Mathematics Commons^™

Distance-Based Graph Invariants Of Trees And The Harary Index, Stephan G. Wagner, Hua Wang, Xiao-Dong Zhang

Mathematical Sciences: Faculty Publications

Introduced in 1947, the Wiener index W(T) = ∑{u,v}⊆V(T) d(u, v) is one of the most thoroughly studied chemical indices. The extremal structures (in particular, trees with various constraints) that maximize or minimize the Wiener index have been extensively investigated. The Harary index H(T) = ∑{u,v}⊆V(T) , introduced in 1993, can be considered as the 'reciprocal analogue' of the Wiener index. From recent studies, it is known that the extremal structures of the Harary index and the Wiener index coincide in many instances, i.e., the graphs that maximize the Wiener index minimize the Harary index and vice versa. In this …

Greedy Trees, Subtrees And Antichains, Eric Ould Dadah Andriantiana, Stephan G. Wagner, Hua Wang

Mathematical Sciences: Faculty Publications

Greedy trees are constructed from a given degree sequence by a simple greedy algorithm that assigns the highest degree to the root, the second-, third-, ... highest degrees to the root's neighbors, and so on.

They have been shown to maximize or minimize a number of different graph invariants among trees with a given degree sequence. In particular, the total number of subtrees of a tree is maximized by the greedy tree. In this work, we show that in fact a much stronger statement holds true: greedy trees maximize the number of subtrees of any given order. This parallels recent …

Exponential Kleisli Monoids As Eilenberg-Moore Algebras, Dirk Hofmann, Frédéric Mynard, Gavin J. Seal

Mathematical Sciences: Faculty Publications

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

On Q-Analogs Of Wostenholme Type Congruences For Multiple Harmonic Sums, Jianqiang Zhao

Mathematical Sciences: Faculty Publications

Multiple harmonic sums are iterated generalizations of harmonic sums. Recently Dilcher has considered congruences involving q-analogs of these sums in depth one. In this paper we shall study the homogeneous case for arbitrary depth by using generating functions and shuffle relations of the q-analog of multiple harmonic sums. At the end, we also consider some non-homogeneous cases.

Proceedings Of The Seventh Annual Meeting Of The Georgia Association Of Mathematics Teacher Educators Introductory Texts

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

Contents of 7th Annual GAMTE Proceedings Front Matter:

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

How Differing Interpretations Of Making Mathematics Fun Influence Teaching Practice, Amanda Sawyer

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

I investigated an experienced teacher and a beginning teacher who held similar beliefs about making mathematics learning fun yet held different interpretations of implementation. I found when a teacher equated fun with problem solving, her classroom practice included activities with higher-level thinking skills. In contrast, a teacher who defined fun as students’ enjoyment layered manipulatives and group work on top of procedures. Therefore, teachers need opportunities to reflect on the nature of student understanding as a precursor to shaping their views of fun.

Negative phrases and attitudes toward mathematics have become commonplace in popular culture. Perhaps in response to this …

The Mathematics Teacher Educator’S Reflection On Edtpa: The Basics And The Complexity Of Georgia’S Implementation, Lynn Stallings, Woong Lim

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

Beginning in fall 2015, Georgia will require that all teacher candidates pass the edTPA, a performance-based assessment, as a requirement for initial certification. With the potential to impact teacher preparation programs in a profound way, the questions and issues related to the implementation of edTPA merit some critical reflection. In this article, we introduce the brief history of the instrument and review the basics of edTPA with a particular interest in middle grades mathematics. Furthermore, drawing upon our experience with edTPA during a pilot program and the extensive knowledge base of edTPA rubrics and scoring process, we present issues related …

Full Newton Step Interior Point Method For Linear Complementarity Problem Over Symmetric Cones, Andrii Berdnikov

College of Graduate Studies: Theses & Dissertations

In this thesis, we present a new Feasible Interior-Point Method (IPM) for Linear Complementarity Problem (LPC) over Symmetric Cones. The advantage of this method lies in that it uses full Newton-steps, thus, avoiding the calculation of the step size at each iteration. By suitable choice of parameters we prove the global convergence of iterates which always stay in the the central path neighborhood. A global convergence of the method is proved and an upper bound for the number of iterations necessary to find ε-approximate solution of the problem is presented.

The Convergence Of Two Algorithms For Compressed Sensing Based Tomography, Xiezhang Li, Jiehua Zhu

Mathematical Sciences: Faculty Publications

The constrained total variation minimization has been developed successfully for image reconstruction in computed tomography. In this paper, the block component averaging and diagonally-relaxed orthogonal projection methods are proposed to incorporate with the total variation minimization in the compressed sensing framework. The convergence of the algorithms under a certain condition is derived. Examples are given to illustrate their convergence behavior and noise performance.

Cycles, The Degree Distance, And The Wiener Index, Daniel Gray, Hua Wang

Mathematical Sciences: Faculty Publications

The degree distance of a graph G is D'(G)=(1/2)∑ⁿ_i=1∑ⁿ_j=1(d_i+d_j)L_{i ,j}, where d_i and d_j are the degrees of vertices v_i, v_j ∈ V (G), and L_i,j is the distance between them. The Wiener index is defined as W(G)=(1/2)∑ⁿi₌₁ ∑ⁿ_j-1L_{i, j}. An elegant result (Gutman; Klein, Mihalic, Plavsic and Trinajstic) is known regarding their correlation, that D'(T)=4W(T)-n(n-1)for a tree T with n vertices. In …

Integer Compositions, Gray Code, And The Fibonacci Sequence, Linus Lindroos

College of Graduate Studies: Theses & Dissertations

In this thesis I show the relation of binary and Gray Code to integer compositions and the Fibonacci sequence through the use of analytic combinatorics, Zeckendorf's Theorem, and generating functions.

Integer Solutions To Optimization Problems And Modular Sequences Of Nexus Numbers, Jeremy T. Davis

College of Graduate Studies: Theses & Dissertations

In this thesis, we examine the use of integers through two ideas. As mathematics teachers, we prefer students not use calculators on assessments. In order to require this, students compute the problems by hand. We take a look at the classic Calculus I optimization box problem while restricting values to integers. In addition, sticking with the integer theme, we take a new look at the nexus numbers. Nexus numbers are extensions of the hex and rhombic dodecahedral numbers. We put these numbers into a sequence, and through a few computations of modular arithmetic, we analyze the sequences and their patterns …

A Non-Parametric Approach To Change-Point Detection In Cross-Asset Correlations, L. Kaili Diamond

College of Graduate Studies: Theses & Dissertations

In this thesis we explore the problem of detecting change-points in cross-asset correlations using a non-parametric approach. We began by comparing and contrasting several common methods for change-point detection as well as methods for measuring correlation. We finally settle on a statistic introduced in early 2012 by Herold Dehling et.al. and test this statistic against real world financial data. We provide the estimated change-point for this data as well as the asymptotic p-value associated with this statistic. Once this process was complete we went on to use simulated data to measure the accuracy, power, and type 1 error associated with …

Compositions, Bijections, And Enumerations, Charles R. Dedrickson Iii

College of Graduate Studies: Theses & Dissertations

In this thesis we give an introduction to colored-compositions of an integer. This is a generalization of traditional integer compositions, and we show a few results for n-color compositions which are analogous to regular compositions with both combinatorial and analytic proofs. We also show several bijections between various types of compositions to certain types of numeric strings, and provide a generalization of a classic bijection between compositions and binary strings.

Generalized Diamond-Alpha Dynamic Opial Inequalities, Nuriye Atasever, Billûr Kaymakçalan, Goran Lesaja, Kenan Taş

Mathematical Sciences: Faculty Publications

We establish some new dynamic Opial-type diamond alpha inequalities in time scales. Our results in special cases yield some of the recent results on Opial's inequality and also provide new estimates on inequalities of this type. Also, we introduce an example to illustrate our result.

Balance With Unbounded Complexes, Edgar E. Enochs, Sergio Estrada, Alina Iacob

Mathematical Sciences: Faculty Publications

Given a double complex X there are spectral sequences with the E₂ terms being either H_I (H_II(X)) or H_II(H_I(X)). But if H_I(X)=H_II(X)=0, then both spectral sequences have all their terms 0. This can happen even though there is nonzero (co)homology of interest associated with X. This is frequently the case when dealing with Tate (co)homology. So, in this situation the spectral sequences may not give any information about the (co)homology of interest. In this article, we …

Formulæ For The Number Of Partitions Of N Into At Most M Parts (Using The Quazi-Polynomial Ansatz), Andrew Sills, Doron Zeilberger

Mathematical Sciences: Faculty Publications

The purpose of this short article is to announce, and briefly describe, a Maple package, PARTITIONS, that (inter alia) completely automatically discovers, and then proves, explicit expressions (as sums of quasi-polynomials) for p_m(n) for any desired m. We do this to demonstrate the power of “rigorous guessing” as facilitated by the quasi-polynomial ansatz.

Generalized Borcea-Voisin Construction, Jimmy Dillies

Mathematical Sciences: Faculty Publications

C. Voisin and C. Borcea have constructed mirror pairs of families of Calabi-Yau threefolds by taking the quotient of the product of an elliptic curve with a K3 surface endowed with a non-symplectic involution. In this paper, we generalize the construction of Borcea and Voisin to any prime order and build three and four dimensional Calabi-Yau orbifolds. We classify the topological types that are obtained and show that, in dimension 4, orbifolds built with an involution admit a crepant resolution and come in topological mirror pairs. We show that for odd primes, there are generically no minimal resolutions and the …

Lecture Hall Sequences, Q-Series, And Asymmetric Partition Identities, Sylvie Corteel, Carla D. Savage, Andrew Sills

Mathematical Sciences: Faculty Publications

We use generalized lecture hall partitions to discover a new pair of q-series identities. These identities are unusual in that they involve partitions into parts from asymmetric residue classes, much like the little Göllnitz partition theorems. We derive a two-parameter generalization of our identities that, surprisingly, gives new analytic counterparts of the little Göllnitz theorems. Finally, we show that the little Göllnitz theorems also involve “lecture hall sequences,” that is, sequences constrained by the ratio of consecutive parts.

The National & State Context Of Stem Education: Past, Present, And Future, Henry M. Huckaby, Kamau Bobb, Charles Kutal, Judy Awong-Taylor

Interdisciplinary STEM Teaching & Learning Conference (2012-2019) (Archived)

N/A

Improving Education Of Mathematics Majors, Goran Lesaja

Mathematical Sciences: Faculty Presentations (1991-2022)

In this talk we will explore and discuss different ways of improving instruction of upper level mathematics classes. Several case studies will be presented, including in Calculus and Operations Research courses. We will also discuss the importance of extracurricular activities in education of mathematics majors. In particular, we will describe activities related to undergraduate mathematics competitions.

Cotorsion Pairs In C(R-Mod), Diego Bravo, Edgar E. Enochs, Alina Iacob, Overtoun M. G. Jenda, Juan Rada

Mathematical Sciences: Faculty Publications

In [8] Salce introduced the notion of a co-torsion pair (A, B) in the category of abelian groups. But his definitions and basic results carry over to more general abelian categories and have proved useful in a variety of settings. In this article we will consider complete cotorsion pairs (C, D)in the category C(R-Mod) of complexes of left R-modules over some ring R.If(C, D) is such a pair, and if C is closed un-der taking suspensions, we will show when we regard K(C) and K(D) as subcategories of the homotopy category K(R-Mod), then the embedding functors K(C) → K(R-Mod) and …

Weighted Generalized Beta Distribution Of The Second Kind, Yuan Ye, Broderick O. Oluyede, Marvis Pararai

Mathematical Sciences: Faculty Publications

In this paper, 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 generalizes the generalized beta distribution of second kind (GB2). The properties of these distributions including behavior of hazard functions, moments, variance, coefficients of variation, skewness and kurtosis are obtained. The moments of other weighted distributions that are related to WGB2 are obtained. Other important …

The Distribution Of Individual Stock Returns In A Modified Black-Scholes Option Pricing Model, Daniel Lee Richey

College of Graduate Studies: Theses & Dissertations

Author's abstract: There have been many attempts to find a model that can accurately price options. These models are built on many assumptions, including which probability distribution stock returns follow. In this paper, we test several distributions to see which best fit the log returns of 20 different companies over a period between November 1, 2006 to October 31, 2011. If a "best" distribution is found, a modified Black-Scholes model will be defined by modifying the Weiner process. We use Monte Carlo simulations to generate estimated prices under specified parameters, and compare these prices to those simulated by the model …

Theoretical Properties Of The Weighted Generalized Rayleigh And Related Distributions, Xueheng Shi, Broderick O. Oluyede, Marvis Pararai

Mathematical Sciences: Faculty Publications

In this paper, a new class of weighted generalization of the Rayleigh distribution is constructed and studied. The statistical properties of these distributions including the behavior of hazard or failure rate and reverse hazard functions, moments, moment generating function, mean, variance, coecient of variation, coecient of skewness, coecient of kurtosis are obtained. Other important properties including entropy (Shannon, beta and generalized) which are measures of the uncertainty in these distributions are also presented.

Theoretical Properties Of The Length-Biased Inverse Weibull Distribution, Jing X. Kersey, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

We investigate the length-biased inverse Weibull (LBIW) distribution, deriving its density function, hazard and reverse hazard functions, and reliability function. The moments, moment-generating function, Fisher information and Shannon entropy are also given. We discuss parameter estimation via the method of moments and maximum likelihood, and hypothesis testing for the LBIW and parent distributions.

On Some Order 6 Non-Symplectic Automorphisms Of Elliptic K3 Surfaces, Jimmy J. Dillies

Mathematical Sciences: Faculty Publications

We classify primitive non-symplectic automorphisms of order 6 on K3 surfaces. We show how their study can be reduced to the study of non-symplectic automorphisms of order 3 and to a local analysis of the fixed loci. In particular, we determine the possible fixed loci and show that when the Picard lattice is fixed, K3 surfaces come in mirror pairs.

Full Newton-Step Interior-Point Method For Linear Complementarity Problems, Goran Lesaja, Antre M. Drummer, Ljiljana Miletić

Mathematical Sciences: Faculty Publications

In this paper we consider an Infeasible Full Newton-step Interior-Point Method (IFNS-IPM) for monotone Linear Complementarity Problems (LCP). The method does not require a strictly feasible starting point. In addition, the method avoids calculation of the step size and instead takes full Newton-steps at each iteration. Iterates are kept close to the central path by suitable choice of parameters. The algorithm is globally convergent and the iteration bound matches the best known iteration bound for these types of methods.

Structure Of Colored Complete Graphs Free Of Proper Cycles, Vincent E. Coll, Colton Magnant, Kathleen Ryan

Mathematical Sciences: Faculty Publications

For a fixed integer m, we consider edge colorings of complete graphs which contain no properly edge colored cycle Cm as a subgraph. Within colorings free of these subgraphs, we establish global structure by bounding the number of colors that can induce a spanning and connected subgraph. In the case of smaller cycles, namely C4,C5, and C6, we show that our bounds are sharp.

Estimation Of Parameters In Weighted Generalized Beta Distribution Of The Second Kind, Yuan Ye, Broderick O. Oluyede, Marvis Pararai

Mathematical Sciences: Faculty Publications

This paper applies the class of weighted generalized beta distribution of the second kind (WGB2) as descriptive models for size distribution of income. The properties of WGB2 including mean, variance, coefficient of variation (CV), coefficient of skewness (CS), coefficient of kurtosis (CK) are presented. Other properties including top-sensitive index, bottom-sensitive index, mean logarithmic deviation (MLD) index and Theil index obtained from generalized entropy (GE) are applied in this paper. WGB2 proved to be in the generalized beta-F family of distributions, and maximum likelihood estimation (MLE) is used to obtain the parameter estimates. WGB2 is fitted to U.S. family income (2001-2009) …