Physical Sciences and Mathematics Commons^™

On Some Differential Equations, Mekki Terbeche, Broderick O. Oluyede

Department of Mathematical Sciences Faculty Publications

This paper investigates Cauchy and Goursat problems for partial differential operators. Successive approximation techniques for partial differential equations and the estimated results are employed to obtain the existence and the uniqueness of the solutions of such problems. An extended Darboux-Goursat-Beudon problem is studied.

Combinatorics Of Ramanujan-Slater Type Identities, James Mclaughlin, Andrew Sills

Department of Mathematical Sciences Faculty Publications

We provide the missing member of a family of four q-series identities related to the modulus 36, the other members having been found by Ramanujan and Slater. We examine combinatorial implications of the identities in this family, and of some of the identities we considered in Identities of the Ramanujan-Slater type related to the moduli 18 and 24.

Authentic Discovery Projects In Statistics, Dianna J. Spence, Robb Sinn

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

We report the activities and findings of a 3-year project, ―Authentic, Career-Specific, Discovery Learning Projects in Introductory Statistics,‖ funded by the National Science Foundation. The project scope includes: 1) development of teaching materials for using discovery learning projects to teach statistics; 2) training secondary teachers to use the materials developed; 3) evaluation of student outcomes, in both content knowledge and attitudes toward statistics; and 4) extending and refining teacher training. With input from an interdisciplinary team of instructors, materials were developed to assist the teacher in facilitating collaborative discovery projects using linear regression techniques and comparison techniques with appropriate t-tests. …

First-Passage Time Models With A Stochastic Time Change In Credit Risk, Hui Li

Theses and Dissertations (Comprehensive)

Many authors have used a time-changed Brownian motion as a model of log-stock returns. Using a Levy process as a stochastic time change, one obtains well known asset price models such as the variance gamma (VG) and normal inverse Gaussian (NIG) models. Following on the heels of these asset price models, it is natural to extend structural credit models by using a time-changed geometric Brownian motion and other jump-diffusion processes to model the value of a firm. To avoid the difficulties that arise in computing the associated first passage time distribution and in analogy to the time-changed Markov chain models, …

A New Elasticity Element Made For Enforcing Weak Stress Symmetry, Bernardo Cockburn, Jay Gopalakrishnan, Johnny Guzmán

Mathematics and Statistics Faculty Publications and Presentations

We introduce a new mixed method for linear elasticity. The novelty is a simplicial element for the approximate stress. For every positive integer k, the row-wise divergence of the element space spans the set of polynomials of total degree k. The degrees of freedom are suited to achieve continuity of the normal stresses. What makes the element distinctive is that its dimension is the smallest required for enforcing a weak symmetry condition on the approximate stress. This is achieved using certain "bubble matrices", which are special divergence-free matrix-valued polynomials. We prove that the approximation error is of order k + …

Polynomial Extension Operators. Part Ii, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

Mathematics and Statistics Faculty Publications and Presentations

Consider the tangential trace of a vector polynomial on the surface of a tetrahedron. We construct an extension operator that extends such a trace function into a polynomial on the tetrahedron. This operator can be continuously extended to the trace space of H(curl ). Furthermore, it satisfies a commutativity property with an extension operator we constructed in Part I of this series. Such extensions are a fundamental ingredient of high order finite element analysis.

The Derivation Of Hybridizable Discontinuous Galerkin Methods For Stokes Flow, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we introduce a new class of discontinuous Galerkin methods for the Stokes equations. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries. We present four ways of hybridizing the methods, which differ by the choice of the globally coupled unknowns. Classical methods for the Stokes equations can be thought of as limiting cases of these new methods.

Unified Hybridization Of Discontinuous Galerkin, Mixed, And Continuous Galerkin Methods For Second Order Elliptic Problems, Bernardo Cockburn, Jay Gopalakrishnan, Raytcho Lazarov

Mathematics and Statistics Faculty Publications and Presentations

We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continu- ous Galerkin, non-conforming and a new, wide class of hybridizable discontinuous Galerkin methods. The distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements. Since the associated matrix is sparse, symmetric and positive definite, these methods can be efficiently implemented. Moreover, the …

Transformée En Échelle De Signaux Stationnaires, Daniel Alpay, Mamadou Mboup

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using the scale transform of a discrete time signal we define a new family of linear systems. We focus on a particular case related to function theory in the bidisk.

Generalized Q-Functions And Dirichlet-To-Neumann Maps For Elliptic Differential Operators, Daniel Alpay, Jussi Behrndt

Mathematics, Physics, and Computer Science Faculty Articles and Research

The classical concept of Q-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized Q-function. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein type formulas for the difference of the resolvents and trace formulas in an H2-framework are obtained.

The Schur Transformation For Nevanlinna Functions: Operator Representations, Resolvent Matrices, And Orthogonal Polynomials, Daniel Alpay, A. Dijksma, H. Langer

Mathematics, Physics, and Computer Science Faculty Articles and Research

A Nevanlinna function is a function which is analytic in the open upper half plane and has a non-negative imaginary part there. In this paper we study a fractional linear transformation for a Nevanlinna function n with a suitable asymptotic expansion at ∞, that is an analogue of the Schur transformation for contractive analytic functions in the unit disc. Applying the transformation p times we find a Nevanlinna function np which is a fractional linear transformation of the given function n. The main results concern the effect of this transformation to the realizations of n and np, by which we …

Krein Systems, Daniel Alpay, I. Gohberg, M. A. Kaashoek, L. Lerer, A. Sakhnovich

Mathematics, Physics, and Computer Science Faculty Articles and Research

In the present paper we extend results of M.G. Krein associated to the spectral problem for Krein systems to systems with matrix valued accelerants with a possible jump discontinuity at the origin. Explicit formulas for the accelerant are given in terms of the matrizant of the system in question. Recent developments in the theory of continuous analogs of the resultant operator play an essential role.

Content Knowledge And Pedagogical Content Knowledge Of Algebra Teachers And Changes In Both Types Of Knowledge As A Result Of Professional Development, Joy W. Black

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

In seeking to improve the mathematics education of all students, it is important to understand the connection between the content knowledge and pedagogical content knowledge of mathematics and how professionals can influence growth in both of these types of knowledge. We do not have an answer about the interplay of content knowledge and pedagogical content knowledge in successful instructional practices in the mathematics classroom. This study involves assessing the content knowledge and pedagogical content knowledge of secondary teachers of Algebra I. In addition, how are these types of knowledge expressed in instructional practices? Last, how do content knowledge, pedagogical content …

Analysis Of Achievement For Understanding Geometry, Annita W. Hunt

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

The purpose of this study was to investigate the effectiveness of a mathematics professional development course. More specifically, in this study we examine whether geometric experiences have an impact on level of performance in mathematics. The van Hiele (Fuys, D., Geddes, D., & Tischler, R., 1988) model of geometric understanding provided a research framework from which to view geometric understanding. This model suggests five levels of understanding that should be taken into consideration when examining levels of geometric thinking: Visual, Descriptive/Analytic, Abstract/Relational, Formal Deduction/Proof, and Rigor. The sample under study was three cohorts of practicing elementary teachers and mathematics coaches …

Proceedings Of The Third Annual Meeting Of The Georgia Association Of Mathematics Teacher Educators Front Matter

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Contents of 3rd Annual GAMTE Proceedings Front Matter:

• Proceedings Committee
• Officers of GAMTE
• Purposes and Goals of GAMTE
• Table of Contents
• Letter from President

Gas-Kinetic Schemes For Direct Numerical Simulations Of Compressible Homogeneous Turbulence, Wei Liao, Yan Peng, Li-Shi Luo

Mathematics & Statistics Faculty Publications

We apply the gas-kinetic scheme (GKS) for the direct numerical simulations (DNSs) of compressible decaying homogeneous isotropic turbulence (DHIT). We intend to study the accuracy, stability, and efficiency of the gas-kinetic scheme for DNS of compressible homogeneous turbulence depending on both flow conditions and numerics. In particular, we study the GKS with multidimensional, quasi-one-dimensional, dimensional-splitting, and smooth-flow approximations. We simulate the compressible DHIT with the Taylor microscale Reynolds number Reλ =72.0 and the turbulence Mach number Ma_t between 0.1 and 0.6. We compute the low-order statistical quantities including the total kinetic energy K (t), the dissipation rate ε (t), …

Third Grade Students' Challenges And Strategies To Solving Mathematical Word Problems, Elizabeth Bernadette

Open Access Theses & Dissertations

This project explores the difficulties and challenges that third grade students face solving mathematical word problems. Three students were asked to be participants and share their knowledge on this topic as well as their work. After extensive interviews it was concluded that the challenges of mathematical word problems include but are not limited to the level of reading comprehension, conceptual understanding of mathematical concepts, and the belief that math is a compilation of computations and unexplainable procedures. The participants provided insight as to what strategies are helpful to students. These strategies include group discussion on problem solving strategies, self-assessment, incorporating …

Multiplicative Riesz Decomposition On The Ring Of Matrices Over A Totally Ordered Field, Julio Cesar Urenda

Open Access Theses & Dissertations

The Riesz Decomposition Theorem for lattice ordered groups asserts that when G is an l-group and when a nonnegative element a is bounded by a product of nonnegative elements b1,...,bn, then a can be decomposed into a product of nonnegative elements b'1,...,b'n, i.e., a = b'1·...·b' n, with the property that b'i ≤ bi for any i = 1,...,n. In this work we characterize all nonnegative matrices for which this decomposition is possible with respect to matrix multiplication. In addition, we show that this result can be applied to ordered semigroups.

Toric Surface Codes And Minkowski Length Of Polygons, Ivan Soprunov, Jenya Soprunova

Mathematics and Statistics Faculty Publications

In this paper we prove new lower bounds for the minimum distance of a toric surface code CP defined by a convex lattice polygon P⊂R2. The bounds involve a geometric invariant L(P), called the full Minkowski length of P. We also show how to compute L(P) in polynomial time in the number of lattice points in P.

The Maximum Of The Maximum Rectilinear Crossing Numbers Of D-Regular Graphs Of Order N, Matthew Alpert, Elie Feder, Heiko Harborth

Publications and Research

We extend known results regarding the maximum rectilinear crossing number of the cycle graph (Cn) and the complete graph (Kn ) to the class of general d-regular graphs Rn,d. We present the generalized star drawings of the d-regular graphs Sn,d of order n where n + d ≡ 1 (mod 2) and prove that they maximize the maximum rectilinear crossing numbers. A star-like drawing of Sn,d for n ≡ d ≡ 0 (mod 2) is introduced and we conjecture that this drawing maximizes the maximum rectilinear crossing numbers, too. We offer a simpler proof of two results initially proved by …

The Orchard Crossing Number Of An Abstract Graph, Elie Feder, David Garber

Publications and Research

.We introduce the Orchard crossing number, which is defined in a similar way to the well-known rectilinear crossing number. We compute the Orchard crossing number for some simple families of graphs. We also prove some properties of this crossing number.

Moreover, we define a variant of this crossing number which is tightly connected to the rectilinear crossing number, and compute it for some simple families of graphs.

Mathematical Modeling Of Quantum Dots With Generalized Envelope Functions Approximations And Coupled Partial Differential Equations, Dmytro Sytnyk

Theses and Dissertations (Comprehensive)

No abstract provided.

A Logistic Approximation To The Cumulative Normal Distribution, Shannon R. Bowling, Mohammad T. Khasawneh, Sittichai Kaewkuekool, Byung R. Cho

Engineering Management & Systems Engineering Faculty Publications

This paper develops a logistic approximation to the cumulative normal distribution. Although the literature contains a vast collection of approximate functions for the normal distribution, they are very complicated, not very accurate, or valid for only a limited range. This paper proposes an enhanced approximate function. When comparing the proposed function to other approximations studied in the literature, it can be observed that the proposed logistic approximation has a simpler functional form and that it gives higher accuracy, with the maximum error of less than 0.00014 for the entire range. This is, to the best of the authors’ knowledge, the …

Simulation Studies On Estimation Of Variance Components For Multilevel Models, Sara Vakilian

Theses and Dissertations (Comprehensive)

With the presence of unequal sampling in a multilevel model, the weight inflated estimators for variance components can be biased even though the use of survey weights results in design consistent estimators of the parameters. In this thesis I will carry out a simulation study to examine the performance of current existing methods and I will examine the resampling method for correcting bias of estimators of variance components of a multilevel model with covariates. This study will be based on these three papers: “Weighting for Unequal Selection Probabilities in Multilevel Models” by D. Pfeffermann , C. J. Skinner, D.J. Holmes, …

Applications Of New Diffusion Models To Barrier Option Pricing And First Hitting Time In Finance, Keang Ly

Theses and Dissertations (Comprehensive)

The main focus of this thesis is in the application of a new family of analytical solvable diffusion models to arbitrage-free pricing exotic financial derivatives, such as barrier options. The family of diffusions is the so-called “Drifted Bessel family” having nonlinear (smile-like) local volatility with multiple adjustable parameters. In particular, the drifted Bessel-K diffusion is used to model asset (stock) price processes under a risk-neutral measure whereby discounted asset price are martingales.

Closed-form spectral expansions for barrier option values are derived within the Bessel-K family of models. This follow from the closed-form spectral expansions for the transition probability …

Dynamics And Rheology Of Biaxial Liquid Crystal Polymers, Sarthok K. Sircar

Theses and Dissertations

In this thesis we derive a hydrodynamical kinetic theory to study the orientational response of a mesoscopic system of nematic liquid crystals in the presence of an external flow field. Various problems have been attempted in this direction. First, we understand the steady-state behavior of uniaxial LCPs under an imposed elongational flow, electric and magnetic field respectively. We show that (1) the Smoluchowski equation can be cast into a generic form, (2) the external field is parallel to one of the eigenvectors of the second moment tensor, and (3) the steady state probability density function is of the Boltzmann type. …

Strict-Dominance Solvability Of Games On Continuous Strategy Spaces, Andrew Campbell Elkington

Theses and Dissertations (Comprehensive)

The concept of strict dominance provides a technique that can be used normatively to predict the play of games based only on the assumption of individual rationality. Such predictions, unlike those based on Nash equilibria, do not depend on players’ beliefs about the behaviour of others. One strategy strictly dominates another if and only if the payoff from the first strategy is strictly greater than the payoff from the second, no matter how the opponent(s) plays. It is possible for iterated elimination of strictly dominated strategies to remove all but a single choice for each player, in which case we …

Models For On-Line Social Networks, Noor Hadi

Theses and Dissertations (Comprehensive)

On-line social networks such as Facebook or Myspace are of increasing interest to computer scientists, mathematicians, and social scientists alike. In such real-world networks, nodes represent people and edges represent friendships between them. Mathematical models have been proposed for a variety of complex real-world networks such as the web graph, but relatively few models exist for on-line social networks.

We present two new models for on-line social networks: a deterministic model we call Iterated Local Transitivity (ILT), and a random ILT model. We study various properties in the deterministic ILT model such as average degree, average distance, and diameter. We …

Complementary Responses To Mean And Variance Modulations In The Perfect Integrate-And-Fire Model, Joanna R. Wares, Todd W. Troyer

Department of Math & Statistics Faculty Publications

In the perfect integrate-and-fire model (PIF), the membrane voltage is proportional to the integral of the input current since the time of the previous spike. It has been shown that the firing rate within a noise free ensemble of PIF neurons responds instantaneously to dynamic changes in the input current, whereas in the presence of white noise, model neurons preferentially pass low frequency modulations of the mean current. Here, we prove that when the input variance is perturbed while holding the mean current constant, the PIF responds preferentially to high frequency modulations. Moreover, the linear filters for mean and variance …

Extending The Skill Test For Disease Diagnosis, Shu-Chuan Lin, Paul H. Kvam, Jye-Chyi Lu

Department of Math & Statistics Faculty Publications

For diagnostic tests, we present an extension to the skill plot introduced by Briggs and Zaretski (Biometrics 2008; 64:250–261). The method is motivated by diagnostic measures for osteopetrosis in a study summarized by Hans et al. (The Lancet 1996; 348:511–514). Diagnostic test accuracy is typically defined using the area (or partial area) under the receiver operator characteristic (ROC) curve. If partial area is used, the resulting statistic can be highly subjective because the focus region of the ROC curve corresponds to a set of low false‐positive rates that are chosen by the experimenter. This paper introduces a more …