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The 12th Annual
Garden State Undergraduate Mathematics Conference
a conference for undergraduate mathematics students
Saturday, April 11, 2015
Monmouth University
400 Cedar Ave
West Long Branch, NJ 07764
Schedule
8:30 – 9:15am
9:30 – 10:30am
10:45 – 12:00pm
12:00 – 1:00pm
1:00 – 2:00pm
2:15 – 3:15pm
3:15 – 3:30pm
3:30 – 4:25pm
4:30 – 5:00pm
Check-in and Breakfast - Bey Hall: Lobby outside Young Auditorium
New Jersey Undergraduate Math Competition Individual part
Bey Hall: Young Auditorium
New Jersey Undergraduate Math Competition Team part
Bey, Edison, Howard Halls. Rooms to be announced at the individual part.
Student Lunch - Bey Hall: Lobby outside Young Auditorium
Student Poster Presentation Session - Edison Hall: Lobby and Edison 117
Student Oral Presentation Sessions - Edison Hall: Edison 120, 121, and 122
Break - Wilson Hall: Foyer
Invited Plenary Presentation
Harmonious Equations: A Mathematical Exploration of Music
David Kung, St. Mary’s College of Maryland
Wilson Auditorium
Recogntion, Awards and Prizes: student presentation and competition
awards; door prizes and silent auction winners (must be present to win).
Wilson Auditorium
Student Oral Presentation Schedule
Time\Room
2:15-2:27
2:30-2:42
2:45-2:57
3:00-3:12
Session 1
Edison 120
Chair: Reem Jaafar,
Scrudato – Montclair
Breault – Muhlenberg
Campbell – Muhlenberg
Session 2
Edison 121
Chair: Shenglan Yuan,
Crawford – Rowan
Logan – Rowan
Labruna – Montclair
Castillo – Montclair
Session 3
Edison 122
Chair: Sarita Nemani,
Tu – NJIT
Hill – SUNY New Paltz
Gates - Rowan
Invited Plenary Presentation
Harmonious Equations: a Mathematical Exploration of Music
David Kung, St. Mary’s College of Maryland
Abstract: Mathematics and music seem to come from different spheres (arts and
sciences), yet they share an amazing array of commonalities. We will explore
these connections by examining the musical experience from a mathematical
perspective. The mathematical study of a single vibrating string unlocks a world
of musical overtones and harmonics-and even explains why a clarinet plays so
much lower than its similar-sized cousin the flute. Calculus, and the related field
of differential equations, shows us how our ears hear differences between two
instruments-what musicians call timbre-even when they play the same note at
the same loudness. Finally, abstract algebra gives modern language to the
structures beneath the surface of Bach's magnificent canons and fugues.
Throughout the talk, mathematical concepts will come to life with musical
examples played by the speaker, an amateur violinist.
Presenter: David Kung is Professor of Mathematics at St. Mary's College of
Maryland. He fell in love with both mathematics and music at a very early age.
More successful with one than the other, he completed three degrees at the
University of Wisconsin – Madison, all in mathematics and none in music.
However, he still enjoys playing violin with students and in the local community
orchestra. His recorded course "How Music and Mathematics Relate" has been a
best-seller for the Teaching Company. Dave co-founded the Emerging Scholars
Program in mathematics at St. Mary's College and has been involved in efforts to
develop similar programs that work with historically-underserved populations
at his own and other institutions. He has authored articles on topics in harmonic
analysis and mathematics education. The recipient of numerous awards,
including the 2006 Teaching Award from his section of the MAA, Dave gave the
2010 Undergraduate Lecture in Mathematics at the Joint Math Meetings. He is
the current Director of the MAA's Project NExT. (adapted from the presenter's
Web site)
Student Poster Presentation Session
1:00 – 2:00 pm. Edison Hall Lobby and Edison 117
Organizers: Joyati Debnath, Winona State University and Lee Collins, County
College of Morris
1. Harris Kittner and Jessica Cobb, Monmouth University
Faculty Advisor: Chengwen Wang
Application of Survival Analysis Techniques in Relation to Dogs with Tracheal
Collapse
Survival analysis refers to a family of statistical techniques dealing with the time it takes
for a specified event to occur: a cure, a death, a relapse, etc. In our research, we used
death as an event for the dogs, after receiving stenting as treatment for tracheal collapse
and looked at the survival time, in days. The different types of analyses are performed in
this study with twenty-six dogs using life tables, Kaplan-Meier methods, and cox
regression procedures. The Kaplan-Meier method aims to describe the survival time for
cases with a graphical representation of the survival rate as a function of time, called the
survivor function. Cox regression procedures aim to determine if survival time is
influenced by some other variable. We attempted to build a model using Cox regression
that could be used as a formula to predict the survival time for dogs after undergoing
the endoluminal stenting procedure. In this presentation, we will discuss how we were
able to build two models that could be used to predict survival time using age of dog at
stent placement, stent diameter, tracheal length, and whether shortening of the stent
occurred. We will also describe what we found through Kaplan-Meier and life tables in
regards to survival times for dogs that fell into different categories.
2. Jessica Kozma, Monmouth University; Faculty Advisor: Richard Bastian
R and the Welch Correction
The Welch Correction is commonly used to correct the degrees of freedom in a Student’s
t-test when the two sample variances are not equal. The question addressed in this
poster is how unequal the variances have to be in order for the Welch Correction to
make a difference. To test this, the statistical software program, R, was used to calculate
the Student’s t-test with and without the Welch Correction for forty samples with
different ratios of unequal variances. The power of each test was used to compare the
results. Overall, it was found that the Welch Correction for unequal variances is more
important under different situations.
3. Vincent Longo, The College of New Jersey; Faculty Advisor: Cynthia Curtis
Knot Invariants from Spanning Surfaces
The Alexander Polynomial is one of the most important and studied well knot invariants
in the field of knot theory. It is defined to be the determinant of the matrix V-t V^T,
where V is a linking matrix of an orientable (2-sided) surface bounded by the knot, V^T
is the transpose of V, and t is a variable. In this presentation we will explain our idea of
defining a similar polynomial using the non-orientable surfaces bounded by the knot.
4. Joseph Ruffo, The College of New Jersey; Faculty Advisor: Andrew Clifford
Fixed Subgroups of Finitely Generated Free Group Automorphisms
The fixed subgroup of a finitely generated free group automorphism is explored. Onedimensional models; known as Gersten Graphs; are utilized to develop a process that
generates the fixed elements of these automorphisms. This process is used to examine
two special classes of Gersten Graphs in order to find the corresponding fixed
subgroups associated with the automorphisms they represent.
5. Rob Rexler and A. Baello, Montclair State University; Faculty Advisor: Aihua Li
Design of Knapsack Cryptosystems Using Certain t-Superincreasing Sequences
Superincreasing sequences have been used widely in designing Knapsack
cryptosystems. Generalizing the concept, we define a new type of sequence, the tsuperincreasing sequence. In this presentation, we report our result on designing
Knapsack cryptosystems with t-superincreasing sequences. We describe methods for
creating t-superincreasing sequences, as well as the use of these sequences to construct
knapsack cryptosystems. This research is funded by MAA NREUP through NSF grants
DMS-1156582 and DMS-1359016.
6. Krista Varanyak, Monmouth University; Faculty Advisor: Richard Bastian
Effect of Missing Data Using Simulations in R
During the summer of 2014 our research group came across a data set with a very small
sample size. The team ran most common statistical tests, the Analysis of Variance
(ANOVA) across different variables, but knew that a small sample size could cause a
Type II error, resulting in a very low power of the test. To explore and model the issue of
depleting sample sizes and missing data as it relates to the F-Statistic and power, the
group selected a data set with N=500, then used the statistical program R to create an
original function to calculate the F-statistic. In this presentation we will discuss our
hypothesis, what we used, what results we obtained and what we were able to analyze
for various percentages of missing data. We will also describe what we were able to
conclude via simulation of missing data and explain what further research needs to be
conducted.
7. Patricia O'Dwyer, Monmouth University
Statistical Analysis of the Robustness of Oysters in the Hudson River Estuary
We report on a study of sites in the Hudson River Estuary that was conducted in order
to help determine potential locations for oyster restoration projects. Robust oysters from
an oyster farm (Copps Island) were used as a control in order to compare the conditions
of the oysters collected at each location. Condition indices used to compare these oysters
were based on parameters such as meat weight, shell weight and shell cavity volume.
Oysters were separated into respective categories based on their size and compared
based on these classifications. We discuss the results of Statistical tests tests that were
used to compare the conditions of oysters and the ratio of males to females by location.
We also describe additional tests on a comparison of condition indices from the oysters
in this study to those previously collected at the same locations in 2009.
8. Gabriel Romero, LaGuardia Community College; Faculty Advisor: Reem Jaafar
Single Molecule Magnets (SMM), investigated since 1981, typically have a large
magnetic anisotropy that impels the spin to point along a specific preferred axis. They
are used in the application of magnetic clusters for quantum information elements and
for magnetic storage devices in the field of quantum computing. We have conducted
Quantum Mechanical calculations of an emerging new class, Bicoordinate First Row
Transition Metal Complex Systems, of molecules, and are in the process of determining
the energetics of the relevant spin states. The ultimate aim is to predict the magnetic
properties, including the thermal barrier height, and, thus, to incorporate modifications
to the coordinating groups in the complexes, in order to increase the anisotropic barrier.
We plan to discuss our research in this presentation.
Student Oral Presentation Sessions
2:15-3:15pm – Edison 120, 121 and 122
Organizers: Joyati Debnath, Winona State University; Lee Collins, County
College of Morris; Jonathan Weisbrod, Burlington County College
Session 1, Edison 120
Chair: Reem Jaafar, LaGuardia Community College
2:15–2:27pm Kaitlyn Scrudato, Montclair State University; Faculty Advisor: Haiyan Su
Analysis of Water Quality in New Jersey
To model the quality of the water at any given time with available predictors, data from
bodies of water across New Jersey from 1999 to 2013 was collected from the database
STORET. The water quality parameters studied were Escherichia coli (e. coli) and
enterococcus with the predictors as Dissolved oxygen (DO), pH, Salinity, Temperature,
Total Dissolved Solids (TDS) and Total Suspended Solids (TSS). Multiple linear
regression was fitted first but did not fit the data well. Logistic regression models
indicated that the odds of having unsafe water (having more than 35 cfu of
enterococcus) for salt water is 0.176705 times the odds of fresh water when all other
values are held constant. To improve the poor fit of the multiple regression models, the
lasso regression method was also used to model the data.
2:30–2:42pm Macauley Breault, Muhlenberg College; Faculty Advisor: William Gryc
Mathematics of Neuroscience
Spike trains are electrical impulses of a single neuron within a discrete amount of time.
The timing between spikes, or the rate at which a neuron spikes, can reveal important
information about how a neuron reacts to a stimulus. Understanding the way neurons
respond to their environment can have important applications such as with brain
computer interfaces. Thus, it is critical that there be an accurate way to model neural
activity for spike trains, such as the rate at which neurons fire within a given time
interval called the intensity function. One means to estimate this function is the
reproducing-kernel Hilbert-space method.
2:45–2:57pm Melanie Campbell, Muhlenberg College;
Faculty Advisor: Allison Davidson
Modeling Fly Decision Making in a Maze
The intention of the mathematics research being presented is to work towards a
probability model of the decision making flies do within the maze and how it is affected
by visual stimuli. The research project studies the decision making pattern of the
populations of flies as they move through the maze. We worked to make a probability
prototype that most accurately models the fly behavior expressed in the experimental
data using simulations that made distinct assumptions about fly decision making.
Additionally, we proposed an experiment which tests how a single fly makes decisions
in a maze and how that compares to the decision making patterns of prior experiments.
Session 2, Edison 121
Chair: Shenglan Yuan, LaGuardia Community College
2:15 – 2:27pm Jennifer Crawford, Rowan University; Faculty Advisor: Hieu D. Nguyen
A New Class of Basis Polynomials Derived from the Generalized Prouhet-ThueMorse Sequence
The famous Prouhet-Thue-Morse (PTM) binary sequence {0; 1; 1; 0; 1; 0; 0; 1; ...} has
important applications in coding theory; number theory; and combinatorics. Its productgenerating function is well-known and has been used to identify sets with equal sums of
like powers. We consider the product-generating function of the PTM sequence
generalized to base p in terms of a new class of basis polynomials. New explicit formulas
are given for these polynomials in terms of eigenvectors and eigenvalues derived from
their recurrence matrices.
2:30 – 2:42pm Brooke Logan, Rowan University; Faculty Advisor: Michael Mossinghoff
Possible dimensions for circulant Hadamard matrices
A circulant Hadamard matrix is an n by n matrix with a number of properties: its rows
are mutually orthogonal, its entries are all positive or negative one, and each row after
the first is a circular shift of the prior row. It is widely conjectured that no circulant
Hadamard matrices exist with order x>4. Many algebraic restrictions are known on the
order of such a matrix, and as a result there are less than 1400 values less than 4x10^26
which have not been eliminated as a possible order of such a matrix. A description of a
project from the Summer@ICERM REU program at Brown University in 2014 will be
presented in which computational methods are used to improve the search for Wieferich
prime pairs. These pairs are fundamental in the construction of possible circulant
Hadamard matrices.
2:45 – 2:57pm Giancarlo Labruna, Montclair State University; Faculty Advisor: Aihua Li
Minimum and Maximum Randic Connectivity Indices from Trees Connected to a
Hexagon
In this research, I study simple graphs that attach a tree with n vertices to one vertex of a
hexagon. The Randic Connectivity Indices (RCI) of such graphs are calculated. The main
goal is to determine which graph has the maximum or minimum RCI. In this
presentation, I will report results from this research, including explicit formulas of RCI
for a class of graphs mentioned above and which graphs have the minimum or
maximum RCI values. The main results are: among a selected class of graphs, the one
with a path as the attached tree has the maximum RCI and the one with a star as the
attached tree produces the minimum RCI. Descriptions of the main result are be
discussed in the presentation.
3:00 – 3:12pm Doralia Castillo and Ma. Karina Soriano, Montclair State University;
Faculty Advisor: Ashwin Vaidya
Metastable States in Terminal Orientation of Symmetric Bodies in a Flow
Symmetric bodies such as cylinders and spheroids, in their terminal stable state, are
known to align their long axis perpendicular to the direction of a flow. This property has
been verified theoretically, experimentally, and numerically. The transition to a terminal
stable state is believed to coincide with the onset of significant inertial effects in a flow.
However, the threshold at which this transition occurs is unknown. In this article, we
conduct an experimental study to examine the nature of the transition of prolate
spheroids and cylinders of various aspect ratios, from their initial to terminal stable
equilibrium. Specifically, our experiments reveal the presence of intermediate
metastable states which are sensitive to the flow’s Reynolds number and physical
attributes of the immersed body, which gradually leads to the stable state. A phase
diagram of Reynolds number versus non-dimensional inertia clearly demarcates the
metastable, stable, and oscillatory states that the bodies undergo in current and shows
consistency between current and past observations.
Session 3, Edison 122
Chair: Sarita Nemani, Georgian Court University
2:15 – 2:27pm Thomas, Tu, New Jersey Institute of Technology;
Faculty Advisor: Richard Moore
Computing Optimal Observer Paths For Inferring an Uncertain Velocity Field
To solve the problem of optimally inferring a velocity field using periodic measurement
data from controlled gliders affected by that field, we apply a Kalman filter and
numerically solve our optimal control equations using relaxation. By parameterizing the
field using a Fourier basis, we develop a model that can be used by the Kalman filter to
assimilate data from each measurement optimally. Then, using the calculus of
variations, we determine partial differential equations for the optimal path to the next
position to take a measurement. Since these equations cannot be solved analytically, we
solve them numerically using a relaxation method.
2:30 – 2:42pm Sean Hill, SUNY New Paltz; Faculty Advisor: Ekaterina Shemyakova
Classifications of Darboux Transformations for Super KdV (The Intrigue of Non
Standard Differential Calculus)
The Darboux Transformation is a method for finding solutions of partial differential
equations. The goal of this project is to extend this method to partial diff. equations
which may depend on fermionic variables. These differential equations appear in the
scope of String Theory. This involves supercommutativity and we refer to this as the
super case. Specifically, we consider the super Korteweg-de Vries (KdV) equation and
the Darboux transformation of the KdV. We have shown that all order-one Darboux
Transformations have a specific form, reminiscent of the classical case. Previously, Liu
et. al. found a family of solutions of this form to the super KdV but it was not shown
that they were the only possible solutions. Our current work is on Darboux
Transformations of order two with hopes to generalize for order n. In this talk, we start
by presenting the peculiarities of non-standard calculus of the super case. We
demonstrate some examples of such in comparison to standard calculus. Finally, we
conclude with our new result and goals for the future.
2:45 – 2:57pm Dante Gates, Rowan University; Faculty Advisor: Hieu D. Nguyen
An Outside Analysis of the Mandelbrot Set
Due to its fractal nature, much about the Mandelbrot set M remains to be understood.
While a series formula has been proven to calculate the area of the M, to date the exact
value of this area remains unknown. The challenge lies in computing the series
coefficients which are recursively defined by a two dimensional sequence. We present
new approximations concerning the 2-adic valuation of the series coefficients. Moreover
we use these coefficients, derived from an analytic homeomorphism defined on the
complement of M to generate high resolution plots of the Mandelbrot set to give an
outside perspective of its fractal boundary.
Conference Organizing Committee
Amanda Beecher, Ramapo College; Lee Collins, County College of Morris; Joyati
Debnath, Winona State University; Katarzyna Kowal, Ramapo College of New Jersey;
Mince John, New Jersey City University; Ken McMurdy, Ramapo College of New Jersey;
A. David Trubatch (Director), Montclair State University; Jonathan Weisbrod,
Burlington County College
Mathematics Competition Committee
Katarzyna Kowal (Co-Director), Ramapo College of New Jersey; Tom Leong, The
University of Scranton; Ken McMurdy (Co-Director), Ramapo College of New Jersey.
David Molnar, Rutgers University; Ken Monks, The University of Scranton; Marek
Slaby, Fairleigh Dickinson University
Mathematics Competition Graders and Proctors
Tom Leong, The University of Scranton; Ken Monks, The University of Scranton, Marek
Slaby, Fairleigh Dickinson University; Emanuel Palsu-Andriescu, Monmouth
University; Nader Goubran, La Guardia CC; Jeremy Russell, TCNJ; Jimmy Mathews,
SUNY Stony Brook; Michael Saks, Rutgers University; Alexander Casti, Fairleigh
Dickinson University; Benjamin Daniels, Rowan University; Vasil Skenderi, St. Joseph
College; Robert Roach, Burlington CC; Chia-Lin Wu, Stockton University; Lee Collins,
CC of Morris; Steve Donahue, Cumberland CC.
Local Arrangements Committee
Bonnie Gold and David Marshall, Monmouth University
The Garden State Undergraduate Mathematics Conference is a function of the New
Jersey section of the Mathematical Association of America. The 2015 GSUMC is made
possible by the support of National Science Foundation through the MAA Regional
Undergraduate Mathematics Conferences program (NSF DMS-0846477), as well as
Monmouth University and the NJ section of the MAA.
The GSUMC organizers thank Department of Mathematics of Monmouth University for
their kind hospitality in hosting the meeting.