Physical Sciences and Mathematics Commons^™

Individual Difference Correlates Of Being Sexually Unrestricted Yet Declining An Hiv Test, Nicholas S. Holtzman, Stephen W. Carden, Stacy W. Smallwood, Janice Steirn, S. Mason Garrison

Department of Mathematical Sciences Faculty Publications

Which individual differences accurately predict one’s decision to get tested for human immunodeficiency virus (HIV), and do individuals who have regular short-term sex get tested at higher rates? Two studies—one lab study (total valid N = 69, with n = 20 who were tested) and one involving a student health center (valid N = 250, n = 4 who were tested)—involved participants (total valid N = 319, with n = 24 who got tested) taking a number of personality and individual difference measures, including the dark triad (Machiavellianism, narcissism, and psychopathy). Then, in both studies, participants had the opportunity to …

K-8 Preservice Teachers’ Statistical Thinking When Determining Best Measure Of Center, Ha Nguyen, Eryn M. Stehr Maher, Gregory Chamblee, Sharon Taylor

Department of Mathematical Sciences Faculty Publications

The purpose of this study was to determine K-8 preservice teacher (PST) candidates’ statistical thinking when selecting the best center representation for the given data. Forty-four PSTs enrolled in a Statistics and Probability for K-8 Teachers course in a university located in the southeastern region of the United States were asked to complete a 2007 National Assessment of Educational Progress test item. All 44 PSTs’ data were qualitatively analyzed for correctness and statistical thinking strategies used. Findings were that most PSTs either incorrectly selected the mean, rather than median, as the best measure of center for the given data or …

Supporting Spatial Reasoning: Identifying Aspects Of Length, Area, And Volume In Textbook Definitions, Eryn M. Stehr Maher, Jia He

Department of Mathematical Sciences Faculty Publications

Length, area, and volume share structural similarities enabling flexibility in reasoning for real-world applications. Deep understanding of structures can help teachers connect these concepts to support their students’ mathematical reasoning and practices involving real-world situations. In mathematics textbooks designed for future teachers, definitions of length, area, and volume vary from procedural (e.g., use a ruler to measure side lengths, use formulas to calculate measures) to conceptual (e.g., construct appropriate n-dimensional units that tessellate the n-dimensional space) to formal (e.g., construct a function mapping qualitative size to a quantity of appropriate units). Most textbooks describe length, area, and volume …

Mathematics Learning, Teaching, And Equity In Policy And Programs: The Case Of Secondary Mathematics Teacher Education In The United States, Eryn M. Stehr Maher, Hyunyi Jung, Jill Newton

Department of Mathematical Sciences Faculty Publications

Professional organizations have provided recommendations for mathematics teaching and learning; however, few studies have investigated the practical integration of those recommendations into mathematics teacher education programs. In this study, we examine how the reported “big ideas” of courses in secondary mathematics teacher education programs emphasized the content and teaching practices necessary for future mathematics teachers, as recommended by policy documents. As part of a larger study, we conducted a series of interviews in secondary mathematics teacher education programs at four universities (names are descriptive pseudonyms): Great Lakes University (GLU), Midwestern Research University (MRU), Midwestern Urban University (MUU), and Southeastern Research …

Teacher Educators Learning With Prospective Teachers: Finding Relevant Mathematics In Our (Their) Lives, Lindsay M. Keazer, Eryn Stehr Maher

Department of Mathematical Sciences Faculty Publications

Two mathematics teacher educators (MTEs) discuss the mathematical contexts generated by prospective teachers (PTs) when pushed to look for relevant mathematics in their lives and communities. Through collaborative teacher action research focused on iterations of collecting, categorizing, and discussing PTs’ mathematical contexts, and posing selected examples for PTs’ own examination, layers of learning occurred for both PTs and MTEs. PTs began to craft more personalized, story-like contexts, seemingly noticing more mathematics in their lives. MTEs were unexpectedly pushed to clarify their thinking about what it means to develop contexts that are authentic and relevant, and to contemplate how their actions …

A Nonlinear Multi-Population Behavioral Model To Assess The Roles Of Education Campaigns, Random Supply Of Aids, And Delayed Art Treatment In Hiv/Aids Epidemics, Divine Wanduku

Department of Mathematical Sciences Faculty Publications

The successful reduction in prevalence rates of HIV in many countries is attributed to control measures such as information and education campaigns (IEC), antiretroviral therapy (ART), and national, multinational and multilateral support providing offcial developmental assistance (ODAs) to combat HIV. However, control of HIV epidemics can be interrupted by limited random supply of ODAs, high poverty rates and low living standards. This study presents a stochastic HIV/AIDS model with treatment assessing the roles of IEC, the supply of ODAs and early treatment in HIV epidemics. The supply of ODAs is assessed via the availability of medical and financial resources leading …

Some Asymptotic Properties Of Seirs Models Withnonlinear Incidence And Random Delays, Divine Wanduku, Broderick O. Oluyede

Department of Mathematical Sciences Faculty Publications

This paper presents the dynamics of mosquitoes and humans with general nonlinear incidence rate and multiple distributed delays for the disease. The model is a SEIRS system of delay differential equations. The normalized dimensionless version is derived; analytical techniques are applied to find conditions for deterministic extinction and permanence of disease. The BRN R0* and ESPR E(e–(μvT1+μT2)) are computed. Conditions for deterministic extinction and permanence are expressed in terms of R0* and E(e–(μvT1+μT2)) and applied to a P. vivax malaria scenario. Numerical results are given.

Exploration Using Without-Replacement Sampling Of Actions Is Sometimes Inferior, Stephen W. Carden, S. Dalton Walker

Department of Mathematical Sciences Faculty Publications

In many statistical and machine learning applications, without-replacement sampling is considered superior to with-replacement sampling. In some cases, this has been proven, and in others the heuristic is so intuitively attractive that it is taken for granted. In reinforcement learning, many count-based exploration strategies are justified by reliance on the aforementioned heuristic. This paper will detail the non-intuitive discovery that when measuring the goodness of an exploration strategy by the stochastic shortest path to a goal state, there is a class of processes for which an action selection strategy based on without-replacement sampling of actions can be worse than with-replacement …

Conceptualizing And Interpreting Mean And Median With Future Teachers, Eryn Stehr Maher, Ha Nguyen, Gregory Chamblee, Sharon Taylor

Department of Mathematical Sciences Faculty Publications

Mathematical Education of Teachers II (METII), echoed by the American Statistical Association publication, Statistical Education of Teachers, recommended teacher preparation programs support future teachers in developing deep understandings of mean and median, such that middle grades teachers may use them to “summarize, describe, and compare distributions” (Conference Board of Mathematical Sciences, 2012, p. 44; Franklin et al., 2015). Georgia Standards of Excellence require statistical reasoning from students beginning as early as 6-7 years old, including interpretation of measures of center and statistical reasoning about best measures of center (Georgia Department of Education, 2015). This level of understanding and interpretation of …

Hodge Theory On Transversely Symplectic Foliations, Yi Lin

Department of Mathematical Sciences Faculty Publications

In this paper, we develop symplectic Hodge theory on transversely symplectic foliations. In particular, we establish the symplectic dδ-lemma for any such foliations with the (transverse) s-Lefschetz property. As transversely symplectic foliations include many geometric structures, such as contact manifolds, co-symplectic manifolds, symplectic orbifolds, and symplectic quasi-folds as special examples, our work provides a unifying treatment of symplectic Hodge theory in these geometries.

As an application, we show that on compact K-contact manifolds, the s-Lefschetz property implies a general result on the vanishing of cup products, and that the cup length of a 2n+1 dimensional compact K-contact manifold with the …

Cahost Facilitating The Johnson-Neyman Technique For Two-Way Interactions In Multiple Regression, Stephen W. Carden, Nicholas Holtzman, Michael Strube

Department of Mathematical Sciences Faculty Publications

When using multiple regression, researchers frequently wish to explore how the relationship between two variables is moderated by another variable; this is termed an interaction. Historically, two approaches have been used to probe interactions: the pick-a-point approach and the Johnson-Neyman (JN) technique. The pick-a-point approach has limitations that can be avoided using the JN technique. Currently, the software available for implementing the JN technique and creating corresponding figures lacks several desirable features–most notably, ease of use and figure quality. To fill this gap in the literature, we offer a free Microsoft Excel 2013 workbook, CAHOST (a concatenation of the first …

Vanishing Of Ext And Tor Over Fiber Products, Saeed Nasseh, Sean Sather-Wagstaff

Department of Mathematical Sciences Faculty Publications

Consider a non-trivial fiber product R=S×kT of local rings S, T with common residue field k. Given two finitely generate R-modules M and N, we show that if TorRi(M,N)=0=TorRi+1(M,N) for some i≥5, then pdR(M)≤1 or pdR(N)≤1. From this, we deduce several consequence, for instance, that R satisfies the Auslander-Reiten Conjecture.

Global Analysis Of A Stochastic Two-Scale Network Human Epidemic Dynamic Model With Varying Immunity Period, Divine Wanduku, G. S. Ladde

Department of Mathematical Sciences Faculty Publications

A stochastic SIR epidemic dynamic model with distributed-time-delay, for a two-scale dynamic population is derived. The distributed time delay is the varying naturally acquired immunity period of the removal class of individuals who have recovered from the infection, and have acquired natural immunity to the disease. We investigate the stochastic asymptotic stability of the disease free equilibrium of the epidemic dynamic model, and verify the impact on the eradication of the disease.

CO-Characterization Of Symplectic And Contact Embeddings And Lagrangian Rigidity, Stefan Müller

Department of Mathematical Sciences Faculty Publications

We present a novel C0-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by J.-C. Sikorav and Y. Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of C0-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from J-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a C0-characterization of contact embeddings …

The Gamma-Generalized Inverse Weibull Distribution With Applications To Pricing And Lifetime Data, Broderick O. Oluyede, Boikanyo Makubate, Divine Wanduku, Ibrahim Elbatal, Valeriia Sherina

Department of Mathematical Sciences Faculty Publications

A new distribution called the gamma-generalized inverse Weibull distribution which includes inverse exponential, inverse Rayleigh, inverse Weibull, Frechet, generalized inverse Weibull, gamma-exponentiated inverse exponential, exponentiated inverse exponential, Zografos and Balakrishnan-generalized inverse Weibull, Zografos and Balakrishnan-inverse Weibull, Zografos and Balakrishnan-generalized inverse exponential, Zografos and Balakrishnan-inverse exponential, Zografos and Balakrishnan-generalized inverse Rayleigh, Zografos and Balakrishnan-inverse Rayleigh, and Zografos and Balakrishnan-Fr'echet distributions as special cases is proposed and studied in detail. Some structural properties of this new distribution including density expansion, moments, Renyi entropy, distribution of the order statistics, moments of the order statistics and L-moments are presented. Maximum likelihood estimation technique is …

Ghost Series And A Motivated Proof Of The Andrews–Bressoud Identities, Shashank Kanade, James Lepowsky, Matthew C. Russell, Andrew Sills

Department of Mathematical Sciences Faculty Publications

We present what we call a “motivated proof” of the Andrews–Bressoud partition identities for even moduli. A “motivated proof” of the Rogers–Ramanujan identities was given by G.E. Andrews and R.J. Baxter, and this proof was generalized to the odd-moduli case of Gordon's identities by J. Lepowsky and M. Zhu. Recently, a “motivated proof” of the somewhat analogous Göllnitz–Gordon–Andrews identities has been found. In the present work, we introduce “shelves” of formal series incorporating what we call “ghost series,” which allow us to pass from one shelf to the next via natural recursions, leading to our motivated proof. We anticipate that …

Gorenstein Projective Precovers, Sergio Estrada, Alina Iacob, Katelyn A. Coggins

Department of Mathematical Sciences Faculty Publications

We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In Sect. 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension) and that are not right coherent.

On Gorenstein Fiber Products And Applications, Saeed Nasseh, Ryo Takahashi, Keller Vandebogert

Department of Mathematical Sciences Faculty Publications

We show that a non-trivial fiber product S×kT of commutative noetherian local rings S,T with a common residue field k is Gorenstein if and only if it is a hypersurface of dimension 1. In this case, both S and T are regular rings of dimension 1. We also give some applications of this result.

Totally Acyclic Complexes, Sergio Estrada, Xianhui Fu, Alina Iacob

Department of Mathematical Sciences Faculty Publications

It is known that over an Iwanaga–Gorenstein ring the Gorenstein injective (Gorenstein projective, Gorenstein flat) modules are simply the cycles of acyclic complexes of injective (projective, flat) modules. We consider the question: are these characterizations only working over Iwanaga–Gorenstein rings? We prove that if R is a commutative noetherian ring of finite Krull dimension then the following are equivalent: 1. R is an Iwanaga–Gorenstein ring. 2. Every acyclic complex of injective modules is totally acyclic. 3. The cycles of every acyclic complex of Gorenstein injective modules are Gorenstein injective. 4. Every acyclic complex of projective modules is totally acyclic. 5. …

Gorenstein Flat And Projective (Pre)Covers, Sergio Estrada, Alina Iacob, Sinem Odabasi

Department of Mathematical Sciences Faculty Publications

We consider a right coherent ring R. We prove that the class of Gorenstein flat complexes is covering in the category of complexes of left R-modules Ch(R). When R is also left n-perfect, we prove that the class of Gorenstein projective complexes is special precovering in Ch(R).

Multiple Solutions With Constant Sign Of A Dirichlet Problem For A Class Of Elliptic Systems With Variable Exponent Growth, Li Yin, Jinghua Yao, Qihu Zhang, Chunshan Zhao

Department of Mathematical Sciences Faculty Publications

We present here, in the system setting, a new set of growth conditions under which we manage to use a novel method to verify the Cerami compactness condition. By localization argument, decomposition technique and variational methods, we are able to show the existence of multiple solutions with constant sign for the problem without the well-known Ambrosetti--Rabinowitz type growth condition. More precisely, we manage to show that the problem admits four, six and infinitely many solutions respectively.

Gorenstein Injective Envelopes And Covers Over Two Sided Noetherian Rings, Alina Iacob

Department of Mathematical Sciences Faculty Publications

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.

A Zariski-Local Notion Of F-Total Acyclicity For Complexes Of Sheaves, Lars Winther Christensen, Sergio Estrada, Alina Iacob

Department of Mathematical Sciences Faculty Publications

We study a notion of total acyclicity for complexes of flat sheaves over a scheme. It is Zariski-local—i.e. it can be verified on any open affine covering of the scheme—and for sheaves over a quasi-compact semi-separated scheme it agrees with the categorical notion. In particular, it agrees, in their setting, with the notion studied by Murfet and Salarian for sheaves over a noetherian semi-separated scheme. As part of the study we recover, and in several cases extend the validity of, recent results on existence of covers and precovers in categories of sheaves. One consequence is the existence of an adjoint …

Minimization And Eulerian Formulation Of Differential Geometry Based Nonpolar Multiscale Solvation Models, Zhan Chen

Department of Mathematical Sciences Faculty Publications

In this work, the existence of a global minimizer for the previous Lagrangian formulation of nonpolar solvation model proposed in [1] has been proved. One of the proofs involves a construction of a phase field model that converges to the Lagrangian formulation. Moreover, an Eulerian formulation of nonpolar solvation model is proposed and implemented under a similar parameterization scheme to that in [1]. By doing so, the connection, similarity and difference between the Eulerian formulation and its Lagrangian counterpart can be analyzed. It turns out that both of them have a great potential in solvation prediction for nonpolar molecules, while …

Extension Groups For Dg Modules, Saeed Nasseh, Sean Sather-Wagstaff

Department of Mathematical Sciences Faculty Publications

Let M and N be differential graded (DG) modules over a positively graded commutative DG algebra A. We show that the Ext-groups ExtiA(M,N) defined in terms of semi-projective resolutions are not in general isomorphic to the Yoneda Ext-groups YExtiA(M,N) given in terms of equivalence classes of extensions. On the other hand, we show that these groups are isomorphic when the first DG module is semi-projective.

Rank Of Submatrices Of The Pascal Matrix, Scott N. Kersey

Department of Mathematical Sciences Faculty Publications

In a previous paper, we derived necessary and sufficient conditions for the invertibility of square submatrices of the Pascal upper triangular matrix. To do so, we established a connection with the two-point Birkhoff interpolation problem. In this paper, we extend this result by deriving a formula for the rank of submatrices of the Pascal matrix. Our formula works for both square and non-square submatrices. We also provide bases for the row and column spaces of these submatrices. Further, we apply our result to one-point lacunary polynomial approximation.

A Generalized Class Of Exponentiated Modified Weibull Distribution With Applications, Shusen Pu, Broderick O. Oluyede, Yuqi Qui, Daniel F. Linder

Department of Mathematical Sciences Faculty Publications

In this paper, a new class of five parameter gamma-exponentiated or generalized modified Weibull (GEMW) distribution which includes exponential, Rayleigh, Weibull, modified Weibull, exponentiated Weibull, exponentiated exponential, exponentiated modified Weibull, exponentiated modified exponential, gamma-exponentiated exponential, gamma-exponentiated Rayleigh, gamma-modified Weibull, gamma-modified exponential, gamma-Weibull, gamma-Rayleigh and gamma-exponential distributions as special cases is proposed and studied. Mathematical properties of this new class of distributions including moments, mean deviations, Bonferroni and Lorenz curves, distribution of order statistics and Renyi entropy are presented. Maximum likelihood estimation technique is used to estimate the model parameters and applications to real data sets presented in order to illustrate …

Equivariant Formality Of Transversely Symplectic Foliations And Frobenius Manifolds, Yi Lin, Xiangdong Yang

Department of Mathematical Sciences Faculty Publications

Consider the Hamiltonian action of a compact connected Lie group on a transversely symplectic foliation whose basic cohomology satisfies the Hard Lefschetz property. We establish an equivariant formality theorem and an equivariant symplectic dδ-lemma in this setting. As an application, we show that there exists a natural Frobenius manifold structure on the equivariant basic cohomology of the given foliation. In particular, this result provides a class of new examples of dGBV-algebras whose cohomology carries a Frobenius manifold structure.

Lefschetz Contact Manifolds And Odd Dimensional Symplectic Geometry, Yi Lin

Department of Mathematical Sciences Faculty Publications

In the literature, there are two different versions of Hard Lefschetz theorems for a compact Sasakian manifold. The first version, due to Kacimi-Alaoui, asserts that the basic cohomology groups of a compact Sasakian manifold satisfies the transverse Lefschetz property. The second version, established far more recently by Cappelletti-Montano, De Nicola, and Yudin, holds for the De Rham cohomology groups of a compact Sasakian manifold. In the current paper, using the formalism of odd dimensional symplectic geometry, we prove a Hard Lefschetz theorem for compact K-contact manifolds, which implies immediately that the two existing versions of Hard Lefschetz theorems are mathematically …

The Log-Logistic Weibull Distribution With Applications To Lifetime Data, Broderick O. Oluyede, Susan Foya, Gayan Warahena-Liyanage, Shujiao Huang

Department of Mathematical Sciences Faculty Publications

In this paper, a new generalized distribution called the log-logistic Weibull (LLoGW) distribution is developed and presented. This distribution contain the log-logistic Rayleigh (LLoGR), log-logistic exponential (LLoGE) and log-logistic (LLoG) distributions as special cases. The structural properties of the distribution including the hazard function, reverse hazard function, quantile function, probability weighted moments, moments, conditional moments, mean deviations, Bonferroni and Lorenz curves, distribution of order statistics, L-moments and Renyi entropy are derived. Method of maximum likelihood is used to estimate the parameters of this new distribution. A simulation study to examine the bias, mean square error of the maximum likelihood estimators …