infinite Series Without Calculus
Abstract: In this talk we will introduce an entirely algebraic proof for the divergence of a well
known infinite series, the harmonic series. From here we will use this technique to prove the
divergence of a more complicated infinite series. Then finally, we will close by talking a little bit
about the properties of infinity that allow for the types of proof demonstrated.
Ashley Colopy, SUNY Brockport
Modeling Snowfall Variation Across New York State
Abstract:
Background: New York State is known for its snowstorms. Forecasters try to predict when the
storms are coming and how much snow they are going to produce. At the end of the season, data
is recorded throughout New York State on the annual snow precipitation for that year.
Objectives: The purpose of this study was to develop statistical models which evaluate the
predictors of variation in snow precipitation levels across New York State.
Methods: We analyzed 100 di↵erent cities across four distinct regions of the state. We used
geographical factors such as altitude, latitude, region and proximity to bodies of water such as
Lake Erie, Lake Ontario and the Atlantic Ocean as predictors in a multiple linear regression model
of the annual snowfall precipitation for the season.
Results: After rigorous analysis, it was found that the geographical factors of altitude, proximity
to the Atlantic Ocean and region are the only factors needed to significantly explain the annual
snowfall precipitation levels across New York State.
Elizabeth Comatos, Hamilton College
Getting to the Root of Newton’s Method
Abstract: Newton provides us with a technique to approximate roots of equations. His algorithm
necessitates an initial guess from which successive approximations are recursively generated. The
resulting sequence converges to the desired root. In this talk we will explore the theory behind
Newton’s Method and mild conditions under which the algorithm is guaranteed to work.
Timothy Cowan, Hamilton College
The Pre-Gaming Paradox: Consuming Alcohol in Anticipation of Further Alcohol Consumption
Abstract: Consuming alcohol in anticipation of attending an event that serves alcohol, or, utilizing
current vernacular, pre-gaming or front-loading, is not a new problem colleges face. Regardless of
the absurdity of this behavior, little has been done to understand and address its impact. Utilizing
the 2012 NESCAC Alcohol Survey, specifically the self-reported results of Hamilton College
students, I will attempt to deconstruct pre-gaming customs and identify consequences with a focus
on drinking’s negative outcomes (drunkenness, loss of memory, sickness and hospital visitation).
Further, I will attempt to identify gender di↵erences, if any, in pre-gaming behavior and its
consequences.
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Spring 2015 MAA Seaway Section Meeting, Colgate University
Kaitlyn Danziger, Niagara University
Mathematical Paradoxes: Exploring Mathematical Phenomena and Explaining the Unexplainable
Abstract: Math for many people is seen as boring, useless, and even too difficult to try to
understand, but by looking at mathematical paradoxes that exist in the world, it is easy to see that
math is used every day and that it can be a fun and interesting subject to explore! Paradoxes are
statements that regardless of sound reasoning from suitable premises, lead to a conclusion that
appears to be ludicrous, logically insupportable, or even self-contradictory. Simply put, the
outcome seems a bit backwards! Where is the root of the paradox? What are possible solutions to
the paradox? In this thesis, many mathematical paradoxes were explored in hopes of answering
these questions and explaining the unexplainable.
Iancu Dima, Ithaca College
The Laplace Transform and its Applications to Special Functions
Abstract: The Laplace transform is one of the most utilized integral transforms in the fields of
applied Mathematics, Physics and Engineering. It is known to have useful operational properties
which have great use in Ordinary Di↵erential Equations. This project aims to review some known
work and provide new results on the application of the Laplace transform on di↵erent special
functions including a particular interest on the Dawson integral.
By combining the known properties of the Laplace transform and the Dawson integral, we hope to
attain new outcomes using di↵erent approaches. Throughout the development of the project, we
will make use of some other operators that will help us in our computations, such as convolution.
The future goals of this project are to investigate other special functions like the Bessel function,
which has applications ranging from electromagnetism to quantum physics.
Robert Huben, Hamilton College
Splitting Some (but not all) Polynomials Over a Field
Abstract: We attempt to construct fields where every polynomial of a low degree splits, but where
there are polynomials of higher degrees that do not split.
Leo’el Jackson, SUNY Potsdam
Derivatives of Fractional Order
Abstract: You have computed the first, second and the third derivative but how about the -3rd
derivative or the 2/3rd derivative or even the pith derivative? We will discuss one way that the
concept of derivative can be extended to be defined for all real numbers. We derive further
simplifications of the definition and use them to compute some derivatives of non integer order.
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Spring 2015 MAA Seaway Section Meeting, Colgate University
Jack Jenkins, SUNY Geneseo
There’s a Glitch in the Matrix!
Abstract: The numerical range of an n ⇥ n matrix M , also known as its field of values, is the
subset W (M ) = {x⇤ M x : x 2 Cn , ||x|| = 1} of the complex plane. Numerical ranges are defined
algebraically, but have rich geometric properties that have been the subject of intense
mathematical research in the last fifteen years, although the origins of the field can be traced back
to work done in the 1950s. It is well known that the numerical range of a matrix is a compact and
convex region. Despite this generic description, depending on the matrix, the convex boundary of
the numerical range can take on a variety of shapes, such as an ellipse, a polygon, or even the
union of flat and curved portions, and the richness of these possibilities increases as the dimension
of the matrix increases. These sets can be segregated into several classes according to properties of
the singular points of certain algebraic curves associated with numerical range boundaries. The
speaker will present a number of his new results on the stability of each class of numerical range
under perturbations of the matrix, particularly in the case where the dimension of the matrix is
four. The types of numerical ranges of matrices with algebraic number entries will also be
addressed from the point of view of matrix perturbations. Finally, the presenter will share some
new observations on the density distribution of output points of the mapping x 7! x⇤ M x near the
numerical range boundary.
Nathan Jue, Ithaca College
Explorations with a Pascal type triangle formed with Fibonacci numbers
Abstract: The subject of this study was a Pascal-type triangle formed by using the Fibonacci
sequence. The Fibonacci sequence is a sequence of numbers starting with 0 and 1, and is generated
by adding the two previous numbers to form the next. For example 0+1=1. 1+1=2. 1+2=3.
2+3=5 etc. One of the main goals in investigating the triangle was to derive a formula to predict
any given number in the triangle using a two-point coordinate system, and hopefully in doing so
acquire some new knowledge about the Fibonacci sequence and the structure of Pascals Triangle.
It is possible to derive predictive formulas for many of the observed rows. When comparing the
formulas for consecutive rows, it is possible to hypothesize a formula for any given number in any
given row. This formula will be the focus of the study.
Douglas Knowles, SUNY Geneseo
Numerical Ranges over Finite Fields
Abstract: Let p be a prime number congruent to 3 modulo 4. We
p in the finite field
p will work
Fp [i] = {a + bi|a, b 2 Fp }, where Fp = {0, 1, ..., p 1} and i = p 1 =
1. Let A be a matrix
with entries from Fp [i]. Let xT x = k. denote the conjugate transpose of x. Consider a number
k 2 Fp [i]. Let Sk be the set of all vectors x with entries in Fp [i] where the product xT x = k. The
author has created a definition of a new concept, the k-numerical range Wk (A), which is the set of
numbers of the form xT Ax, for all x in Sk . We investigate the properties of these k-numerical
ranges, and explore the fundamental di↵erences between W0 (A) and Wk (A) for nonzero k. We will
then discuss our pioneering work in classifying the shapes W1 (A) can take. This includes the
author’s proof that W1 (A) can be a union of pairwise disjoint lines.
Taylor Kremis, SUNY Brockport
Vector Methods for Inequalities
Abstract: We will show how certain inequalities can be obtained either as optimization problems
using Lagrange multipliers or as consequences of some geometric constructions using vectors.
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Spring 2015 MAA Seaway Section Meeting, Colgate University
Stephanie Lash, SUNY Brockport
Numbers that Show Up at the End of their Powers
Abstract: All powers of 25 end with 25. We will discuss how to find all such numbers and also
some that show up at the end of some of their powers but not at the end of all.
Julia Martin, SUNY Oswego
Mad Vet Semigroup Isomorphic to the Group Zn
Abstract: Each “Mad Vet” scenario is associated with a semigroup and a directed graph. The
directed graph can tell us if the semigroup is actually a group and if so, linear algebra based on the
directed graph can tell us the structure of the group.
But what if we want to find a “Mad Vet” scenario to go along with a group chosen in advance? In
this talk I’ll explain what a “Mad Vet” scenario is and how I found a scenario to fit my group of
choice: the cyclic group of order n where the integer n is greater than or equal to 1.
Britney Mazzetta, Ithaca College
Expressing Egyptian Fractions with Hexagonal Numbers
Abstract: The ancient Egyptians expressed fractions as a sum of distinct unit fractions such as
1
1
1
2 + 5 + 7 . Each fraction in the expression has a numerator equal to 1 and a distinct denominator.
The denominator is a positive number and each fraction has a di↵erent denominator. All fractions
are expressed as a sum of distinct unit fractions. For example, the Egyptian fraction that would
result from the above fractions listed is 27
35 . These fractions had continued to be used by other
civilizations into the middle ages. Eventually, a transformation occurred and the decimal notation,
that is popularly used today, has replaced those fractions. However, Egyptian fractions continue to
be studied today in number theory, recreational mathematics, and in the study of the history of
ancient mathematics. In this presentation, after discussing some properties of Egyptian fractions,
including its history, we will demonstrate how to express Egyptian fractions using hexagonal
numbers, which were studied by Ancient Greeks and are still popular today.
Lauren Miranda, SUNY Oneonta
Napoleon-Like Theorems
Abstract: Napoleon’s Theorem states that if we construct equilateral triangles externally on the
sides of any arbitrary triangle, the centroids of those equilateral triangles also form an equilateral
triangle. As it turns out, a coordinate proof of this result is relatively difficult to write, and a
synthetic proof is almost impossible to handle. As such, we will present the geometric
transformation concept and also present a proof of Napoleon’s Theorem, as well as proofs of some
related results, based on transformation theory.
Kelly Mitchell, SUNY Potsdam
A Natural Proof of the Chain Rule
Abstract: The typical proof of the chain rule for a composition of two real-valued functions
overlooks the possibility that f (x) = f (c), causing the proof to be flawed. Following a paper of
Stephen Kenton, we will apply certain natural techniques to show that, with the addition of some
small steps, the chain rule proof still holds in this case.
Anthony Morse, SUNY Brockport
Counting the Number of Real Roots of Random Polynomials
Abstract: Though solving high degree univariate random polynomials and determining the precise
real roots out of all the roots are some of the most important problems in mathematics, science
and engineering, it has remained a highly challenging problem in computational mathematics. By
combining an efficient implementation of the Lindsey-Fox algorithm that can compute roots of
high degree univariate random polynomials and Smale’s alpha theorem which can certify if a given
numerical root in the quadratic convergence region of a nearby exact solution, we obtain the
certified real roots and certified counting of the number of real roots of the polynomials.
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Spring 2015 MAA Seaway Section Meeting, Colgate University
Samantha Paradis, SUNY Brockport
Rebecca Buranich, SUNY Brockport
Using Inequalities to Solve Equations in Integers
Abstract: Many equations in integers can be reduced to checking cases. However, if we have a
single equation and many variables, this may result in a huge number of cases. It is then important
to use some methods to limit the possibilities.
Cameron Perry, SUNY Potsdam
Perfect Sets
Abstract: Using a simple game and the concept of a perfect set, we will analyze the game and
prove that the interval from 0 to 1 is an uncountable set.
Ujjwal Pradhan, Hamilton College
Approximating ⇡ using Bu↵on’s Needle
Abstract: In 1777, French Naturalist Bu↵on posed a question about the probability that a needle
of length l will land on a line, given a floor with equally spaced parallel lines at distance d apart.
This problem has multiple conditions. In this talk, we will use the case where the length of needle
is not greater than distance d to approximate the number ⇡. We will also explore some applications
of Bu↵on’s Needle Problem.
Kenny Ro↵o, SUNY Oswego
A Necessary Set of Turns to Solve a Rubik’s Cube
Abstract: Upon playing with a Rubik’s Cube it becomes apparent that the only way to manipulate
the puzzle is to turn one of the six faces. Intuitively, it would seem that it would be necessary to
use these six types of turns to put the puzzle back in the solved state. In this presentation, a proof
that not 6, but only 5 of these turns are necessary to solve a Rubik’s Cube will be presented. No
prior knowledge of Rubik’s Cubes will be assumed.
Miranda Russell, SUNY Potsdam
The Set of Sublimits of a Sequence
Abstract: Let (an) be any sequence of real numbers. L is a sublimit of (an) if some subsequence of
(an ) converges to L. Let S(an ) be the set of sublimits of (an). We will prove that S(an ) is a closed
set and discuss related problems.
Joe Scutaro, SUNY Potsdam
Irrational Numbers
Abstract: A real number is irrational if it cannto be expressed as ab where a is an integer and b is a
natural number. We examine the irrationality of several important numbers like e and ⇧ and what
it takes to prove they are irrational.
Jatinder Sharma, SUNY Potsdam
Continuous Nowhere-Di↵erentiable Functions
Abstract: We know that if the function is continuous everywhere it is not necessarily di↵erentiable
everywhere, in fact there are some functions are continuous everywhere and nowhere di↵erentiable.
1
X
1
The most well-known of these functions are variations of f (x) =
cos (3nx) or
2n
n=0
1
X 1
f (x) =
g(2nx) where g(x) is the distance from x to the nearest integer. Although these
2n
n=0
examples are straightforward, they are difficult to visualize or even sketch their graphs. Hidefumi
Katsuura took that into account and came up with his own function that is not reasonably difficult
to graph and can be seen intuitively as to why it works. One unique aspect of this function is that
it contains smaller versions of the function within itself making it self-similar.
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Spring 2015 MAA Seaway Section Meeting, Colgate University
Rosalie Siciliano, SUNY Fredonia
Bu↵on’s Rectangle Problem
Abstract: In the 18th century, Georges-Louis Leclerc, Comte de Bu↵on posed the question of the
probability of a needle of length l hitting a line when it is thrown onto a plane that contains equally
spaced parallel lines. In this talk, I will present a solution to the Bu↵on Needle Problem where the
needles are replaced with rectangles. The solution to the problem of finding the probability of a
needle hitting a line when it is thrown on to a grid of squares will also be presented.
Minhoon Sohn, Hamilton College
Divisibility Tests in Other Bases
Abstract: This talk will briefly overview divisibility tests. Then we will look at uncommon
divisibility tests in other bases and explore possible applications.
Nicole Stephens, SUNY Potsdam
Analysis of Power Series Using Abel’s Theorem
Abstract: Abel’s theorem is a powerful tool for proving a series is uniformly convergent. It applies
to many series not easily handled with typical Calculus 2 techniques. We examine the proof of
Abel’s Theorem in addition to Dirichlet’s test and Abel’s test.
John Tucker, SUNY Fredonia
Cavalieri’s Principle for Surface Area
Abstract: Historically, Cavalieri’s Principle provided a method for computing volumes before the
invention of calculus. Unfortunately, the ideas employed by Cavalieri and his contemporaries did
not deal with surface area. This talk will use calculus techniques to investigate the surface areas of
a number of transformed objects and present some results in the spirit of Cavalieri.
Nicholas Vassos, Hamilton College
How Do Four Years At College Change Students? - An Analysis Of A NESCAC Alcohol Survey
Abstract: Students’ views, behaviors, and preferences are subject to change as they progress
through their college careers, influenced by their peers, their faculty, and their schools’
environments. Using Hamilton College self-reported student data from the Spring 2012 NESCAC
Alcohol Survey, I will attempt to identify areas of change and of non-change, focusing particularly
on respondents’ alcohol-related social behavior. I will also address the issues and concerns of using
data from a non-random sample to make inferences.
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Spring 2015 MAA Seaway Section Meeting, Colgate University
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Ryan Vogt, Rochester Institute of Technology
Implicit Method for Solving a Nonlinear Parabolic Thin-Film Equation
Abstract: The Navier-Stokes equations of fluid mechanics model the motion of viscous fluids.
Solving these equations in closed form by way of analytical methods is usually not possible.
Numerical methods for the Navier-Stokes equations are complicated in nature. While finite
element methods are often used to compute solutions, other methods are also applicable in some
cases. The two-dimensional Navier-Stokes equations may be reduced under the assumption that
the fluid exist as a thin film of lubricant. Assuming that the film’s thickness and its gradient are
small compared to other length scales in the problem, the leading order approximation of the
Navier-Stokes equations yields the one-dimensional lubrication equation. The lubrication equation
appears below, together with the initial and boundary conditions that will be considered.
✓ ✓ 3
◆◆
@h 1 @
@ h @h
+
h3
=0
(1)
@t
3 @x
@x3
@x
✓
◆
8x
h(x, 0) = 1 + .02 cos p
(2)
2
3
@h
@ h
=
=0
(3)
@x x=0
@x3 x=0
@h
@x
p
x=2 2⇡
=
@3h
@x3
p
x=2 2⇡
=0
(4)
An implicit numerical scheme for computing solutions of this model will be presented. Since the
equation is nonlinear, finite di↵erencing does not lead to a conventional linear system of equations
of the form Aun+1 = b. Rather, a large nonlinear system of equations, f (un+1 ) = 0, must be
solved in each time step. Roots of nonlinear algebraic systems of this form are computed using
Newton’s multidimensional root finding method. The numerical method and computed solutions
will be presented.