The results presented here are concerned with questions of decomposability of multiplicative semigroups of matrices with nonnegative entries. Chapter 1 covers some preliminary results which become useful in the remainder of the exposition. Chapters 2 and 3 constitute an exposition of some recent known results on special semigroups. Chapter 2 explores conditions for decomposability of semigroups in terms of conditions derived from linear functionals and in Chapter 3, we give a complete proof of an extension of the celebrated Perron-Frobenius Theorem. No originality is claimed for the results in Chapters 2 and 3. In Chapter 4, we present some new results on sufficient conditions for finiteness of semigroups of matrices.

We study the degree structures of the strong reducibilities $(\leq_{ibT})$ and $(\leq_{cl})$, as well as $(\leq_{rK})$ and $(\leq_{wtt})$. We show that any noncomputable c.e. set is part of a uniformly c.e. copy of $(\BQ,\leq)$ in the c.e. cl-degrees within a single wtt-degree; that there exist uncountable chains in each of the degree structures in question; and that any countable partially-ordered set can be embedded into the cl-degrees, and any finite partially-ordered set can be embedded into the ibT-degrees. We also offer new proofs of results of Barmpalias and Lewis-Barmpalias concerning the non-existence of cl-maximal sets.

It is known that almost all numbers are transcendental in the sense of Lebesgue measure. However there is no simple rule to separate transcendental numbers from algebraic numbers. Today research in this direction is about establishing new transcendence criteria for new families of transcendental numbers. By applying a recent refinement of Subspace Theorem, Boris Adamczewski and Yann Bugeaud determined new transcendence criteria for real numbers which we shall present in this thesis. Published only three years ago, their articles explore combinatorial, algorithmic and dynamic approaches in discussing the notion of complexity of both continued fraction and b-adic expansions of a certain class of real numbers. The condition on the expansions are those of being stammering and non-eventually periodic. Taking together these articles give a well-structured picture of the interrelationships between sequence characteristics of expansion (i.e. complexity, periodicity, type of generator) and algebraic characteristics of number itself (i.e. class, transcendency).

The classical interpolation theorem for the open complex unit disk, due to Nevanlinna and Pick in the early 20th century, gives an elegant criterion for the solvability of the problem as an eigenvalue problem. In the 1960s, Sarason reformulated problems of this type firmly in the language of operator theoretic function theory. This thesis will explore connections between interpolation problems on various domains (both single and several complex variables) with the viewpoint that Sarason’s work suggests. In Chapter 1, some essential preliminaries on bounded operators on Hilbert space and the functionals that act on them will be presented, with an eye on the various ways distances can be computed between operators and a certain type of ideal. The various topologies one may define on B(H) will play a prominent role in this development. Chapter 2 will introduce the concept of a reproducing kernel Hilbert space, and a distinguished operator algebra that we associate to such spaces know as the multiplier algebra. The various operator theoretic properties that multiplier algebras enjoy will be presented, with a particular emphasis on their invariant subspace lattices and the connection to distance formulae. In Chapter 3, the Nevanlinna-Pick problem will be invested in general for any repro- ducing kernel Hilbert space, with the basic heuristic for distance formulae being presented. Chapter 4 will treat a large class of reproducing kernel Hilbert spaces associated to measure spaces, where a Pick-like theorem will be established for many members of this class. This approach will closely follow similar results in the literature, including recent treatments by McCullough and Cole-Lewis-Wermer. Reproducing kernel Hilbert spaces where the analogue of the Nevanlinna-Pick theorem holds are particularly nice. In Chapter 5, the operator theory of these so-called complete Pick spaces will be developed, and used to tackle certain interpolation problems where additional constraints are imposed on the solution. A non-commutative view of interpola- tion will be presented, with the non-commutative analytic Toeplitz algebra of Popescu and Davidson-Pitts playing a prominent role. It is often useful to consider reproducing kernel Hilbert spaces which arise as natural products of other spaces. The Hardy space of the polydisk is the prime example of this. A general commutative and non-commutative view of such spaces will be presented in Chapter 6, using the left regular representation of higher-rank graphs, first introduced by Kribs-Power. A recent factorization theorem of Bercovici will be applied to these algebras, from which a Pick-type theorem may be deduced. The operator-valued Pick problem for these spaces will also be discussed. In Chapter 7, the various tools developed in this thesis will be applied to two related problems, known as the Douglas problem and the Toeplitz corona problem.

Thesis (M.A.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 82-86).

It is known that almost all numbers are transcendental in the sense of Lebesgue measure. However there is no simple rule to separate transcendental numbers from algebraic numbers. Today research in this direction is about establishing new transcendence criteria for new families of transcendental numbers. By applying a recent refinement of Subspace Theorem, Boris Adamczewski and Yann Bugeaud determined new transcendence criteria for real numbers which we shall present in this thesis. Published only three years ago, their articles explore combinatorial, algorithmic and dynamic approaches in discussing the notion of complexity of both continued fraction and b-adic expansions of a certain class of real numbers. The condition on the expansions are those of being stammering and non-eventually periodic. Taking together these articles give a well-structured picture of the interrelationships between sequence characteristics of expansion (i.e. complexity, periodicity, type of generator) and algebraic characteristics of number itself (i.e. class, transcendency).

The main theme of this thesis is studying classes of structures with respect to various measurements of complexity. We will briefly discuss the notion of computable dimension, while the breadth of the paper will focus on calculating the Turing ordinal and the back-and-forth ordinal of various classes, along with an exploration of how these two ordinals are related in general. Computable structure theorists study which computable dimensions can be realized by structures from a given class. Using a structural characterization of the computably categorical equivalence structures due to Calvert, Cenzer, Harizanov and Morozov, we prove that the only possible computable dimension of an equivalence structure is 1 or ω. In 1994, Jockusch and Soare introduced the notion of the Turing ordinal of a class of structures. It was unknown whether every computable ordinal was the Turing ordinal of some class. Following the work of Ash, Jocksuch and Knight, we show that the answer is yes, but, as one might expect, the axiomatizations of these classes are complex. In 2009, Montalban defined the back-and-forth ordinal of a class using the back-and-forth relations. Montalban, following a result of Knight, showed that if the back-and-forth ordinal is n+1, then the Turing ordinal is at least n. We will prove a theorem stated by Knight that extends the previous result to all computable ordinals and show that if the back-and-forth ordinal is α (infinite) then the Turing ordinal is at least α. It is conjectured at present that if a class of structures is relatively nice then the Turing ordinal and the back-and-forth ordinal of the class differ by at most 1. We will present many examples of classes having axiomatizations of varying complexities that support this conjecture; however, we will show that this result does not hold for arbitrary Borel classes. In particular, we will prove that there is a Borel class with infinite Turing ordinal but finite back-and-forth ordinal and show that, for each positive integer d, there exists a Borel class of structures such that the Turing ordinal and the back-and-forth ordinal of the class are both finite and differ by at least d.

The classical interpolation theorem for the open complex unit disk, due to Nevanlinna and Pick in the early 20th century, gives an elegant criterion for the solvability of the problem as an eigenvalue problem. In the 1960s, Sarason reformulated problems of this type firmly in the language of operator theoretic function theory. This thesis will explore connections between interpolation problems on various domains (both single and several complex variables) with the viewpoint that Sarason’s work suggests. In Chapter 1, some essential preliminaries on bounded operators on Hilbert space and the functionals that act on them will be presented, with an eye on the various ways distances can be computed between operators and a certain type of ideal. The various topologies one may define on B(H) will play a prominent role in this development. Chapter 2 will introduce the concept of a reproducing kernel Hilbert space, and a distinguished operator algebra that we associate to such spaces know as the multiplier algebra. The various operator theoretic properties that multiplier algebras enjoy will be presented, with a particular emphasis on their invariant subspace lattices and the connection to distance formulae. In Chapter 3, the Nevanlinna-Pick problem will be invested in general for any repro- ducing kernel Hilbert space, with the basic heuristic for distance formulae being presented. Chapter 4 will treat a large class of reproducing kernel Hilbert spaces associated to measure spaces, where a Pick-like theorem will be established for many members of this class. This approach will closely follow similar results in the literature, including recent treatments by McCullough and Cole-Lewis-Wermer. Reproducing kernel Hilbert spaces where the analogue of the Nevanlinna-Pick theorem holds are particularly nice. In Chapter 5, the operator theory of these so-called complete Pick spaces will be developed, and used to tackle certain interpolation problems where additional constraints are imposed on the solution. A non-commutative view of interpola- tion will be presented, with the non-commutative analytic Toeplitz algebra of Popescu and Davidson-Pitts playing a prominent role. It is often useful to consider reproducing kernel Hilbert spaces which arise as natural products of other spaces. The Hardy space of the polydisk is the prime example of this. A general commutative and non-commutative view of such spaces will be presented in Chapter 6, using the left regular representation of higher-rank graphs, first introduced by Kribs-Power. A recent factorization theorem of Bercovici will be applied to these algebras, from which a Pick-type theorem may be deduced. The operator-valued Pick problem for these spaces will also be discussed. In Chapter 7, the various tools developed in this thesis will be applied to two related problems, known as the Douglas problem and the Toeplitz corona problem.

Thesis (M.A.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 82-86). Also available in print.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1973. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.