Mathematics Commons^™

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.

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.

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...

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 …

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.

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 …

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.

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.

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.

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).

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.

Primitive Ideals Of Semigroup Graded Rings, Hema Gopalakrishnan

Mathematics Faculty Publications

Prime ideals of strong semigroup graded rings have been characterized by Bell, Stalder and Teply for some important classes of semigroups. The prime ideals correspond to certain families of ideals of the component rings called prime families. In this paper, we define the notion of a primitive family and show that primitive ideals of such rings correspond to primitive families of ideals of the component rings.

Mathematics Placement Test: Helping Students Succeed, Norma Rueda, Carole Sokolowski

Mathematics Faculty Publications

A study was conducted at Merrimack College in Massachusetts to compare the grades of students who took the recommended course as determined by their mathematics placement exam score and those who did not follow this recommendation. The goal was to decide whether the mathematics placement exam used at Merrimack College was effective in placing students in the appropriate mathematics class. During five years, first-year students who took a mathematics course in the fall semester were categorized into four groups: those who took the recommended course, those who took an easier course than recommended, those who took a course more difficult …

A Random Boolean Network Model And Deterministic Chaos, Mihaela Teodora Matache, Jack Heidel

Mathematics Faculty Publications

This paper considers a simple Boolean network with N nodes, each node’s state at time t being determined by a certain number of parent nodes, which may vary from one node to another. This is an extension of a model studied by Andrecut and Ali ( [5]) who consider the same number of parents for all nodes. We make use of the same Boolean rule as the authors of [5], provide a generalization of the formula for the probability of finding a node in state 1 at a time t and use simulation methods to generate consecutive states of the …

A Fixed Point Theorem For Analytic Functions, Valentin Matache

Mathematics Faculty Publications

We prove that each analytic self-map of the open unit disk which interpolates between certain n-tuples must have a fixed point.

Norms Of Linear-Fractional Composition Operators, Paul S. Bourdon, E. E. Fry, Christopher Hammond, C. H. Spofford

Mathematics Faculty Publications

No abstract provided.

On Classifying Finite Edge Colored Graphs With Two Transitive Automorphism Groups, Thomas Q. Sibley

Mathematics Faculty Publications

This paper classifies all finite edge colored graphs with doubly transitive automorphism groups. This result generalizes the classification of doubly transitive balanced incomplete block designs with λ=1 and doubly transitive one-factorizations of complete graphs. It also provides a classification of all doubly transitive symmetric association schemes.

Inspiring Students To Study And Learn Mathematics Using Technology, Ma. Louise Antonette N. De Las Peñas, Wei-Chi Yang

Mathematics Faculty Publications

In this paper, we focus on the advantages of studying mathematics from analytical, graphical and geometric perspectives. Using Casio’s ClassPad 300 ([1]), we present activities on problems involving abstract concepts and real life applications. These activities viewed from the analytical, graphical, tabular and geometric points of view, promote creative ways to facilitate the learning of mathematics that inspire students with varying levels of ability. Consequently, students develop a profound appreciation and a deeper understanding of mathematics.

Fixed Point Theorems For Infinite Dimensional Holomorphic Functions, Lawrence A. Harris

Mathematics Faculty Publications

This talk discusses conditions on the numerical range of a holomorphic function defined on a bounded convex domain in a complex Banach space that imply that the function has a unique fixed point. In particular, extensions of the Earle-Hamilton Theorem are given for such domains. The theorems are applied to obtain a quantitative version of the inverse function theorem for holomorphic functions and a distortion form of Cartan's uniqueness theorem.

Combinatorial Identities Deriving From The N-Th Power Of A 2 X 2 Matrix, James Mclaughlin

Mathematics Faculty Publications

In this paper we give a new formula for the n-th power of a 2 × 2 matrix. More precisely, we prove the following: Let A = (a b c d) be an arbitrary 2 × 2 matrix, T = a + d its trace, D = ad − bc its determinant and define yn : = b X n/2c i=0 (n − i i )T n−2i (−D) i . Then, for n ≥ 1, A n = (yn − d yn−1 b yn−1 c yn−1 yn − a yn−1) . We use this formula together with an existing formula …

A Theorem On Divergence In The General Sense For Continued Fractions, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of such continued fractions. We apply this theorem to two general classes of q continued fraction to show, that if G(q) is one of these continued fractions and |q| > 1, then either G(q) converges or does not converge in the general sense. We also show that if the odd and even parts of the continued fraction K∞n=1an/1 converge to …

On The Divergence Of The Rogers-Ramanujan Continued Fraction On The Unit Circle, Douglas Bowman, James Mclaughlin

Mathematics Faculty Publications

This paper is an intensive study of the convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number t ∈ (0, 1) be denoted by [0, a1(t), a2(t), · · · ] and let the i-th convergent of this continued fraction expansion be denoted by ci(t)/di(t). Let S = {t ∈ (0, 1) : ai+1(t) ≥ φ di(t) infinitely often}, where φ = (√ 5 + 1)/2. Let YS = {exp(2πit) : t ∈ S}. It is shown that if y ∈ YS then the Rogers-Ramanujan continued fraction, R(y), diverges at y. S is an …

Palindrome-Polynomials With Roots On The Unit Circle, John Konvalina, Valentin Matache

Mathematics Faculty Publications

Given a polynomial f(x) of degree n, let fr(x) denote its reciprocal, i.e., fr(x) = xnf(1=x). If a polynomial is equal to its reciprocal, we call it a palindrome since the coefficients are the same when read backwards or forwards. In this mathematical note we show that palindromes whose coefficients satisfy a certain magnitude-condition must have a root on the unit circle...

Homology Of Holomorphs Of Free Groups, Craig A. Jensen

Mathematics Faculty Publications

Let Fn be the free group on n letters and let Aut(Fn), Out(Fn) denote the automorphism group and the outer automorphism group of Fn. In this paper the purpose is to obtain some new results on stability and to calculate the mod-p homology of the holomorph of Fn for odd primes in dimensions 1 and 2, and the rational homology in dimensions from 1 through 5.

Automorphisms Of Free Groups With Boundaries, Craig A. Jensen, Nathalie Wahl

Mathematics Faculty Publications

The automorphisms of free groups with boundaries form a family of groups An,k closely related to mapping class groups, with the standard automorphisms of free groups as An,0 and (essentially) the symmetric automorphisms of free groups as A0,k. We construct a contractible space Ln,k on which An,k acts with finite stabilizers and finite quotient space and deduce a range for the virtual cohomological dimension of An,k. We also give a presentation of the groups and calculate their first homology group.

Deconstructing Bases: Fair, Fitting, And Fast Bases, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.