Physical Sciences and Mathematics Commons^™

Distances Between Composition Operators, Valentin Matache

Mathematics Faculty Publications

Composition operators Cϕ induced by a selfmap ϕ of some set S are operators acting on a space consisting of functions on S by composition to the right with ϕ, that is Cϕf = f ◦ ϕ. In this paper, we consider the Hilbert Hardy space H2 on the open unit disk and find exact formulas for distances kCϕ − Cψk between composition operators. The selfmaps ϕ and ψ involved in those formulas are constant, inner, or analytic selfmaps of the unit disk fixing the origin.

Solving Ramanujan's Differential Equations For Eisenstein Series Via A First Order Riccati Equation, James M. Hill, Bruce C. Berndt, Timothy Huber

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper we prove that Ramanujan's differential equations for the Eisenstein series P, Q, and R are invariant under a simple one-parameter stretching group of transformations. Using this, we show that the three differential equations may be reduced to a first order Riccati differential equation, the solution of which may be represented in terms of hypergeometric functions. The resulting formulas allow for the derivation of parametric representations of P, Q, and R, analogous to representations in Ramanujan's second notebook. In contrast, in the classical approach, one first needs to derive the fundamental formula connecting theta functions with elliptic integrals. …

Stochastic Comparisons Of Parallel Systems When Component Have Proportional Hazard Rates, Subhash C. Kochar, Maochao Xu

Mathematics and Statistics Faculty Publications and Presentations

Let Χ₁, … Χ_n be independent random variables with Χᵢ having survival function Fλᵢ, i=1, … , n, and let Y₁, … , Yn be a random sample with common population survival distribution Fλ, where λ = Σᵢ=₁nλᵢl n. Let Χn:n and Yn:n denote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that Xn:n is greater than Yn:n in terms of likelihood ratio order. It is also proved that the sample range Χn:n - Χ₁:n is larger than Yn n:n - Y₁:n according to reverse hazard rate ordering. These two results …

Some Recent Results On Stochastic Comparisons And Dependence Among Order Statistics In The Case Of Phr Model, Subhash C. Kochar, Maochao Xu

Mathematics and Statistics Faculty Publications and Presentations

This paper reviews some recent results on stochastic orders and dependence among order statistics when the observations are independent and follow the proportional hazard rates model.

The Topology Of Surface Mediatrices, James Bernhard, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

Given a pair of distinct points p and q in a metric space with distance d, the mediatrix is the set of points x such that d(x,p)=d(x,q). In this paper, we examine the topological structure of mediatrices in connected, compact, closed 2-manifolds whose distance function is inherited from a Riemannian metric. We determine that such mediatrices are, up to homeomorphism, finite, closed simplicial 1-complexes with an even number of incipient edges emanating from each vertex. Using this and results from [J.J.P. Veerman, J. Bernhard, Minimally separating sets, mediatrices and Brillouin spaces, Topology Appl., in press], we give the classification …

Microscopic-Macroscopic Simulations Of Rigid-Rod Polymer Hydrodynamics: Heterogeneity And Rheochaos, M. Gregory Forest, Ruhai Zhou, Qi Wang

Mathematics & Statistics Faculty Publications

Rheochaos is a remarkable phenomenon of nematic (rigid-rod) polymers in steady shear, with sustained chaotic fluctuations of the orientational distribution of the rod ensemble. For monodomain dynamics, imposing spatial homogeneity and linear shear, rheochaos is a hallmark prediction of the Doi-Hess theory [M. Doi, J. Polym. Sci. Polym. Phys. Ed., 19 (1981), pp. 229-243; M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, London, New York, 1986; S. Hess, Z. Naturforsch., 31 (1976), pp. 1034-1037. The model behavior is robust, captured by second-moment tensor approximations G. Rienäcker, M. Kröger, and S. Hess, Phys. Rev. …

Portfolio Selection In Gaussian And Non-Gaussian Worlds, Jing Wang

Theses and Dissertations (Comprehensive)

In the case of minimizing risk with a given level of expected return, we discuss the portfolio selection problem with the asset returns are characterized by a Gaussian distribution and heavy tailed distribution.

More specifically, under the Gaussian assupmtion, we give the explicit solutions to the problems of minimizing risk variance, CaR and EaR respectively. When a compound Poisson process is assumed, we derive explicit solutions to the variance, CaR and EaR. Furthermore, we give the explicit soultion for the CaR when a Lévy distribution is considered.

For the more realistic process-normal inverse process, we are able to obtain the …

Partial Separability And Partial Additivity For Orderings Of Binary Alternatives, Md. Abul Bashar

Theses and Dissertations (Comprehensive)

In Multiple-Criteria Decision Analysis (MCDA), a good way to find the best alternative is to construct a value function that represents a Decision Maker’s (DM) preferences. For multidimensional alternatives, an additive value function is easiest to work with because it assesses the alternatives in a simple and transparent manner. A DM’s preferences over consequences on a subset of the set of criteria may or may not depend on consequences on the rest of the criteria. Preferences that are free from all such interdependence are said to be separable. The existence of an additive value function implies separability and, when consequences …

Closure Under Transfinite Extensions, Edgar Enochs, Alina Iacob, Overtoun Jenda

Department of Mathematical Sciences Faculty Publications

The closure under extensions of a class of objects in an abelian category is often an important property of that class. Recently the closure of such classes under transfinite extensions (both direct and inverse) has begun to play an important role in several areas of mathematics, for example, in Quillen's theory of model categories and in the theory of cotorsion pairs. In this paper we prove that several important classes are closed under transfinite extensions.

A Note On Relations For Reliability Measures In Zero-Adjusted Models, Broderick O. Oluyede, Mavis Pararai

Department of Mathematical Sciences Faculty Publications

In this note we examine and study relations in zero-adjusted models. Relations for reliability measures in the adjusted and unadjusted models are established and appropriate comparisons including the relative error are presented. The relative error is shown to be a decreasing function of the counts. Some inequalities and comparisons for weighted zero-adjusted models are established.

The Mathematical Preparation Of Secondary School Teachers, Kelly W. Edenfield

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

In the summer of 2007, a group of doctoral students at the University of Georgia gathered to discuss the mathematical preparation of secondary teachers. The group used Mathematics for High School Teachers: An Advanced Perspective by Usiskin, Peressini, Marchisotto, and Stanley (2003) as the catalyst for the discussion. Participants agreed that future teachers need opportunities to examine high school and college mathematics differently from the way they had as students, with specific emphasis on connections, representations, and history. Features of this text that were highlighted in the discussions were the attention topics with commonly held misconceptions, the historical rationales and …

On Energy And Expected Uncertainty Measures In Weighted Distributions, Broderick O. Oluyede, Mekki Terbeche

Department of Mathematical Sciences Faculty Publications

In this note, bounds and inequalities for the comparisons of weighted energy functions, entropy, and discrimination information measures and their unweighted counterparts are presented. Inequalities for weighted expected uncertainty, cross-entropy or discrimination information measures are also presented. A useful result on the convergence of the weighted kernel density informational energy estimates is given and some informational energy applications presented.

Digit Reversal Without Apology, Lara Pudwell

Lara K. Pudwell

No abstract provided.

Graphing Transformations: Does Order Make A Difference?, John Hawkins

John B. Hawkins

No abstract provided.

Open Source Surveys With Asset, Bert Wachsmuth

Bert Wachsmuth

No abstract provided.

Symbolization Of Generating Functions; An Application Of The Mullin–Rota Theory Of Binomial Enumeration, Tian-Xiao He, Peter J.S. S, Leetsch C. Hsu

Tian-Xiao He

We have found that there are more than a dozen classical generating functions that could be suitably symbolized to yield various symbolic sum formulas by employing the Mullin–Rota theory of binomial enumeration. Various special formulas and identities involving well-known number sequences or polynomial sequences are presented as illustrative examples. The convergence of the symbolic summations is discussed.

Fourier Transform Of Bernstein–Bézier Polynomials, Tian-Xiao He, Charles K. Chui, Qingtang Jiang

Tian-Xiao He

Explicit formulae, in terms of Bernstein–Bézier coefficients, of the Fourier transform of bivariate polynomials on a triangle and univariate polynomials on an interval are derived in this paper. Examples are given and discussed to illustrate the general theory. Finally, this consideration is related to the study of refinement masks of spline function vectors.

Two Number-Theoretic Problems That Illustrate The Power And Limitations Of Randomness, Andrew Shallue

Andrew Shallue

This thesis contains work on two problems in algorithmic number theory. The first problem is to give an algorithm that constructs a rational point on an elliptic curve over a finite field. A fast and easy randomized algorithm has existed for some time. We prove that in the case where the finite field has characteristic 2, there is a deterministic algorithm with the same asymptotic running time as the existing randomized algorithm.

Construction Of Biorthogonal B-Spline Type Wavelet Sequences With Certain Regularities, Tian-Xiao He

Tian-Xiao He

No abstract provided.

The Abacus Of Universal Logics, Rudolf Kaehr

Rudolf Kaehr

No abstract provided.

The Sheffer Group And The Riordan Group, Tian-Xiao He, Peter J.S. Shiue, Leetsch C. Hsu

Tian-Xiao He

We define the Sheffer group of all Sheffer-type polynomials and prove the isomorphism between the Sheffer group and the Riordan group. An equivalence of the Riordan array pair and generalized Stirling number pair is also presented. Finally, we discuss a higher dimensional extension of Riordan array pairs.