ASPiRE 2015 mathematics conference at Florida Gulf Coast University

1. ABSTRACTS
Academic Building 7 (AB7)
Florida Gulf Coast University
Saturday, Feb. 07, 2015

2. New Upper Bounds on the Distance Domination Numbers of Grids.
Michael Farina. Undergraduate Student, Florida Gulf Coast University.
Abstract: In his 1992 Ph.D. thesis Chang identified an efficient way to
dominate m×n grid graphs and conjectured that his construction gives the
most efficient dominating sets for relatively large grids. In 2011 Goncalves,
Pinlou, Rao, and Thomasse' proved Chang's conjecture, establishing a closed
formula for the domination number of a grid. In March 2013 Fata, Smith and
Sundaram established upper bounds for the k-distance domination numbers
of grid graphs by generalizing Chang's construction of dominating sets to k-
distance dominating sets. In this paper we improve the upper bounds
established by Fata, Smith, and Sundaram for the k-distance domination
numbers of grids. We also will unveil an example of how intuitive this section
of mathematics can be with a game for people to explore the idea in a fun
way.
Counting Symmetric Fullerene Patches with 4 Pentagons.
Armando Grez. Undergraduate Student, Florida Gulf Coast University.
Abstract: This study examines a method for constructing fullerene patches in
the hexagonal tessellation of the plane. We extend a result of Graves and
Graves (2013) by producing an exact process for drawing fullerene patches
with 4 pentagonal faces embedded in them. Attempts to formulate a closed
equation for the number of symmetric patches, up to isomorphism, are
shown. These include direct counting by predetermining the placement of two
pentagonal faces and noting graphical symmetries that cause overcounting,
using the Principle of Inclusion-Exclusion approach and by recoordinatizing
our depiction from Coxeter coordinates to Euclidean in order to analytically
count the patches within the largest convex polygon interior to all the
boundary lines we construct.

3. Patrolling Domination numbers.
Gabriel Guillen. Undergraduate Student, Florida Gulf Coast University.
Abstract: I have researched Patrolling Domination numbers. Where I have
define along with Dr.Insko an upper and lower bound. We also define the
patrolling numbers for specific graphs and grids.
Universal Cycles of Posets.
Molly Honecker. Undergraduate Student, Florida Gulf Coast University.
Abstract: Working with Dr. Katie Johnson and Daniel DePrisco, we worked
to find a deeper relation of Universal Cycles and Posets. Poset, which is short
for a partially ordered set, is a set with a relation such that it is reflexive and
transitive, and also antisymmetric. A universal cycle involves looking at a set
through a window of a certain size, k. Cycling through the set one k − window
at a time leads to every possible element of size k exactly once. Through our
work, we looked to find whether universal cycles can exist in posets, and if
not, whether a collection of specific posets can meet this qualification.
Complex Mappings and a Refinement of the Valence of Harmonic
Polynomials. Hunter Jackson. Undergraduate Student, University of South Florida.
Abstract: In this paper, we provide a comprehensive background on the
study of the valence and properties of complex polynomials that are harmonic
in the plane. This includes derivation from the Fundamental Theorem of
Algebra, an introduction to Bezout's Theorem and one of the most recent
developments, a counter-example to Wilmshurst's Conjecture provided by
Khavinson, Swiatek, et al. From there, we will study the orientation of
harmonic polynomials as well as determine the number of roots for a
canonical example of a harmonic polynomial including contour and caustic
plots. Finally, we will discuss the open questions currently being studied and
will answer the question: Is the Fundamental Theorem of Algebra still true
for Harmonic Polynomials?

4. Robot learning by mimicking Humans with Kinect Sensors.
Christie Mauretour. Undergraduate Student, Florida Gulf Coast University.
Abstract: Robotic devices have become an integral part of human lives.
Robots operate on people in operating rooms, clean our floors, assist in
military tasks and surveillance, and are present in countless factories across
the globe. In the future, interactions between robots and humans will need to
be more robust as they perform more human-like tasks. since human
interactions have a wide range of variability and are not limited to basic
commands and sensor technology is steadily improving, these sensors are
giving us a new way to interact with robots. Any kind of future human-robot
coexistence will need to have some way of teaching" the robot, by
demonstration, to perform a certain task. Depth sensors, such as the
KINECT by Microsoft, allow us to mimic robot learning by mapping human
poses and movements to equivalent robotic poses. Interpreting this KINECT
data effectively is a constrained optimization problems known as inverse
kinematics or motion retargeting. We will apply various optimization,
regression, and linear algebra techniques to map human poses onto a robot.
The robot we will be using is the DARwIn - OP (Dynamic Anthropomorphic
Robot with Intelligence - Open Platform), due to its relatively simple joint
structure and it's similarity to the human body..
Modeling a Smallpox Outbreak.
Jean Merone. Undergraduate Student, Florida Gulf Coast University.
Abstract: Smallpox was declared eradicated in 1979 by the World Health
Organization.Since then, only two laboratories in the U.S and Russia
safeguard variants of the virus and vaccination programs have ceased. To
model the possibility of an outbreak, we developed a series of diff
erential equations creating a SVEIRD model. It subdivides the population
into Susceptible, Vaccinated, Exposed, Infectious, Recovered, and Dead.
Simulating a city population of a million, we model the results on MATLAB.
We investigate vaccination campaigns to lower the mortality.

5. Results on Defective Triangular Matrices.
Orion Moore. Undergraduate Student, Florida Gulf Coast University.
Abstract: In this presentation, we consider the properties of defective
matrices, which tend to be more difficult to perform certain calculations with
than nondefective matrices. We use as example the matrix exponential. We
explore the properties which can guarantee the defectivity of a triangular
matrix, and formalize our findings as a theorem. We conclude by discussing
the proof of the theorem, using familiar concepts from linear algebra as a
starting point.
Bound States in the Radiation Continuum for Periodic Structures.
Joseph Park. Undergraduate Student, University of Florida.
Abstract: All optical data-processing could diminish the limitations of
computational power, a pervasive problem in computational research. The
biggest obstacle is developing an optical analog of a transistor. My research
advisor, mathematical physicist Professor Sergei Shabanov, has made
significant progress toward this end investigating bound states of
electromagnetic waves in the radiation continuum. It was proved that the
interaction between trapped electromagnetic modes can lead to scattering
resonances of negligible width, which are the bound states in the radiation
continuum first discovered in quantum systems by von Neumann and
Wigner. It was then shown in a double array of subwavelength dielectric
cylinders that by varying the spatial parameters toward the critical value,
the near field can be amplified in certain regions. The present study is the
generalized system of an arbitrary number of arrays, two parallel 2D lattices
of spherical scatterers, and analogous systems for elastodynamic and/or
acoustic waves. The main fields of study involved are mathematical physics,
scattering theory, functional analysis, operator theory, electromagnetism,
acoustics and elastodynamics. Other potential applications include large
amplification of electromagnetic fields within photonic structures and, hence,
enhancement of nonlinear phenomena, impurity detection, biosensing, as
well as perfect filters and waveguides for a particular frequency.

6. Peak Sets of Coxeter Groups of Classical Lie Types.
Darleen Perez-Lavin. Graduate Student, Florida Gulf Coast University.
Abstract: We say that a permutation 𝜋 = 𝜋1 𝜋2 ⋯ 𝜋 𝑛 in the symmetric group
𝑆 𝑛 has a 𝑝𝑝𝑝𝑝 at index 𝑖 if 𝜋𝑖−1 < 𝜋𝑖 < 𝜋𝑖+1 and we let
𝑃( 𝜋) = { 𝑖 | 𝑖 𝑖𝑖 𝑎 𝑝𝑝𝑝𝑝 𝑜𝑜 𝜋}.$ Given a set 𝑆 of positive integers we let 𝑃 (𝑆; 𝑛)
denote the subset of 𝑆 𝑛 consisting of all permutations 𝜋, where 𝑃( 𝜋) = 𝑆. In
2013, Billey, Burdzy and Sagan proved that |𝑃(𝑆, 𝑛)| = 𝑝( 𝑛)2{𝑛−|𝑆|−1}
, where
𝑝(𝑛) is a polynomial of degree 𝑚𝑚𝑚(𝑆) − 1. In 2014, Casto et, al. considered
the Coxeter group of type 𝐵 as the group of signed permutations on 𝑛 letters
and showed that |𝑃_𝐵(𝑆; 𝑛)| = 𝑝( 𝑛)2{2𝑛−|𝑆|−1}
. We partition the set 𝑃(𝑆; 𝑛) into
subsets of elements which end with an ascent or a descent and provide
recursive formulas for the cardinalities of these subsets. We then embed the
Coxeter groups of Lie type 𝐵𝑛 and 𝐷 𝑛 into 𝑆2𝑛 and use the partitioning of
𝑃(𝑆; 𝑛) to describe and enumerate the sets 𝑃� 𝐵(𝑆;𝑛) , 𝑃� 𝐵(𝑆∪{𝑛};𝑛), 𝑃� 𝐷(𝑆;𝑛) and
𝑃� 𝐷(𝑆∪{𝑛};𝑛). Furthermore, these results lead to a collection of interesting
identities as special cases of our enumerative formulas for 𝑃� 𝐵(𝑆;𝑛) and 𝑃� 𝐷(𝑆;𝑛).
Seeing Stars: Polynomials and Gravitational Lenses.
Andres Saez. Undergraduate Student, University of South Florida.
Abstract: Gravitational lenses are created when a large mass, like a galaxy
or black hole, sits in the straight-line path between us, the observers, and a
star. This will cause multiple images of the same star to form in the night
sky. In this talk, we will discuss how these may be modelled using rational
functions, as well as extensions of this same idea to polynomials and an
investigation into a generalized fundamental theorem of algebra. This
extension, stemming from the work of A.S. Wilmshurst, would give us an
upper bound on the number of zeroes of harmonic polynomials.

7. The Mathematics and Art Connection- Symmetry Group
Classification Algorithms. Julia Seay. Florida Atlantic University.
Abstract: The beauty and harmony of symmetry is highly prevalent in nature
and in our everyday lives. While it may be obvious to the naked eye that
various symmetric objects have distinct patterns and structures, it behooves
the mathematician to quantify and classify these distinctions found in
symmetries to make use of them in various fields. In mathematics, we use
numbers to determine size or amount; however, to determine symmetries, we
turn to what is known as a group. For this study, we will investigate the
presence of two symmetry groups in particular: frieze and wallpaper groups.
These groups describe a symmetric pattern known as a tessellation, or a
tiling of the plane, in which the same figure is repeated in the plane with no
overlaps or gaps. Upon observation of these patterns in artwork by graphic
artist M.C. Escher and in local Florida art and architecture, we will then
mathematically classify these symmetries using frieze and wallpaper groups.
Furthermore, we will confirm our mathematical classifications through our
symmetry group classification algorithm, executed through a new dynamic
programming-language known as Julia. Upon implementing this algorithm
through the Julia language, we will seek to make improvements on existing
symmetry group classification algorithms. In using art, mathematics and
computer science to achieve our goals, we hope to help eliminate the
dichotomy between the arts and sciences.
Collisions of Random Walks on a Graph.
Rade Stoisavljevic. Graduate Student, Florida Gulf Coast University.
Abstract: The Cops and Robbers game was first introduced by Nowakowski,
Winkler, and Quillot about thirty years ago. In this game, cop and robber are
placed on two different vertices of a graph G. During each round, players
have to move to an adjacent vertex. Players follow no set strategy; hence,
they take random walks. The question is "What is the probability that the
robber will still be free after 'm' rounds of the game?" We show that this
depends solely on the starting position of the players and we give both
recursive and closed formulas for complete graphs, complete bipartite graphs,
friendship graphs, windmills, and cycles.

8. A Proof for Using Net Cash Flows and Descartes Rule of Signs to
Determine a Small Upper Bound for the Number of Yeilds in
Discounted Cash flow Analysis. Philip Tirino, Florida Gulf Coast University.
Abstract: Discounted Cash Flow Analysis is a method of determining the
value of a long term investment with regard to both time and money in terms
of a single internal rate of return, similar to assigning an interest rate to the
investment. In certain situations the standard method of DCFA can yield
multiple rates. I say standard because this is the method used by the
Financial Accounting Standards Board in FASB-13. For the purposes of this
paper, we may simply assume yielding multiple rates is bad. If we apply
Descartes Rule to the function used to generate the internal rate or return we
get an upper bound for the number of possible rates, but it is far from the
least upper bound and hence not very meaningful. If we could instead look for
sign changes in the net cash flows we would get a much more meaningful
upper bound for the number of possible rates. The purpose of this research is
to first prove the validity of this method and second, examine the meaning of
multiple rates so we can better understand the situation at hand when it
arises.
An Alternate Method of Integration with Absolute Value
Functions. Joshua Witte. Undergraduate Student, Florida SouthWestern State College.
Abstract: We present an alternative approach to calculus with absolute value
functions. In this study we define absolute value functions and their
associated compositions, arguing that the redefined forms are easier to
handle and present accurate representations of their corresponding integrals.