In the usual definition of a Euclidean domain, a ring has a norm function whose codomain is the positive integers. It was noticed by Motzkin in 1949 that the codomain could be replaced by any well-ordered set. This motivated the study of transfinite Euclidean domains in which the codomain of the norm function is replaced by the class of ordinals. We prove that there exists a (transfinitely valued) Euclidean Domain with Euclidean order type for every indecomposable ordinal. Modifying the construction, we prove that there exists a Euclidean Domain with no multiplicative norm. Following a definition of Clark and Murty, we define a set of admissible primes. We develop an algorithm that can be used to find sets of admissible primes in the ring of integers of quadratic extensions of the rationals and provide some examples.

O objetivo da dissertação é provar que toda álgebra, sobre o corpo dos números reais, algébrica e absolutamente valuada é de dimensão nita, e portanto isótopa a D . Observamos que H é a álgebra real dos Quatérnios e D R , C , H ou a álgebra real dos Octônios. A demonstração do resultado é feita gradualmente, considerando inicialmente álgebras reais absolutamente valuadas algébrica com unidade, a seguir com unidade e nalmente, algébrica. Na demonstração do teorema será necessário combinar resultados não triviais de álgebras não associativas, análise funcional, álgebras de Banach e técnicas de ultraprodutos de espaços normados. As álgebra absolutamente valuadas não são necessariamente associativas. Abraham Adrian 1947 mostrou que R , C , H e D são as únicas álgebras reais absolutamente valuadas dimensão nita e com unidade; o mesmo Albert dois anos depois, em 1949 , caracterizou Albert em de essas mesmas álgebras como as únicas que são absolutamente valuadas algébricas e com unidade sobre os reais. Em 1960 Fred B. Wright e Kazimierz Urbanik provaram que R , C , D são as únicas álgebra reais absolutamente valuadas e com unidade. Recentemente, em 1997 , Kaidi El-Amin, Maria Isabel Ramírez e Ángel Rodríguez Palacios mostraram que H e toda álgebra real absolutamente valuadas e algébrica é isótopa a uma de estas quatro. Nosso objetivo é desenvolver e unicar os resultados obtidos nestes 4 trabalhos. / Our goal here is to study the absolute valued algebraic real algebras. In order to reach our intention, we regard an absolute valued real algebra and on which one we impose: First, such one is nite-dimensional algebra; second; such one is algebraic algebra; third, such one is with unity; and in the end such one is algebraic algebra. In the latter case, our aim, it needs of certain classic results of functional analysis and others one of Banach algebras; then, we reach that such one real algebra is isotope to one of the classical absolute valued real algebras algebra and D R , C , H or D . Where H is the Quaternions real is the Octonions real algebra. The absolute valued algebras are not necessarily associative. Abraham Adrian Albert was the rst mathematician considering absolute valued algebras in a context not necessarily associative. In 1947 , he proved that any nite-dimensional absolute valued real algebra with unit element is isomorphic to either real eld H or the Octonions algebra D . Two years R , the complex eld C , the Quaternions algebra later, he demonstrated that R , C , H and D are the unique absolute valued algebraic real algebras with unit element. Recently, in 1997 , Kaidi El-Amin, Maria Isabel Ramírez and Ángel Rodríguez Palacios proved that any absolute valued algebraic real algebra is nite-dimensional.

As quantum computers edge closer to viability, it becomes necessary to create logic synthesis and minimization algorithms that take into account the particular aspects of quantum computers that differentiate them from classical computers. Since quantum computers can be functionally described as reversible computers with superposition and entanglement, both advances in reversible synthesis and increased utilization of superposition and entanglement in quantum algorithms will increase the power of quantum computing. One necessary component of any practical quantum computer is the computation of irreversible functions. However, very little work has been done on algorithms that synthesize and minimize irreversible functions into a reversible form. In this thesis, we present and implement a pair of algorithms that extend the best published solution to these problems by taking advantage of Product-Sum EXOR (PSE) gates, the reversible generalization of inhibition gates, which we have introduced in previous work [1,2]. We show that these gates, combined with our novel synthesis algorithms, result in much lower quantum costs over a wide variety of functions as compared to our competitors, especially on incompletely specified functions. Furthermore, this solution has applications for milti-valued and multi-output functions.

The results of this thesis are motivated by the investigation of abstract Cauchy problems. Our primary contribution is encapsulated in two new theorems. The first main theorem is a generalisation of a result of E. M. Stein. In particular, we show that every symmetric diffusion semigroup acting on a complex-valued Lebesgue space has a tensor product extension to a UMD-valued Lebesgue space that can be continued analytically to sectors of the complex plane. Moreover, this analytic continuation exhibits pointwise convergence almost everywhere. Both conclusions hold provided that the UMD space satisfies a geometric condition that is weak enough to include many classical spaces. The theorem is proved by showing that every symmetric diffusion semigroup is dominated by a positive symmetric diffusion semigoup. This allows us to obtain (a) the existence of the semigroup's tensor extension, (b) a vector-valued version of the Hopf--Dunford--Schwartz ergodic theorem and (c) an holomorphic functional calculus for the extension's generator. The ergodic theorem is used to prove a vector-valued version of a maximal theorem by Stein, which, when combined with the functional calculus, proves the pointwise convergence theorem. The second part of the thesis proves the existence of abstract Strichartz estimates for any evolution family of operators that satisfies an abstract energy and dispersive estimate. Some of these Strichartz estimates were already announced, without proof, by M. Keel and T. Tao. Those estimates which are not included in their result are new, and are an abstract extension of inhomogeneous estimates recently obtained by D. Foschi. When applied to physical problems, our abstract estimates give new inhomogeneous Strichartz estimates for the wave equation, extend the range of inhomogeneous estimates obtained by M. Nakamura and T. Ozawa for a class of Klein--Gordon equations, and recover the inhomogeneous estimates for the Schr??dinger equation obtained independently by Foschi and M. Vilela. These abstract estimates are applicable to a range of other problems, such as the Schr??dinger equation with a certain class of potentials.

<p>This thesis deals with different aspects towards many-valued unification which have been studied in the scope of category theory. The main motivation of this investigation comes from the fact that in logic programming, classical unification has been identified as the provision of coequalizers in Kleisli categories of term monads. Continuing in that direction, we have used categorical instrumentations to generalise the classical concept of a term. It is expected that this approach will provide an appropriate formal framework for useful developments of generalised terms as a basis for many-valued logic programming involving an extended notion of terms.</p><p>As a first step a concept for generalised terms has been studied. A generalised term is given by a composition of monads that again yields a monad, i.e. compositions of powerset monads with the term monad provide definitions for generalised terms. A composition of monads does, however, not always produce a monad. In this sense, techniques for monads composition provide a helpful tool for our concerns and therefore the study of these techniques has been a focus of this research.</p><p>The composition of monads make use of a lot of equations. Proofs become complicated, not to mention the challenge of understanding different steps of the equations. In this respect, we have studied visual techniques and show how a graphical approach can provide the support we need.</p><p>For the purpose of many-valued unification, similarity relations, generalised substitutions and unifiers have been defined for generalised terms.</p>

Thesis (M.A.)--Memorial University of Newfoundland, 2005. / Bibliography: leaves 64-70.

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A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process - the conditioned multitype Feller branching diffusion - are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too .

This thesis deals with different aspects towards many-valued unification which have been studied in the scope of category theory. The main motivation of this investigation comes from the fact that in logic programming, classical unification has been identified as the provision of coequalizers in Kleisli categories of term monads. Continuing in that direction, we have used categorical instrumentations to generalise the classical concept of a term. It is expected that this approach will provide an appropriate formal framework for useful developments of generalised terms as a basis for many-valued logic programming involving an extended notion of terms. As a first step a concept for generalised terms has been studied. A generalised term is given by a composition of monads that again yields a monad, i.e. compositions of powerset monads with the term monad provide definitions for generalised terms. A composition of monads does, however, not always produce a monad. In this sense, techniques for monads composition provide a helpful tool for our concerns and therefore the study of these techniques has been a focus of this research. The composition of monads make use of a lot of equations. Proofs become complicated, not to mention the challenge of understanding different steps of the equations. In this respect, we have studied visual techniques and show how a graphical approach can provide the support we need. For the purpose of many-valued unification, similarity relations, generalised substitutions and unifiers have been defined for generalised terms.

In spite of its 150 years history, the problem of classifying all finite-dimensional division algebras over a field k is still unsolved whenever k is not algebraically closed. The present thesis concerns some different aspects of this problem, and the related problems of classifying all composition and absolute valued algebras. A tripartition of the class of all fields is given, based on the dimensions in which division algebras over a field exist. Moreover, all finite-dimensional flexible real division algebras are classified. This class includes in particular all finite-dimensional commutative real division algebras, of which two different classifications, along different lines, are presented. It is shown that every vector product algebra has dimension zero, one, three or seven, and that its isomorphism type is determined by its adherent quadratic form. This yields a new and elementary proof for the corresponding, classical result for unital composition algebras. A rotation in a Euclidean space is an orthogonal map that locally acts as a plane rotation with a fixed angle. All pairs of rotations in finite-dimensional Euclidean spaces are classified up to orthogonal similarity. A description of all composition algebras having an LR-bijective idempotent is given. On the basis of this description, all absolute valued algebras having a one-sided unity or a non-zero central idempotent are classified.