Applied Mathematics | Open Access Articles | Digital Commons Network™

Padé Spline Functions, Tian-Xiao He

Scholarship

We present here the definition of Pad´e spline functions, their expressions, and the estimate of the remainders of pad´e spline expansions. Some algorithms are also given.

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Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

Mathematics and Statistics Faculty Publications and Presentations

In this series of papers, we construct operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. These extensions are novel in that they have certain polynomial preservation properties important in the analysis of high order finite elements. This part of the series is devoted to introducing our new technique for constructing the extensions, and its application to the case of polynomial extensions from H^½(∂K) into H¹(K), for any tetrahedron K.

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Asymptotic And Numerical Techniques For Resonances Of Thin Photonic Structures, Jay Gopalakrishnan, Shari Moskow, Fadil Santosa

Mathematics and Statistics Faculty Publications and Presentations

We consider the problem of calculating resonance frequencies and radiative losses of an optical resonator. The optical resonator is in the form of a thin membrane with variable dielectric properties. This work provides two very different approaches for doing such calculations. The first is an asymptotic method which exploits the small thickness and high index of the membrane. We derive a limiting resonance problem as the thickness goes to zero, and for the case of a simple resonance, find a first order correction. The limiting problem and the correction are in one less space dimension, which can make the approach …

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A Mixed Method For Axisymmetric Div-Curl Systems, Dylan M. Copeland, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

We present a mixed method for a three-dimensional axisymmetric div-curl system reduced to a two-dimensional computational domain via cylindrical coordinates. We show that when the meridian axisymmetric Maxwell problem is approximated by a mixed method using the lowest order Nédélec elements (for the vector variable) and linear elements (for the Lagrange multiplier), one obtains optimal error estimates in certain weighted Sobolev norms. The main ingredient of the analysis is a sequence of projectors in the weighted norms satisfying some commutativity properties.

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On The Spectra Of Certain Directed Paths, Carlos Martins Da Fonseca, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

We describe the eigenpairs of special kinds of tridiagonal matrices related to problems on traffic on a one-lane road. Some numerical examples are provided.

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Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

The finite element method when applied to a second order partial differential equation in divergence form can generate operators that are neither Hermitian nor definite when the coefficient function is complex valued. For such problems, under a uniqueness assumption, we prove the continuous dependence of the exact solution and its finite element approximations on data provided that the coefficients are smooth and uniformly bounded away from zero. Then we show that a multigrid algorithm converges once the coarse mesh size is smaller than some fixed number, providing an efficient solver for computing discrete approximations. Numerical experiments, while confirming the theory, …

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Symmetric Presentations Of Finite Groups, Joshua Anthony Roche

Theses Digitization Project

Symmetric presentations of groups allow us to represent, and manipulate, group elements in a manner that is typically more convenient than conventional techniques; in this sense, symmetric presentations are particularly useful in the study of large finite groups.

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Inverse Wavelet Reconstruction For Resolving The Gibbs Phenomenon, Nataniel Greene

Publications and Research

The Gibbs phenomenon refers to the lack of uniform convergence which occurs in many orthogonal basis approximations to piecewise smooth functions. This lack of uniform convergence manifests itself in spurious oscillations near the points of discontinuity and a low order of convergence away from the discontinuities. Here we describe a numerical procedure for overcoming the Gibbs phenomenon called the inverse wavelet reconstruction method. The method takes the Fourier coefficients of an oscillatory partial sum and uses them to construct the wavelet coefficients of a non-oscillatory wavelet series.

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An Order Model For Infinite Classical States, Joe Mashburn

Mathematics Faculty Publications

In 2002 Coecke and Martin (Research Report PRG-RR-02-07, Oxford University Computing Laboratory,2002) created a model for the finite classical and quantum states in physics. This model is based on a type of ordered set which is standard in the study of information systems. It allows the information content of its elements to be compared and measured. Their work is extended to a model for the infinite classical states. These are the states which result when an observable is applied to a quantum system. When this extended order is restricted to a finite number of coordinates, the model of Coecke and …

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Qualitative Properties Of Nonlinear Volterra Integral Equations, Muhammad Islam, Jeffrey T. Neugebauer

Mathematics Faculty Publications

In this article, the contraction mapping principle and Liapunov's method are used to study qualitative properties of nonlinear Volterra equations of the form x(t)=a(t)−∫^t₀C(t,s)g(s,x(s))ds,t≥0. In particular, the existence of bounded solutions and solutions with various L^p properties are studied under suitable conditions on the functions involved with this equation.

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Models Of Phototransduction In Rod Photoreceptors, Harihar Khanal, Vasilios Alexiades

Publications

Phototransduction is the process by which photons of light generate an electrical response in retinal rod and cone photoreceptors, thereby initiating vision. We compare the electrical response in salamander rods from increasingly more (spacialy) detailed models of phototransduction: 0-dimensional (bulk), 1-dimensional (longitudinal), 2-dimensional (axisymmetric), and 3-dimensional (with incisures). We discuss issues of finding physical parameters for simulation and validation of models, and also present some computational experiments for rods with geometry of mouse and human photoreceptors.

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On Existence And Uniqueness Results For The Bbm Equation With Arbitrary Forcing Terms, Timothy A. Smith

Publications

The problem of classical solutions for the regularized long-wave equation is considered where various additional forcing terms are introduced which are often required for physical modifications in the wave theory. Sufficient conditions of solvability and existence are established and then these conditions are related to the structure of the forcing terms under consideration.

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Trace Forms Of Abelian Extensions Of Number Fields, Karli Smith

LSU Doctoral Dissertations

This dissertation is concerned with providing a description of certain symmetric bilinear forms, called trace forms, associated with finite normal extensions N/K of an algebraic number field K, with abelian Galois group Gal(N/K). These abelian trace forms are described up to Witt equivalence, that is, they are described as elements in the Witt ring W(K). Complete descriptions are obtained when the base field K has exactly one dyadic prime and either no real embeddings or one real embedding. For these fields K, the set of abelian trace forms is closed under multiplication in the Witt ring W(K).

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Rational Approximation Schemes For Solutions Of Abstract Cauchy Problems And Evolution Equations, Patricio Gabriel Jara

LSU Doctoral Dissertations

In this dissertation we study time and space discretization methods for approximating solutions of abstract Cauchy problems and evolution equations in a Banach space setting. Two extensions of the Hille-Phillips functional calculus are developed. The first result is the Hille-Phillips functional calculus for generators of bi-continuous semigroups, and the second is a C-regularized version of the Hille-Phillips functional calculus for generators of C-regularized semigroups. These results are used in order to study time discretization schemes for abstract Cauchy problems associated with generators of bi-continuous semigroups as well as C-regularized semigoups. Stability, convergence results, and error estimates for rational approximation schemes …

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Differential Geometry In Cartesian Closed Categories Of Smooth Spaces, Martin Laubinger

LSU Doctoral Dissertations

The main categories of study in this thesis are the categories of diffeological and Fr\"olicher spaces. They form concrete cartesian closed categories. In Chapter 1 we provide relevant background from category theory and differentiation theory in locally convex spaces. In Chapter 2 we define a class of categories whose objects are sets with a structure determined by functions into the set. Fr\"olicher's $M$-spaces, Chen's differentiable spaces and Souriau's diffeological spaces fall into this class of categories. We prove cartesian closedness of the two main categories, and show that they have all limits and colimits. We exhibit an adjunction between the …

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On An Integrable Two-Component Camassa-Holm Shallow Water System, Adrian Constantin, Rossen Ivanov

Articles

The interest in the Camassa-Holm equation inspired the search for various generalizations of this equation with interesting properties and applications. In this letter we deal with such a twocomponent integrable system of coupled equations. First we derive the system in the context of shallow water theory. Then we show that while small initial data develop into global solutions, for some initial data wave breaking occurs. We also discuss the solitary wave solutions. Finally, we present an explicit construction for the peakon solutions in the short wave limit of system.

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Algebraic Discretization Of The Camassa-Holm And Hunter-Saxton Equations, Rossen Ivanov

Articles

The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the H1 and H.1 right-invariant metrics correspondingly. There is an analogy to the Euler equations in hydrodynamics, which describe geodesic flow for a right-invariant metric on the infinitedimensional group of diffeomorphisms preserving the volume element of the domain of fluid flow and to the Euler equations of rigid body whith a fixed point, describing geodesics for a left-invariant metric on SO(3). The CH and HS equations are integrable bi-hamiltonian equations and one of their Hamiltonian structures is associated to the …

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Eigenvalue Comparisons For Boundary Value Problems Of The Discrete Elliptic Equation, Jun Ji, Bo Yang

Faculty Articles

In this paper we study a boundary value problem for a discrete elliptic equation. The focus will be on the structure of the spectrum of this problem and the existence of a positive eigenvector corresponding to the smallest eigenvalue. Comparison results for the eigenvalues are also established as the coefficients of the problem changes.

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Estimates Of Positive Solutions For Higher Order Right Focal Boundary Value Problem, Bo Yang

Faculty Articles

We consider the (p;n - p) right focal boundary value problem. A new set of upper and lower estimates of positive solutions for the boundary value problem are obtained. These estimates implement and improve the ones in the literature.

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Origin Of Conductive Surface Layer In Annealed Zno, David C. Look, B. Claflin, Helen Smith

Mathematics and Statistics Faculty Publications

The highly conductive surface layers found in nearly all as-grown or annealed bulk ZnO wafers are studied by temperature-dependent Hall-effect and secondary-ion mass spectroscopy (SIMS) measurements. In this work, we have used annealing in N₂ at 900 degrees C, and forming gas (5% H₂ in N₂) at 600 degrees C, to cause a large enough surface conduction that SIMS measurements can be reliably employed. The increased near-surface donor density, as determined from two-layer Hall-effect modeling, is consistent with an increased near-surface concentration of Al, Ga, and In atoms, resulting from diffusion. There is no evidence for …

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Algorithm-Independent Optimal Input Fluxes For Boundary Identification In Thermal Imaging, Kurt Bryan, Lester Caudill

Department of Math & Statistics Faculty Publications

An inverse boundary determination problem for a parabolic model, arising in thermal imaging, is considered. The focus is on intelligently choosing an effective input heat flux, so as to maximize the practical effectiveness of an inversion algorithm. Three different methods, based on different interpretations of the term “effective", are presented and analyzed, then demonstrated through numerical examples. It is noteworthy that each of these flux-selection methods is independent of the particular inversion algorithm to be used.

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Use Of Trees By The Texas Ratsnake (Elaphe Obsoleta) In Eastern Texas, Josh B. Pierce, Robert R. Fleet, Lance Mcbrayer, D. Craig Rudolph

Faculty Publications

We present information on the use of trees by Elaphe obsoleta (Texas Ratsnake) in a mesic pine-hardwood forest in eastern Texas. Using radiotelemetry, seven snakes (3 females, 4 males) were relocated a total of 363 times from April 2004 to May 2005, resulting in 201 unique locations. Snakes selected trees containing cavities and used hardwoods and snags for a combined 95% of arboreal locations. Texas Ratsnake arboreal activity peaked during July and August, well after the peak of avian breeding activity, suggesting arboreal activity involves factors other than avian predation.

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Dynamic Equations With Piecewise Continuous Argument, Christian Keller

Masters Theses

"We extend the theory of differential equations with piecewise continuous argument to general time scales. Linear and quasi-linear systems of functional dynamic equations with alternating retarding and advanced argument will be investigated and conditions for globally asymptotic stability of those systems will be stated and proven. Furthermore, oscillation criteria for linear first-order equations with piecewise continuous argument will be established"--Abstract, page iii.

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Super Fuzzy Matrices And Super Fuzzy Models For Social Scientists, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Amal

Branch Mathematics and Statistics Faculty and Staff Publications

The concept of supermatrix for social scientists was first introduced by Paul Horst. The main purpose of his book was to introduce this concept to social scientists, students, teachers and research workers who lacked mathematical training. He wanted them to be equipped in a branch of mathematics that was increasingly valuable for the analysis of scientific data. This book introduces the concept of fuzzy super matrices and operations on them. The author has provided only those operations on fuzzy supermatrices that are essential for developing super fuzzy multi expert models. We do not indulge in labourious use of suffixes or …

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Methods In Industrial Biotechnology For Chemical Engineers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Industrial Biotechnology is an interdisciplinary topic to which tools of modern biotechnology are applied for finding proper proportion of raw mix of chemicals, determination of set points, finding the flow rates etc., This study is significant as it results in better economy, quality product and control of pollution. The authors in this book have given only methods of industrial biotechnology mainly to help researchers, students and chemical engineers. Since biotechnology concerns practical and diverse applications including production of new drugs, clearing up pollution etc. we have in this book given methods to control pollution in chemical industries as it has …

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Towards Universal Cover Decoding, Nathan Axvig, Deanna Dreher, Katherine Morrison, Eric T. Psota, Lance C. Pérez, Judy L. Walker

Department of Mathematics: Faculty Publications

Low complexity decoding of low-density paritycheck (LDPC) codes may be obtained from the application of iterative message-passing decoding algorithms to the bipartite Tanner graph of the code. Arguably, the two most important decoding algorithms for LDPC codes are the sum-product decoder and the min-sum (MS) decoder. On a bipartite graph without cycles (a tree), the sum-product decoder minimizes the probability of bit error, while the min-sum decoder minimizes the probability of word error [9]. While the behavior of sum-product and min-sum is easily understood when operating on trees, their behavior becomes much more difficult to characterize when the Tanner graph …

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Ldpc Codes From Voltage Graphs, Christine A. Kelley, Judy L. Walker

Department of Mathematics: Faculty Publications

Several well-known structure-based constructions of LDPC codes, for example codes based on permutation and circulant matrices and in particular, quasi-cyclic LDPC codes, can be interpreted via algebraic voltage assignments. We explain this connection and show how this idea from topological graph theory can be used to give simple proofs of many known properties of these codes. In addition, the notion of abelianinevitable cycle is introduced and the subgraphs giving rise to these cycles are classified. We also indicate how, by using more sophisticated voltage assignments, new classes of good LDPC codes may be obtained.

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Average Min-Sum Decoding Of Ldpc Codes, Nathan Axvig, Deanna Dreher, Katherine Morrison, Eric T. Psota, Lance C. Pérez, Judy L. Walker

Department of Mathematics: Faculty Publications

Simulations have shown that the outputs of minsum (MS) decoding generally behave in one of two ways: the output either eventually stabilizes at a codeword or eventually cycles through a finite set of vectors that may include both codewords and non-codewords. This inconsistency in MS across iterations has significantly contributed to the difficulty in studying the performance of this decoder. To overcome this problem, a new decoder, average min-sum (AMS), is proposed; this decoder outputs the average of the min-sum output vectors over a finite set of iterations. Simulations comparing MS, AMS, linear programming (LP) decoding, and maximum likelihood (ML) …

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Algebraic Properties Of Edge Ideals, Rachelle R. Bouchat

University of Kentucky Doctoral Dissertations

Given a simple graph G, the corresponding edge ideal I_G is the ideal generated by the edges of G. In 2007, Ha and Van Tuyl demonstrated an inductive procedure to construct the minimal free resolution of certain classes of edge ideals. We will provide a simplified proof of this inductive method for the class of trees. Furthermore, we will provide a comprehensive description of the finely graded Betti numbers occurring in the minimal free resolution of the edge ideal of a tree. For specific subclasses of trees, we will generate more precise information including explicit formulas for …

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Mathematical Models: A Generalization From Population Biology And Time Scales, Martin Eggensperger

Theses and Dissertations

In this work, the theory of time scales calculus that we will employ is developed, as derivatives, integrals, and fundamental results are introduced. When necessary for subsequent discussion, other results are presented. The Hilger Complex Plane, the time scale exponential function, and its properties are given, as is a generalization of the variation of constants formula. The discrete and continuous Malthus population model, and the discrete and continuous Verhulst population models are investigated and analyzed. A unified and extended version of both models is developed and criteria for stability of critical solutions is presented. We investigate the relationship between both …

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Full-Text Articles in Applied Mathematics

Padé Spline Functions, Tian-Xiao He

Scholarship

Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

Mathematics and Statistics Faculty Publications and Presentations

Asymptotic And Numerical Techniques For Resonances Of Thin Photonic Structures, Jay Gopalakrishnan, Shari Moskow, Fadil Santosa

Mathematics and Statistics Faculty Publications and Presentations

A Mixed Method For Axisymmetric Div-Curl Systems, Dylan M. Copeland, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

On The Spectra Of Certain Directed Paths, Carlos Martins Da Fonseca, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

Symmetric Presentations Of Finite Groups, Joshua Anthony Roche

Theses Digitization Project

Inverse Wavelet Reconstruction For Resolving The Gibbs Phenomenon, Nataniel Greene

Publications and Research

An Order Model For Infinite Classical States, Joe Mashburn

Mathematics Faculty Publications

Qualitative Properties Of Nonlinear Volterra Integral Equations, Muhammad Islam, Jeffrey T. Neugebauer

Mathematics Faculty Publications

Models Of Phototransduction In Rod Photoreceptors, Harihar Khanal, Vasilios Alexiades

Publications

On Existence And Uniqueness Results For The Bbm Equation With Arbitrary Forcing Terms, Timothy A. Smith

Publications

Trace Forms Of Abelian Extensions Of Number Fields, Karli Smith

LSU Doctoral Dissertations

Rational Approximation Schemes For Solutions Of Abstract Cauchy Problems And Evolution Equations, Patricio Gabriel Jara

LSU Doctoral Dissertations

Differential Geometry In Cartesian Closed Categories Of Smooth Spaces, Martin Laubinger

LSU Doctoral Dissertations

On An Integrable Two-Component Camassa-Holm Shallow Water System, Adrian Constantin, Rossen Ivanov

Articles

Algebraic Discretization Of The Camassa-Holm And Hunter-Saxton Equations, Rossen Ivanov

Articles

Eigenvalue Comparisons For Boundary Value Problems Of The Discrete Elliptic Equation, Jun Ji, Bo Yang

Faculty Articles

Estimates Of Positive Solutions For Higher Order Right Focal Boundary Value Problem, Bo Yang

Faculty Articles

Origin Of Conductive Surface Layer In Annealed Zno, David C. Look, B. Claflin, Helen Smith

Mathematics and Statistics Faculty Publications

Algorithm-Independent Optimal Input Fluxes For Boundary Identification In Thermal Imaging, Kurt Bryan, Lester Caudill

Department of Math & Statistics Faculty Publications

Use Of Trees By The Texas Ratsnake (Elaphe Obsoleta) In Eastern Texas, Josh B. Pierce, Robert R. Fleet, Lance Mcbrayer, D. Craig Rudolph

Faculty Publications

Dynamic Equations With Piecewise Continuous Argument, Christian Keller

Masters Theses

Super Fuzzy Matrices And Super Fuzzy Models For Social Scientists, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Amal

Branch Mathematics and Statistics Faculty and Staff Publications

Methods In Industrial Biotechnology For Chemical Engineers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Towards Universal Cover Decoding, Nathan Axvig, Deanna Dreher, Katherine Morrison, Eric T. Psota, Lance C. Pérez, Judy L. Walker

Department of Mathematics: Faculty Publications

Ldpc Codes From Voltage Graphs, Christine A. Kelley, Judy L. Walker

Department of Mathematics: Faculty Publications

Average Min-Sum Decoding Of Ldpc Codes, Nathan Axvig, Deanna Dreher, Katherine Morrison, Eric T. Psota, Lance C. Pérez, Judy L. Walker

Department of Mathematics: Faculty Publications

Algebraic Properties Of Edge Ideals, Rachelle R. Bouchat

University of Kentucky Doctoral Dissertations

Mathematical Models: A Generalization From Population Biology And Time Scales, Martin Eggensperger

Theses and Dissertations