Mathematics Commons^™

Lipschitz Continuity Of The Best Approximation Operator In Vector-Valued Chebyshev Approximation, Martin Bartelt, John Swetits

Mathematics & Statistics Faculty Publications

When G is a finite dimensional Haar subspace of C(X, R^k), the vector-valued continuous functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X, R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (H˝older) condition of order 1/2. This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1.

Algebraic Properties Of Edge Ideals, Rachelle R. Bouchat

University of Kentucky Doctoral Dissertations

Given a simple graph G, the corresponding edge ideal I_G is the ideal generated by the edges of G. In 2007, Ha and Van Tuyl demonstrated an inductive procedure to construct the minimal free resolution of certain classes of edge ideals. We will provide a simplified proof of this inductive method for the class of trees. Furthermore, we will provide a comprehensive description of the finely graded Betti numbers occurring in the minimal free resolution of the edge ideal of a tree. For specific subclasses of trees, we will generate more precise information including explicit formulas for …

Proceedings Of The Second 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 2nd Annual GAMTE Proceedings Front Matter:

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

Fixed-Effect Estimation Of Highly-Mobile Production Technologies, William Clinton Horrace, Kurt E. Schnier

Center for Policy Research

Revised from November 2006 and July 2007. We consider fixed-effect estimation of a production function where inputs and outputs vary over time, space, and cross-sectional unit. Variability in the spatial dimension allows for time-varying individual effects, without parametric assumptions on the effects. Asymptotics along the spatial dimension provide consistency and normality of the marginal products. A finite-sample example is provided: a production function for bottom-trawler fishing vessels in the flatfish fisheries of the Bering Sea. We find significant spatial variability of output (catch) which we exploit in estimation of a harvesting function.

Semiparametric Deconvolution With Unknown Error Variance, William C. Horrace, Christopher F. Parmeter

Center for Policy Research

Deconvolution is a useful statistical technique for recovering an unknown density in the presence of measurement error. Typically, the method hinges on stringent assumptions about the nature of the measurement error, more specifically, that the distribution is *entirely* known. We relax this assumption in the context of a regression error component model and develop an estimator for the unknown density. We show semi-uniform consistency of the estimator and provide Monte Carlo evidence that demonstrates the merits of the method.

A Manifold Structure For The Group Of Orbifold Diffeomorphisms Of A Smooth Orbifold, Joseph E. Borzellino, Victor Brunsden

Mathematics

For a compact, smooth C^r orbifold (without boundary), we show that the topological structure of the orbifold diffeomorphism group is a Banach manifold for 1 ≤ r < ∞ and a Fréchet manifold if r = ∞. In each case, the local model is the separable Banach (Fréchet) space of C^r (C ^∞, resp.) orbisections of the tangent orbibundle.

A Fundamental Region On Two Copies Of The Hyperbolic Plane, Alyssa Jesse Soenksen

Honors Program Theses

Hyperbolic geometry is a beautiful, non-Euclidean space that hosts spectacular patterns and infinite designs. To learn about this space, this paper will focus on how linear fractional transformations act on this space and the patterns that reveal themselves. To expand on this, I will construct a fundamental domain under these mappings and explore the coding of closed geodesics on the fundamental domain, which requires an understanding of continued fractions. My research will then be applied to two copies of the hyperbolic plane. My goal is to understand fundamental regions in this space and eventually the geodesics.

This paper is intended …

Sir Francis Galton, Sandra J. Peart, David M. Levy

Jepson School of Leadership Studies articles, book chapters and other publications

Cousin to Charles Darwin and a talented statistician, Sir Francis Galton had an influence on social science that was profound. His major contributions to mathematical statistics included the initial development of quantiles and linear regression techniques. Along with F. Y. Edgeworth and Karl Pearson, he developed general techniques of multiple regression and correlation analysis, statistical devices that serve as substitutes for experiments in social science. Galton had a major impact on economics, and with W. R. Greg, was instrumental in creating the “science” of eugenics.

An Action For A Classical String, The Equation Of Motion And Group Invariant Classical Solutions, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

A string action which is essentially a Willmore functional is presented and studied. This action determines the physics of a surface in Euclidean three space which can be used to model classical string configurations. By varying this action an equation of motion for the mean curvature of the surface is obtained which is shown to govern certain classical string configurations. Several classes of classical solutions for this equation are discussed from the symmetry group point of view and an application is presented.

A Fragment On Euler's Constant In Ramanujan's Lost Notebook, Bruce C. Berndt, Timothy Huber

School of Mathematical & Statistical Sciences Faculty Publications

A formula for Euler’s constant found in Ramanujan’s lost notebook and also in a problem he submitted to the Journal of the Indian Mathematical Society is proved and discussed.

A Numerical Analysis Approach For Estimating The Minimum Traveling Wave Speed For An Autocatalytic Reaction, Erika Blanken

Electronic Theses and Dissertations

This thesis studies the traveling wavefront created by the autocatalytic cubic chemical reaction A + 2B → 3B involving two chemical species A and B, where A is the reactant and B is the auto-catalyst. The diffusion coefficients for A and B are given by DA and DB. These coefficients differ as a result of the chemical species having different size and/or weight. Theoretical results show there exist bounds, v* and v*, depending on DB/DA, where for speeds v ≥ v*, a traveling wave solution exists, while for speeds v < v*, a solution does not exist. Moreover, if DB ≤ DA, and v* and v* are similar to one another and in the order of DB/DA when it is small. On the other hand, when DA ≤ DB there exists a minimum speed vmin, such that there is a traveling wave solution if the speed v > vmin. The determination of vmin is very important in determining …

Fractal Interpolation, Gayatri Ramesh

Electronic Theses and Dissertations

This thesis is devoted to a study about Fractals and Fractal Polynomial Interpolation. Fractal Interpolation is a great topic with many interesting applications, some of which are used in everyday lives such as television, camera, and radio. The thesis is comprised of eight chapters. Chapter one contains a brief introduction and a historical account of fractals. Chapter two is about polynomial interpolation processes such as Newton s, Hermite, and Lagrange. Chapter three focuses on iterated function systems. In this chapter I report results contained in Barnsley s paper, Fractal Functions and Interpolation. I also mention results on iterated function system …

Analysis Of Kolmogorov's Superposition Theorem And Its Implementation In Applications With Low And High Dimensional Data., Donald Bryant

Electronic Theses and Dissertations

In this dissertation, we analyze Kolmogorov's superposition theorem for high dimensions. Our main goal is to explore and demonstrate the feasibility of an accurate implementation of Kolmogorov's theorem. First, based on Lorentz's ideas, we provide a thorough discussion on the proof and its numerical implementation of the theorem in dimension two. We present computational experiments which prove the feasibility of the theorem in applications of low dimensions (namely, dimensions two and three). Next, we present high dimensional extensions with complete and detailed proofs and provide the implementation that aims at applications with high dimensionality. The amalgamation of these ideas is …

Comparing Assessment Methods As Predictors Of Student Learning In Undergraduate Mathematics, Nichole Shorter

Electronic Theses and Dissertations

This experiment was designed to determine which assessment method: continuous assessment (in the form of daily in-class quizzes), cumulative assessment (in the form of online homework), or project-based learning, best predicts student learning (dependent upon posttest grades) in an undergraduate mathematics course. Participants included 117 university-level undergraduate freshmen enrolled in a course titled "Mathematics for Calculus". Initially, a multiple regression model was formulated to model the relationship between the predictor variables (the continuous assessment, cumulative assessment, and project scores) versus the outcome variable (the posttest scores). However, due to the possibility of multicollinearity present between the cumulative assessment predictor variable …

On The Reproducing Kernel Hilbert Spaces Associated With The Fractional And Bi-Fractional Brownian Motions, Daniel Alpay, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

We present decompositions of various positive kernels as integrals or sums of positive kernels. Within this framework we study the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions. As a tool, we define a new function of two complex variables, which is a natural generalization of the classical Gamma function for the setting we consider.

Rational Functions Associated To The White Noise Space And Related Topics, Daniel Alpay, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

Motivated by the hyper-holomorphic case we introduce and study rational functions in the setting of Hida’s white noise space. The Fueter polynomials are replaced by a basis computed in terms of the Hermite functions, and the Cauchy-Kovalevskaya product is replaced by the Wick product.