Mathematics Commons^™

Generating Minimal Pair-Wise Covering Test Suites, Luis C. Gutierrez ^, Martine Ceberio *

COURI Symposium Abstracts, Spring 2012

Software is ubiquitous and needs to be reliable. Software testing therefore plays an important role in software development. Proper testing a software system informs about its quality and reliability so as to prevent unexpected behavior during system execution. One of the methods to prevent failures consists in testing a system under different input values, but when all possible input values are tested, an impractical number of test cases might result. In software testing, pair-wise testing is a combinatorial technique which uses combination of pair input values to generate test cases. Using pair-wise testing dramatically reduces the number of test cases, …

Simplicity Is Worse Than Theft: A Constraint-Based Explanation Of A Seemingly Counter-Intuitive Russian Saying, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, simplified models, models that enable us to gauge the quality of different decisions reasonably well, lead to far-from-optimal situations when used in searching for an optimal decision. There is even an appropriate Russian saying: simplicity is worse than theft. In this paper, we provide a mathematical explanation of this phenomenon.

Kinematic Spaces And De Vries Algebras: Towards Possible Physical Meaning Of De Vries Algebras, Olga Kosheleva, Francisco Zapata

Departmental Technical Reports (CS)

Traditionally, in physics, space-times are described by (pseudo-)Riemann spaces, i.e., by smooth manifolds with a tensor metric field. However, in several physically interesting situations smoothness is violated: near the Big Bang, at the black holes, and on the microlevel, when we take into account quantum effects. In all these situations, what remains is causality -- an ordering relation. To describe such situations, in the 1960s, geometers H. Busemann and R. Pimenov and physicists E. Kronheimer and R. Penrose developed a theory of kinematic spaces. Originally, kinematic spaces were formulated as topological ordered spaces, but it turned out that kinematic …

High-Order Central Finite-Volume Schemes For Atmospheric Modeling, Kiran Kumar Katta

Open Access Theses & Dissertations

Atmospheric numerical modeling has been going through drastic changes over the past decade, mainly to utilize the massive computing capability of the petascale systems. This obliges the modelers to develop grid systems and numerical algorithms that facilitate exceptional level of scalability on these systems. The numerical algorithms that can address these challenges should have the local properties such as the high on-processor operation count and minimum parallel communication i.e., high parallel efficiency. They should also satisfy the following properties such as inherent local and global conservation, high-order accuracy, geometric flexibility, non-oscillatory advection and positivity preservation properties. The goal of this …

Analysis Of Intermittence And Log-Periodicity Of Foreign Exchange Rates Near A Crash, Arturo Casillas

Open Access Theses & Dissertations

Many believe that financial indices near a crash exhibit a type of critical point characterized by log-periodic signatures. Models have been developed based on these ideas in an attempt to mathematically characterize financial data about to crash. Few of these models consider the property of intermittency. Intermittency is a concept borrowed from fluid dynamics that essentially implies that a system alternates between a stable, or predictable, state and unstable state. One model that attempts to characterize crashes incorporates intermittency in the form of log-stationary intervals. It models the asset price as a step function that follows an underlying power law. …

The Hammer-And-Nail Phenomenon In Mathematics Education, Kien H. Lim

Departmental Papers (Math)

"For a person with a hammer, everything looks like a nail" is a proverb that can be used to highlight the phenomenon that students tend to rely on familiar ideas as opposed to taking time to think about and analyse a problem. Presented in this theoretical paper is the usefulness of the hammer-and-nail metaphor, other related theoretical constructs, pedagogical causes of student impulsive behaviours, and pedagogical suggestions for addressing them.

An Immersed Interface Method For A 1d Poroelasticity Problem With Discontinuous Coefficients, Maranda Lee Bean

Open Access Theses & Dissertations

Poroelastic models deal with the coupling of changes in stress and fluid pressure in some porous medium. For physical reasons, some of the coefficients used in the models may have discontinuities. This project focuses on applying the immersed interface method to the one dimensional Biot model, in order to handle these discontinuities. This method is applied on a staggered grid. Using the immersed interface method allows us to alter only a small number of irregular grid points surrounding a discontinuity. For most of the grid points, a standard finite difference method is used. However at the irregular grid points, the …

Inverse Semigroups And Inverse Categories, Alexandra Macedo

Open Access Theses & Dissertations

The theory of inverse semigroups forms a major part of semigroup theory. This theory has deep connections with important mathematical disciplines, not only classical ones such as geometry, functional analysis, number theory, but also with more recent theories: the theory of algorithms, graph theory, the mathematical theory of automata, etc. The importance of inverse semigroups, first and foremost, is that they form an abstract class of algebraic structures which are isomorphic to semigroups of partial bijections.

This thesis is organized in two parts, inverse semigroups in Part 1, and inverse categories (that arises if we apply a basic property of …

The Topology Of Statistical Convergence, Khdiga Kalifa Tabib

Open Access Theses & Dissertations

A sequence {xn} is said to be statistically convergent to ℓ provided that "almost all" of the values of {xn} are arbitrarily close to ℓ. One can also define what is meant by statistical limit point, statistical limit superior, statistical limit inferior of a sequence and so forth and thus create a theory of convergence that includes ordinary convergence. In this work we investigate all these concepts and prove some new results. We also introduce a topology defined by this new convergence which we call statistical topology. Then we prove that both the statistical topology and the regular topology are …

A Multimodal Freight Collaborative Hub Location And Network Design Problem, Jiri Tylich

Open Access Theses & Dissertations

The study presents an analytical framework to explore the rail-road collaborative paradigm.

New collaborative technologies have been developed in recent years and they offer a potential solutions and opportunities for collaboration among all modes of transportation. The most progressive technologies that could fulfill the gap in rail-road collaborative paradigm are identified and presented in this research.

The research deals with current state and possible development of collaboration of rail and highway modes of transportation, referred to as rail-road collaboration. Multimodal transportation is the shipment of goods in a single transportation unit. The longest part of the route takes place by …

Lifting And Wavelets: A Factorization Method, Adrian Delgado

Open Access Theses & Dissertations

We begin with the concept of a discrete wavelet transformation. We begin with a scaling function satisfying a certain number of properties to be able to implement the wavelet transformation. We will then require translates of the scaling function to obey the same set of properties and this set will form what is called a Multiresolution Analysis. We then switch the ideas to the Fourier transform domain were we get equivalent results. However, instead of choosing a scaling function with coefficients to satisfy a set of properties, we will work backwards and construct a scaling function based on having a …

Comparative Investigation Of Cognitive And Pragmatic Thinking Modes Linear Algebra, Enayatollah Kalantarian

Open Access Theses & Dissertations

The analysis focused on the presence of different thinking modes found on the interview and responses to questions related to linear independence, span, and spanning sets of two students taking their first linear algebra course at the college level. The findings provide insight on how first year linear algebra students move from one thinking mode to another. The focal point of this research was to document and discern the similarities and differences between student thinking modes displayed on their interviews and class assignments.

Based on the analysis, it was concluded that the level of exposure of the two students to …

Assessing Impulsive-Analytic Disposition: The Likelihood-To-Act Survey And Other Instruments, Kien H. Lim, Amy E. Wagler

Departmental Papers (Math)

The likelihood-to-act (LtA) survey is a 32-item instrument that measures impulsive and analytic dispositions in solving math problems. In this research report, we compare it to other instruments related to the impulsive-analytic construct such as Frederick’s Cognitive Reflection Test (CRT) and the Barratt Impulsive Scale in terms of mean scores, Cronbach alpha values, and correlation values. Both LtA-Impulsive and LtA-Analytic subscales have acceptable reliabilities of 0.79 and 0.83 respectively. The LtA-Analytic and LtA-Difference (analytic-impulsive difference) correlated well with other the Need for Cognition subscale and CRT scores. The correlations involving LtA-Impulsive subscale were unexpected and call for further investigation.

Impulsive-Analytic Disposition In Mathematical Problem Solving: A Survey And A Mathematics Test, Kien H. Lim, Amy E. Wagler

Departmental Papers (Math)

The Likelihood-to-Act (LtA) survey and a mathematics test were used in this study to assess students’ impulsive-analytic disposition in the context of mathematical problem solving. The results obtained from these two instruments were compared to those obtained using two widely-used scales: Need for Cognition (NFC) and Barratt Impulsivity Scale (BIS). The exhibited correlations of the LtA scores with the NFC, BIS, and a math test provide evidence of the criterion validity of the analytic LtA items, and suggests further revision of the impulsive LtA items to improve the overall measurement validity of the LtA scale. Students LtA scores were found …

Dihedral Cayley Directed Strongly Regular Graphs, Jose Jonathan Gamez

Open Access Theses & Dissertations

A graph is a directed strongly regular graph (DSRG) if and only if the number of paths of length 2 from x to y is: λ, if there is an edge from x to y; μ, if there is not an edge from x to y (with x not equal to y); and t, if x = y. For every vertex in G, the in-degree and out-degree is k. The number of vertices in G is denoted by v. If G is a group and S a subset of G, then the …

Partial Orders For Representing Uncertainty, Causality And Decision Making: General Properties, Operations, And Algorithms, Francisco Adolfo Zapata

Open Access Theses & Dissertations

One of the main objectives of science and engineering is to help people select the most beneficial decisions. To make these decisions, we must know people's preferences, we must have the information about different possible consequences of different decisions. Since information is never absolutely accurate and precise, we must also have information about the degree of certainty of different parts on information. All these types of information naturally lead to partial orders:

- For preferences, a <= b means that b is preferable to a. This relation is used in decision theory.

- For events, a <= b means that a can influence b. This causality relation is one of the fundamental notions of physics, especially of physics of space-time.

* For uncertain statements, a <= b means that a is less certain than b. This relation is used in logics describing uncertainty, such as fuzzy logic.

In each of these areas, there is abundant research about studying the corresponding partial orders. …

Cognitive Constructs In Linear Algebra; Metaphors, Metonymies, Modes, Ruben Carrizales

Open Access Theses & Dissertations

Analysis focused on the presence of different thinking modes, metonymies, and metaphors found on the interview responses to questions related to linear independence, span, and spanning sets of three students, A12, A22 and C3, taking their first linear algebra course at the college level. Findings provide insight into how first year linear algebra students move from one thinking mode to another, and the kind of metonymies and metaphors are used to construct new knowledge. The main purpose of this research was to discover and analyze the presence of the different modes of thinking and metonymy/metaphors- in the reasoning of these …

The Isomorphisms Between The Upper And Lower Triangular Matrix Algebras, Zahi Fawaz

Open Access Theses & Dissertations

Since matrix equations with triangular matrices are easier to solve, the triangular matrices are very important in mathematics. For instance, the LU decomposition gives an algorithm to decompose any invertible matrix A into two triangular factors: a lower triangle matrix L and an upper triangle matrix U. Moreover, the inverse of a triangular matrix is also triangular. Also the product of two lower triangular matrices produces a lower triangular matrix, the same apply for upper triangular matrices. A lot of important notions, such as the determinant, the eigenvalue problem and many others are easy to handle when we are working …

A Characterization Of Directly Ordered Subspaces Of R^N, Jennifer Del Valle

Open Access Theses & Dissertations

In the finite dimensional ordered vector space Rⁿ, we consider the standard positive cone to be the set Rⁿ₊ = {x : x ≥ 0} for an element x in Rⁿ. Given a subspace V of Rⁿ, we define the positive cone of V to be the intersection of V with Rⁿ₊. The cone V₊ is said to be generating if V = V₊ − V₊, that is, if any vector v in V can be expressed as the difference of two vectors, v = …

Metaphors, Metonymies, Modes And Linear Algebra, Persis Samanta Beaven

Open Access Theses & Dissertations

The analysis focused on the presence of different thinking modes, metonymies, and metaphors found on the interview responses to questions related to linear independence, span, and spanning sets of four students taking their first linear algebra course at the college level. The findings provide insight of how first year linear algebra students move from one thinking mode to another and what kind of metonymies and metaphors are used to construct new knowledge. The main purpose of this research was to document and determine the main characteristics that categorize the four students' reasoning.

Optimal Control Applied To A Discrete Time Influenza Model, Paula Andrea Gonzalez Parra

Open Access Theses & Dissertations

For the last decades, mathematical epidemiological models have been used to understand the dynamics of infectious diseases and guide public health policy.

In particular, several continuous models have been considered to study influenza outbreaks and their controls policies. However, most epidemiological data is discrete; therefore, a discrete formulation is more convenient to compare collected data with the output of the model. We introduce a discrete time model in order to study optimal control strategies for influenza transmission.

In our model, we divide the population into four classes: susceptible, infectious, treated, and recovered individuals.

In particular, we evaluate the potential effect …

A Reduced Order Parameter Estimation Technique Using Orthonormal Wavelets, Miguel Hernandez Iv

Open Access Theses & Dissertations

In this work we introduce methods for model order reduction using orthonormal wavelets. Specifically, we propose techniques and algorithms for wavelet-based reduced order parameter estimation using orthonormal wavelets for solving nonlinear least squares problems. Approaches for parameter reduction using the one dimensional and two dimensional wavelet transforms are presented using multiple levels of decomposition.

The performance of wavelet-based reduced order parameter estimation is tested using a suite of orthonormal wavelets on a groundwater hydraulics model provided by the U.S. Army Corps of Engineers. Using the hydraulics model, the goal is to find a least-squares solution of the permeability field given …

Estimating Statistical Characteristics Under Interval Uncertainty And Constraints: Mean, Variance, Covariance, And Correlation, Ali Jalal-Kamali

Open Access Theses & Dissertations

In many practical situations, we have a sample of objects of a given type. When we measure the values of a certain quantity x for these objects, we get a sequence of values x1, . . . , xn. When the sample is large enough, then the arithmetic mean E of the values xi is a good approximation for the average value of this quantity for all the objects from this class. Other expressions provide a good approximation to statistical characteristics such as variance, covariance, and correlation.

The values xi come from measurements, and measurement is never absolutely accurate.

Often, …

Signature Matrices: The Eigenvalue Problem, Valeria Aguirre Holguin

Open Access Theses & Dissertations

Dealing with matrices can give us a hard time, especially when their dimension is too big, but we also well know how valuable information a matrix may carry, and that is why we study them. When a matrix has a significant number of zeroes we realize how much easier all calculations are. For instance, the product will be simpler to calculate, the determinant, the inverse and even the eigenvalue problem. This thesis provides the description and behavior of a very special kind of matrices which we call signature matrices. A particular feature of these matrices lies in the fact that …

Incidence Functions, Yiyu Liao

Open Access Theses & Dissertations

In the mid 1960's, the incidence algebra was introduced in the seminal paper of Gian-Carlo Rota. He addressed the importance of the Mobius function in combinatorics. In particular, the incidence algebra of a locally finite poset plays an essentially unifying role in the theory of the Mobius function. One of the significant generalizations is the incidence algebra of a Mobius category introduced by Pierre Leroux. With the help from Mobius category, it was exciting to be able to extend the combinatorial results more broadly than just on posets. Before attempting to study this generalization of the Mobius function, we have …

Use Of Cognitive Constructs In Linear Algebra, Azucena Zamora

Open Access Theses & Dissertations

Thesis analyzed the presence of two cognitive entities-- modes of thinking and metonymies and metaphors- in the reasoning of three students enrolled in a first year matrix algebra course at a southwest university via the responses given to a set of eight questions asked during one-on-one interviews. The findings of our analysis shed light on the cognitive constructs that students employ to form their understanding of linear algebra concepts. Furthermore, our findings provide clues about the ways in which students move from one mode of thinking to another in the context of varying levels of exposure to graphical, algebraic, and …

Examples Of Hyperbolic Knots With Distance 3 Toroidal Surgeries In The 3-Sphere, Cesar Garza

Open Access Theses & Dissertations

By the work of Thurston, any surgery on a hyperbolic knot in the 3-sphere produces a hyperbolic 3-manifold except in at most finitely many cases. So far, the figure-8 knot seems to be the best candidate for a hyperbolic knot with the most (8) non-trivial exceptional surgeries. In recent years, much progress has been made in the classification of hyperbolic knots admitting more than one exceptional toroidal surgery. In fact, such classification is known for toroidal surgeries with distance at least 4.

We give a classification of hyperbolic knots in $S^3$ admitting two toroidal surgeries at distance 3, whose slopes …

Exponential Dichotomy Of Ode's, Nada Farid Al-Hanna

Open Access Theses & Dissertations

I present an elementary Functional Analytic proof of the roughness of Exponential Dichotomy of Ordinary Differential Equations (with exponential growth) on an arbitrary Banach Space.

A Characterization Of Pseudo-Orders In The Ring Zn, Jorge Ivan Vargas

Open Access Theses & Dissertations

Partially ordered sets are widely studied in algebra. The theory of lattice-ordered and partially-ordered rings is an important case. In this paper, we consider a more general relation: pseudo orders, which are relations that are reflexive and antisymmetric, but are not necessarily transitive. Analogously to partially-ordered rings, we study pseudo-ordered rings. We present a general characterization for pseudo-ordered rings, and then we focus on a specific structure: the ring Zn. Having the constraints of this ring, we are able to give the conditions under which Zn can be pseudo-ordered.

Asymptotic Approximation Of The Free Boundary For The American Put Near Expiry, Walt E. Bales

Open Access Theses & Dissertations

An introduction to boundary value problems for the heat operator will focus on the Dirichlet and Neumann problems in bounded domains. Following an overview of the basic financial principles of derivative securities, a derivation of the Black-Scholes formulae will be given. Partial differential equations as free boundary problems will then be discussed and applied to modeling the American Put option. A method of solution close to expiry will then be investigated.