Applied Mathematics Commons^™

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.

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.

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 …

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.

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' …

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.

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.

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.

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.

Collected Papers Vol. Iii, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

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 …

Self-Consistency Algorithms, Thaddeus Tarpey

Mathematics and Statistics Faculty Publications

The k-means algorithm and the principal curve algorithm are special cases of a self-consistency algorithm. A general self-consistency algorithm is described and results are provided describing the behavior of the algorithm for theoretical distributions, in particular elliptical distributions. The results are used to contrast the behavior of the algorithms when applied to a theoretical model and when applied to finite datasets from the model. The algorithm is also used to determine principal loops for the bivariate normal distribution.

Hopf Bifurcation In Models For Pertussis Epidemiology, Herbert W. Hethcote, Yi Li, Zhujun Jing

Mathematics and Statistics Faculty Publications

Pertussis (whooping cough) incidence in the United States has oscillated with a period of about four years since data was first collected in 1922. An infection with pertussis confers immunity for several years, but then the immunity wanes, so that reinfection is possible. A pertussis reinfection is mild after partial loss of immunity, but the reinfection can be severe after complete loss of immunity. Three pertussis transmission models with waning of immunity are examined for periodic solutions. Equilibria and their stability are determined. Hopf bifurcation of periodic solutions around the endemic equilibrium can occur for some parameter values in two …

A Construction Of Orthogonal Compactly-Supported Multiwavelets On $\R^{2}$, Bruce Kessler

Mathematics Faculty Publications

This paper will provide the general construction of the continuous, orthogonal, compactly-supported multiwavelets associated with a class of continuous, orthogonal, compactly-supported scaling functions that contain piecewise linears on a uniform triangulation of $\R^2$. This class of scaling functions is a generalization of a set of scaling functions first constructed by Donovan, Geronimo, and Hardin. A specific set of scaling functions and associated multiwavelets with symmetry properties will be constructed.

Generalized Averages For Solutions Of Two-Point Dirichlet Problems, Philip Korman, Yi Li

Mathematics and Statistics Faculty Publications

For very general two-point boundary value problems we show that any positive solution satisfies a certain integral relation. As a consequence we obtain some new uniqueness and multiplicity results.

Random Fluctuations Of Convex Domains And Lattice Points, Alex Iosevich, Kimberly Kinateder

Mathematics and Statistics Faculty Publications

In this paper, we examine a random version of the lattice point problem.

Axiomatic Approach For Quantification Of Image Resolution, Ge Wang, Yi Li

Mathematics and Statistics Faculty Publications

Image resolution is the primary parameter for performance characterization of any imaging system. In this work, we present an axiomatic approach for quantification of image resolution, and demonstrate that a good image resolution measure should be proportional to the standard deviation of the point spread function of an imaging system.

Stability Of Self-Similar Solutions For Van Der Waals Driven Thin Film Rupture, Thomas P. Witelski, Andrew J. Bernoff

All HMC Faculty Publications and Research

Recent studies of pinch-off of filaments and rupture in thin films have found infinite sets of first-type similarity solutions. Of these, the dynamically stable similarity solutions produce observable rupture behavior as localized, finite-time singularities in the models of the flow. In this letter we describe a systematic technique for calculating such solutions and determining their linear stability. For the problem of axisymmetric van der Waals driven rupture (recently studied by Zhang and Lister), we identify the unique stable similarity solution for point rupture of a thin film and an alternative mode of singularity formation corresponding to annular “ring rupture.”

Codes Over Rings From Curves Of Higher Genus, José Felipe Voloch, Judy L. Walker

Department of Mathematics: Faculty Publications

We construct certain error-correcting codes over finite rings and estimate their parameters. These codes are constructed using plane curves and the estimates for their parameters rely on constructing “lifts” of these curves and then estimating the size of certain exponential sums.

THE purpose of this paper is to 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 the dimension of trace codes over rings (generalizing work of van der Vlugt over fields and some results on lifts of affin curves from field of characteristic p …

Variational Principles For Average Exit Time Moments For Diffusions In Euclidean Space, Kimberly Kinateder, Patrick Mcdonald

Mathematics and Statistics Faculty Publications

Let D be a smoothly bounded domain in Euclidean space and let X_t be a diffusion in Euclidean space. For a class of diffusions, we develop variational principles which characterize the average of the moments of the exit time from D of a particle driven by X_t, where the average is taken overall starting points in D.

Positive Solutions To Semilinear Problems With Coefficient That Changes Sign, Nguyen Phuong Cac, Juan A. Gatica, Yi Li

Mathematics and Statistics Faculty Publications

No abstract provided.

Tracking The Trochoid On Safari, Andrew Simoson

ACMS Conference Proceedings 1999

While on sabbatical in Dar es Salaam in 1997–98, I was tasked to be the department’s seminar director and discovered that a dearth of speakers roam about in east Africa; so I opted to set up a lecture tour for myself to include Zambia, Zimbabwe, and Botswana as well as Tanzania—and spoke about the envelope of curves generated by a series of line segments obtained when two runners attached by an ideal bungee cord (the line segments) proceed about a circular track. Herein we describe life in east Africa and show that those envelopes are trochoids.

Revolutions In Mathematics (Book Review), Kevin Vander Meulen

ACMS Conference Proceedings 1999

A review of Revolutions in Mathematics, edited by Donald Gillies, Clarendon Press, Oxford 1992.

Lewis Carroll: Author, Mathematician, And Christian, David L. Neuhouser

ACMS Conference Proceedings 1999

Although a Christian, an author, and a mathematician, Charles Letwidge Dodgson (better known as Lewis Carroll) wrote very few works in which these three aspects of his person was present. The only examples of him merging these interests are in Sylvie and Bruno and Sylvie and Bruno Concluded. This paper will explore what motivated him to make these works and whether or not they were successful.