## REU Symposium Spring 2013 |
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## Mathematics Department
## Friday April 26 Session - 2pm-4pm - LCB 32302:00pm-02:20pm, Alessandro Gondolo, Characterization and Synthesis of Rayleigh-damped elastodynamic networksMentor: Fernando Guevara VasquezAbstract:
We consider damped elastodynamic networks where the damping matrix is
assumed to be a non-negative linear combination of the stiffness and mass
matrices (Rayleigh damping). We give here a characterization of the
frequency response of such networks. We also answer the synthesis question
for such networks, i.e., how to construct a Rayleigh damped elastodynamic
network with a given frequency response. The characterization and
synthesis questions are sufficient and necessary conditions for a
frequency dependent matrix to be the frequency response of a Rayleigh
damped network. We only consider the non-planar case. 02:20pm-02:40pm, Ben DeViney, Suction Feeding on Multiple PreyMentor: Tyler SkorczewskiAbstract:
Suction feeding is one of the most common forms of predation among
aquatic feeding vertebrates. There are several instances in nature and
technology that show numerous bodies in close proximity working together
to reduce the effects of drag. Here we simulate a suction feeding attack
on multiple prey items in an attempt to understand how the fluid dynamics
changes between single and multiple prey cases. To model a suction feeding
attack, we numerically find solutions to the incompressible Navier-Stokes
equations on Chimera overset grids. In comparing the prey capture times
across various cases, our results show that additional prey have no
effect. From these results, we can conclude that suction feeding
mechanisms are precise and highly localized to the point where multiple
prey items have no effect on the suction attack.02:40pm-03:00pm, Alex Burringo, Carli Edwards, Trevor Myrick, Mathematical Modeling of Molecular MotorsMentors: Parker Childs and Ross MagiAbstract:
Kinesin and dynein are molecular motor proteins that transport cargo in
opposite directions along microtubules in cells. They are of interest in
biophysics because of their roles in DNA replication and intracellular
transport. Despite considerable experimental research concerning single
motor dynamics, a clear theoretical explanation for motor procession along
cellular microtubule filaments is not present, especially when many motors
are present. Current research shows that it is common for multiple dynein
and kinesin motors to be attached to a single cargo. However, there is
much speculation as to how the cargo moves given this expected tug-of-war
possibility. To help reconcile current theories for procession, we created
a stochastic tug-of-war model of kinesin and dynein dynamics to simulate
data for comparison with current experimental data. Much of the semester
was spent coding the molecular movement of dynein and kinesin, with
evaluation of stepping probabilities, diffusion force effects, cargo-motor
attachment forces, effect of multiple motors, motor-microtubule attachment
forces, and the addition of temperature variations. Because of the
stochastic nature of the code, it was run numerous times in parallel on a
math department compute cluster. Evaluation of model effectiveness will be
done by statistical comparison of the simulated and experimental results,
specifically by Dr. Vershinin in the Physics Department, to aid in the
continuing research of molecular motor dynamics.03:00pm-03:20pm, Brady Thompson, Binary Quadratic FormsMentor: Gordan SavinAbstract:
Binary quadratic forms are quadratic forms in two variables having the
form $f(x, y) = ax^2 + bxy + cy^2$. In this talk we'll discuss the
reduction, automorphisms, and finiteness reduced forms of a given
discriminant. The main objective will be to present an interesting proof
about the relationship between automorphisms of forms and solutions to the
Pell equation.03:20pm-03:40pm, Steven Sullivan, A Trace Formula for $G_2$Mentor: Gordan SavinAbstract:
An $n$-dimensional representation of a group $G$ on a vector space $V$ is a
homomorphism from $G$ to $GL(V)$. For our purposes, we consider an
irreducible representation to be a representation which cannot be
decomposed into the direct sum of smaller-dimensional representations.
Let $H$ be a subgroup of $G$. The way in which irreducible representations of
$G$ decompose into irreducible representations of $H$ is called branching. In
order to calculate such branching, one must first obtain a trace formula
for each conjugacy class of $H$ in irreducible representations of $G$. In
this talk, we summarize the obstacles which must be overcome when
attempting to calculate such trace formulas for the conjugacy classes of
$G_2(2)$ in irreducible representations of $G_2$. Then we show the methods we
used to overcome these challenges and find the trace formulas for the
sixteen conjugacy classes.03:40pm-04:00pm, Wyatt Mackey, On the inverse Galois problem for symmetric groups and quaternionsMentor: Dan CiubotaruAbstract:
The Galois group was introduced in relation to the question of solvability
by radicals of polynomial equations. The inverse Galois problem asks if,
given a finite group $G$, there exists a field extension of rational
numbers such that the Galois group equals $G$. This projects investigate
explicit field extensions when $G$ is the quaternionic group $Q_8$ or a
symmetric group.## Monday April 29 Session - 2pm-4pm - JWB 33302:00pm-02:20pm, Sean Quinonez, Street fighters: a model of conflict in Tetramorium caespitumMentor: Fred AdlerAbstract:
Mathematical models provide a powerful tool for describing and analyzing
interactions between populations of organisms. Of interest are
interactions among eusocial populations, where simple rules at the
individual level translate into complex patterns of interaction within and
between colonies. We present a model that describes conflict between
colonies of the ant Tetramorium caespitum, an ideal study species due to
their abundance and status as an urbanized species. Conflicts between T.
caespitum colonies arise during seasonal competition for territory. Our
mathematical model tracks flow of ants in each of two colonies as they
recruit and travel to a conflict, where ants search and grapple with
enemies. The model will examine two mechanisms of motivation, and study
how these mechanisms control the location and size of the battle. These
recruitment mechanisms introduce delays that can produce oscillations in
the location and make-up of the battle, model predictions that correspond
to empirical measurements of ant battles.02:20pm-02:40pm, Skip Fowler, Mathematical modeling of epidemics using a probabilistic graphMentor: Fred AdlerAbstract:
In order to predict the number of survivors, we study a stochastic model
of an epidemic that includes with transitions infection and recovery.
This approach can be represented as a graph with probabilities on the
edges between states. The distribution of the number of survivors can be
found from the set of probabilities associated with states where the
number of infected people equals $0$. We find that this probability
distribution breaks into two distinct components. We compute the
probability of rapid epidemic extinction with many survivors, $Q$, using
methods from stochastic process theory. If the epidemic takes hold, we
use an SIR model without vital dynamics, a deterministic set of
differential equations describing the spread of an epidemic, which
approaches zero only when the epidemic begins and ends. We compare the
mean of the probabilities for states after $Q$ to determine if the mean
matches the end point of the epidemic in the deterministic SIR model, and
check whether their distribution is approximately normal. We developed
computer simulations to calculate and plot the probabilities and compare
with the mathematical analysis of $Q$ and the SIR model. Our discoveries
during the course of this project include finding the Catalan numbers and
Catalan's triangle within the graph structure and finding a knights move
based on the number of steps to any state within the graph structure. The
probabilities after $Q$ are not normal due to increasing kurtosis as
population size and infection rate are increased, although the mean is
well-approximated by the differential equation. We are searching for a
higher dimensional system that captures this deviation from normality.02:40pm-03:00pm, Kyle Zortman, Modeling the Progression to Invasive Cervical CancerMentor: Fred AdlerAbstract:
This talk will discuss the biological development of cervical cancer in
the body and the difficulties of stochastically modeling this process.
Because progression of cervical cancer is time and stage dependent, simple
stochastic modeling approaches fail. A solution is then proposed using a
modification of the Gillespie Algorithm. Simulation will be compared to
real data and more deterministic solutions. Finally, the system will be
analyzed for sensitivity and applications will be discussed.03:00pm-03:20pm, Boya Song, Horizontal fluid transport through Arctic melt pondsMentor: Ken GoldenAbstract:
The albedo or reflectance of the sea ice pack is an important parameter
in climate modeling. Being able to more accurately predict sea ice albedo
can significantly increase the reliability of climate model projections.
Sea ice albedo is closely related to the evolution of melt ponds on the
surface of Arctic sea ice. In the process of melt pond evolution, a
critical component is the drainage of surface melt water through holes and
cracks that exist naturally in ice floes. The speed and range of the
drainage is related to how easy it is for fluid to flow horizontally over
the ice. This horizontal fluid conductivity can be modeled using random
resistor networks. The horizontal conductivity of the melt pond
configurations plays a key role in modeling melt pond evolution, and
therefore sea ice albedo. In this lecture we will discuss efforts at
mapping melt pond configurations onto random graphs, and then developing
efficient algorithms to calculate the effective conductivity of these
random graphs.03:20pm-03:40pm, Brady Bowen, Random Fourier Surfaces: Applications to Arctic Melt Pond ModelingMentor: Ken GoldenAbstract:
Arctic melt ponds are important in climate models because they modify the
level of reflectance, or albedo, of the sea ice pack. We decided to model
this system using a two dimensional random Fourier series expansion. Level
sets of these random surfaces give shapes that closely resemble the melt
ponds found in the Arctic. Random Fourier surfaces also have multiple
controlled variables that can be changed to determine which length scale
dominates the generated surface. We modified these constants in order to
obtain a better fitting match of the observed melt ponds. We then
calculated area-perimeter data in order to find at what point the
generated shapes went through a fractal dimension transition, which is
exhibited by real data obtained from the Arctic melt ponds. It was also
found that changing the variables which characterize the Fourier surfaces
altered where this fractal dimension shift occurred and how fast the
fractal dimension of the shapes changed.For any questions, please contact the REU Director:Fernando Guevara Vasquez |
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