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We study various quotient maps between frames which are defined by stipulating that they satisfy certain conditions on the cozero parts of their domains and codomains. By way of example, we mention that C-quotient and C -quotient maps (as defined by Ball and Walters- Wayland [7]) are typical of the types of homomorphisms we consider in the initial parts of the thesis. To be little more precise, we study uplifting quotient maps, C1- and C2-quotient maps and show that these quotient maps possess some properties akin to those of a C-quotient maps. The study also focuses on R - and G - quotient maps and show, amongst other things, that these quotient maps coincide with the well known C - quotient maps in mildly normal frames. We also study quasi-F frames and give a ring-theoretic characterization that L is quasi-F precisely when the ring RL is quasi-B´ezout. We also show that quasi-F frames are preserved and reflected by dense coz-onto R -quotient maps. We characterize normality and some of its weaker forms in terms of some of these quotient maps. Normality is characterized in terms of uplifting quotient maps, -normally separated frames in terms of C1-quotient maps and mild normality in terms of R - and G -quotient maps. Finally we define cozero complemented frames and show that they are preserved and reflected by dense z#- quotient maps. We end by giving ring-theoretic characterizations of these frames. / Mathematical Science / D. Phil. (Mathematics)

Thesis (MSc)--University of Stellenbosch, 2003. / ENGLISH ABSTRACT: Let A and B be unital complex Banach algebras with identities 1 and I' respectively. A linear map T : A -+ B is invertibility preserving if Tx is invertible in B for every invertible x E A. We say that T is unital if Tl = I'. IfTx2 = (TX)2 for all x E A, we call T a Jordan homomorphism. We examine an unsolved problem posed by 1. Kaplansky: Let A and B be unital complex Banach algebras and T : A -+ B a unital invertibility preserving linear map. What conditions on A, Band T imply that T is a Jordan homomorphism? Partial motivation for this problem are the Gleason-Kahane-Zelazko Theorem (1968) and a result of Marcus and Purves (1959), these also being special instances of the problem. We will also look at other special cases answering Kaplansky's problem, the most important being the result stating that if A is a von Neumann algebra, B a semi-simple Banach algebra and T : A -+ B a unital bijective invertibility preserving linear map, then T is a Jordan homomorphism (B. Aupetit, 2000). For a unital complex Banach algebra A, we denote the spectrum of x E A by Sp (x, A). Let a(x, A) denote the union of Sp (x, A) and the bounded components of <C \ Sp (x, A). We denote the spectral radius of x E A by p(x, A). A unital linear map T between unital complex Banach algebras A and B is invertibility preserving if and only if Sp (Tx, B) C Sp (x, A) for all x E A. This leads one to consider the problems that arise when, in turn, we replace the invertibility preservation property of T in Kaplansky's problem with Sp (Tx, B) = Sp (x, A) for all x E A, a(Tx, B) = a(x, A) for all x E A, and p(Tx, B) = p(x, A) for all x E A. We will also investigate some special cases that are solutions to these problems. The most important of these special cases says that if A is a semi-simple Banach algebra, B a primitive Banach algebra with minimal ideals and T : A -+ B a surjective linear map satisfying a(Tx, B) = a(x, A) for all x E A, then T is a Jordan homomorphism (B. Aupetit and H. du T. Mouton, 1994). / AFRIKAANSE OPSOMMING: Gestel A en B is unitale komplekse Banach algebras met identiteite 1 en I' onderskeidelik. 'n Lineêre afbeelding T : A -+ B is omkeerbaar-behoudend as Tx omkeerbaar in B is vir elke omkeerbare element x E A. Ons sê dat T unitaal is as Tl = I'. As Tx2 = (TX)2 vir alle x E A, dan noem ons T 'n Jordan homomorfisme. Ons ondersoek 'n onopgeloste probleem wat deur I. Kaplansky voorgestel is: Gestel A en B is unitale komplekse Banach algebras en T : A -+ B is 'n unitale omkeerbaar-behoudende lineêre afbeelding. Watter voorwaardes op A, B en T impliseer dat T 'n Jordan homomorfisme is? Gedeeltelike motivering vir hierdie probleem is die Gleason-Kahane-Zelazko Stelling (1968) en 'n resultaat van Marcus en Purves (1959), wat terselfdertyd ook spesiale gevalle van die probleem is. Ons salook na ander spesiale gevalle kyk wat antwoorde lewer op Kaplansky se probleem. Die belangrikste van hierdie resultate sê dat as A 'n von Neumann algebra is, B 'n semi-eenvoudige Banach algebra is en T : A -+ B 'n unitale omkeerbaar-behoudende bijektiewe lineêre afbeelding is, dan is T 'n Jordan homomorfisme (B. Aupetit, 2000). Vir 'n unitale komplekse Banach algebra A, dui ons die spektrum van x E A aan met Sp (x, A). Laat cr(x, A) die vereniging van Sp (x, A) en die begrensde komponente van <C \ Sp (x, A) wees. Ons dui die spektraalradius van x E A aan met p(x, A). 'n Unitale lineêre afbeelding T tussen unit ale komplekse Banach algebras A en B is omkeerbaar-behoudend as en slegs as Sp (Tx, B) c Sp (x, A) vir alle x E A. Dit lei ons om die probleme te beskou wat ontstaan as ons die omkeerbaar-behoudende eienskap van T in Kaplansky se probleem vervang met Sp (Tx, B) = Sp (x, A) vir alle x E A, O"(Tx, B) = O"(x, A) vir alle x E A en p(Tx, B) = p(x, A) vir alle x E A, onderskeidelik. Ons salook 'n paar spesiale gevalle van hierdie probleme ondersoek. Die belangrikste van hierdie spesiale gevalle sê dat as A 'n semi-eenvoudige Banach algebra is, B 'n primitiewe Banach algebra met minimale ideale is, en T : A -+ B 'n surjektiewe lineêre afbeelding is sodanig dat O"(Tx, B) = O"(x, A) vir alle x E A, dan is T 'n Jordan homomorfisme (B. Aupetit en H. du T. Mouton, 1994).

This work deals with some aspects of universal topological and metric dynamic behavior of iterated maps of the interval.

Orientador: José Marcio Machado / Banca: Gilberto Aparecido Pratavieira / Banca: Manoel Ferreira Borges Neto / Resumo: As funções hipercomplexas do tipo zn, n natural, têm uma dilatação linear K uniformemente limitada em um domínio simplesmente conexo D, então podem ser classificadas de funções K-quaseconformes. Procuramos aqui quantificar K e verificar suas dependências. Para tanto, as generalizações de zn foram necessárias e obtidas, originando para z escrito em coordenadas esféricas, polinômios em função de um raio r. / Abstract: The hypercomplex functions of zn type, natural n, have a linear dilatation K, uniformly limited in a connected domain D, so they can be classified in K-quasiconformal functions. We try here to quantify K and check its dependancy. To enable this, the generalizations of zn were necessary and obtained be-forehand, originating for z written in spherical coordenates, polynomial according to a radial r. / Mestre

by Ye Yuanling. / "August 2000." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 82-85). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Nonlinear, local and highly parallel algorithms can perform several simple but important visual computations. Specific classes of algorithms can be considered in an abstract way. I study here the class of polynomial algorithms to exemplify some of the important issues for visual processing like linear vs. nonlinear and local vs. global. Polynomial algorithms are a natural extension of Perceptrons to time dependent grey level images.. Although they share most of the limitations of Perceptrons, they are powerful parallel computational devices. Several of their properties are characterized and especially (a) their equivalence with Perceptrons for geometrical figures and (b) the synthesis of non-linear algorithms (mappings) via associative learning. Finally, the paper considers how algorithms of this type could be implemented in nervous hardware, in terms of synaptic interactions strategically located in a dendritic tree.

We have designed the Standard Map Machine(SMM) as an answer to the intensive computational requirements involved in the study of chaotic behavior in nonlinear systems. The high-speed and high-precision performance of this computer is due to its simple architecture specialized to the numerical computations required of nonlinear systems. In this report, we discuss the design and implementation of this special-purpose machine.

The purpose of the thesis is to develop a new visualization method for Gene Ontologiesand hierarchical clustering. These are both important tools in biology andmedicine to study high-throughput data such as transcriptomics and metabolomicsdata. Enrichment of ontology terms in the data is used to identify statistically overrepresentedontology terms, that give insight into relevant biological processes orfunctional modules. Hierarchical clustering is a standard method to analyze andvisualize data to nd relatively homogeneous clusters of experimental data points.Both methods support the analysis of the same data set, but are usually consideredindependently. However, often a combined view such as: visualizing a large data setin the context of an ontology under consideration of a clustering of the data.The result of the current work is a user-friendly program that combines twodi erent views for analysing Gene Ontology and Cluster simultaneously. To makeexplorations of such a big data possible we developed new visualization approach.

En esta tesis, presentamos un nuevo enfoque para validar mappings entre esquemas de datos que permite al diseñador comprobar si el mapping satisface o no ciertas propiedades deseables. La respuesta que obtiene el diseñador no se limita a un simple valor booleano, sino que dependiendo del resultado de la comprobación obtendrá un ejemplo/contraejemplo que ilustre ese resultado, o bien se le indicará el conjunto de restricciones de integridad de los esquemas y formulas del mapping responsables de ese resultado. Una de las características principales de nuestro enfoque es que es capaz de tratar una clase muy expresiva de mappings y esquemas relacionales. En particular, nuestro enfoque es capaz de tratar con formulas de mapping consistentes en inclusiones y igualdades de consultas, además de permitir el uso de negaciones y comparaciones aritméticas tanto en las propias formulas del mapping como en las vistas definidas en los esquemas. Nuestro enfoque también permite tratar restricciones de integridad, las cuales pueden estar definidas no solo sobre las tablas sino también sobre las vistas de los esquemas. Dado que razonar sobre este tipo de mappings y esquemas es, desafortunadamente, indecidible, proponemos realizar un test de terminación previo a la validación del mapping. Si el test de terminación da una respuesta positiva, entonces podremos estar seguros de que la posterior comprobación de la propiedad deseable correspondiente terminará. Finalmente, también extendemos nuestro enfoque más allá del caso relacional y lo aplicamos al contexto de mappings entre esquemas XML. / In this thesis, we present a new approach to the validation of mappings between data schemas. It allows the designer to check whether the mapping satisfies certain desirable properties. The feedback that our approach provides to the designer is not only a Boolean answer, but either a (counter)example for the (un)satisfiability of the tested property, or the set of mapping assertions and schema constraints that are responsible for that (un)satisfiability. One of the main characteristics of our approach is that it is able to deal with a very expressive class of relational mapping scenarios; in particular, it is able to deal with mapping assertions in the form of query inclusions and query equalities, and it allows the use of negation and arithmetic comparisons in both the mapping assertions and the views of the schemas; it also allows for integrity constraints, which can be defined not only over the base relations but also in terms of the views. Since reasoning on the class of mapping scenarios that we consider is, unfortunately, undecidable, we propose to perform a termination test as a pre-validation step. If the answer of the test is positive, then checking the corresponding desirable property will terminate. We also go beyond the relational setting and study the application of our approach to the context of mappings between XML schemas.