Physical Sciences and Mathematics Commons^™

Isolation And Component Structure In Spaces Of Composition Operators, Christopher Hammond, Barbara D. Maccluer

Mathematics Faculty Publications

We establish a condition that guarantees isolation in the space of composition operators acting between H ^p (B _N ) and H ^q (B _N ), for 0 < p ≤ ∞, 0 < q < ∞, and N ≥ 1. This result will allow us, in certain cases where 0 < q < p ≤ ∞, completely to characterize the component structure of this space of operators.

A Convergence Theorem For Continued Fractions Of The Form K_{N=1}^{\Infty}A_{N}/1, James Mclaughlin, Nancy Wyshinski

Mathematics Faculty Publications

In this paper we present a convergence theorem for continued fractions of the form K∞n=1an/1. By deriving conditions on the an which ensure that the odd and even parts of K∞n=1an/1 converge, these same conditions also ensure that they converge to the same limit. Examples will be given.

On Convergence To The Denjoy-Wolff Point, P. S. Bourdon, Valentin Matache, J. H. Shapiro

Mathematics Faculty Publications

For holomorphic selfmaps of the open unit disc U that are not elliptic automorphisms, the Schwarz Lemma and the Denjoy-Wolff Theorem combine to yield a remarkable result: each such map φ has a (necessarily unique) "Denjoy-Wolff point"...

On Multiscale Approaches To 3-Dimensional Modeling Of Morphogenesis, Rajiv Chaturvedi, Chengbang Huang, Bogdan Kazmierczak, T. Schneider, Jesus A. Izaguirre, Tilmann Glimm, H. George E. Hentschel, Stuart (Stuart A.) Newman, Mark Alber, James A. Glazier

Mathematics Faculty Publications

In this paper we present the foundation of a unified, object-oriented, three-dimensional (3D) biomodeling environment, which allows us to integrate multiple submodels at scales from subcellular to tissues and organs. Our current implementation combines a modified discrete model from statistical mechanics, the Cellular Potts Model (CPM), with a continuum reaction-diffusion (RD) model and a state automaton with well-defined conditions for cell differentiation transitions to model genetic regulation. This environment allows us to rapidly and compactly create computational models of a class of complex developmental phenomena. To illustrate model development, we simulate a simplified version of the formation of the skeletal …

Ramanujan And The Regular Continued Fraction Expansion Of Real Numbers, James Mclaughlin, Nancy Wyshinski

Mathematics Faculty Publications

In some recent papers, the authors considered regular continued fractions of the form [a0; a, · · · , a | {z } m , a2 , · · · , a2 | {z } m , a3 , · · · , a3 | {z } m , · · · ], where a0 ≥ 0, a ≥ 2 and m ≥ 1 are integers. The limits of such continued fractions, for general a and in the cases m = 1 and m = 2, were given as ratios of certain infinite series. However, these formulae can be derived …

The Norm Of A Composition Operator With Linear Symbol Acting On The Dirichlet Space, Christopher Hammond

Mathematics Faculty Publications

We obtain a representation for the norm of a composition operator on the Dirichlet space induced by a map of the form φ(z)=az+b. We compare this result to an upper bound for ‖Cφ‖ that is valid whenever φ is univalent. Our work relies heavily on an adjoint formula recently discovered by Gallardo-Gutiérrez and Montes-Rodríguez.

Balanced Scaling Vectors Using Linear Combinations Of Existing Scaling Vectors, Bruce Kessler

Mathematics Faculty Publications

The majority of the research done into creating balanced multiwavelets has involved establishing a series of conditions on the mask of the new scaling vector by solving a large nonlinear system. The result is a completely different new function vector solution to the dilation equation with the new matrix coefficients. The research presented here will show a way to use previously-constructed orthonormal scaling vectors to generate equivalent orthonormal scaling vectors that are balanced up to the approximation order of the previous scaling vector. The technique uses linear combinations of the integer translates of the previous-constructed scaling vector.

The Cohomology Of Right Angled Artin Groups With Group Ring Coefficients, Craig A. Jensen

Mathematics Faculty Publications

The cohomology of a right-angled Artin group with group ring coefficients is explicitly presented in terms of the cohomology of its defining flag complex. 2000 Mathematics Subject Classification 20F36 (primary), 57M07 (secondary).

Proper Actions Of Automorphism Groups Of Free Products Of Finite Groups, Craig A. Jensen, Yuqing Chen, Henry H. Glover

Mathematics Faculty Publications

If G is a free product of finite groups, let ΣAut₁(G) denote all (necessarily symmetric) automorphisms of G that do not permute factors in the free product. We show that a McCullough–Miller and Gutiérrez–Krstić derived (also see Bogley–Krstić) space of pointed trees is an EΣAut₁(G)-space for these groups.

On The Supersymmetry Group Of The Classical Bose-Fermi Oscillator,", Tilmann Glimm, Rudolf Schmid

Mathematics Faculty Publications

Applying the concept of a momentum map for supersymplectic supervectorspaces to the one-dimensional Bose-Fermi oscillator, we show that the largest symmetry group that admits a momentum map is the identity component of the intersection of the orthosymplectic group OSp(2|2) and the group of supersymplectic transformations. This gives a systematic characterization of a certain class of odd supersymmetry transformations that were originally introduced in an ad hoc way.

Connectivity Of Random K-Nearest-Neighbor Graphs, Paul Balister, Béla Bollobás, Amites Sarkar, Mark Walters

Mathematics Faculty Publications

Let P be a Poisson process of intensity one in a square S_n of area n. We construct a random geometric graph G_n,k by joining each point of Pto its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074logn then the probability that G_n,k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774logn, then the probability that G_n,k is connected tends to 1 as n → ∞. They conjectured that the …

Review Of: Newton Methods For Nonlinear Problems: Affine Invariance And Adaptive Algorithms, By P. Deuflhard, Tjalling Ypma

Mathematics Faculty Publications

In the context of solving nonlinear equations, the term "affine invariance" was introduced to describe the fact that when a function F: Rⁿ → Rⁿ is transformed to G = AF ,where A is an invertible matrix, then the equation F(x) = 0 has the same solutions as G(x) = 0, and the Newton iterates X_^k+1 = X_k-F'(X_k)^-1F(X_k) remain unchanged when F is replaced by G. The idea was that this property of Newton's method should …

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.

Positive Operators And Maximum Principles For Ordinary Differential Equations, Paul W. Eloe

Mathematics Faculty Publications

We show an equivalence between a classical maximum principle in differential equations and positive operators on Banach Spaces. Then we shall exhibit many types of boundary value problems for which the maximum principle is valid. Finally, we shall present extended applications of the maximum principle that have arisen with the continued study of the qualitative properties of Green’s functions.

Real Numbers With Polynomial Continued Fraction Expansions, James Mclaughlin, Nancy Wyshinski

Mathematics Faculty Publications

In this paper we show how to apply various techniques and theorems (including Pincherle’s theorem, an extension of Euler’s formula equating infinite series and continued fractions, an extension of the corresponding transformation that equates infinite products and continued fractions, extensions and contractions of continued fractions and the Bauer-Muir transformation) to derive infinite families of in-equivalent polynomial continued fractions in which each continued fraction has the same limit. This allows us, for example, to construct infinite families of polynomial continued fractions for famous constants like π and e, ζ(k) (for each positive integer k ≥ 2), various special functions evaluated at …

Powers Of A Matrix And Combinatorial Identities, James Mclaughlin, B. Sury

Mathematics Faculty Publications

In this article we obtain a general polynomial identity in k variables, where k ≥ 2 is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a k × k matrix. Finally, we use these results to derive various combinatorial identities.

Compact Topologically Torsion Elements Of Topological Abelian Groups, Peter Loth

Mathematics Faculty Publications

In this note, we prove that in a Hausdorff topological abelian group, the closed subgroup generated by all compact elements is equal to teh closed subgroup generated by all compact elements which are topologically p-torsion for some prime p. In particular, this yields a new, short solution to a question raised by Armacost [A]. Using Pontrjagin duality, we obtain new descriptions of the identity component of a locally compact abelian group.

Queuing Systems With Multiple Fbm-Based Traffic Models, Mihaela Teodora Matache, Valentin Matache

Mathematics Faculty Publications

A multiple fractional Brownian motion (FBM)-based traffic model is considered. Various lower bounds for the overflow probability of the associated queueing system are obtained. Based on a probabilistic bound for the busy period of an ATM queueing system associated with a multiple FBM-based input traffic, a minimal dynamic buffer allocation function (DBAF) is obtained and a DBAF-allocation algorithm is designed. The purpose is to create an upper bound for the queueing system associated with the traffic. This upper bound, called a DBAF, is a function of time, dynamically bouncing with the traffic. An envelope process associated with the multiple FBM-based …

Asynchronous Random Boolean Network Model Based On Elementary Cellular Automata Rule 126, Mihaela Teodora Matache, Jack Heidel

Mathematics Faculty Publications

This paper considers a simple Boolean network with N nodes, each node’s state at time tbeing determined by a certain number k of parent nodes, which is fixed for all nodes. The nodes, with randomly assigned neighborhoods, are updated based on various asynchronous schemes. We make use of a Boolean rule that is a generalization of rule 126 of elementary cellular automata. We provide formulas for the probability of finding a node in state 1 at a time t for the class of asynchronous random Boolean networks (ARBN) in which only one node is updated at every time step, and …

A Short Proof Of A Characterization Of Inner Functions In Terms Of The Composition Operators They Induce, Valentin Matache

Mathematics Faculty Publications

The paper contains a new proof for the sufficiency in Joel H. Shapiro’s recent characterization of inner functions...

Existence, Uniqueness And Constructive Results For Delay Differential Equations, Paul W. Eloe, Youssef N. Raffoul, Christopher C. Tisdell

Mathematics Faculty Publications

Here, we investigate boundary-value problems (BVPs) for systems of second-order, ordinary, delay-differential equations. We introduce some differential inequalities such that all solutions (and their derivatives) to a certain family of BVPs satisfy some a priori bounds. The results are then applied, in conjunction with topological arguments, to prove the existence of solutions. We then apply earlier abstract theory of Petryshyn to formulate some constructive results under which solutions to BVPs for systems of second-order, ordinary, delay-differential equations are A-solvable and may be approximated via a Galerkin method. Finally, we provide some differential inequalities such that solutions to our equations are …

Convergent Sequences Of Composition Operators, Valentin Matache

Mathematics Faculty Publications

Composition operators Cφ on the Hilbert Hardy space H² over the unit disk are considered.

The Use And Influence Of Technology In Mathematics Education, Ma. Louise Antonette N. De Las Peñas, Wei-Chi Yang

Mathematics Faculty Publications

The use of various types of technologies in the classroom and examinations is growing rapidly and is strongly influencing teaching and learning practices. In this paper, we will look at particular situations on how various technologies such as numerically capable calculators, graphics calculators, and technological tools that are CAS enabled or have CAS with Dynamic Geometry, impact students' learning. We also discuss briefly the educational opportunities that are made available by the emergence of graphics calculators with capabilities of handling electronic learning activities, such as Casio’s Class Pad (see [1]) and Casio’s 9860 graphics calculator.

Operator Self-Similar Processes On Banach Spaces, Mihaela Teodora Matache, Valentin Matache

Mathematics Faculty Publications

Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determinedmultiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0, that is, their norm is less than a nonnegative constant times such a power-function, provided that the linear space spanned by the expectations has category …