Physical Sciences and Mathematics Commons^™

Flipped Calculus: A Study Of Student Performance And Perceptions, Lori Beth Ziegelmeier, Chad M. Topaz

Lori Beth Ziegelmeier

No abstract provided.

Flipped Calculus: A Study Of Student Performance And Perceptions, Lori Beth Ziegelmeier, Chad M. Topaz

Chad M. Topaz

No abstract provided.

Form Domains And Eigenfunction Expansions For Differential Equations With Eigenparameter Dependent Boundary Conditions, Branko Ćurgus, Paul Binding

Branko Ćurgus

Form domains are characterized for regular 2n-th order differential equations subject to general self-adjoint boundary conditions depending affinely on the eigenparameter. Corresponding modes of convergence for eigenfunction expansions are studied, including uniform convergence of the first n - 1 derivatives.

An Unexpected Limit Of Expected Values, Branko Ćurgus, Robert I. Jewett

Branko Ćurgus

Let t⩾0. Select numbers randomly from the interval [0,1] until the sum is greater than t . Let α(t) be the expected number of selections. We prove that α(t)=et for 0⩽t⩽1. Moreover, . This limit is a special case of our asymptotic results for solutions of the delay differential equation f′(t)=f(t)-f(t-1) for t>1. We also consider four other solutions of this equation that are related to the above selection process.

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

Catherine Kublik

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 …

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

Catherine Kublik

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.

An Implicit Interface Boundary Integral Method For Poisson’S Equation On Arbitrary Domains, Catherine Kublik, Nicolay M. Tanushev, Richard Tsai

Catherine Kublik

We propose a simple formulation for constructing boundary integral methods to solve Poisson’s equation on domains with smooth boundaries defined through their signed distance function. Our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain, where the integrations are carried out in the level set framework using an appropriate Jacobian. By the coarea formula, the algorithm operates in the Euclidean space and does not require any explicit parameterization of the boundaries. We present numerical results in two and three dimensions.

Lyapunov Functionals That Lead To Exponential Stability And Instability In Finite Delay Volterra Difference Equations, Catherine Kublik, Youssef Raffoul

Catherine Kublik

We use Lyapunov functionals to obtain sufficient conditions that guarantee exponential stability of the zero solution of the finite delay Volterra difference equation. Also, by displaying a slightly different Lyapunov functional, we obtain conditions that guarantee the instability of the zero solution. The highlight of the paper is the relaxing of the condition |a(t)| < 1. Moreover, we provide examples in which we show that our theorems provide an improvement of some recent results.

Foliations And Global Inversion, Eduardo C. Balreira

Eduardo Cabral Balreira

We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism f : M → Rn is bijective if and only if Hn−1(M) = 0 and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometric constructions involving foliations and tools from intersection theory. This topological result generalizes in finite dimensions the classical analytic theorem of Hadamard-Plastock, including its recent improvement by Nollet-Xavier. The main theorem also relates to a conjecture of the aforementioned authors, involving the well known Jacobian Conjecture in …

Stability Of Hyperbolic And Nonhyperbolic Fixed Points Of One-Dimensional Maps, Fozi M. Dannan, Saber Elaydi, Vadim Ponomarenko

Saber Elaydi

We present a complete theory for the stability of non-hyperbolic fixed points of one-dimensional continuous maps. As well as we give simple criteria for the global stability of general maps without using derivatives.

Global Stability Of Cycles: Lotka-Volterra Competition Model With Stocking, Saber Elaydi, Abdul-Aziz Yakubu

Saber Elaydi

In this article, we prove that in connected metric spaces k - cycles are not globally attracting (where k>2). We apply this result to a two species discrete-time Lotka-Volterra competion model with stocking. In particular, we show that an k-cycle cannot be the ultimate life-history of evolution of all population sizes. This solves Yakubu's conjecture but the question on the structure of the boundary of the basins of attraction of the locally stable n-cycles is still open.

On The Topology Of The Permutation Pattern Poset, Peter R. W. Mcnamara, Einar Steingrímsson

Peter R. W. McNamara

A Note On Irreducibility And Weak Covering Properties, Joe Mashburn

Joe D. Mashburn

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 irreducible can be easily …

Sobriety In Delta Not Sober, Joe Mashburn

Joe D. Mashburn

We will show that the space delta not sober defined by Coecke and Martin is sober in the Scott topology, but not in the weakly way below topology.

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

Joe D. Mashburn

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 …

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

Joe D. Mashburn

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 …

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

Joe D. Mashburn

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 …

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

Joe D. Mashburn

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.

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

Joe D. Mashburn

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, …

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

Joe D. Mashburn

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.

A Spectral Order For Infinite Dimensional Quantum Spaces, Joe Mashburn

Joe D. Mashburn

In this paper we extend the spectral order of Coecke and Martin to infinite-dimensional quantum states. Many properties present in the finite-dimensional case are preserved, but some of the most important are lost. The order is constructed and its properties analysed. Most of the useful measurements of information content are lost. Shannon entropy is defined on only a part of the model, and that part is not a closed subset of the model. The finite parts of the lattices used by Birkhoff and von Neumann as models for classical and quantum logic appear as subsets of the models for infinite …

An Order Model For Infinite Classical States, Joe Mashburn

Joe D. Mashburn

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 …

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

Joe D. Mashburn

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 which have this …

Existence And Stability Of Periodic Orbits Of Periodic Difference Equations With Delays, Ziyad Alsharawi, James Angelos, Saber Elaydi

Saber Elaydi

In this paper, we investigate the existence and stability of periodic orbits of the p-periodic difference equation with delays xn = f(n−1, xn−k). We show that the periodic orbits of this equation depend on the periodic orbits of p autonomous equations when p divides k. When p is not a divisor of k, the periodic orbits depend on the periodic orbits of gcd(p, k) nonautonomous p gcd(p,k) - periodic difference equations. We give formulas for calculating the number of different periodic orbits under certain conditions. In addition, when p and k are relatively prime integers, we introduce what we call …

On The Stochastic Beverton-Holt Equation With Survival Rates, Paul H. Bezandry, Toka Diagana, Saber Elaydi

Saber Elaydi

The paper studies a Beverton-Holt difference equation, in which both the recruitment function and the survival rate vary randomly. It is then shown that there is a unique invariant density, which is asymptotically stable. Moreover, a basic theory for random mean almost periodic sequence on Z+ is given. Then, some suffcient conditions for the existence of a mean almost periodic solution to the stochastic Beverton-Holt equation are given.

Population Models With Allee Effect: A New Model, Saber Elaydi, Robert J. Sacker

Saber Elaydi

In this paper we develop several mathematical models of Allee effects. We start by defining the Allee effect as a phenomenon in which individual fitness increases with increasing density. Based on this biological assumption, we develop several fitness functions that produce corresponding models with Allee effects. In particular, a rational fitness function yields a new mathematical model that is our focus of study. Then we study the dynamics of 2-periodic systems with Allee effects and show the existence of an asymptotically stable 2-periodic carrying capacity.

Bifurcation And Invariant Manifolds Of The Logistic Competition Model, Malgorzata Guzowska, Rafael Luis, Saber Elaydi

Saber Elaydi

In this paper we study a new logistic competition model. We will investigate stability and bifurcation of the model. In particular, we compute the invariant manifolds, including the important center manifolds, and study their bifurcation. Saddle-node and period doubling bifurcation route to chaos is exhibited via numerical simulations.

General Allee Effect In Two-Species Population Biology, G Livadiotis, Saber Elaydi

Saber Elaydi

The main objective of this work is to present a general framework for the notion of the strong Allee effect in population models, including competition, mutualistic, and predator–prey models. The study is restricted to the strong Allee effect caused by an inter-specific interaction. The main feature of the strong Allee effect is that the extinction equilibrium is an attractor. We show how a ‘phase space core’ of three or four equilibria is sufficient to describe the essential dynamics of the interaction between two species that are prone to the Allee effect. We will introduce the notion of semistability in planar …

Quasisymmetric (K; L)-Hook Schur Functions, Sarah K. Mason, Elizabeth Niese

Elizabeth Niese

We introduce a quasisymmetric generalization of Berele and Regev's (k,l)-hook Schur functions. These quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. The quasisymmetric hook Schur functions can be defined as the generating function for a certain set of composition tableaux on two alphabets. We will look at the combinatorics of the quasisymmetric hook Schur functions, including an analogue of the RSK algorithm and a generalized Cauchy Identity.

Incompressibility And Global Inversion, Eduardo C. Balreira

Eduardo Cabral Balreira

Given a local diffeomorphism f : ℝn → ℝn, we consider certain in- compressibility conditions on the parallelepiped D f (x) ([0, 1]n) which imply that the pre-image of an affine subspace is non-empty and has trivial homotopy groups. These conditions are then used to establish criteria for f to be globally invertible, generalizing in all dimensions the previous results of M. Sabatini.