Mathematics Commons^™

Dynamical Invariants And The Fluid Impulse In Plasma Models, Martin Michalak

Electronic Theses and Dissertations

Much progress has been made in understanding of plasmas through the use of the MHD equations and newer models such as Hall MHD and electron MHD. As with most equations of fluid behavior, these equations are nonlinear, and no general solutions can be found. The use of invariant structures allows limited predictions of fluid behavior without requiring a full solution of the underlying equations. The use of gauge transformation can allow the creation of new invariants, while differential geometry offers useful tools for constructing additional invariants from those that are already known. Using these techniques, new geometric, integral and topological …

Creating A User Satisfaction Index From A Parsimonious Survey Instrument, Brian Barthel

All Graduate Theses, Dissertations, and Other Capstone Projects

In this paper we present a comprehensive method for creating a user satisfaction index using a survey instrument. First we construct a parsimonious survey instrument, using the PageRank Centrality, to measure attributes of user satisfaction. Then confirmatory factor analysis is applied to extract ``weights'' on the questions that are used in a linear model of computing the user satisfaction index. Throughout the paper an analysis of an existing data set is implemented to illustrate the proposed method. In addition the validity of the confirmatory factor model is tested using bootstrap sampling.

Dynamic Sequent Calculus For The Logic Of Epistemic Actions And Knowledge, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

"Dynamic Logics (DLs) form a large family of nonclassical logics, and perhaps the one enjoying the widest range of applications. Indeed, they are designed to formalize change caused by actions of diverse nature: updates on the memory state of a computer, displacements of moving robots in an environment, measurements in models of quantum physics, belief revisions, knowledge updates, etc. In each of these areas, DL-formulas express properties of the model encoding the present state of affairs, as well as the pre- and post-conditions of a given action. Actions are semantically represented as transformations of one model into another, encoding the …

Uniform Gaussian Bounds For Subelliptic Heat Kernels And An Application To The Total Variation Flow Of Graphs Over Carnot Groups, Luca Capogna, Giovanna Citti, Maria Manfredini

Mathematics Sciences: Faculty Publications

In this paper we study heat kernels associated with a Carnot group G, endowed with a family of collapsing left-invariant Riemannian metrics σϵ which converge in the Gromov- Hausdorff sense to a sub-Riemannian structure on G as ϵ rarr; 0. The main new contribution are Gaussian-Type bounds on the heat kernel for the σϵ metrics which are stable as ϵ rarr; 0 and extend the previous time-independent estimates in [16]. As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in a step two Carnot group (G; σϵ ). We establish …

Billey's Formula In Combinatorics, Geometry, And Topology, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

In this expository paper we describe a powerful combinatorial formula and its implications in geometry, topology, and algebra. This formula first appeared in the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey discovered it independently five years later, and it played a prominent role in her work to evaluate certain polynomials closely related to Schubert polynomials. Billey's formula relates many pieces of Schubert calculus: the geometry of Schubert varieties, the action of the torus on the flag variety, combinatorial data about permutations, the cohomology of the flag variety and of the Schubert varieties, and the combinatorics of …

A Generalization Of The Turaev Cobracket And The Minimal Self-Intersection Number Of A Curve On A Surface, Patricia Cahn

Mathematics Sciences: Faculty Publications

Goldman and Turaev constructed a Lie bialgebra structure on the free Zmodule generated by free homotopy classes of loops on a surface. Turaev conjectured that his cobracket ∆(α) is zero if and only if α is a power of a simple class. Chas constructed examples that show Turaev’s conjecture is, unfortunately, false. We define an operation µ in the spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams. The Turaev cobracket factors through µ, so we can view µ as a generalization of ∆. We show that Turaev’s conjecture holds when ∆ is replaced with µ. We also show that µ(α) …

Intersections Of Loops And The Andersen–Mattes–Reshetikhin Algebra, Patricia Cahn, Vladimir Chernov

Mathematics Sciences: Faculty Publications

Given two free homotopy classes α₁,α₂ of loops on an oriented surface, it is natural to ask how to compute the minimum number of intersection points # (α₁,α₂) of loops in these two classes.

We show that, for α₁≠α₂, the number of terms in the Andersen–Mattes–Reshetikhin Poisson bracket of α₁ and α₂ is equal to # (α₁,α₂). Chas found examples showing that a similar statement does not, in general, hold for the Goldman Lie bracket of α₁ and α₂.

The …

A Convex Optimization Algorithm For Sparse Representation And Applications In Classification Problems, Reinaldo Sanchez Arias

Open Access Theses & Dissertations

In pattern recognition and machine learning, a classification problem refers to finding an algorithm for assigning a given input data into one of several categories. Many natural signals are sparse or compressible in the sense that they have short representations when expressed in a suitable basis. Motivated by the recent successful development of algorithms for sparse signal recovery, we apply the selective nature of sparse representation to perform classification. Any test sample is represented in an overcomplete dictionary with the training sample as base elements. A given test sample can be expressed as a linear combination of only those training …

Generalizations Of Dirichlet Convolution, Juan Carlos Villarreal

Open Access Theses & Dissertations

The thesis reviews Dirichlet convolution of arithmetic functions and some generalizations. First, we present the main properties of the Dirichlet convolution on the set of arithmetic functions and some characterizations of prime numbers via arithmetic functions are given. Then we explore the K-convolution and the regular convolutions and finally we extend the study of Dirichlet convolution in categories. We give also an example for the evaluation of the Mobius function of a Mobius category.

Introduction To International Perspectives On Problem Solving Research In Mathematics Education, Luis Moreno Armella, Manuel Santos-Trigo

The Mathematics Enthusiast

Any field of research and innovation must be exposed to revisions, criticisms and to an intense scrutiny not only to discuss the state of the art but, hopefully, to identify prospective changes and new areas of study and exploration as well.

Reflections On Problem Solving Theory And Practice, Alan H. Schoenfeld

The Mathematics Enthusiast

In this article, the author reflects on the current state of mathematical problem solving, both in theory and in instruction. The impact of the book Mathematical Problem solving (Schoenfeld, 1985) is also discussed, along with implications of problem solving today with the advent of 21st century technologies.

Cognitive Processes Developed By Students When Solving Mathematical Problems Within Technological Environments, Fernando Barrera-Mora, Aarón Reyes-Rodríguez

The Mathematics Enthusiast

In this paper we document and discuss how the use of digital technologies in problem solving activities can help students to develop mathematical competences; particularly, we analyze the characteristics of reasoning that students develop as a result of using Cabri Geometry software in problem solving. We argue that the dynamical nature of representations constructed with Cabri, and the availability of measure tools integrated to it are important elements that enhance students’ ability to think mathematically and foster the implementation of several heuristic strategies in problem solving processes.

Mathematical Problem Solving In Training Elementary Teachers From A Semiotic Logical Approach, Martín Socas, Josefa Hernández

The Mathematics Enthusiast

The aim of this article is to consider the professional knowledge and competences of mathematics teachers in compulsory education, and to propose basic tasks and activities in an initial training programme in the framework of a global proposal for “Immersion” in the curriculum of the educational phase which the trainee teacher would go on to work in. Problem-solving, in this context, is considered as being an inherent part of mathematics and this is described in terms of problem-solving, establishing connections between concepts, operations and implicit processes in the mathematical activity (conceptual field) and their relationships problem-solving; and it is assumed …

Becoming Aware Of Mathematical Gaps In New Curricular Materials: A Resource-Based Analysis Of Teaching Practice, José Guzman, Carolyn Kieran

The Mathematics Enthusiast

The study featured in this article, with its central focus on resources-in-use, draws upon salient aspects of the documentational approach of didactics. It includes an a priori analysis of the curricular resources being used by a teacher for the first time, followed by detailed in situ observations of the unfolding of her teaching practice involving these resources. The central mathematical problem of the lesson being analyzed deals with families of polynomial functions. The analysis highlights the teacher’s growing awareness of the mathematical gaps in the resources she is using, which we conjecture to be a first step for her in …

Developing Problem Solving Experiences In Practical Action Projects, François Pluvinage

The Mathematics Enthusiast

Problem solving doubtless is an essential element of mathematical learning, so that mathematics educators often are satisfied when finding situations that lead their students to such activity. But in many cases, the chosen situations and the ways to guide students' works are not sufficiently analyzed from a didactic point of view. Our goal in the present analysis is to underline the possible ways for managing the situations, and to exhibit the parameters that educators have at their disposition within their role as mediator between students and mathematical knowledge and know-how.

Framing The Use Of Computational Technology In Problem Solving Approaches, Manuel Santos-Trigo, Matías Camacho Machín

The Mathematics Enthusiast

Mathematical tasks are key ingredient to foster teachers and students’ development and construction of mathematical thinking. The use of distinct computational tools offers teachers a variety of ways to represent and explore mathematical tasks which often extends problem solving approaches based on the use of paper and pencil. We sketch a framework to characterize ways of reasoning that emerge as result of using computational technology to solve a task that involves dealing with variation phenomena.

Thoughts About Research On Mathematical Problem- Solving Instruction, Frank K. Lester Jr.

The Mathematics Enthusiast

In this article, the author, who has written extensively about mathematical problem solving over the past 40 years, discusses some of his current thinking about the nature of problem-solving and its relation to other forms of mathematical activity. He also suggests several proficiencies teachers should acquire in order for them to be successful in helping students become better problem solvers and presents a framework for research on problem-solving instruction. He closes the article with a list of principles about problem-solving instruction that have emerged since the early 1970s.

Developing The Art Of Seeing The Easy When Solving Problems, Alfinio Flores, Jaclyn Braker

The Mathematics Enthusiast

For Leonardo da Vinci “saper vedere”, that is, knowing how to see, or having the art to see, was the key to unlocking the secrets of the visible world. Saper vedere included a precise sensory intuitive faculty as well as artistic imagination (Heydenreich, 1954) which were at the root of Leonardo’s inventiveness and creativity. According to Leonardo, to understand, you only have to see things properly (Bramly 1994, p. 264). Knowing how to see is also important in mathematics. The Italian mathematician Bruno de Finetti (1967) stresses this importance in his book on “Saper vedere” in mathematics. He highlights several …

Spectral Analysis Of Randomly Generated Networks With Prescribed Degree Sequences, Clifford Davis Gaddy

Theses and Dissertations

Network science attempts to capture real-world phenomenon through mathematical models. The underlying model of a network relies on a mathematical structure called a graph. Having seen its early beginnings in the 1950's, the field has seen a surge of interest over the last two decades, attracting interest from a range of scientists including computer scientists, sociologists, biologists, physicists, and mathematicians. The field requires a delicate interplay between real-world modeling and theory, as it must develop accurate probabilistic models and then study these models from a mathematical perspective. In my thesis, we undertake a project involving computer programming in which we …

Applications Of The Lopsided Lovász Local Lemma Regarding Hypergraphs, Austin Tyler Mohr

Theses and Dissertations

The Lovász local lemma is a powerful and well-studied probabilistic technique useful in establishing the possibility of simultaneously avoiding every event in some collection. A principle limitation of the lemma's application is that it requires most events to be independent of one another. The lopsided local lemma relaxes the requirement of independence to negative dependence, which is more general but also more difficult to identify. We will examine general classes of negative dependent events involving maximal matchings of uniform hypergraphs, partitions of sets, and spanning trees of complete graphs. The results on hypergraph matchings (together with the configuration model of …

Coloring Pythagorean Triples And A Problem Concerning Cyclotomic Polynomials, Daniel White

Theses and Dissertations

One may easily show that there exist $O( \log n)$-colorings of $\{1,2, \ldots, n\}$ such that no Pythagorean triple with elements $\le n$ is monochromatic. In Chapter~\ref{CH:triples}, we investigate two analogous ideas. First, we find an asymptotic bound for the number of colors required to color $\{1,2,\ldots ,n\}$ so that every Pythagorean triple with elements $\le n$ is $3$-colored. Afterwards, we examine the case where we allow a vanishing proportion of Pythagorean triples with elements $\le n$ to fail to have this property.

Unrelated, in 1908, Schur raised the question of the irreducibility over $\Q$ of polynomials of the form …

Deducing Vertex Weights From Empirical Occupation Times, David Collins

Theses and Dissertations

We consider the following problem arising from the study of human problem solving: Let $G$ be a vertex-weighted digraph with marked "start" and "end" vertices. Suppose that a random walker begins at the start vertex, steps to neighbors of vertices with probability proportional to their weights, and stops upon reaching the end vertex. Could one deduce the weights from the paths that many such walkers take? We analyze an iterative numerical solution to this reconstruction problem, in particular, given the empirical mean occupation times of the walkers. We then consider the existence of a choice of weights for a given …

Selected Research In Covering Systems Of The Integers And The Factorization Of Polynomials, Joshua Harrington

Theses and Dissertations

In 1960, Sierpi\'{n}ski proved that there exist infinitely many odd positive integers $k$ such that $k\cdot 2^n+1$ is composite for all positive integers $n$. Such integers are known as Sierpi\'{n}ski numbers. Letting $f(x)=ax^r+bx+c\in\mathbb{Z}[x]$, Chapter 2 of this document explores the existence of integers $k$ such that $f(k)2^n+d$ is composite for all positive integers $n$. Chapter 3 then looks into a polynomial variation of a similar question. In particular, Chapter~\ref{CH:FH} addresses the question, for what integers $d$ does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ with $f(1)\neq -d$ such that $f(x)x^n+d$ is reducible for all positive integers $n$. The last two chapters of …

Generalizations Of Sperner's Theorem: Packing Posets, Families Forbidding Posets, And Supersaturation, Andrew Philip Dove

Theses and Dissertations

Sperner's Theorem is a well known theorem in extremal set theory that gives the size of the largest antichain in the poset that is the Boolean lattice. This is equivalent to finding the largest family of subsets of an $n$-set, $[n]:=\{1,2,\dots,n\}$, such that the family is constructed from pairwise unrelated copies of the single element poset. For a poset $P$, we are interested in maximizing the size of a family $\mathcal{F}$ of subsets of $[n]$, where each maximally connected component of $\mathcal{F}$ is a copy of $P$, and finding the extreme configurations that achieve this value. For instance, Sperner showed …

The Compact Implicit Integration Factor Scheme For The Solution Of Allen-Cahn Equations, Meshack K. Kiplagat

Theses and Dissertations

In this thesis we apply the compact implicit integration factor (cIIF) scheme towards solving the Allen-Cahn equations with zero-flux or periodic boundary conditions. The Allen-Cahn equation is a second-order nonlinear PDE which has been the focus of many applications spanning a wide range of fields, such as in material science where it was first introduced to model the phase separation of two metallic alloys, and in biology to study population dynamics, just to name a few. The compact implicit integration method works by first transforming the PDE into a system of ODEs by discretizing the spatial derivatives using the central …

Optimization In Non-Parametric Survival Analysis And Climate Change Modeling, Iuliana Teodorescu

USF Tampa Graduate Theses and Dissertations

Many of the open problems of current interest in probability and statistics involve complicated data

sets that do not satisfy the strong assumptions of being independent and identically distributed. Often,

the samples are known only empirically, and making assumptions about underlying parametric

distributions is not warranted by the insufficient information available. Under such circumstances,

the usual Fisher or parametric Bayes approaches cannot be used to model the data or make predictions.

However, this situation is quite often encountered in some of the main challenges facing statistical,

data-driven studies of climate change, clinical studies, or financial markets, to name a few. …

Nonlinear Techniques For Stochastic Systems Of Differential Equations, Tadesse G. Zerihun

USF Tampa Graduate Theses and Dissertations

Two of the most well-known nonlinear methods for investigating nonlinear dynamic processes in sciences and engineering are nonlinear variation of constants parameters and comparison method. Knowing the existence of solution process, these methods provide a very powerful tools for investigating variety of problems, for example, qualitative and quantitative properties of solutions, finding error estimates between solution processes of stochastic system and the corresponding nominal system, and inputs for the designing engineering and industrial problems. The aim of this work is to systematically develop mathematical tools to undertake the mathematical frame-work to investigate a complex nonlinear nonstationary stochastic systems of differential …

G-Strands And Peakon Collisions On Diff(R), Darryl Holm, Rossen Ivanov

Articles

A G-strand is a map g : R x R --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. Some G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that G-strands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of G-strands when G = Diff( …

The Gordon Game, Anthony Frenk

EWU Masters Thesis Collection

In 1992, about 30 years after Gordon introduced group sequencings to construct row-complete Latin squares, John Isbell introduced the idea of competitive sequencing, the Gordon Game. Isbell investigated the Gordon Game and found solutions for groups of small order. The purpose of this thesis is to analyze the Gordon Game and develop a brute force method of determining solutions to the game for all groups of order 12 (up to isomorphism) as well as for abelian groups of order less than 20. The method used will be a depth first search program written in MATLAB. Consequently, group representation using matrices …

On Characterizations Of Exponential Distribution Through Order Statistics And Record Values With Random Shifts, M. Ahsanullah, Imtiyaz A. Shah, George Yanev

School of Mathematical & Statistical Sciences Faculty Publications

Distributional relations of the form Y d= X +T where X, Y, and T are record values or order statistics and the random translator T is independent from X are considered. Characterizations of the exponential distribution when the ordered random variables are non-neighboring are proved. Corollaries for Pareto and power function distributions are also derived.