Physical Sciences and Mathematics Commons^™

Quantum States Localized On Lagrangian Submanifolds, François Ziegler

Department of Mathematical Sciences Faculty Presentations

Let X be a symplectic manifold and Aut(L) the automorphism group of a Kostant-Souriau line bundle on X. *Quantum states for X*, as defined by J.-M. Souriau in the 1990s, are certain positive-definite functions on Aut(L) or, less ambitiously, on any “large enough” subgroup G of Aut(L). This definition has two major drawbacks: when G = Aut(L) there are no known examples; and when G is a Lie subgroup the notion is far from selective enough. In this talk I’ll introduce the concept of a quantum state *localized at Y *, where Y is a coadjoint orbit of a subgroup …

Functions On Adjacent Vertex Degrees Of Trees With Given Degree Sequence, Hua Wang

Department of Mathematical Sciences Faculty Publications

In this note we consider a discrete symmetric function f(x, y) where f(x; a) + f(y, b) ≥ f(y, a) + f(x, b) for any x ≥ y and a ≥ b, associated with the degrees of adjacent vertices in a tree. The extremal trees with respect to the corresponding graph invariant, defined as Σ _uv∈E(T) f(deg(u), deg(v)), are characterized by the “greedy tree” and “alternating greedy tree”. This is achieved through simple generalizations of previously used ideas on similar questions. As special cases, the already known extremal structures of the Randić index follow as corollaries. The extremal structures for …

Symplectic Mackey Theory, Francois Ziegler

Department of Mathematical Sciences Faculty Publications

Many years ago Kazhdan, Kostant and Sternberg defined the notion of inducing a hamiltonian action from a Lie subgroup. In this paper, we develop the attendant imprimitivity theorem and Mackey analysis in the full generality needed to deal with arbitrary closed normal subgroups.

A Case Study Of How Ninth Grade Mathematics Students Construct Knowledge During A Productive Failure Model, Amy F. Westbrook Dr.

Georgia Educational Research Association Conference

The purpose of this qualitative study was to explain how ninth grade mathematics students at a rural high school in Georgia constructed knowledge through student talk when problem solving using Kapur’s (2012) productive failure design. An embedded case study design was used to understand how a group of students constructed knowledge through their use of talk, persistence during the task, and use of prior knowledge while working on a productive failure modeled task. Triangulation resulted from the collected data from multiple sources, which included videotaping, interviewing, and analyzing student artifacts. Utilization of the constructivist perspectives of Vygotsky (1934/1962), Piaget (1971), …

Synthetic Lethality As A Promising Approach For Targeted Cancer Prevention, Wei Tu, Hua Wang, Guang Peng

Department of Mathematical Sciences Faculty Publications

Carcinogenesis is recognized as a multistep process. It occurs over a relative long span of time, which offers intervention opportunities for cancer prevention [1] . Using drugs to prevent cancer rather than treat cancer is the major research goal in the field of ‘chemoprevention’. Tremendous research efforts have been devoted toward using natural, synthetic or biological agents to prevent, suppress or delay the initiation and or the progression of premalignant cells to cancer [1] . However a big challenge for effective cancer prevention is to identify chemoprevention agents with demonstrable efficacy and safety for healthy general …

Fast Inverse Distance Weighting-Based Spatiotemporal Interpolation: A Web-Based Application Of Interpolating Daily Fine Particulate Matter Pm2.5 In The Contiguous U.S. Using Parallel Programming And K-D Tree, Lixin Li, Travis Losser, Charles Yorke, Reinhard E. Piltner

Department of Mathematical Sciences Faculty Publications

Epidemiological studies have identified associations between mortality and changes in concentration of particulate matter. These studies have highlighted the public concerns about health effects of particulate air pollution. Modeling fine particulate matter PM2.5exposure risk and monitoring day-to-day changes in PM2.5 concentration is a critical step for understanding the pollution problem and embarking on the necessary remedy. This research designs, implements and compares two inverse distance weighting (IDW)-based spatiotemporal interpolation methods, in order to assess the trend of daily PM2.5 concentration for the contiguous United States over the year of 2009, at both the census block group level and county level. …

Epistasis In Predator-Prey Relationships, Iuliia Inozemtseva, James P. Braselton

Department of Mathematical Sciences Faculty Publications

Epistasis is the interaction between two or more genes to control a single phenotype. We model epistasis of the prey in a two-locus two-allele problem in a basic predator-prey relationship. The resulting model allows us to examine both population sizes as well as genotypic and phenotypic frequencies. In the context of several numerical examples, we show that if epistasis results in an undesirable or desirable phenotype in the prey by making the particular genotype more or less susceptible to the predator or dangerous to the predator, elimination of undesirable phenotypes and then genotypes occurs.

Fejér And Suffridge Polynomials In The Delayed Feedback Control Theory, Dmitriy Dmitrishin, Anna Khamitova, Anatolii Korenovskyi, Alexander M. Stokolos

Department of Mathematical Sciences Faculty Publications

A remarkable connection between optimal delayed feedback control (DFC) and complex polynomial mappings of the unit disc is established. The explicit form of extremal polynomials turns out to be related with the Fejer polynomials. The constructed DFC can be used to stabilize cycles of one-dimensional non-linear discrete systems.

Existence Of Positive Solutions For P(X)-Laplacian Equations With A Singular Nonlinear Term, Jingjing Liu, Qihu Zhang, Chunshan Zhao

Department of Mathematical Sciences Faculty Publications

In this article, we study the existence of positive solutions for the p(x)-Laplacian Dirichlet problem −∆_p(x)u = λf(x, u) in a bounded domain Ω ⊂ R^N. The singular nonlinearity term f is allowed to be either f(x, s) → +∞, or f(x, s) → +∞ as s → 0+ for each x ∈ Ω. Our main results generalize the results in [15] from constant exponents to variable exponents. In particular, we give the asymptotic behavior of solutions of a simpler equation which is useful for finding supersolutions of differential equations with variable exponents, which is of independent …

Eigenvalue Estimates Of Laplacians Defined By Fractal Measures, Sze-Man Ngai

Department of Mathematical Sciences Faculty Presentations

We study various lower and upper estimates for the first eigenvalue of Dirichlet Laplacians defined by positive Borel measures on bounded open subsets of Euclidean spaces. These Laplacians and the corresponding eigenvalue estimates differ from classical ones in that the defining measures can be singular. By using properties of self-similar measures, such as Strichartz's second-order self-similar identities, we improve some of the eigenvalue estimates.

Generalizations Of The Inverse Weibull And Related Distributions With Applications, Broderick O. Oluyede, Tao Yang

Broderick O. Oluyede

In this paper, the generalized inverse Weibull distribution including the exponentiated or proportional reverse hazard and Kumaraswamy generalized inverse Weibull distributionsare presented. Properties of these distributions including the behavior of the hazard and reverse hazard functions, moments, coefficients of variation, skewness, andkurtosis, entropy, Fisher information matrix are studied. Estimates of the model parameters via method of maximum likelihood (ML), and method of moments (MOM) are presented for complete and censored data. Numerical examples are also presented.

Generalizations Of The Inverse Weibull And Related Distributions With Applications, Broderick O. Oluyede, Tao Yang

Department of Mathematical Sciences Faculty Publications

In this paper, the generalized inverse Weibull distribution including the exponentiated or proportional reverse hazard and Kumaraswamy generalized inverse Weibull distributionsare presented. Properties of these distributions including the behavior of the hazard and reverse hazard functions, moments, coefficients of variation, skewness, andkurtosis, entropy, Fisher information matrix are studied. Estimates of the model parameters via method of maximum likelihood (ML), and method of moments (MOM) are presented for complete and censored data. Numerical examples are also presented.

Localized Quantum States, François Ziegler

François Ziegler

Let X be a symplectic manifold and Aut(L) the automorphism group of a Kostant-Souriau line bundle on X. *Quantum states for X*, as defined by J.-M. Souriau in the 1990s, are certain positive-definite functions on Aut(L) or, less ambitiously, on any "large enough" subgroup G of Aut(L). This definition has two major drawbacks: when G=Aut(L) there are no known examples; and when G is a Lie subgroup the notion is, as we shall see, far from selective enough. In this paper we introduce the concept of a quantum state *localized at Y*, where Y is a coadjoint orbit of a …

Localized Quantum States, Francois Ziegler

Department of Mathematical Sciences Faculty Publications

Let X be a symplectic manifold and Aut(L) the automorphism group of a Kostant-Souriau line bundle on X. *Quantum states for X*, as defined by J.-M. Souriau in the 1990s, are certain positive-definite functions on Aut(L) or, less ambitiously, on any "large enough" subgroup G of Aut(L). This definition has two major drawbacks: when G=Aut(L) there are no known examples; and when G is a Lie subgroup the notion is, as we shall see, far from selective enough. In this paper we introduce the concept of a quantum state *localized at Y*, where Y is a coadjoint orbit of a …

Analysis Of Resource Allocation Through Game Theory, Brittney L. Benzio

Honors College Theses

Game theory presents a set of decision-makers in a model in order to simulate how they will interact according to a set of rules. The game is set up with a set of players, actions, and preferences. The model allows each player to, in some way, be affected by the actions of all players. Nash equilibrium illustrates that the best action for any given player depends on the other players’ actions, and so, each player must make some assumption about what the competition will do. The goal of this project is to model situations of different car companies to improve …

On The Boundary Blow-Up Solutions Of P(X)-Laplacian Equations With Gradient Terms, Yuan Liang, Qihu Zhang, Chunshan Zhao

Department of Mathematical Sciences Faculty Publications

In this paper we investigate boundary blow-up solutions of the problem

⎧⎩⎨⎪⎪−△p(x)u+f(x,u)=ρ(x,u)+K(|x|)|∇u|δ(|x|) in Ω, u(x)→+∞ as d(x, ∂Ω)→0,

where −△p(x)u=−div(|∇u|p(x)−2∇u) is called p(x)-Laplacian. The existence of boundary blow-up solutions is proved and the singularity of boundary blow-up solution is also given for several cases including the case of ρ(x,u) being a large perturbation (namely, ρ(x,u(x))f(x,u(x))→1 as x→∂Ω). In particular, we do not have the comparison principle.

Odd Fibbinary Numbers And The Golden Ratio, Linus Lindroos, Andrew Sills, Hua Wang

Department of Mathematical Sciences Faculty Publications

The fibbinary numbers are positive integers whose binary representation contains no consecutive ones. We prove the following result: If the jth odd fibbinary is the nth odd fibbinary number, then j={nɸ²]-1.

Weighted Inverse Weibull Distribution: Statistical Properties And Applications, Valeriia Sherina, Broderick O. Oluyede

Department of Mathematical Sciences Faculty Publications

In this paper, the weighted inverse Weibull (WIW) class of distributions is proposed and studied. This class of distributions contains several models such as: length-biased, hazard and reverse hazard proportional inverse Weibull, proportional inverse Weibull, inverse Weibull, inverse exponential, inverse Rayleigh, and Fr´echet distributions as special cases. Properties of these distributions including the behavior of the hazard function, moments, coefficients of variation, skewness, and kurtosis, R´enyi entropy and Fisher information are presented. Estimates of the model parameters via method of maximum likelihood (ML) are presented. Extensive simulation study is conducted and numerical examples are given.

Multiscale Geometric Modeling Of Macromolecules I: Cartesian Representation, Kelin Xia, Xin Feng, Zhan Chen, Yiying Tong, Guo-Wei Wei

Zhan Chen

This paper focuses on the geometric modeling and computational algorithm development of biomolecular structures from two data sources: Protein Data Bank (PDB) and Electron Microscopy Data Bank (EMDB) in the Eulerian (or Cartesian) representation. Molecular surface (MS) contains non-smooth geometric singularities, such as cusps, tips and self-intersecting facets, which often lead to computational instabilities in molecular simulations, and violate the physical principle of surface free energy minimization. Variational multiscale surface definitions are proposed based on geometric flows and solvation analysis of biomolecular systems. Our approach leads to geometric and potential driven Laplace–Beltrami flows for biomolecular surface evolution and formation. The …

Multiscale Geometric Modeling Of Macromolecules I: Cartesian Representation, Kelin Xia, Xin Feng, Zhan Chen, Yiying Tong, Guo-Wei Wei

Department of Mathematical Sciences Faculty Publications

This paper focuses on the geometric modeling and computational algorithm development of biomolecular structures from two data sources: Protein Data Bank (PDB) and Electron Microscopy Data Bank (EMDB) in the Eulerian (or Cartesian) representation. Molecular surface (MS) contains non-smooth geometric singularities, such as cusps, tips and self-intersecting facets, which often lead to computational instabilities in molecular simulations, and violate the physical principle of surface free energy minimization. Variational multiscale surface definitions are proposed based on geometric flows and solvation analysis of biomolecular systems. Our approach leads to geometric and potential driven Laplace–Beltrami flows for biomolecular surface evolution and formation. The …

Caep, Nctm, And Secondary Mathematics Program Revisions, Dianna J. Spence

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

Eight assessments were developed for CAEP (formerly NCATE) and NCTM recognition of our secondary mathematics program. These assessments include internship work samples, field evaluations, and candidate portfolios addressing content knowledge, pedagogical methods, and mathematics technology. Based on data collected from these assessments, alongside ongoing evaluation of the program, several curriculum and program revisions were implemented, including: 1) development of mathematics content-specific courses in classroom management, assessment, and secondary curriculum; 2) restructuring of a senior seminar course in mathematics education; and 3) an increased content focus in probability and statistics. The adoption of new NCTM standards in the CAEP review process …

A Mathematics Teacher’S Journey Of Identity Construction And Change, Anthony B. Stinson

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

Despite some gains, improving mathematics instruction remains an area of concern in the United States. The implementation of the Common Core Standards and the challenge of teaching the 21st Century student require mathematics teachers to examine their pedagogy to determine if they need to change or improve their practices. This paper provides a personal account of my journey when determining my identity as a mathematics teacher and how constructing my identity helped in changing and improving my practices as a mathematics teacher. The study was done using autoethnography, a burgeoning research method, and identity theory. This study has the goals …

A Decomposition Of Gallai Multigraphs, Alexander Halperin, Colton Magnant, Kyle Pula

Department of Mathematical Sciences Faculty Publications

An edge-colored cycle is rainbow if its edges are colored with distinct colors. A Gallai (multi)graph is a simple, complete, edge-colored (multi)graph lacking rainbow triangles. As has been previously shown for Gallai graphs, we show that Gallai multigraphs admit a simple iterative construction. We then use this structure to prove Ramsey-type results within Gallai colorings. Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decomposition into directed trees.

Building A Better Bijection Between Classes Of Compositions, James D. Diffenderfer

Department of Mathematical Sciences Faculty Publications

A bijective proof is given for the following theorem: The number of compositions of n into parts congruent to a (mod b) equals the number of compositions of n + b - a into parts congruent to b (mod a) that are greater than or equal to b. The bijection is then shown to preserve palindromicity.

Note On Rainbow Connection In Oriented Graphs With Diameter 2, Rebecca Holliday, Colton Magnant, Pouria Salehi Nowbandegani

Department of Mathematical Sciences Faculty Publications

In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter $2$. For a tournament $T$ of diameter $2$, we show $2 \leq \overrightarrow{rc}(T) \leq 3$. Furthermore, we provide a general upper bound on the rainbow $k$-connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of $k^{th}$ diameter $2$ has rainbow $k$-connection number at most approximately $k^{2}$.

Rainbow Generalizations Of Ramsey Theory - A Dynamic Survey, Shinya Fujita, Colton Magnant, Kenta Ozeki

Department of Mathematical Sciences Faculty Publications

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.

Rainbow Generalizations Of Ramsey Theory - A Dynamic Survey, Shinya Fujita, Colton Magnant, Yaping Mao, Kenta Ozeki

Theory and Applications of Graphs

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.

Note On Rainbow Connection In Oriented Graphs With Diameter 2, Rebecca Holliday, Colton Magnant, Pouria Salehi Nowbandegani

Theory and Applications of Graphs

In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter 2. For a tournament Τ of diameter 2, we show 2 ≤ → rc (Τ) ≤ 3. Furthermore, we provide a general upper bound on the rainbow κ-connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of κ^thdiameter 2 has rainbow κ-connection number at most approximately κ².

Proceedings Of The Eighth 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

• GAMTE 2014 Officers
• GAMTE 2014 Conference Committee
• GAMTE 2014 Proceedings Committee
• Purposes and Goals of GAMTE
• Letter from the President

Design Considerations For Visually-Aided Discussion Prompts: Emphasizing Mathematical Reasoning In Teacher Education, Anne Marie S. Marshall, Kadian M. Callahan

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

The availability and familiarity of online discussion tools create new instructional options that teacher educators can use to foster prospective teachers’ understanding of mathematics. In particular, online discussion blogs provide an avenue through which teacher educators can press prospective teachers to explore mathematical concepts and share their mathematical reasoning with peers. Furthermore, by incorporating visual stimulations as a design component of these discussion blogs, prospective teachers can make sense of and respond to others’ ideas about mathematical concepts with greater clarity. This paper shares preliminary findings of a research study that examined the extent to which the design of a …