Mathematics Commons^™

Non-Isomorphic Real Simple Lie Algebras Of The Same Complex Type And Character, Ian M. Anderson

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Complex simple Lie algebras are classified by their root types. The type of a real simple Lie algebra is the root type of the associated complex algebra. The character of a real simple Lie algebra is the signature of its Killing form.

For many root types, the character is sufficient to uniquely classify the corresponding real Lie algebras. However, one should not take this statement to be literally true – there are a few cases where the character does not suffice to distinguish all possible real forms.

In this worksheet we will show that the 2 real non-isomorphic Lie algebras …

The Game Of Thrones: A Study Of Power Networks And How They Change, Trevor Williams

Research on Capitol Hill

No abstract provided.

A Rank 7 Pfaffian System On A 15-Dimensional Manifold With F4 Symmetry Algebra, Ian M. Anderson

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Let I be a differential system on a manifold M. The infinitesimal symmetry algebra of I is the set of all vectors fields X on M such that preserve I. In this worksheet we present an example, due to E. Cartan of a rank 7 Pfaffian system on a 15-dimensional manifold whose infinitesimal symmetry algebra is the split real form of the exceptional Lie algebra f4 .

Cartan Involutions And Cartan Decompositions Of A Semi-Simple Lie Algebra, Ian M. Anderson

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In this worksheet we shall review the basic definitions and properties of Cartan involutions and Cartan decompositions and illustrate these using the DifferentialGeometry software package for Lie algebras.

Individual-Based Modeling: Mountain Pine Beetle Seasonal Biology In Response To Climate, Jacques Regniere, Barbara J. Bentz, James A. Powell, Remi St-Amant

Mathematics and Statistics Faculty Publications

Over the past decades, as significant advances were made in the availability and accessibility of computing power, individual-based models (IBM) have become increasingly appealing to ecologists (Grimm 1999). The individual-based modeling approachprovides a convenient framework to incorporate detailed knowledge of individuals and of their interactions within populations (Lomnicki 1999). Variability among individuals is essential to the success of populations that are exposed to changing environments, and because natural selection acts on this variability, it is an essential component of population performance. © Springer International Publishing Switzerland 2015.

Cartan Subalgebras, Compact Roots And The Satake Diagram For Su(2, 2), Ian M. Anderson

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In this worksheet we use the 15-dimensional real Lie algebra su(2, 2) to illustrate some important points regarding the general structure theory and classification of real semi-simple Lie algebras.

1. Recall that a real semi-simple Lie algebra g is called a compact Lie algebra if the Killing form is negative definite. The Lie algebra g is compact if and only if all the root vectors for any Cartan subalgebra are purely imaginary. However, if the root vectors are purely imaginary for some choice of Cartan subalgebra it is not necessarily true that the Lie algebra is compact.

2. A real …

Jordan Algebras And The Exceptional Lie Algebra F4, Ian M. Anderson

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This worksheet analyzes the structure of the Jordan algebra J(3, O) and its split and exceptional versions. The algebra of derivations is related to the exceptional Lie algebra f4.

Linear Operators That Preserve Graphical Properties Of Matrices: Isolation Numbers, Leroy B. Beasley, Seok-Zun Song, Young Bae Jun

Mathematics and Statistics Faculty Publications

Let A be a Boolean {0, 1} matrix. The isolation number of A is the maximum number of ones in A such that no two are in any row or any column (that is they are independent), and no two are in a 2 × 2 submatrix of all ones. The isolation number of A is a lower bound on the Boolean rank of A. A linear operator on the set of m × n Boolean matrices is a mapping which is additive and maps the zero matrix, O, to itself. A mapping strongly preserves a set, S, if it …

Citizen Science Reveals Widespread Negative Effects Of Roads On Amphibian Distributions, Bradley J. Cosentino, David M. Marsh, Kara S. Jones, Joseph J. Apodaca, Christopher Bates, Jessica Beach, Karen H. Beard, Kelsie Becklin, Jane Margaret Bell, Christopher Crockett, George Fawson, Jennifer Fjelsted, Elizabeth A. Forys, Kristen S. Genet, Melanie Grover, Jaimie Holmes, Katherine Indeck, Nancy E. Karraker, Eran S. Kilpatrick, Tom A. Langen, Stephen G. Mugel, Alessandro Molina, James R. Vonesh, Ryan J. Weaver, Anisha Willey

Wildland Resources Faculty Publications

Landscape structure is important for shaping the abundance and distribution of amphibians, but prior studies of landscape effects have been species or ecosystem-specific. Using a large-scale, citizen science-generated database, we examined the effects of habitat composition, road disturbance, and habitat split (i.e. the isolation of wetland from forest by intervening land use) on the distribution and richness of frogs and toads in the eastern and central United States. Undergraduates from nine biology and environmental science courses collated occupancy data and characterized landscape structure at 1617 sampling locations from the North American Amphibian Monitoring Program. Our analysis revealed that anuran species …

Network Applications And The Utah Homeless Network, Michael A. Snyder

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Graph theory is the foundation on which social network analysis (SNA) is built. With the flood of "big data," graph theoretical concepts and their linear algebraic counterparts are essential tools for analysis in the burgeoning field of network data analysis, in which SNA is a subset. Here we begin with an overview of SNA. We then discuss the common descriptive measures taken on network data as well as proposing new measures specific to homeless networks. We also define a new data structure which we call the location sequence matrix. This data structure makes certain computational network analyses particularly easy. …

Traditional Lecture Versus An Activity Approach For Teaching Statistics: A Comparison Of Outcomes, Jennifer L. Loveland

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Many educational researchers have proposed teaching statistics with less lecture and more active learning methods. However, there are only a few comparative studies that have taught one section of statistics with lectures and one section with activity-based methods; of those studies, the results are contradictory. To address the need for more research on the actual effectiveness of active learning methods in introductory statistics, this research study was undertaken.

An introductory, university level course was divided into two sections. One section was taught entirely with traditional lecture. The other section was taught using active learning methods and a minimal amount of …

Hamiltonicity, Pancyclicity, And Cycle Extendability In Graphs, Deborah C. Arangno

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

A significant portion of Graph Theory is devoted to determining the characteristics which guarantee the existence of long cycles.

Long cycles have roles in applications to civil engineering, chemistry, and communications, among many others, but the problem, in and of itself, of determining whether a graph has a cycle of some fixed and typically large length is one of the most important problems of both pure Mathematics and Computer Science.

A cycle containing all the vertices of the graph is called a Hamiltonian cycle, and a graph which possesses such a cycle is said to be Hamiltonian. If …

Statistical Modeling, Exploration, And Visualization Of Snow Water Equivalent Data, James Beguah Odei

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Due to a continual increase in the demand for water as well as an ongoing regional drought, there is an imminent need to monitor and forecast water resources in the Western United States. In particular, water resources in the Intermountain West rely heavily on snow water storage. Thus, the need to improve seasonal forecasts of snowpack and considering new techniques would allow water resources to be more effectively managed throughout the entire water-year. Many available models used in forecasting snow water equivalent (SWE) measurements require delicate calibrations.

In contrast to the physical SWE models most commonly used for forecasting, we …

Multiplicity Results Of Periodic Solutions For Two Classes Of Nonlinear Problems, Kazuya Hata

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

We investigate the existences and qualitative properties of periodic solutions of the following two classes of nonlinear differential equations:

I) (Special) Relativistic Pendulum Equations (RPEs);

II) (2-coupled) Gross-Pitaevskii Equations (GPEs).

The pendulum equation describes the motion of a pendulum. According to Special Relativity, which was published by A. Einstein in 1905, causality is more fundamental than constant time-space, thus time will flow slower and space will distort to keep causality if the speed of motion is near the speed of light. In such high speed situations, the pendulum equation needs to be revised due to Special Relativity. The revised equation …

Family-Wise Error Rate Control In Quantitative Trait Loci (Qtl) Mapping And Gene Ontology Graphs With Remarks On Family Selection, Garrett Saunders

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

One of the great aims of statistics, the science of collecting, analyzing, and interpreting data, is to protect against the probability of falsely rejecting an accepted claim, or hypothesis, given observed data stemming from some experiment. This is generally known as protecting against a Type I Error, or controlling the Type I Error rate. The extension of this protection against Type I Errors to the situation where thousands upon thousands of hypotheses are examined simultaneously is known as multiple hypothesis testing. This dissertation presents an improvement to an existing multiple hypothesis testing approach, the Focus Level method, specific to gene …

Modeling Asset Volatility Using Various Resources, Isaac G. Blackhurst

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Volatility is of central interest in modern financial econometrics. This thesis evaluates three different methods of measuring volatility.

High Cognitive Test Item Development And Implementation, Ashley Salisbury

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

In secondary math classrooms there has been a movement toward discovery and problem solving based instruction. This type of instruction requires teachers to teach on what is often considered a higher level of cognition and allows students to discover more ideas and concepts on their own as opposed to traditional lecture style. Teachers with well thought-out examples, questions, and activities provide students with tools to solve problems on their own requiring students to make mathematical discoveries and connections. These skills not only benefit students in their math class but are analytical skills students can use throughout their lives.

With this …

The Optimal Release Of Sterile Males In Pest Management, Sergio Ramirez

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

With an application of the sterile insect technique, it is our goal to know the optimal rate of production of sterile males in order to control an insect pest invasion and protect crops. Starting with a fundametal relationship between costs of production and feeding rates, we construct an objective functional. We then use the relationships between the dierent stages of female reproduction to build the state equations. Once, we have dened the optimal control problem, we use the simulated annealing algorithm to determine the amount of sterile males to be released at each discrete time event. Multiple calculations are made …

The Spacetime Geometry Of A Null Electromagnetic Field, Charles G. Torre

Presentations and Publications

We give a set of local geometric conditions on a spacetime metric which are necessary and sufficient for it to be a null electrovacuum, that is, the metric is part of a solution to the Einstein–Maxwell equations with a null electromagnetic field. These conditions are restrictions on a null congruence canonically constructed from the spacetime metric, and can involve up to five derivatives of the metric. The null electrovacuum conditions are counterparts of the Rainich conditions, which geometrically characterize non-null electrovacua. Given a spacetime satisfying the conditions for a null electrovacuum, a straightforward procedure builds the null electromagnetic field from …

Stability Of Traveling-Wave Solutions For A Schrodinger System With Power-Type Nonlinearities, Nghiem Nguyen, Rushun Tian, Zhi-Qiang Wang

Mathematics and Statistics Faculty Publications

In this article, we consider the Schrodinger system with powertype nonlinearities, (Formula presented) where j = 1,...,m, u_j are complex-valued functions of (x, t) 2 R^N+1, a, b are real numbers. It is shown that when b > 0, and a + (m - 1)b > 0, for a certain range of p, traveling-wave solutions of this system exist, and are orbitally stable.

A Shortcut For Multiple Testing On The Directed Acyclic Graph Of Gene Ontology, Garrett Saunders, John R. Stevens, S. Clay Isom

Mathematics and Statistics Faculty Publications

Background: Gene set testing has become an important analysis technique in high throughput microarray and next generation sequencing studies for uncovering patterns of differential expression of various biological processes. Often, the large number of gene sets that are tested simultaneously require some sort of multiplicity correction to account for the multiplicity effect. This work provides a substantial computational improvement to an existing familywise error rate controlling multiplicity approach (the Focus Level method) for gene set testing in high throughput microarray and next generation sequencing studies using Gene Ontology graphs, which we call the Short Focus Level.

Results: The Short Focus …

Rainich-Type Conditions For Null Electrovacuum Spacetimes Ii, Charles G. Torre

Research Vignettes

In this second of two worksheets I continue describing local Rainich-type conditions which are necessary and sufficient for the metric to define a null electrovacuum. In other words, these conditions, which I will call the null electrovacuum conditions, guarantee the existence of a null electromagnetic field such that the metric and electromagnetic field satisfy the Einstein-Maxwell equations. When it exists, the electromagnetic field is easily constructed from the metric. In this worksheet I consider the null electrovacuum conditions which apply when a certain null geodesic congruence determined by the metric is twisting. I shall illustrate the these conditions using a …

Gr 20 Workshop, Warsaw, July 2013, Ian M. Anderson, Charles G. Torre

Presentations

These are the Maple worksheets used at the Differential Geometry in Maple Workshop, which was held at the 20th International Conference on General Relativity and Gravitation, in Warsaw, July 2013.

There are 6 worksheets which can be downloaded from the list of files below.

The Spacetime Geometry Of A Null Electromagnetic Field, Charles G. Torre

Presentations and Publications

We give a set of local geometric conditions on a spacetime metric which are necessary and sufficient for it to be a null electrovacuum, that is, the metric is part of a solution to the Einstein-Maxwell equations with a null electromagnetic field. These conditions are restrictions on a null congruence canonically constructed from the spacetime metric, and can involve up to five derivatives of the metric. The null electrovacuum conditions are counterparts of the Rainich conditions, which geometrically characterize non-null electrovacua. Given a spacetime satisfying the conditions for a null electrovacuum, a straightforward procedure builds the null electromagnetic field from …

Tournament Directed Graphs, Sarah Camille Mousley

Undergraduate Honors Capstone Projects

Paired comparison is the process of comparing objects two at a time. A tournament in Graph Theory is a representation of such paired comparison data. Formally, an n-tournament is an oriented complete graph on n vertices; that is, it is the representation of a paired comparison, where the winner of the comparison between objects x and y (x and y are called vertices) is depicted with an arrow or arc from the winner to the other.

In this thesis, we shall prove several results on tournaments. In Chapter 2, we will prove that the maximum number of vertices …

Particle Dynamics And Resistivity Characteristics In Bifurcated Current Sheets, Tushar Andriyas

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Charged particle chaos and its collective effects in different magnetic geometries are investigated in a sequence of various numerical experiments. The fields generated by the particles as a result of interaction with the background electric and magnetic fields is not accounted for in the simulation. An X-line is first used to describe the geometry of the magnetotail prior to magnetic reconnection and a study of the behavior of charged particles is done from a microscopic viewpoint. Another important geometry in the magnetotail prior to substorm onset is Bifurcated Current Sheet. The same analysis is done for this configuration. The existence …

Students Rise To The Challenge Of Modeling Yeast Growth Despite Sour Hiccups From Imperfect Data, Alicia Caldwell

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Students rarely receive the opportunity to experience a learning activity involving mathematical modeling. This paper describes a lab in which students in an Applied Mathematics in Biology course observe the growth of Saccharomyces cerevisiae, a yeast strain, in differing sugar concentrations for use in learning modeling. They parameterized the logistic equation and an alternate model, which they themselves constructed, based on the data they collected. I participated in this lab as a student in 2012 then observed and reviewed the work of other students involved. I found that students gained a deeper understanding of limiting factors and the role of …

Concurrent Neurological And Behavioral Assessment Of Number Line Estimation Performance In Children And Adults, Joseph Michael Baker

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Understanding the brain’s response to common math-learning activities may help improve math education. For example, by imaging the brains of typically developing children and adults throughout a number line estimation task, it is possible to establish a baseline of what “typical” brains do in such situations. Thus, comparisons may be made to determine the degree to which brain functioning differs between typical and atypical math learners. Moreover, by identifying methods that may increase the brain’s response to real-world math activities, it may be possible to improve the math learning process for typical and atypical learners alike.

Brain imaging devices such …

Dot Product Graphs And Their Applications To Ecology, Sean Bailey

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

We will introduce a new tool to visualize the comparison between different birds. This tool will allow users to use any number of measurable traits to see relationships between different birds, both individually and collectively.

Processing And Manipulation Of Data Collected From The Educational On-Line Game Refraction, Xiaotian Dai

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

A team of students, artists, and researchers at the Center for Game Science at the

University of Washington are trying to create video games that can discover optimal

pathways for learning. They have focused so far on early mathematics education,

including topics such as fractions and algebra, which are some of the main bottlenecks

preventing students from pursuing a career in science. As a result, the educational

on-line game \Refraction" was created, which is aimed at students who start learning

fraction computations. When the students are playing the game online, all the data

and information, such as mouse movements and …