Applied Mathematics Commons^™

A Critical Look At Self-Dual Codes, Judy L. Walker

Department of Mathematics: Faculty Publications

We investigate self-dual codes from a structural point of view. In particular, we study properties of critical indecomposable codes which appear in the spectrum of a self-dual code. As an application of the results we obtain, we revisit the study of self-dual codes of dimension at most 10.

In the late 1950’s, Slepian [4] became the first to take an abstract approach to the study of error-correcting codes. He introduced a structure theory for binary linear codes, developing in particular the idea of an indecomposable code; that is, a code which is not isomorphic to a nontrivial direct sum of …

Efficient Traitor Tracing Algorithms Using List Decoding, Alice Silverberg, Jessica Staddon, Judy L. Walker

Department of Mathematics: Faculty Publications

We use powerful new techniques for list decoding error-correcting codes to efficiently trace traitors. Although much work has focused on constructing traceability schemes, the complexity of the tracing algorithm has received little attention. Because the TA tracing algorithm has a runtime of O(N) in general, where N is the number of users, it is inefficient for large populations.We produce schemes for which the TA algorithm is very fast. The IPP tracing algorithm, though less efficient, can list all coalitions capable of constructing a given pirate. We give evidence that when using an algebraic structure, the ability to …

Two Photon Absorption In Chromophore Doped Solid Matrices, S.C. Mancas, Michael Canva, Yves Levy, Kathleen A. Richardson, Giselle Roger

Publications

Over the past decades organic materials have shown an important potential for applications in the field of nonlinear optics. Two-photon absorbing materials can be optically addressed in three dimensions of space, which make them unique for many new applications, including 3D displays, optical memories, bio-sensors, etc. Fluorescent organic chromophores can be synthesized with structures especially optimized for this nonlinear optical property. Yet, for some applications, they have to be incorporated in solid state matrices. We especially investigate hybrid organic/inorganic doped matrices synthesized by solgel process. However , the linear transmission for such molecules is often significantly less than unity. Two-photon …

The Range Of The Iterated Matrix Adjoint Operator, Tze-Jang Chen, Jenn-Tsann Lin, C. H. Cooke

Mathematics & Statistics Faculty Publications

The following inverse problem is considered: for a given n × n real matrix B, does there exist a real matrix A such that where the classical adjoint operation is intended? The rank of B and the number of applications of the adjoint operator determine the character of this general inverse problem for the iterated adjoint operator. Thus, for given B, the question of interest is whether or not B lies in the range of the iterated matrix adjoint operator. Maple V R5 is used as an aid to obtain results indicated here. (©) 2001 Elsevier Science Ltd. …

Normal Forms, Canonical Forms, And Invariants Of Single Input Nonlinear Systems Under Feedback, Issa Amadou Tall, Witold Respondek

Miscellaneous (presentations, translations, interviews, etc)

We study the feedback group action on single-input nonlinear control systems. We follow an approach of Kang and Krener based on analysing, step by step, the action of homogeneous transformations on the homogeneous part of the system. We construct a dual normal form and dual invariants with respect to those obtained by Kang. We also propose a canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms coincide. We give an explicit construction of transformations bringing the system to its normal, dual normal, and canonical form.

Commuting Self-Adjoint Extensions Of Symmetric Operators Defined From The Partial Derivatives, Palle Jorgensen, Steen Pedersen

Mathematics and Statistics Faculty Publications

We consider the problem of finding commuting self-adjoint extensions of the partial derivatives {(1/i)(∂/∂x_j):j=1,...,d} with domain C^∞_c(Ω) where the self-adjointness is defined relative to L²(Ω), and Ω is a given open subset of R^d.

The Expected Wet Period Of Finite Dam With Exponential Inputs, Eui Yong Lee, Kimberly Kinateder

Mathematics and Statistics Faculty Publications

We use martingale methods to obtain an explicit formula for the expected wet period of the finite dam of capacity V, where the amounts of inputs are i.i.d exponential random variables and the output rate is one, when the reservoir is not empty. As a consequence, we obtain an explicit formula for the expected hitting time of either 0 or V and a new expression for the distribution of the number of overflows during the wet period, both without the use of complex analysis.

Alternative Principal Components Regression Procedures For Dendrohydrologic Reconstructions, Hugo G. Hidalgo, Thomas C. Piechota, John A. Dracup

Civil and Environmental Engineering and Construction Faculty Research

Streamflow reconstruction using tree ring information (dendrohydrology) has traditionally used principal components analysis (PCA) and stepwise regression to form a transfer function. However, PCA has several procedural choices that may result in very different reconstructions. This study assesses the different procedures in PCA-based regression and suggests alternative procedures for selection of variables and principal components. Cross-validation statistics are presented as an alternative for independently testing and identifying the optimal model. The objective is to use these statistics as a measure of the model's performance to find a conceptually acceptable model with a low prediction error and the fewest number of …

Stochastic Hybrid Control, A. Bensoussan, J. L. Menaldi

Mathematics Faculty Research Publications

The objective of this paper is to study the stochastic version of a previous paper of the authors, in which hybrid control for deterministic systems was considered. The modelling is quite similar to the deterministic case. We have a system whose state is composed of a continuous part and a discrete part. They are affected by a continuous type control and an impulse control. The dynamics is moreover perturbed by noise, also a continuous and a discrete noise process. The Markovian character of the state process is preserved. We develop the model and show how the dynamic programming approach leads …

A Nonlinear Parabolic Equation Modelling Surfactant Diffusion, Xinfu Chen, Chaocheng Huang, Jennifer Zhao

Mathematics and Statistics Faculty Publications

An initial-boundary value problem for nonlinear parabolic equations modelling surfactant diffusions is investigated. The boundary conditions are of nonlinear adsorptive types, and the initial value has a single point jump. We study the well-posedness of the problem, the convergence of a numerical scheme, and the regularity as well as quantitative behaviour of solutions.

An Efficient Method For Band Structure Calculations In 3d Photonic Crystals, David C. Dobson, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

A method for computing band structures for three-dimensional photonic crystals is described. The method combines a mixed finite element discretization on a uniform grid with a fast Fourier transform preconditioner and a preconditioned subspace iteration algorithm. Numerical examples illustrating the behavior of the method are presented.

Solvability Of A Parabolic Boundary Value Problem With Internal Jump Condition, Kurt M. Bryan, Lester Caudill

Mathematical Sciences Technical Reports (MSTR)

We examine a model for the propagation of heat through a one-dimensional object with an interior ''flaw". The flaw is modeled as a nonlinear relationship between the flux and temperature jump at an interior point of the object. Under realistic hypotheses, the resulting nonlinear initial boundary value problem is shown to have a unique and suitably smooth solution.

The Structure Of Free Semigroup Algebras, Kenneth R. Davidson, Elias Katsoulis, David R. Pitts

Department of Mathematics: Faculty Publications

A free semigroup algebra is WOT-closed algebra generated by an n-tuple of isometries with pairwise orthogonal ranges. The interest in these algebras arises primarily from two of their interesting features. The first is that they provide useful information about unitary invariants of representations of the Cuntz-Toeplitz algebras. The second is that they form a class of nonself-adjoint operator algebras which are of interest in their own right. This class contains a distinguished representative, the "non-commutative Toeplitz algebra", which is generated by the left regular representation of the free semigroup on n letters and denoted . This paper provides a general …

Euclidean Weights Of Codes From Elliptic Curves Over Rings, José Felipe Voloch, Judy L. Walker

Department of Mathematics: Faculty Publications

We construct certain error-correcting codes over finite rings and estimate their parameters. For this purpose, we need to develop some tools, notably an estimate for certain exponential sums and some results on canonical lifts of elliptic curves. These results may be of independent interest.

A code is a subset of Aⁿ, where A is a finite set (called the alphabet). Usually A is just the field of two elements and, in this case, one speaks of binary codes. Such codes are used in applications where one transmits information through noisy channels. By building redundancy into the code, transmitted …

Tiling As A Loop Parallelization Technique, Hoda Ahmed Khalil

Archived Theses and Dissertations

No abstract provided.

Finite Element Model Updating Using Antiresonant Frequencies, Keith W. Jones

Theses and Dissertations

The applications of antiresonant frequencies to finite element (FE) model updating are few and usually limited to numerical examples. This work uses antiresonant frequencies in the model updating of an experimental structure and analyzes the physical correctness of the updated model by using it to detect damage. Antiresonant frequencies were used in the FE model updating of a six-meter aluminum truss. The model used rigid links to model welded and bolted joints. Rigid link dimensions were used as parameters in an iterative update based on eigenvalue and antiresonance sensitivities. The first update used 11 natural frequencies and 21 antiresonant frequencies …

Singular Solutions To A Nonlinear Elliptic Boundary Value Problem Originating From Corrosion Modeling, Kurt M. Bryan, Michael Vogelius

Mathematical Sciences Technical Reports (MSTR)

We consider a nonlinear elliptic boundary value problem on a planar domain. The exponential type nonlinearity in the boundary condition is one that frequently appears in the modeling of electrochemical systems. For the case of a disk we construct a family of exact solutions that exhibit limiting logarithmic singularities at certain points on the boundary. Based on these solutions we develop two criteria that we believe predict the possible locations of the boundary singularities on quite general domains.

An Analysis Of The Theory Of Functions Of One Real Variable, Robert J. Reed

Inquiry: The University of Arkansas Undergraduate Research Journal

Few undergraduates are aware that the Riemann integral taught in introductory calculus courses has only limited application-essentially this integral can be used only to integrate continuous functions over intervals. The necessity to integrate a broader class of functions over a wider range of sets that arises in many applications motivates the theory of abstract integration and functional analysis. The founder of this theory was the French mathematician Henri Lebesgue, who in 1902 defined the "Lebesgue measure" of subsets of the real line. The purpose of this project is to elucidate the theory of abstract measure spaces and of important spaces …

Selection Of Curricular Topics Using Extensions Of Quality Function Deployment, Paul Kauffmann, Abel Fernandez, Charles Keating, Derya Jacobs, Resit Unal

Engineering Management & Systems Engineering Faculty Publications

Decision science can be an effective tool for enhancing organizational participation during strategic and complex decision making. This involvement develops a group consensus for relating organizational goals and the methods to achieve them. This paper describes an application of Quality Function Deployment (QFD) to define curricular topics that meet program objectives. Based on the ability of QFD to establish relationships, the model identifies the most important topics and quantifies their impact on meeting program goals. The model was developed to support restructuring of a Masters of Engineering Management degree program. The model supported decisions in selecting and prioritizing the required …

Rotary Honing: A Variant Of The Taylor Paint-Scraper Problem, Christopher Hills, H. Moffatt

Articles

The three-dimensional Row in a corner of fixed angle α induced by the rotation in its plane of one of the boundaries is considered. A local similarity solution valid in a neighbourhood of the centre of rotation is obtained and the streamlines are shown to be closed curves. The effects of inertia are considered and are shown to be significant in a small neighbourhood of the plane of symmetry of the flow. A simple experiment confirms that the streamlines are indeed nearly closed; their projections on planes normal to the line of intersection of the boundaries are precisely the 'Taylor' …

Collected Papers Vol. Iii, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Tight Bounds On The Algebraic Connectivity Of A Balanced Binary Tree, Jason J. Molitierno, Michael Neumann, Bryan L. Shader

Mathematics Faculty Publications

In this paper, quite tight lower and upper bounds are obtained on the algebraic connectivity, namely, the second-smallest eigenvalue of the Laplacian matrix, of an unweighted balanced binary tree with k levels and hence n = 2^k - 1 vertices. This is accomplished by considering the inverse of a matrix of order k - 1 readily obtained from the Laplacian matrix. It is shown that the algebraic connectivity is 1/(2^k - 2k + 3) + 0(1/2^2k).

Oif Spaces, Zoltan Balogh, Harold Bennett, Dennis Burke, Gary Gruenhage, David Lutzer, Joe D. Mashburn

Mathematics Faculty Publications

A base β of a space X is called an OIF base when every element of B is a subset of only a finite number of other elements of β. We will explore the fundamental properties of spaces having such bases. In particular, we will show that in T2 spaces, strong OIF bases are the same as uniform bases, and that in T3 spaces where all subspaces have OIF bases, compactness, countable compactness, or local compactness will give metrizability.

Mortar Estimates Independent Of Number Of Subdomains, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

The stability and error estimates for the mortar finite element method are well established. This work examines the dependence of constants in these estimates on shape and number of subdomains. By means of a Poincar´e inequality and some scaling arguments, these estimates are found not to deteriorate with increase in number of subdomains.

Separation Property Of Solutions For A Semilinear Elliptic Equation, Yi Liu, Yi Li, Yinbin Deng

Mathematics and Statistics Faculty Publications

In this paper, we study the following elliptic problem[formula]where K(x) is a given function in C^α(ⁿ\0) for some fixed α∈(0, 1), p>1 is a constant. Some existence, monotonicity and asymptotic expansion at infinity of solutions of (*) are discussed.

Two-Groups With Few Conjugacy Classes, Nigel Boston, Judy L. Walker

Department of Mathematics: Faculty Publications

An old question of Brauer asking how fast numbers of conjugacy classes grow is investigated by considering the least number c_n of conjugacy classes in a group of order 2ⁿ. The numbers c_n are computed for n ≤ 14 and a lower bound is given for c₁₅. It is observed that c_n grows very slowly except for occasional large jumps corresponding to an increase in coclass of the minimal groups G_n. Restricting to groups that are 2-generated or have coclass at most 3 allows us to extend these computations.

Translation Theorems For Fourier-Feynman Transforms And Conditional Fourier-Feynman Transforms, Seung Jun Change, Chull Park, David Skough

Department of Mathematics: Faculty Publications

Translation theorems for Wiener integrals were given by Cameron and Martin in [3] and by Cameron and Graves in [2]. Translation theorems for analytic Feynman integrals were given by Cameron and Storvick in [4], [7] and translation theorems for Feynman integrals on abstract Wiener and Hilbert spaces were given by Chung and Kang in [12].

Codes And Curves, Judy L. Walker

Department of Mathematics: Faculty Publications

When information is transmitted, errors are likely to occur. Coding theory examines effi cient ways of packaging data so that these errors can be detected, or even corrected. The traditional tools of coding theory have come from combinatorics and group theory. Lately, however, coding theorists have added techniques from algebraic geometry to their toolboxes. In particular, by re-interpreting the Reed- Solomon codes, one can see how to defi ne new codes based on divisors on algebraic curves. For instance, using modular curves over fi nite fi elds, Tsfasman, Vladut, and Zink showed that one can defi ne a sequence of …

Multigrid For The Mortar Finite Element Method, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

A multigrid technique for uniformly preconditioning linear systems arising from a mortar finite element discretization of second order elliptic boundary value problems is described and analyzed. These problems are posed on domains partitioned into subdomains, each of which is independently triangulated in a multilevel fashion. The multilevel mortar finite element spaces based on such triangulations (which need not align across subdomain interfaces) are in general not nested. Suitable grid transfer operators and smoothers are developed which lead to a variable Vcycle preconditioner resulting in a uniformly preconditioned algebraic system. Computational results illustrating the theory are also presented.

Upper Bounds To The Clique Width Of Graphs, Bruno Courcelle, Stephan Olariu

Computer Science Faculty Publications

Hierarchical decompositions of graphs are interesting for algorithmic purposes. Many NP complete problems have linear complexity on graphs with tree-decompositions of bounded width. We investigate alternate hierarchical decompositions that apply to wider classes of graphs and still enjoy good algorithmic properties. These decompositions are motivated and inspired by the study of vertex-replacement context-free graph grammars. The complexity measure of graphs associated with these decompositions is called clique width. In this paper we bound the clique width of a graph in terms of its tree width on the one hand, and of the clique width of its edge complement on …