College of Arts and Sciences (AU-CAS)http://aladinrc.wrlc.org:80/handle/1961/51302014-08-23T11:32:38Z2014-08-23T11:32:38ZTechnical Report No. 2014-4: Continuous Selections of Modulus of Continuity (AU-Cas-Mathstats)Moskey, Adamhttp://aladinrc.wrlc.org:80/handle/1961/165272014-04-15T07:45:21Z2010-01-01T00:00:00ZTechnical Report No. 2014-4: Continuous Selections of Modulus of Continuity (AU-Cas-Mathstats)
Moskey, Adam
The main result of this thesis deals with continuous functions on metric spaces. Specifically,
we show that given a continuous function f from a metric space (M1; d1) to a metric space (M2; d2), there is a function such that is continuous. The
above result was established by Enayat in [Ena00] using the machinery of partitions of unity.
Our expository account of Enayat’s paper contains a substantial body of results in general
topology, including a thorough discussion of paracompact spaces and partitions of unity.
2010-01-01T00:00:00ZTechnical Report No. 2014-3: Ultraproducts and their applications (AU-CAS-MathStats)Purcell, Amandahttp://aladinrc.wrlc.org:80/handle/1961/165012014-04-15T07:42:49Z2013-12-01T00:00:00ZTechnical Report No. 2014-3: Ultraproducts and their applications (AU-CAS-MathStats)
Purcell, Amanda
An ultraproduct is a mathematical construction used primarily in abstract algebra and model theory to create a new structure by reducing a product of a family of existing structures using a class of objects referred to as filters. This thesis provides a rigorous construction of ultraproducts and investigates some of their applications in the fields of mathematical logic, nonstandard analysis, and complex analysis. An introduction to basic set theory is included and used as a foundation for the ultraproduct construction. It is shown how to use this method on a family of models of first order logic to construct a new model of first order logic, with which one can produce a proof of the Compactness Theorem that is both elegant and robust. Next, an ultraproduct is used to offer a bridge between intuition and the formalization of nonstandard analysis by providing concrete infinite and infinitesimal elements. Finally, a proof of the Ax-Grothendieck Theorem is provided in which the ultraproduct and other previous results play a critical role. Rather than examining one in depth application, this text features ultraproducts as tools to solve problems across various disciplines.
To access this Technical Report, please refer to the following link: http://aladinrc.wrlc.org//handle/1961/15286
2013-12-01T00:00:00ZTechnical Report No. 2013-1: Social Network Analysis – Statistical Applications (AU-CAS-MathStats)Rebatta Sun Han, Anahihttp://aladinrc.wrlc.org:80/handle/1961/163732014-04-05T07:43:42Z2012-12-01T00:00:00ZTechnical Report No. 2013-1: Social Network Analysis – Statistical Applications (AU-CAS-MathStats)
Rebatta Sun Han, Anahi
A weighted social network data type is analyzed using two different methods. The first method used is the Exponential Random Graph Models (ERGM), which is a model used on social network analysis (SNA) that includes as parameters the structural characteristics of the network and the network’s nodes attributes. However, ERGM does not take into account the weights associated with the network’s edges. The second method used is the Cumulative Logistic Regression, which incorporates the weights associated to the network’s edges, but it doesn’t take into account the network’s structural characteristics. Both methods are illustrated using a weighted one-mode data.
Technical Report No. 2013-1, 46 pages
2012-12-01T00:00:00ZTechnical Report No. 2014-2: Sheaf Invariants for Information Systems (AU-CAS-MathStats)Robinson, Michaelhttp://aladinrc.wrlc.org:80/handle/1961/163462014-04-04T07:44:11Z2012-01-01T00:00:00ZTechnical Report No. 2014-2: Sheaf Invariants for Information Systems (AU-CAS-MathStats)
Robinson, Michael
The primary objective of this project was to construct, classify, and exploit
invariants for discriminating information systems that are based on abstracted
structural descriptions. This main objective was split into three smaller objectives:
(1) Construct invariants for information systems that exploit coarse and multiscale structural specifications about their underlying network or communication topology,
(2) Classify the semantic and dynamic features of the systems that these in-
variants consider, and
(3) Exploit the classification results to provide actionable design and analysis
rules that can be incorporated into experimental and simulation workflows.
All of these objectives were met. Several interesting (and potentially important) discoveries were made as a result of the project. These discoveries have
been reported to the scientifi c community, and they are being written as articles
for archival journals. In addition, the Principal Investigator (PI), Prof. Michael
Robinson, completed a draft of a manuscript entitled Topological Signal Processing that can be used to teach the techniques discovered on this project to beginning
graduate student researchers. Finally, Prof. Robinson's research group grew from
one student at the start of the project to ve students (partially funded by this
project), partially as a means to apply the new algorithmic techniques discovered
on this program.
Technical Report No. 2014-2, 37 pages
2012-01-01T00:00:00ZTechnical Report No. 2014-1: Measuring Ocean Winds from Space Using a Radar Satellite (AU-CAS-MathStats)Robinson, MichaelDehart, MorganHubler, MattVerdi, MarkZhu, Zhuhttp://aladinrc.wrlc.org:80/handle/1961/163452014-04-05T07:43:01Z2013-11-01T00:00:00ZTechnical Report No. 2014-1: Measuring Ocean Winds from Space Using a Radar Satellite (AU-CAS-MathStats)
Robinson, Michael; Dehart, Morgan; Hubler, Matt; Verdi, Mark; Zhu, Zhu
The goal of this project is to develop and validate image processing algorithms
for measuring wind direction over the ocean. We plan to use wind truth data from
(1) oceanographic buoys and other anchored sensors and (2) wind measurements
from other satellite sensors to validate our SAR-derived wind direction estimates.
Both of these sources of truth data are of necessarily lower resolution than what
is available from TerraSAR-X. It is worth noting that buoy validation is already
routinely done against the ASCAT sensor [14]. Our collection campaign centers on
several buoys described in the NOAA’s database [20]. We are using the resulting
concurrent data stream for validation of spectral shape, but have not completely
validated the recovery of wind direction from the image.
Technical Report No. 2014-1, 34 pages
2013-11-01T00:00:00ZTechnical Report No. 2013-6: Coefficients of Transformations in Multidimensional Quantum Harmonic Oscillators (AU-CAS-MathStats)Volz, Travis J.http://aladinrc.wrlc.org:80/handle/1961/163442014-04-04T13:40:45Z2013-11-06T00:00:00ZTechnical Report No. 2013-6: Coefficients of Transformations in Multidimensional Quantum Harmonic Oscillators (AU-CAS-MathStats)
Volz, Travis J.
When one represents a physical system of harmonic oscillators, it is possible to represent the system in many ways.
For a system with three degrees of freedom, one could represent a single particle in three dimensions which would be
equivalent to three particles in one dimension. The main idea here is that in each case, the total degrees of freedom must
be equal for different representations of a given system. There are many other ways that someone can represent a physical
system in three dimensions. The system can be represented in Cartesian coordinates, cylindrical coordinates, or spherical
coordinates, just to name a few. Then within each of these representations, one could imagine rotating the coordinate
systems or scaling them differently. As a result, one specific state of the multidimensional quantum harmonic oscillator
can be represented in many different ways. The purpose of this research project is to calculate different coefficients for
translating from one representation to another.
Technical Report No. 2013-6, 8 pages
2013-11-06T00:00:00ZTechnical Report No. 2013-5: Journeys Through Non-Euclidean Geometries (AU-CAS-MathStats)LaRochelle, Raymondhttp://aladinrc.wrlc.org:80/handle/1961/163432014-04-04T13:41:34Z2013-05-03T00:00:00ZTechnical Report No. 2013-5: Journeys Through Non-Euclidean Geometries (AU-CAS-MathStats)
LaRochelle, Raymond
The Uniformization Theorem from the study of Riemann surfaces gives a fundamental
understanding of the geometry of our world. The theorem tells us that there are three different
types of geometries for orientable surfaces: Euclidean (flat), spherical, and hyperbolic.
We live in all three at once; it all depends on scale. We travel through our communities as if
we were on a flat surface, since the scale is small enough not to notice the curvature of the
earth. At this level, we experience Euclidean (or “flat”) geometry. If we increase the scale
further, as we do when we fly planes or track satellites, we experience spherical geometry.
Furthermore, due to Einstein’s theory of relativity, space itself exhibits a negative curvature,
which means at the largest scales we experience hyperbolic geometry. On these larger scales,
Euclidean geometry does not accurately measure angles and lengths. While Euclidean geometry
is easily visualized by students, spherical and hyperbolic geometries prove to be more
challenging. This paper provides new ways to visualize these geometries.
Felix Klein and colleagues created a research program that studied geometry in a new
way, known as the Erlangen Program. The Erlangen Program used projective geometry as
the unifying frame of all other geometries and group theory to abstract and organize our
geometric knowledge. The groups studied are the groups of functions called isometries –
bijective maps that preserve length. For example, in Euclidean geometry the isometries
are rotations, reflections, and translations. Of course, spherical and hyperbolic functions
isometries are different from the ones we are familiar with in flat geometry. Consequently,
distances and angles are not the same. My project addresses this, with computer programs
and graphics that represent spherical and hyperbolic worlds. I have created visual tools for
understanding the different geometries. My goal was to give any interested person, from an
elementary school student to an academic, the ability to understand Euclidean, spherical,
and hyperbolic worlds.
Technical Report No. 2013-5, 43 pages
2013-05-03T00:00:00ZTechnical Report No. 2013-4: Automorphisms of Models of Arithmetic (AU-CAS-MathStats)Cassel, Michael H.http://aladinrc.wrlc.org:80/handle/1961/163422014-04-04T13:42:12Z2013-05-09T00:00:00ZTechnical Report No. 2013-4: Automorphisms of Models of Arithmetic (AU-CAS-MathStats)
Cassel, Michael H.
The goal of this thesis is to exposit the key results of the subject,
with an eye towards presenting two open problems related to
the structure of the groups of automorphisms of a model of arithmetic.
Section 2 will present preliminary results and definitions,
discussing the axioms of Peano Arithmetic (PA), and deriving key
theorems in the model theory of arithmetic. Section 3 will discuss
recursive saturation, which is the key to building automorphisms
of models of arithmetic. Essentially, recursive saturation is all
about building models “just rich enough” to have many elements
that are free to move under an automorphism. Finally, Section 4 will cover results at the cutting edge1 of the subject, and present
two open problems.
Technical Report No. 2013-4, 45 pages
2013-05-09T00:00:00Z