Applied Mathematics Commons^™

Iterated Aluthge Transforms: A Brief Survey, Jorge Antezana, Enrique R. Pujals, Demetrio Stojanoff

Publications and Research

Given an r × r complex matrix T, if T = U|T| is the polar decomposition of T, then the Aluthge transform is defined by

∆(T) = |T|^1/2U|T|^1/2.

Let ∆ⁿ(T) denote the n-times iterated Aluthge transform of T, i.e. ∆⁰(T) = T and ∆ⁿ(T) = ∆(∆ⁿ⁻¹(T)), n ∈ N. In this paper we make a brief survey on the known properties and applications of …

Nearly-Hamiltonian Structure For Water Waves With Constant Vorticity, Adrian Constantin, Rossen Ivanov, Emil Prodanov

Articles

We show that the governing equations for two-dimensional gravity water waves with constant non-zero vorticity have a nearly-Hamiltonian structure, which becomes Hamiltonian for steady waves.

A Symbolic Operator Approach To Several Summation Formulas For Power Series Ii, Tian-Xiao He, Peter Shiue, L. C. Hsu

Scholarship

Here expounded is a kind of symbolic operator method that can be used to construct many transformation formulas and summation formulas for various types of power series including some old ones and more new ones.

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.

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.

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.

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 …

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.

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.

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 …

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.

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.

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 …

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.

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 …

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 …

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 …

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.

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) …

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 …

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 …

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, …

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.

A New Family Of Pr Two Channel Filter Banks, Jian-Ao Lian

Applications and Applied Mathematics: An International Journal (AAM)

A new family of multidimensional dimensional (MD) perfect reconstruction (PR) two channel filter banks with finite impulse response (FIR) filters induced from systems of biorthogonal MD scaling functions and wavelets are introduced. One of the advantages of this construction is that the biorthogonal scaling functions and wavelets are easy to establish due to the interpolatory property of the scaling functions to start with. The other advantage is that all filters can be centrosymmetric or bi-linear phase. Examples of two dimensional (2D) bi-linear phase PR twochannel FIR filter banks will be demonstrated.

Bidimensional Pr Qmf With Fir Filters, Jian-Ao Lian

Applications and Applied Mathematics: An International Journal (AAM)

Multidimensional perfect reconstruction (PR) quadrature mirror filter (QMF) banks with finite impulse response (FIR) filters induced from systems of biorthogonal multivariate scaling functions and wavelets are investigated. In particular, bivariate scaling functions and wavelets with dilation as an expansive integer matrix whose determinant is two in absolute value are considered. Demonstrative quincunxial examples are explicitly given and new FIR filters are constructed.

The Substitution Theorem For Semilinear Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang

Articles and Preprints

In this article we establish a substitution theorem for semilinear stochastic evolution equations (see's) depending on the initial condition as an infinite-dimensional parameter. Due to the infinitedimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techniques together with new estimates on the underlying stochastic semiflow. Applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations (spde's) with anticipating initial conditions and non-ergodic stationary solutions. In particular, our result gives a new existence theorem for solutions of semilinear Stratonovich spde's with anticipating …

What Allows Teachers To Extend Student Thinking During Whole-Group Discussions, Nesrin Cengiz

Dissertations

Research indicates that extending students' mathematical thinking during whole-group discussions is challenging, even for the most experienced teachers. That is, it is challenging for teachers to help students move beyond their initial mathematical observations and solutions during whole-group discussions. To better understand this phenomena, the teaching of six experienced elementary school teachers, who had been teaching aStandards-based curriculum for several years and had participated in a multi-year professional development project focused on that curriculum, is explored in this study. In particular, two issues are addressed: what it looks like to extend student thinking during whole-group discussions and how …

Relative Pareto Minimizers To Multiobjective Problems: Existence And Optimality Conditions, Truong Q. Bao, Boris S. Mordukhovich

Mathematics Research Reports

In this paper we introduce and study enhanced notions of relative Pareto minimizers to constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers to general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient …

Image Reconstruction In Multi-Channel Model Under Gaussian Noise, Veera Holdai, Alexander Korostelev

Mathematics Research Reports

The image reconstruction from noisy data is studied. A nonparametric boundary function is estimated from observations in N independent channels in Gaussian white noise. In each channel the image and the background intensities are unknown. They define a non-identifiable nuisance "parameter" that slows down the typical minimax rate of convergence. The large sample asymptotics of the minimax risk is found and an asymptotically optimal estimator for boundary function is suggested.

A Hyperbolic Two -Step Model Based Finite Difference Method For Studying Thermal Deformation In A Micro Thin Film Heated By Ultrashort -Pulsed Lasers, Tianchan Niu

Doctoral Dissertations

Heat transport through micro thin films plays a very important role in microtechnology applications. Many microelectronic devices have metal thin films as their key components. Microscale heat transfer is also important for the thermal processing of materials, including laser micromachining, laser patterning, laser synthesis and laser surface hardening. Hence, studying the thermal behavior of thin films is essential for predicting the performance of a microelectronic device or for obtaining the desired microstructure. Recently, it has become very popular to use ultrashort-pulsed lasers in thermal processing, which lasers have pulse durations of the order of subpicoseconds to femtoseconds, and these kinds …