Digital Commons Network^™

On Some Range Inclusions, Mihaela Anca Poplicher

Doctoral Dissertations

The de Branges-Rovnyak spaces have been introduced by Louis de Branges and James Rovnyak, and have been studied by several authors, in particular by Donald Sarason. The object of this thesis is to study more extensively some properties of this class of spaces. In particular, which of these spaces are invariant under the action of which composition operators, or adjoints of composition operators.

Some known facts and a couple of new properties are included in Chapter 1. There it is shown that the norm of the kernel function for evaluation at a point in the unit disc (in the de …

Composition Operators On Riemann Surfaces, Ioana Crenguta Mihaila

Doctoral Dissertations

General background. Composition operators are defined on a Hilbert (or Banach) spaces of complex valued functions defined on some set X. For the big majority of cases the set X is the unit disc in the complex plane, and the space of functions is one of the Hardy or Bergman spaces (weighted or not). This is due, without doubt, to the richness of those spaces, and the high degree of interest in them. There have been also important papers on the study of the Hardy spaces on the unit ball of n-dimensional complex space.

I have been working with my …

Profiles Of Reform In The Teaching Of Calculus: A Study Of The Implementation Of Materials Developed By The Calculus Consortium Based At Harvard (Cch) Curriculum Project, Alice Darien Lauten

Doctoral Dissertations

The research question addressed in this study is: What profiles of interpretation and implementation of reform in the teaching of calculus emerge from data obtained from mathematics faculty members using Calculus Consortium Based at Harvard (CCH) Curriculum Project materials? Site liaisons from mathematics departments using CCH Curriculum Project materials in 117 academic institutions, consisting of 13 secondary schools, 30 two-year colleges, 19 doctoral and research universities, and 55 other colleges and universities, completed Initial and Site Liaison Surveys. Site liaisons and 266 other instructors from 117 academic institutions completed a Faculty Survey. Six clustering scales were developed from the survey …

Extensions Of Bialgebras And Their Cohomological Description, Mark Lloyd Bochert

Doctoral Dissertations

This paper develops the theory of crossed product Hopf algebras of pairs of arbitrary Hopf algebras. The theory generalizes the crossed products of (Maj90), the Abelian crossed products of (Hof94) and the crossed product algebras of (BCM86). First, conditions are given on the structures involved that are shown to be equivalent to the existence of the crossed product. Next, a bisimplicial object is found that gives a cohomological description of the conditions. Cleft extensions of pairs of arbitrary Hopf algebras are then defined. These generalize the cleft extension algebras of (Swe68) and the Abelian cleft extensions of (By93); they are …

Orbit-Reflexivity, Michael James Mchugh

Doctoral Dissertations

Suppose $H$ is a separable, infinite dimensional Hilbert space and $T$ and $S$ are bounded linear transformations on $H$. Suppose that if $Sx\in\{x, ,Tx ,T\sp2x,...\}\sp{-}$ for every $x$ implies that $S\in\{1, T, T\sp2,...\}\sp{-SOT}$ then $T$ is orbit-reflexive. Many operators are proven to be orbit-reflexive, including analytic Toeplitz operators and subnormal operators with cyclic vectors.

Suppose that if $Sx\in\{\gamma x : x\in H, \gamma\in\doubc\}\sp{-}$ for every $x$, implies that $S\in\{\gamma T\sp{n} : n\ge0, \lambda\in\doubc\}\sp{-SOT}$ then $T$ is $\doubc$-orbit-reflexive. Many operators are shown to be $\doubc$-orbit-reflexive. $\doubc$-orbit-reflexivity is shown to be the same as reflexivity for algebraic operators.

Extending The National Council Of Teachers Of Mathematics' "Recognizing And Recording Reform In Mathematics Education" Documentation Project Through Cross-Case Analyses, Loren Phaffle Johnson

Doctoral Dissertations

The primary emphasis of this study was to broaden the understanding of data collected in the National Council of Teachers of Mathematics' (NCTM) Recognizing and Recording Reform in Mathematics Education (R$\sp3$M) project in order to more fully clarify the processes by which reform in mathematics education was occurring across five high school sites. A secondary emphasis was to develop a model of doing cross-case analyses and identifying those methodological elements and linkages that could be applied generally in large-scale studies of this sort.

R$\sp3$M documenters obtained data that resulted from interviews of mathematics teachers, administrators, and students; classroom observations; and …

Part I Synthesis, Functionalization And Metal Complexation Of Polyamine Macrocycles Part Ii Synthesis And Dynamic Nmr Studies Of Bicyclic Ureas, Daniel Cope Hill

Doctoral Dissertations

Part I. The syntheses and characterizations of novel bicyclic tetraamines (shown below) have been accomplished. The complexation of bicyclic tetraamines (R = CH$\sb3)$ with Li$\sp+$ and Na$\sp+$ have been studied by NMR spectroscopy. These tetraamines are strong bases and selective Li$\sp+$ binders. The parent "cross-bridged cyclam" (R = H; X = Y = CH$\sb2)$ was functionalized with a variety of ligating groups containing heteroatoms. Several of the ligands were investigated as possible radiopharmaceutical ($\rm\sp{99m}Tc$) imaging agent precursors.

Syntheses of non-adjacent selectively functionalized tetraazacycloalkanes have also been developed.

Part II. Synthesis and dynamic NMR studies of intramolecular transamidation of bicyclic urea …

An Investigation Into Students' Conceptual Understandings Of The Graphical Representation Of Polynomial Functions, Judith Ellen Curran

Doctoral Dissertations

Mathematics educators are realizing the impact that technology is having on the way mathematical functions can be represented and manipulated. The increased use of graphing technology in the classroom is paralleled by an increased emphasis on the role of the graphical representation of a function to solve problems. These changes together with a recognition of the significance and complexity of developing a rich understanding of the graphical representations of polynomial functions are the motivation behind this research.

The study was designed to explore students' conceptual understandings of the graphs of polynomial functions. Guided by a constructivist approach to conceptual change, …

Decomposable Functions And Universal C*-Algebras, Llolsten Kaonga

Doctoral Dissertations

This paper deals with universal $C\sp\*$-algebras generated by matricial relations on the generators, for example, the universal $C\sp\*$-algebra with generators $a\sb{ij}, 1 \leq i,j \leq n$, subject to the condition that the matrix ($a\sb{ij}$) be normal and have spectrum in a designated compact subset ${\cal K}$ of the complex plane.

The main thrust of the paper is to compute the K-groups of some of these $C\sp\*$-algebras and to determine when they contain non-trivial projections. In the above example, we show that the K-groups of the algebra coincide with the topological K-groups of the set ${\cal K}$. We show, in general, …

Stability Properties For The Constant Of Hyperreflexivity, Ileana Ionascu

Doctoral Dissertations

Let H be a separable, complex, Hilbert space and let ${\cal B}(H$) be the algebra of all (bounded linear) operators on H. We define a function$$\kappa:{\cal B}(H) \to \lbrack 1,\infty\rbrack;\qquad \kappa(T) = K({\cal A}\sb{w}(T)),\qquad \forall T \in {\cal B}(H),$$where ${\cal A}\sb{w}(T$) is the unital weakly closed algebra generated, in ${\cal B}(H$), by T, and $K({\cal A}\sb{w}(T$)) is the constant of hyperreflexivity of ${\cal A}\sb{w}(T$). If H is finite-dimensional, we show that $\kappa$ is continuous at $T \in {\cal B}(H$) if and only if T is non-reflexive or has dimH distinct eigenvalues (Theorem 2.6). An auxiliary result (Theorem 2.1) states that …

On The Berezin Symbol, Semra Kilic-Bahi

Doctoral Dissertations

Let ${\cal H}$ be a functional Hilbert space of analytic functions on a complex domain $\Omega,$ with the normalized reproducing kernel function $k\sb{z},\ z\in\Omega.$ If A is a linear map of ${\cal H}$ into itself, the Berezin symbol, A, of A is defined on $\Omega$ by $\tilde{A}(z) = \langle Ak\sb{z},\ k\sb{z}\rangle.$ The purpose of this research is to study how the properties of an operator are reflected in the properties of its Berezin symbol. In summary, I have (1) studied the properties of the Berezin symbol as a complex-valued function; (2) characterized multiplication operators, induced by a multiplier of ${\cal …

Reflexive Subspaces And Lattices Of Pairs Of Projections, Deborah Narang

Doctoral Dissertations

Consider the sets ${\cal P}\sb{\cal H}$ and ${\cal P}\sb{\cal K}$ of the projections onto closed subspaces of Hilbert spaces ${\cal H}$ and $\cal K$ respectively. From the usual partial orders (based upon set containment) on $\cal P\sb{\cal K}$ and $\cal P\sb{\cal H}$, we can define a partial order on $\cal P\sb{\cal K}\times\cal P\sb{\cal H}$ by ($Q\sb1,P\sb1)\le(Q\sb2, P\sb2)$ if and only if $P\sb1\le P\sb2$ and $Q\sb2\le Q\sb1.$ Then the map $\alpha : \cal P\sb{\cal K}\times \cal P\sb{\cal H}\to \cal P\sb{\cal K\oplus\cal H}$ given by $\alpha(Q,P)=(1-Q)\oplus P$ is an order-preserving map. In particular, if $\cal L\subseteq\cal P\sb{\cal K\times\cal H}$ is a lattice, …

Hankel Operators On Hilbert Spaces, Pachara Wanpen

Doctoral Dissertations

In this paper we consider the Hankel operators from two points of view. On one hand the Hankel operator is induced by the coefficient sequence $a\sb0,a\sb1,a\sb2,\...$ and operates on a Hilbert space $H\sp2(\beta)$ with $\Sigma\sbsp{n=0}{\infty}\ \beta(n)\sp2 < \infty.$ In this situation we can find necessary conditions and sufficient conditions for the Hankel operator to be bounded. However, with compactness and Hilbert-Schmidt we can get only sufficient conditions. On the other hand we look at the Hankel operator $H\sb{f,\alpha}$ and little Hankel operator $h\sb{f,\alpha},$ with symbol function f, that operates on a weighted Bergman space. In this case we can determine bounded, compact, Hilbert Schmidt, or trace class operators of the Hankel operator $H\sb{f}$ and $h\sb{f,\alpha}.$ We also give a good estimate of bounded norm of little Hankel operators with a particular symbol function $f = z\bar g$ where g is in the Bloch space.

Approximate Equivalence Invon Neumann Algebras, Hui-Ru Ding

Doctoral Dissertations

In this paper we investigate approximate equivalence in von Neumann algebras. We find a necessary and sufficient condition for two normal operators to be approximately equivalent in any von Neumann algebra ${\cal R}$ acting on a separable Hilbert space H with unitaries in ${\cal R}.$ For the approximate equivalence of two unital representations from a given C$\*$-algebra to any von Neumann algebra acting on a separable Hilbert space, we find the necessary condition for the general case. Finally we investigate an interesting class of C$\*$-algebras, closed under direct sum, direct limit and quotient map, which contains C(X) and $M\sb{n}(A),$ where …

Finite Groups As A Generalization Of Vector Spaces Through The Use Of Splitting Systems, Joseph Kirtland

Doctoral Dissertations

The structure of a finite group is investigated through a geometry induced by the splitting systems of the group. The method is based on the one used to induce a geometry on a finite dimensional vector space over a finite field and as a result, concepts related to the special and projective linear group are extended to arbitrary groups. One major by-product is the classification of solvable multiprimitive groups of arbitrary derived length. This leads to a necessary and sufficient condition for a solvable nC-group to be multiprimitive.

Beliefs, Autonomy, And Mathematical Knowledge, Judy Ann Rector

Doctoral Dissertations

The purpose of this study was to investigate the apparent effects of students' beliefs about mathematics and autonomy on their learning of mathematics. The study utilized a multiple-case study design with analysis by and across cases. The cases represented six high school students enrolled in either Algebra II or Algebra II/Trigonometry. Data was collected in three phases: (a) classroom observations and assessment of the teacher's perception of her role in the learning process, (b) an assessment of students' beliefs about mathematics and autonomy, and (c) an assessment of students' newly formed mathematical constructs on functions.

The beliefs' assessment included observing …

Distance Function Constructions In Topological Spaces, Laurie Jean Sawyer

Doctoral Dissertations

This work investigates the use of distance function constructions in the study of semimetrizable spaces, especially as this relates to developable, K-semimetrizable and 1-continuously semimetrizable spaces.

A distance function for X is a nonnegative, symmetric, real-valued function d: X x X $\to$ $\IR$ such that d(p,q) = 0 iff p = q. A distance function d is developable iff, when d(x$\sb{\rm n}$,p) $\to$ 0 and d(y$\sb{\rm n}$,p) $\to$ 0, then d(x$\sb{\rm n}$,y$\sb{\rm n}$) $\to$ 0; and d is a K-distance function iff whenever d(x$\sb{\rm n}$,p) $\to$ 0, d(y$\sb{\rm n}$,q) $\to$ 0 and d(x$\sb{\rm n}$,y$\sb{\rm n}$) $\to$ 0, then p = …

Invariant Operator Ranges Of Operator Algebras, D Benjamin Mathes

Doctoral Dissertations

Given a norm closed unital algebra ${\cal A}$ of operators on Hilbert space, a sublattice of Lat$\sb{1/2}{\cal A}$ is identified, which we denote Lat$\sb{\rm cb}{\cal A}$. It is proved that Lat$\sb{\rm cb}{\cal A}$ is lattice isomorphic to Lat$\sb{1/2}{\cal A}\otimes{\cal K}$ (where ${\cal K}$ denotes the ideal of compact operators on an infinite dimensional separable Hilbert space). This isomorphism is used to prove theorems describing Lat$\sb{\rm cb}{\cal A}$ by carrying over know results concerning Lat$\sb{1/2}{\cal A}\otimes{\cal K}$. It is then shown that Lat$\sb{\rm cb}{\cal A}$ can always be written as the set of ranges of operators that intertwine the algebra with …

The Role Of Error Analysis, Diagnostic Grading Procedures, And Student Reflection In First Semester Calculus Learning (Epistemology, Metacognition, Inquiry), Karen Jean Geuther

Doctoral Dissertations

The present study is an integrative study designed to explore the nature of student difficulties within the context of a first semester college calculus course. The techniques of error analysis were used to identify and categorize the student difficulties. Insights gained from this categorization served as the basis for the design and development of calculus grader preparation materials implemented with a subgroup of undergraduate calculus graders. These grader preparation materials emphasized grader diagnosis of student difficulties and student reflection on errors and process. Current preparation procedures served as the control on the basis of which comparisons and evaluations were made. …

Operator Ranges Of Shifts And C*-Algebras (Strange Range, Quasi-Similarity, Lattice), Charles Lucien Roy

Doctoral Dissertations

It is shown that Lat(, 1/2)(H('(INFIN))(S)), the invariant operator ranges of the commutant of the unilateral shift, is a proper sub-lattice of the lattice of invariant operator ranges of the unilateral shift, S. The notion of a strange operator range for S of order n where n (ELEM) is introduced and it is demonstrated that there exist strange ranges for S of every order. This is done by deriving an operator range condition which is sufficient to insure that a pair of quasi-similar compressions of shifts really be similar. A set of operator ranges which forms a sub-lattice of Lat(, …

Effects Of Field-Dependence/Independence And Sex On Patterns Of Achievement And Grading In A First-Semester Calculus Course (Cognitive Style, Remedial Mathematics, Grading Bias, Appraisal, College), Timothy John Kelly

Doctoral Dissertations

The purpose of this study was twofold: first, to investigate effects of field-dependence/independence (f-d/i), sex, and different types of audiovisual remedial instruction on mathematics achievement (Experiment I); second to investigate the effects of student and grader cognitive style (f-d/i) and sex on calculus test grades (Experiment II).

One hundred fifty first-semester calculus students, identified by an algebra/trigonometry pretest as requiring trigonometry remediation, served as subjects for Experiment I. As measures of f-d/i, the Group Embedded Figures Test (GEFT) and The Miniaturized Rod and Frame Test (MRFT) were administered to the subjects, who were apportioned into six experimental groups, representing three …

Some Problems In Bayesian Inference In A Nonclassical Setting (Operational Statistics, Betting Rates, Coherence), Carolyn Margaret Magness

Doctoral Dissertations

The Bayesian inference strategy is studied within the framework of operational statistics (generalized sample spaces). Conditions necessary and sufficient for consistency and strict consistency of a complete betting rate assignment are derived. A constructive proof is given of the existence of a unique second-order probability which induces a given consistent and countably additive complete betting rate assignment. Continuity of the Bayesian inference strategy is investigated. Asymptotic properties of the Bayesian inference strategy are examined both at the level of betting rates and at the level of second-order probabilities.

Covariance Structural Models For Mathematics Achievement And Participation: An Investigation Of Sex Differences At The Level Of College Calculus Using Factorial Modeling, Thomas Patrick Dick

Doctoral Dissertations

The purpose of this study was to develop covariance structural models that would explain, in terms of orthogonal latent variables, the correlations observed among mathematics achievement and participation measures and related cognitive and affective variables. A random sample of college calculus students (N = 268; 124 females and 144 males) was administered: (1) a battery of cognitive tests, including measures of spatial-visual ability, field independence/field dependence, and logical reasoning; and (2) a battery of affective scales, including measures of attitude toward mathematics, confidence, perceived usefulness of mathematics, effectance motivation, and locus of control. Measures of previous academic achievement and participation …

Statistical Methods For Analysis Of Cancer Chemoprevention Experiments, Stephen Michael Kokoska

Doctoral Dissertations

The experimental systems studied in this dissertation are designed to investigate the effect of diet on incidence rates of cancer. These investigations involve the chemical induction of tumors in experimental animals in order to test the chemopreventative effects of various substances. Both tumor number and rate of tumor development are important in evaluating the effects of a chemopreventative agent. This is made difficult, when multiple tumors occur, by the confounding of the number of induced tumors and their time of detection. This confounding occurs because experiments are terminated before all induced tumors have been detected. Fewer tumors observed in one …

On A Conjugate Class Of Subgroups Determined By A Formation, Mark Challis Hofmann

Doctoral Dissertations

This thesis is an investigation of the interrelationships between a formation f, a finite solvable group G, and G(,f) the residual of f in G. This study is developed by introducing the f-subgroups. It is proven that the f-subgroups of G form a characteristic conjugacy class of CAR-subgroups of G. Moreover these subgroups generate G(,f). As a result, G is an element of the formation f if and only if an f-subgroup is equal to the identity subgroup.

It is established that an f-subgroup is a product of known subgroups of the f-residual. The covering and avoidance properties of f-subgroups …

The Construction Of A Logical-Empirical Structure Of Knowledge For Differential Calculus Using A Theoretical Framework Based Upon Learning Hierarchy Theory And Order Theory, James Martin Devecchi

Doctoral Dissertations

The principal result of this study is a logically and empirically valid Structure of Knowledge for Differential Calculus constructed using a methodology based on Learning Hierarchy Theory (LHT) and Order Theory (OT). A review of existing research into calculus learning is presented and properties of the strucutre of knowledge which address deficiencies in existing knowledge discussed. LHT is developed and techniques for the empirical validation and generation of hierarchies are discussed. Included in this discussion is a detailed analysis of the important properties of OT.

The generation of the structure of knowledge consisted of two phases: the logical analysis and …

Spatial Ability, Mathematics Achievement, And Spatial Training In Male And Female Calculus Students, Joan Ferrini Mundy

Doctoral Dissertations

The primary purpose of this study was to investigate the effects of spatial training upon calculus achievement, spatial visualization, and tendency and ability to visualize solids of revolution in calculus, for college males and females. A random sample of 250 first-semester calculus students at the University of New Hampshire was apportioned into seven experimental groups, each containing males and females. Students were pretested on precalculus background, spatial visualization, and calculus background, and were assigned to one of three treatment conditions: audiovisual spatial training (AV), audio-vidual-tactual spatial training (AVT), and a control condition. The spatial training materials were organized into six …

An Investigation Of The Interactions Of Student Ability Profiles And Instruction In Heuristic Strategies With Problem Solving Performance And Problem Sorting Schemes, Martha Louise Hunt

Doctoral Dissertations

No abstract provided.

Applications Of Algebra In College General Chemistry: A Curriculum Development, Enid Loretta Reeder Burrows

Doctoral Dissertations

No abstract provided.

Contractions With Infinite Defect Index, Kenneth Robert Wadland

Doctoral Dissertations

No abstract provided.