Applied Mathematics Commons^™

Some Generalized Trigonometric Sine Functions And Their Applications, Dongming Wei, Yu Liu, Mohamed B. Elgindi

Mathematics Faculty Publications

In this paper, it is shown that D. Shelupsky's generalized sine function, and various general sine functions developed by P. Drabek, R. Manasevich and M. Otani, P. Lindqvist, including the generalized Jacobi elliptic sine function of S. Takeuchi can be defined by systems of first order nonlinear ordinary differential equations with initial conditions. The structure of the system of differential equations is shown to be related to the Hamilton System in Lagrangian Mechanics. Numerical solutions of the ODE systems are solved to demonstrate the sine functions graphically. It is also demonstrated that the some of the generalized sine functions can …

On The Global Solvability Of A Class Of Fourth-Order Nonlinear Boundary Value Problems, M.B.M. Elgindi, Dongming Wei

Mathematics Faculty Publications

In this paper we prove the global solvability of a class of fourth-order nonlinear boundary value problems that govern the deformation of a Hollomon’s power-law plastic beam subject to an axial compression and nonlinear lateral constrains. For certain ranges of the acting axial compression force, the solvability of the equations follows from the monotonicity of the fourth order nonlinear differential operator. Beyond these ranges the monotonicity of the operator is lost. It is shown that, in this case, the global solvability may be generated by the lower order nonlinear terms of the equations for a certain type of constrains.

Travelling Wave Solutions Of Burgers' Equation For Gee-Lyon Fluid Flows, Dongming Wei, Ken Holladay

Mathematics Faculty Publications

In this work we present some analytic and semi-analytic traveling wave solutions of generalized Burger' equation for isothermal unidirectional flow of viscous non-Newtonian fluids obeying Gee-Lyon nonlinear rheological equation. The solution of Burgers' equation for Newtonian flow as a special case. We also derive estimates of shock thickness for non-Newtonian flows.

Independent Dominating Sets In Triangle-Free Graphs, Wayne Goddard, Jeremy Lyle

Faculty Publications

The independent domination number of a graph is the smallest cardinality of an independent set that dominates the graph. In this paper we consider the independent domination number of triangle-free graphs. We improve several of the known bounds as a function of the order and minimum degree, thereby answering conjectures of Haviland.

Asymptotic Reliability Rheory Of K-Out-Of-N Systems, Nuria Torrado, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

We formulate a theory that allows us to formulate a simple criterion that ensures that two k-out-of-n systems A and are not ordered. If the systems fail the criterion, it does not follow they are ordered. Thus the theory only serves to avoid some a priori useless comparisons: when neither A nor can be said to be better than the other. The power of the theory lies in its wide potential applicability: the assumptions involve very weak estimates on the asymptotic behavior (as t→0 and as t→∞) of the constituent survival probabilities. We include examples.

Lattice Residuability, Philip Theodore Thiem

Masters Theses

"Residuated lattices form the basis of certain kinds of logical interpretations. Also, complete commutative integral zero-bounded residuated lattices are used as a set of truth values for fuzzy logic values, Which are more general than the traditional bounded interval introduced by Zadeh. At times, it is important to know whether or not the lattice can be residuated in the first place. This thesis reviews the literature in lattice residuability and adds more observations. Specifically, (1) bounded chains and top-residuated lattices are show [sic] to be residuable, and (2) additional conditions necessary for residuability are established"--Abstract, page iii.

Constrained Optimal Control For A Multi-Group Discrete Time Influenza Model, Paula Andrea Gonzalez Parra

Open Access Theses & Dissertations

During the last decades, mathematical epidemiological models have been used to understand the dynamics of infectious diseases and guide public health policy. In particular, several continuous models have been considered to study single in uenza outbreaks and the impact of dierent control policies. In this dissertation, a discrete time model is introduced in order to study optimal control strategies for in uenza transmission; since epidemiological data is collected on discrete units of time, a discrete formulation is more ecient. From a mathematical point of view, continuous time model are easier to analyze, however, the numerical solution of discrete-time models is …

Graham's Variety And Perverse Sheaves On The Nilpotent Cone, Amber Russell

LSU Doctoral Dissertations

In recent work, Graham has defined a variety which maps to the nilpotent cone, and which shares many properties with the Springer resolution. However, Graham's map is not an isomorphism over the principal orbit, and for type A in particular, its fibers have a nice relationship with the fundamental groups of the nilpotent orbits. The goal of this dissertation is to determine which simple perverse sheaves appear when the Decomposition Theorem for perverse sheaves is applied in Graham's setting for type A, and to begin to answer this question in the other types as well. In Chapter 1, we give …

Operational Methods For Evolution Equations, Lee Gregory Windsperger

LSU Doctoral Dissertations

This dissertation refines and further develops numerical methods for the inversion of the classical Laplace transform and explores the effectiveness of these methods when applied (a) to an asymptotic generalization of the Laplace transform for generalized functions and (b) to the numerical approximation of solutions of ill-posed evolution equations (e.g. backwards in time problems).

Chapter 1 of the dissertation reviews some of the key features of asymptotic Laplace transform theory and its application to evolution equations. Although some of the statements and results contain slight modifications and improvements, the material presented in Chapter 1 is known …

Subgradient Formulas For Optimal Control Problems With Constant Dynamics, Lingyan Huang

LSU Doctoral Dissertations

In this thesis our fi_x000C_rst concern is the study of the minimal time function corresponding to control problems with constant convex dynamics and closed target sets. Unlike previous work in this area, we do not make any nonempty interior or calmness assumptions and the minimal time functions is generally non-Lipschitzian. We show that the Proximal and Fréchet subgradients of the minimal time function are computed in terms of normal vectors to level sets. And we also computed the subgradients of the minimal time function in terms of the F-projection. Secondly, we consider the value function for Bolza Problem in optimal …

Paley-Wiener Theorem For Line Bundles Over Compact Symmetric Spaces, Vivian Mankau Ho

LSU Doctoral Dissertations

We generalize a Paley-Wiener theorem to homogeneous line bundles $L_\chi$ on a compact symmetric space U/K with $\chi$ a nontrivial character of K. The Fourier coefficients of a $\chi$-bi-coinvariant function f on U are defined by integration of f against the elementary spherical functions of type $\chi$ on U, depending on a spectral parameter $\mu$, which in turn parametrizes the $\chi$-spherical representations $\pi$ of U. The Paley-Wiener theorem characterizes f with sufficiently small support in terms of holomorphic extendability and exponential growth of their $\chi$-spherical Fourier transforms. We generalize Opdam's estimate for the hypergeometric functions in a bigger domain with …

C0 Interior Penalty Methods For Cahn-Hilliard Equations, Shiyuan Gu

LSU Doctoral Dissertations

In this work we study C0 interior penalty methods for Cahn-Hilliard equations. In Chapter 1 we introduce Cahn-Hilliard equations and the time discretization that leads to linear fourth order boundary value problems. In Chapter 2 we review related fundamentals of finite element methods and multigrid methods. In Chapter 3 we formulate the discrete problems for linear fourth order boundary value problems with the boundary conditions of the Cahn-Hilliard type, which are called C0 interior penalty methods, and we carry out the convergence analysis. In Chapter 4 we consider multigrid methods for the C0 interior penalty methods. We present two smoothing …

Some Tracking Problems For Aerospace Models With Input Constraints, Aleksandra Gruszka

LSU Doctoral Dissertations

We study tracking controller design problems for key models of planar vertical takeoff and landing (PVTOL) aircraft and unmanned air vehicles (UAVs). The novelty of our PVTOL work is the global boundedness of our controllers in the decoupled coordinates, the positive uniform lower bound on the thrust controller, the applicability of our work to cases where the velocity measurements may not be available, the uniform global asymptotic stability and uniform local exponential stability of our closed loop tracking dynamics, the generality of our class of trackable reference trajectories, and the input-to-state stability of the controller performance under actuator errors of …

Hypercube Diagrams For Knots, Links, And Knotted Tori, Ben Mccarty

LSU Doctoral Dissertations

For a knot K the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. Examples of knots for which the cube number detects chirality are presented. There is also a Legendrian version of this invariant called the Legendrian cube number. We will show that the Legendrian cube number distinguishes the Legendrian left hand torus knots with maximal Thurston-Bennequin number and maximal rotation number from the Legendrian left hand torus knots with maximal Thurston-Bennequin number and minimal rotation number. Finally, there is a generalization of cube …

On The Witt Groups Of Schemes, Jeremy Allen Jacobson

LSU Doctoral Dissertations

We consider two questions about the Witt groups of schemes: the first is the question of finite generation of the shifted Witt groups of a smooth variety over a finite field; the second is the Gersten conjecture. Regarding the first, we prove that the shifted Witt groups of curves and surfaces are finite, and that finite generation of the motivic cohomology groups with mod 2 coefficients implies finite generation of the Witt groups. Regarding the second, we prove the Gersten conjecture for the Witt groups in the case of a local ring that is essentially smooth over a discrete valuation …

Resonance And Double Negative Behavior In Metamaterials, Yue Chen

LSU Doctoral Dissertations

In this work, a generic class of metamaterials is introduced and is shown to exhibit frequency dependent double negative effective properties. We develop a rigorous method for calculating the frequency intervals where either double negative or double positive effective properties appear and show how these intervals imply the existence of propagating Bloch waves inside sub-wavelength structures. The branches of the dispersion relation associated with Bloch modes are shown to be explicitly determined by the Dirichlet spectrum of the high dielectric phase and the generalized electrostatic spectra of the complement. For numerical purposes, we consider a metamaterial constructed from a sub-wavelength …

A Numerical Investigation Of Apéry-Like Recursions And Related Picard-Fuchs Equations, Maiia J. Bakhova

LSU Doctoral Dissertations

In this work we investigate a generalization of a recursion which was used by Apery in his proof of irrationality of the zeta function values at 2 and 3. It is a continuation of the work of Zagier , who considered generalization of the first equation and numerically investigated it. The study is made for two generalizations of the second equation, one used the mirror symmetry idea from the theory of Calabi-Yau varieties and another worked with recursion. There were discovered connections between them.

High Tech High Touch: Lessons Learned From Project Haiti 2011, Yan Tang, Marc Compere, Yung Lun Wong, Jared Anthony Coleman, Matthew Charles Selkirk

Publications

In this paper, we will share our experiences and lessons learned from a design project for providing clean water to a Haitian orphanage (Project Haiti 2011). Supported by funds from a renewable energy company and the university president’s office, five engineering students and two faculty members from Embry-Riddle Aeronautical University successfully designed and installed a solar powered water purification system for an orphanage located in Chambellan, Haiti. This paper discusses the unique educational experiences gained from unusual design constraints, such as ambiguity of existing facilities due to limited communication, logistics of international construction at a remote village location, and cross-cultural …

Multi-Disciplinary Hands-On Desktop Learning Modules And Modern Pedagogies, Bernard J. Van Wie, David B. Thiessen, Marc Compere, Ximena Toro, Jennifer C. Adam, Et Al.

Publications

Our team’s research focuses on fundamental problems in undergraduate education in terms of how to expand use of well researched, yet still “new”, teaching pedagogies of ‘sensing’ or ‘hands-on’, ‘active’ and ‘problem-based learning’ within engineering courses. It is now widely accepted that traditional lectures ARE NOT best for students – yet that is what the community almost universally does.

To address this issue we are developing new Desktop Learning Modules (DLMs) that contain miniaturized processes with a uniquely expandable electronic system to contend with known sensor systems/removable cartridges, as well as, unknown expansions to the project. We have shown that …

The Generalised Zakharov-Shabat System And The Gauge Group Action, Georgi Grahovski

Articles

The generalized Zakharov–Shabat systems with complex-valued non-regular Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studied. This study includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations, solvable by the inverse scattering method, and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures. The results are illustrated on the example of the multi-component nonlinear Schr¨odinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type …

Eulerian Polynomials And B-Splines, Tian-Xiao He

Scholarship

Here presented is the interrelationship between Eulerian polynomials, Eulerian fractions and Euler-Frobenius polynomials, Euler-Frobenius fractions, B-splines, respectively. The properties of Eulerian polynomials and Eulerian fractions and their applications in B-spline interpolation and evaluation of Riemann-zeta function values at odd integers are given. The relation between Eulerian numbers and B-spline values at knot points are also discussed.

The Head And Tail Conjecture For Alternating Knots, Cody Armond

LSU Doctoral Dissertations

The colored Jones polynomial is an invariant of knots and links, which produces a sequence of Laurent polynomials. In this work, we study new power series link invariants, derived from the colored Jones polynomial, called its head and tail. We begin with a brief survey of knot theory and the colored Jones polynomial in particular. In Chapter 3, we use skein theory to prove that for adequate links, the n-th leading coefficient of the N-th colored Jones polynomial stabilizes when viewed as a sequence in N. This property allows us to define the head and tail for adequate links. In …

The New Stochastic Integral And Anticipating Stochastic Differential Equations, Benedykt Szozda

LSU Doctoral Dissertations

In this work, we develop further the theory of stochastic integration of adapted and instantly independent stochastic processes started by Wided Ayed and Hui-Hsiung Kuo in [1,2]. We provide a first counterpart to the Itô isometry that accounts for both adapted and instantly independent processes. We also present several Itô formulas for the new stochastic integral. Finally, we apply the new Itô formula to solve a linear stochastic differential equations with anticipating initial conditions.

Momcmc: An Efficient Monte Carlo Method For Multi-Objective Sampling Over Real Parameter Space, Yaohang Li

Computer Science Faculty Publications

In this paper, we present a new population-based Monte Carlo method, so-called MOMCMC (Multi-Objective Markov Chain Monte Carlo). for sampling in the presence of multiple objective functions in real parameter space. The MOMCMC method is designed to address the "multi-objective sampling" problem, which is not only of interest in exploring diversified solutions at the Pareto optimal front in the function space of multiple objective functions, but also those near the front. MOMCMC integrates Differential Evolution (DE) style crossover into Markov Chain Monte Carlo (MCMC) to adaptively propose new solutions from the current population. The significance of dominance is taken into …

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 …

R0 Analysis Of A Spatiotemporal Model For A Stream Population, H. W. Mckenzie, Y. Jin, J. Jacobsen, M. A. Lewis

Department of Mathematics: Faculty Publications

Water resources worldwide require management to meet industrial, agricultural, and urban consumption needs. Management actions change the natural flow regime, which impacts the river ecosystem. Water managers are tasked with meeting water needs while mitigating ecosystem im- pacts. We develop process-oriented advection-diffusion-reaction equations that couple hydraulic flow to population growth, and we analyze them to assess the effect of water flow on population persistence. We present a new mathematical framework, based on the net reproductive rate R₀ for advection-diffusion-reaction equations and on related measures. We apply the measures to popula- tion persistence in rivers under various flow regimes. This …

Fuzzy Linguistic Topological Spaces, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Amal

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Cardinal Functions And Integral Functions, Florentin Smarandache, Mircea Selariu, Marian Nitu

Branch Mathematics and Statistics Faculty and Staff Publications

This paper presents the correspondences of the eccentric mathematics of cardinal and integral functions and centric mathematics, or ordinary mathematics. Centric functions will also be presented in the introductory section, because they are, although widely used in undulatory physics, little known. In centric mathematics, cardinal sine and cosine are dened as well as the integrals. Both circular and hyperbolic ones. In eccentric mathematics, all these central functions multiplies from one to innity, due to the innity of possible choices where to place a point. This point is called eccenter S(s;") which lies in the plane of unit circle UC(O;R = …

Quantization And Discretization At Large Scales, Florentin Smarandache, Victor Christianto, Pavel Pintr

Branch Mathematics and Statistics Faculty and Staff Publications

The ongoing search of extrasolar planets is one of the most attractive fields of research in astrophysics and astronomy. Up to now, 360 extrasolar planets have been discovered near stars with similar mass as the Sun. There is also discovery related to the so-called Earth-like planets. With regards to these discoveries, one intriguing question is whether there is relationship between orbit distance of the planets and their stars. Various formulas have been suggested since 1990s, and they suggest that there may be reason to accept quantization of distances of those planets both in our solar system and also in extrasolar …

Innovative Uses Of Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy, Indra Venkatbabu

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors bring out the innovative applications of matrices defined, described and developed by them. Here they do not include the natural product on matrices newly described and defined by them in the book on ‘natural product ×n on matrices’.

This book is organized into seven chapters. The first one is introductory in nature. In the second chapter authors give the unique and new way of analyzing the data which is time dependent. We construct three types of matrices called Average Time Dependent data matrix (ATD matrix), Refined Time Dependent Data matrix (RTD matrix) and Combined Effective Time …