Mathematics Commons^™

Infeasible Full-Newton-Step Interior-Point Method For The Linear Complementarity Problems, Antré Marquel Drummer

College of Graduate Studies: Theses & Dissertations

In this tesis, we present a new Infeasible Interior-Point Method (IPM) for monotone Linear Complementarity Problem (LPC). The advantage of the method is that it uses full Newton-steps, thus, avoiding the calculation of the step size at each iteration. However, by suitable choice of parameters the iterates are forced to stay in the neighborhood of the central path, hence, still guaranteeing the global convergence of the method under strict feasibility assumption. The number of iterations necessary to find -approximate solution of the problem matches the best known iteration bounds for these types of methods. The preliminary implementation of the method …

Homogeneous Symplectic Manifolds Of The Galilei Group, Michael S. Davis

College of Graduate Studies: Theses & Dissertations

In this thesis we classify all symplectic manifolds admitting a transitive, 2-form preserving action of the Galilei group G. Using the moment map and a theorem of Kirillov-Kostant-Souriau, we reduce the problem to that of classifying the coadjoint orbits of a central extension of G discovered by Bargmann. We then develop a systematic inductive technique to construct a cross section of the coadjoint action. The resulting symplectic orbits are interpreted as the manifolds of classical motions of elementary particles with or without spin, mass, and color.

Properties Of Weighted Generalized Beta Distribution Of The Second Kind, Yuan Ye

College of Graduate Studies: Theses & Dissertations

Author's abstract: In this thesis, a new class of weighted generalized beta distribution of the second kind (WGB2) is presented. The construction makes use of the conservability approach' which includes the size or length-biased distribution as a special case. The class of WGB2 is used as descriptive models for the distribution of income. The results that are presented generalize the generalized beta distribution of second kind (GB2). The properties of these distributions including behavior of pdf, cdf, hazard functions, moments, mean, variance, coefficient of variation (CV), coefficient of skewness (CS), coefficient of kurtosis (CK) are obtained. The moments of other …

Compositions, Partitions, And Fibonacci Numbers, Andrew Sills

Mathematical Sciences: Faculty Publications

A bijective proof is given for the following theorem: the number of compositions of n into odd parts equals the number of compositions of n+ 1 into parts greater than one. Some commentary about the history of partitions and compositions is provided.

Multiplicative Noise For Masking Numerical Microdata Data With Constraints, Anna Oganian

Mathematical Sciences: Faculty Publications

Before releasing databases which contain sensitive information about individuals, statistical agencies have to apply Statistical Disclosure Limitation (SDL) methods to such data. The goal of these methods is to minimize the risk of disclosure of the confidential information and at the same time provide legitimate data users with accurate information about the population of interest. SDL methods applicable to the microdata (i. e. collection of individual records) are often called masking methods. In this paper, several multiplicative noise masking schemes are presented. These schemes are designed to preserve positivity and inequality constraints in the data together with the vector of …

Using Performance Assessments To Connect Fractions And Rational Expressions: Noyce Scholars As Mentors To Pre-Service Elementary Teachers, Joy W. Darley, Michelle Cawthorn, Marlynn Griffin, Brian P. Koehler, James Lobue

Mathematical Sciences: Faculty Presentations (1991-2022)

At Georgia Southern University, Noyce scholars are not only being mentored, but they are also serving as mentors to preservice elementary teachers. One topic that proves to be problematic for many students is the conceptual understanding of fractions and rational expressions. Since our Noyce scholars with mathematics degrees will be teaching algebra, it is important that they are fluent with the arithmetic to algebra connection. In addition, it is crucial that these mathematics majors become stakeholders in mathematics education at the elementary school level.

Performance assessments can provide the structure necessary for assisting pre-service elementary teachers in firmly establishing the …

Weak Type Inequalities For Maximal Operators Associated To Double Ergodic Sums, Paul Hagelstein, Alexander M. Stokolos

Mathematical Sciences: Faculty Publications

Given an approach region Γ ∈ Z₊² and a pair U, V of commuting nonperiodic measure preserving transformations on a probability space (Ω, Σ, μ), it is shown that either the associated multiparameter ergodic averages of any function in L¹(Ω) converge a.e. or that, given a positive increasing function ϕ on [0,∞) that is o(log x) as x → ∞, there exists a function g ∈ Lϕ(L)(Ω) whose associated multiparameter ergodic averages fail to converge a.e.

Spectral Asymptotics Of Laplacians Associated To One-Dimensional Iterated Function Systems With Overlaps, Sze-Man Ngai

Mathematical Sciences: Faculty Publications

We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

Polynomial Generalizations Of Two-Variable Ramanujan Type Identities, James Mclaughlin, Andrew Sills

Mathematical Sciences: Faculty Publications

We provide finite analogs of a pair of two-variable q-series identities from Ramanujan's lost notebook and a companion identity.

Kernel-Based Interior-Point Methods For Cartesian P*(Κ)-Linear Complementarity Problems Over Symmetric Cones, Goran Lesaja

Mathematical Sciences: Faculty Publications

We present an interior point method for Cartesian P*(k)-Linear Complementarity Problems over Symmetric Cones (SCLCPs). The Cartesian P*(k)-SCLCPs have been recently introduced as the generalization of the more commonly known and more widely used monotone SCLCPs. The IPM is based on the barrier functions that are defined by a large class of univariate functions called eligible kernel function which have recently been successfully used to design new IPMs for various optimization problems. Eligible barrier (kernel) functions are used in calculating the Nesterov-Todd search directions and the default step-size which leads to a very good complexity results for the method. For …

On Diamond-Alpha Dynamic Equations And Inequalities, Nuriye Atasever

College of Graduate Studies: Theses & Dissertations

In view of the recently developed theory of calculus for dynamic equations on time scales (which unifies discrete and continuous systems), in this project we give some of the basics of the extension of the theory to the combined delta (forward) and nabla (backward) derivatives. In this set up the newly developed theory of diamond-alpha derivatives are analyzed through some equation and inequality properties. In particular Opial type Diamond-alpha dynamic Inequalities are discussed in this context and recently developed results and their improved versions are given in this work.

On Weighted Distributions And Mean Advantage Over Inferiors Functions, Broderick O. Oluyede, Norou Diawara

Mathematical Sciences: Faculty Publications

In this note, some fundamental results including relationship be-tween weighted distribution functions and mean advantage over inferi-ors functions are established. Ordering of reliability and/or distribution functions via mean advantage over inferiors functions and related func-tions for parent and weighted reliability functions are presented. Some applications and examples are given.

Unified Analysis Of Kernel-Based Interior-Point Methods For P *(Κ)-Lcp, Goran Lesaja, C. Roos

Mathematical Sciences: Faculty Publications

We present an interior-point method for the P∗(κ)-linear complementarity problem (LCP) that is based on barrier functions which are defined by a large class of univariate functions called eligible kernel functions. This class is fairly general and includes the classical logarithmic function and the self-regular functions, as well as many non-self-regular functions as special cases. We provide a unified analysis of the method and give a general scheme on how to calculate the iteration bounds for the entire class. We also calculate the iteration bounds of both long-step and short-step versions of the method for several …

Core Compactness And Diagonality In Spaces Of Open Sets, Francis Jordan, Frédéric D. Mynard

Mathematical Sciences: Faculty Publications

We investigate when the space OX of open subsets of a topological space X endowed with the Scott topology is core compact. Such conditions turn out to be related to infraconsonance of X, which in turn is characterized in terms of coincidence of the Scott topology of OX × OX with the product of the Scott topologies of OX at (X,X). On the other hand, we characterize diagonality of OX endowed with the Scott convergence and show that this space can be diagonal without being pretopological. New examples are provided to clarify the relationship between pretopologicity, topologicity and diagonality of …

Long Path Lemma Concerning Connectivity And Independence Number, Shinya Fujita, Alexander Halperin, Colton Magnant

Mathematical Sciences: Faculty Publications

We show that, in a k-connected graph G of order n with α(G)=α, between any pair of vertices, there exists a path P joining them with

|P|≥min{n,(k−1)(n−k)/α +k}.

This implies that, for any edge e∈E(G), there is a cycle containing e of length at least

min{n,(k−1)(n−k)/α +k}.

Moreover, we generalize our result as follows: for any choice S of s≤k vertices in G, there exists a tree T whose set of leaves is S with

|T|≥min{n,(k−s+1)(n−k)/α +k}.

Independence Number And Disjoint Theta Graphs, Shinya Fujita, Colton Magnant

Mathematical Sciences: Faculty Publications

The goal of this paper is to find vertex disjoint even cycles in graphs. For this purpose, define a θ-graph to be a pair of vertices u,v with three internally disjoint paths joining u to v. Given an independence number α and a fixed integer k, the results contained in this paper provide sharp bounds on the order f(k,α) of a graph with independence number α(G)≤α which contains no k disjoint θ-graphs. Since every θ-graph contains an even cycle, these results provide k disjoint even cycles in graphs of order at least f(k,α)+1. We also discuss the relationship between this …

Meander Graphs And Frobenius Seaweed Lie Algebras, Colton Magnant, Vincent E. Coll, Anthony Giaquinto

Mathematical Sciences: Faculty Publications

The index of a seaweed Lie algebra can be computed from its associated meander graph. We examine this graph in several ways with a goal of determining families of Frobenius (index zero) seaweed algebras. Our analysis gives two new families of Frobenius seaweed algebras as well as elementary proofs of known families of such Lie algebras.

On An Identity Of Gessel And Stanton And The New Little Göllnitz Identities, Carla D. Savage, Andrew Sills

Mathematical Sciences: Faculty Publications

We show that an identity of Gessel and Stanton [I. Gessel, D. Stanton, Applications of q-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983) 197, Eq. (7.24)] can be viewed as a symmetric version of a recent analytic variation of the little Göllnitz identities. This is significant, since the Göllnitz–Gordon identities are considered the usual symmetric counterpart to little Göllnitz theorems. Is it possible, then, that the Gessel–Stanton identity is part of an infinite family of identities like those of Göllnitz–Gordon?

Toward this end, we derive partners and generalizations of the Gessel–Stanton identity. We show that …

Hybrid Proofs Of The Q-Binomial Theorem And Other Identities, Dennis Eichhorn, James Mclaughlin, Andrew Sills

Mathematical Sciences: Faculty Publications

We give "hybrid" proofs of the q-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version.

We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities due to Ramanujan.

Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities.

Rademacher-Type Formulas For Restricted Partition And Overpartition Functions, Andrew Sills

Mathematical Sciences: Faculty Publications

A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented, framed by several biographical anecdotes.

Note On Gradient Estimates Of Heat Kernel For Schrödinger Operators, Shijun Zheng

Mathematical Sciences: Faculty Publications

Let H = -Δ+V be a Schrödinger operator on Rⁿ. We show that gradient estimates for the heat kernel of H with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for L^p and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists V in the Schwartz class such that the long time gradient heat …

Theoretical Properties And Estimation In Weighted Weibull And Related Distributions, Ryan Roman

College of Graduate Studies: Theses & Dissertations

The Weibull distribution is a well known and common distribution. In this thesis, theoretical properties of weighted Weibull distributions are presented. Properties of the non-weighted Weibull distribution are also reiterated for comparison. The probability density functions, cumulative distribution functions, survival functions, hazard functions and reverse hazard functions are given for each distribution. In addition, Glaser's Lemma is applied to determine the behavior of the hazard functions. The standardized moments, differential entropy, Fisher information and results based on the likelihood function are given for each distribution as well. These results are also shown for the Rayleigh distribution, a special case of …

LP Estimates For The Maximal Functions, Alexander M. Stokolos

Mathematical Sciences: Faculty Presentations (1991-2022)

This talk was given during the Jozef Marcinkiewicz Centenary Conference.

On The Characteristic Polynomial Of Regular Linear Matrix Pencil, Yan Wu, Phillip Lorren

Mathematical Sciences: Faculty Publications

Linear matrix pencil, denoted by (A,B), plays an important role in control systems and numerical linear algebra. The problem of finding the eigenvalues of (A,B) is often solved numerically by using the well-known QZ method. Another approach for exploring the eigenvalues of (A,B) is by way of its characteristic polynomial, P(λ)=A − λB. There are other applications of working directly with the characteristic polynomial, for instance, using Routh-Hurwitz analysis to count the stable roots of P(λ) and transfer function representation of control systems governed by differential-algebraic equations. In this paper, we …

Analysis Of Discrete Data Under Order Restrictions, Jeff Campbell

College of Graduate Studies: Theses & Dissertations

Strategies for the analysis of discrete data under order restrictions are discussed. We consider inference for sequences of binomial populations, and the corresponding risk difference, relative risk and odds ratios. These concepts are extended to deal with independent multinomial populations. Natural orderings such as stochastic ordering and cumulative ratio probability ordering are discussed. Methods are developed for the estimation and testing of differences between binomial as well as multinomial populations under order restrictions. In particular, inference for sequences of ordered binomial probabilities and cumulative probability ratios in multinomial populations are presented. Closed-form estimates of the multinomial parameters under order restrictions …

A Unified Theory Of Function Spaces And Hyperspaces: Local Properties, Szymon Dolecki, Frédéric D. Mynard

Mathematical Sciences: Faculty Publications

Many classically used function space structures (including the topology of pointwise convergence, the compact-open topology, the Isbell topology and the continuous convergence) are induced by a hyperspace structure counterpart. This scheme is used to study local properties of function space structures on C(X,R), such as character, tighntess, fan-tightness, strong fan-tightness, the Fr{\'e}chet property and some of its variants. Under mild conditions, local properties of C(X,R) at the zero function correspond to the same property of the associated hyperspace structure at X. The latter is often easy to characterize in terms of covering properties …

Investigating Mathematical Literacy Through Teacher Language, Alyson Lischka

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

Communication about mathematical concepts using appropriate terminology is a standard established by the National Council of Teachers of Mathematics. However, international test results show that United States’ students are lagging in mathematical literacy. This case study analyzes the ways in which instructors use language to help students move toward conceptual understanding of mathematical vocabulary. Three mathematics education professors at a mid-size four year institution were observed teaching math classes to students enrolled in elementary or secondary certification programs. Collected data included: audio-recorded observations and field notes, lesson artifacts such as quizzes and handouts, and audio-recorded interviews with each participant. Findings …

Weighted Inverse Weibull And Beta-Inverse Weibull Distribution, Jing Xiong Kersey

College of Graduate Studies: Theses & Dissertations

The weighted inverse Weibull distribution and the beta-inverse Weibull distribution are considered. Theoretical properties of the inverse Weibull model, weighted inverse Weibull distribution including the hazard function, reverse hazard function, moments, moment generating function, coefficient of variation, coefficient of skewness, coefficient of kurtosis, Fisher information and Shanon entropy are studied. The estimation for the parameters of the length-biased inverse Weibull distribution via maximum likelihood estimation and method of moment estimation techniques are presented, as well as a test for the detection of length-biasedness in the inverse Weibull model. Furthermore, the beta-inverse Weibull distribution which is a weighted distribution is presented, …

Some Implications Of Chu's 10Ψ10 Generalization Of Bailey's 6Ψ6 Summation Formula, James Mclaughlin, Andrew Sills, Peter Zimmer

Mathematical Sciences: Faculty Publications

Lucy Slater used Bailey's 6ψ6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type.

In the present paper we apply the same techniques to Chu's 10ψ10 generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs.

In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new …

Rademacher-Type Formulas For Partitions And Overpartitions, Andrew Sills

Mathematical Sciences: Faculty Publications

A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of n and the Zuckerman formula for the Fourier coefficients of ϑ4_0 | τ_−1 is presented.