Applied Mathematics Commons^™

Algorithms For Area Preserving Flows, Catherine Kublik, Selim Esedoglu, Jeffrey A. Fessler

Mathematics Faculty Publications

We propose efficient and accurate algorithms for computing certain area preserving geometric motions of curves in the plane, such as area preserving motion by curvature. These schemes are based on a new class of diffusion generated motion algorithms using signed distance functions. In particular, they alternate two very simple and fast operations, namely convolution with the Gaussian kernel and construction of the distance function, to generate the desired geometric flow in an unconditionally stable manner. We present applications of these area preserving flows to large scale simulations of coarsening.

Coarser Connected Metrizable Topologies, Lynne Yengulalp

Mathematics Faculty Publications

We show that every metric space, X, with w(⩾) c has a coarser connected metrizable topology.

Periodic Solutions Of Neutral Delay Integral Equations Of Advanced Type, Muhammad Islam, Nasrin Sultana, James Booth

Mathematics Faculty Publications

We study the existence of continuous periodic solutions of a neutral delay integral equation of advanced type. In the analysis we employ three fixed point theorems: Banach, Krasnosel'skii, and Krasnosel'skii-Schaefer. Krasnosel'skii-Schaefer fixed point theorem requires an a priori bound on all solutions. We employ a Liapunov type method to obtain such bound.

Linearly Ordered Topological Spaces And Weak Domain Representability, Joe Mashburn

Mathematics Faculty Publications

It is well known that domain representable spaces, that is topological spaces that are homeomorphic to the space of maximal elements of some domain, must be Baire. In this paper it is shown that every linearly ordered topological space (LOTS) is homeomorphic to an open dense subset of a weak domain representable space. This means that weak domain representable spaces need not be Baire.

Coarsening In High Order, Discrete, Ill-Posed Diffusion Equations, Catherine Kublik

Mathematics Faculty Publications

We study the discrete version of a family of ill-posed, nonlinear diffusion equations of order 2n. The fourth order (n=2) version of these equations constitutes our main motivation, as it appears prominently in image processing and computer vision literature. It was proposed by You and Kaveh as a model for denoising images while maintaining sharp object boundaries (edges). The second order equation (n=1) corresponds to another famous model from image processing, namely Perona and Malik's anisotropic diffusion, and was studied in earlier papers. The equations studied in this paper are high order analogues of the Perona-Malik equation, and like the …

Forced Oscillations Of The Korteweg-De Vries Equation On A Bounded Domain And Their Stability, Muhammad Usman, Bingyu Zhang

Mathematics Faculty Publications

It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. The observation has been confirmed mathematically in the context of the damped Kortewg-de Vries (KdV) equation and the damped Benjamin-Bona-Mahony (BBM) equation. In this paper we intend to show the same results hold for the pure KdV equation (without the damping terms) posed on a bounded domain. Consideration is given to the initial-boundary-value problem

uu_xu_xxx 0 < x < 1, t > 0, (*)

It is shown …

Coarser Connected Topologies And Non-Normality Points, Lynne Yengulalp

Mathematics Faculty Publications

We investigate two topics, coarser connected topologies and non-normality points. The motivating question in the first topic is:

Question 0.0.1. When does a space have a coarser connected topology with a nice topological property? We will discuss some results when the property is Hausdorff and prove that if X is a non-compact metric space that has weight at least c, then it has a coarser connected metrizable topology. The second topic is concerned with the following question:

Question 0.0.2. When is a point y ∈ β X\X a non-normality point of β X\X? We will discuss the question in the …

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.

A Spectral Order For Infinite Dimensional Quantum Spaces: A Preliminary Report, Joe Mashburn

Mathematics Faculty Publications

In 2002 Coecke and Martin created a Bayesian order for the finite dimensional spaces of classical states in physics and used this to define a similar order, the spectral order on the finite dimensional quantum states. These orders gave the spaces a structure similar to that of a domain. This allows for measuring information content of states and for determining which partial states are approximations of which pure states. In a previous paper the author extended the Bayesian order to infinite dimensional spaces of classical states. The order on infinite dimensional spaces retains many of the characteristics important to physics, …

A Comparison Of Three Topologies On Ordered Sets, Joe Mashburn

Mathematics Faculty Publications

We introduce two new topologies on ordered sets: the way below topology and weakly way below topology. These are similar in definition to the Scott topology, but are very different if the set is not continuous. The basic properties of these three topologies are compared. We will show that while domain representable spaces must be Baire, this is not the case with the new topologies.

Stability Properties Of Linear Volterra Integrodifferential Equations With Nonlinear Perturbation, Muhammad Islam, Youssef Raffoul

Mathematics Faculty Publications

A Lyapunov functional is employed to obtain conditions that guarantee stability, uniform stability and uniform asymptotic stability of the zero solution of a scalar linear Volterra integrodifferential equation with nonlinear perturbation.

Boundedness And Stability In Nonlinear Delay Difference Equations Employing Fixed Point Theory, Muhammad Islam, Ernest Yankson

Mathematics Faculty Publications

In this paper we study stability and boundedness of the nonlinear difference equation

x(t+1)=a(t)x(t)+c(t)Δx(t−g(t))+q(x(t),x(t−g(t))).

In particular we study equi-boundedness of solutions and the stability of the zero solution of this equation. Fixed point theorems are used in the analysis.

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.

On The Decomposition Of Order-Separable Posets Of Countable Width Into Chains, Gary Gruenhage, Joe Mashburn

Mathematics Faculty Publications

partially ordered set X has countable width if and only if every collection of pairwise incomparable elements of X is countable. It is order-separable if and only if there is a countable subset D of X such that whenever p, q ∈ X and p < q, there is r ∈ D such that p ≤ r ≤ q. Can every order-separable poset of countable width be written as the union of a countable number of chains? We show that the answer to this question is "no" if there is a 2-entangled subset of IR, and "yes" under the Open Coloring Axiom.

Stability Properties And Integrability Of The Resolvent Of Linear Volterra Equations, Muhammad Islam, Paul W. Eloe

Mathematics Faculty Publications

Integrability of the resolvent and the stability properties of the zero solution of linear Volterra integrodifferential systems are studied. In particular, it is shown that, the zero solution is uniformly stable if and only if the resolvent is integrable in some sense. It is also shown that, the zero solution is uniformly asymptotically stable if and only if the resolvent is integrable and an additional condition in terms of the resolvent and the kernel is satisfied. Finally, the integrability of the resolvent is obtained under an explicit condition.

A Note On Reordering Ordered Topological Spaces And The Existence Of Continuous, Strictly Increasing Functions, Joe Mashburn

Mathematics Faculty Publications

The origin of this paper is in a question that was asked of the author by Michael Wellman, a computer scientist who works in artificial intelligence at Wright Patterson Air Force Base in Dayton, Ohio. He wanted to know if, starting with Rn and its usual topology and product partial order, he could linearly reorder every finite subset and still obtain a continuous function from Rⁿ into R that was strictly increasing with respect to the new order imposed on Rⁿ. It is the purpose of this paper to explore the structural characteristics of ordered topological spaces …

On Infinite Delay Integral Equations Having Nonlinear Perturbations, Muhammad Islam

Mathematics Faculty Publications

The existence of bounded solutions and periodic solutions is studied for a system of infinite delay integral equations having nonlinear perturbations. An equivalent system of equations is obtained in terms of the resolvent kernel. Then the existence results are shown for the equivalent equations. Contraction principle, Schauder’s fixed point theorem, and monotone method are used in the study.

Pixley-Roy Hyperspaces Of Ω-Graphs, Joe Mashburn

Mathematics Faculty Publications

The techniques developed by Wage and Norden are used to show that the Pixley-Roy hyperspaces of any two ω-graphs are homeomorphic. The Pixley-Roy hyperspaces of several subsets of Rⁿ are also shown to be homeomorphic.

Periodic Solutions Of Volterra Integral Equations, Muhammad Islam

Mathematics Faculty Publications

Consider the system of equations

x(t)=f(t)+∫−∞tk(t,s)x(s)ds,           (1)

and

x(t)=f(t)+∫−∞tk(t,s)g(s,x(s))ds.        (2)

Existence of continuous periodic solutions of (1) is shown using the resolvent function of the kernel k. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1) it is necessary that the resolvent of k is integrable in some sense. For a scalar convolution kernel k some explicit conditions are derived to determine whether or not the resolvent of k is integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1) and (2) are obtained using the …

A Note On Irreducibility And Weak Covering Properties, Joe Mashburn

Mathematics Faculty Publications

A space X is irreducible if every open cover of X has a minimal open refinement. Interest in irreducibility began when Arens and Dugendji used this property to show that metacompact countably compact spaces are compact. It was natural, then, to find out what other types of spaces would be irreducible and therefore compact in the presence of countable compactness or Lindelof in the presence of N₁-compactness. …

It is shown in this paper that T₁ δθ -refinable spaces and T1 weakly δθ-refinable spaces are irreducible. Since examples of Lindelof spaces that are neither T₁ nor …

The Least Fixed Point Property For Ω-Chain Continuous Functions, Joe Mashburn

Mathematics Faculty Publications

A partially ordered set P is ω-chain complete if every countable chain (including the empty set) in P has a supremum. … Notice that an ω-chain continuous function must preserve order. P has the (least) fixed point property for ω-chain continuous functions if every ω-chain continuous function from P to itself has (least) fixed point.

It has been shown that a partially ordered set does not have to be ω-chain complete to have the least fixed point property for ω-chain continuous functions. This answers a question posed by G. Plotkin in 1978. I.I. Kolodner has shown that an ω-chain complete …

Dissertation: The Least Fixed Point Property For Ω-Chain Continuous Functions, Joe Mashburn

Mathematics Faculty Publications

The basic definitions are given in the first section, including those for ω-chain continuity, ω-chain completeness, and the least fixed point property for ω-chain continuous functions. Some of the relations between completeness and fixed point properties in partially ordered sets are stated and it is briefly shown how the question basic to the dissertation arises.

In the second section, two examples are given showing that a partially ordered set need not be ω-chain complete to have the least fixed point property for ω-chain continuous functions.

Retracts are discussed in section 3, where it is seen that they are not sufficient …

Three Counterexamples Concerning Ω-Chain Continuous Functions And Fixed-Point Properties, Joe Mashburn

Mathematics Faculty Publications

A partially ordered set is ω-chain complete if, for every countable chain, or ω-chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an ω-chain complete partially ordered set has a least element, denoted by 0.