[pt] Subjacente à teoria dos números ordinais transfinitos de Cantor, há uma perspectiva finitista. Segundo tal perspectiva, Deus pode bem ordenar o infinito usando, para tanto, de procedimentos similares ao ato de contar, entendido como o ato de bem ordenar o finito. Desta maneira, um diálogo natural entre Cantor e Dedekind torna-se possível, dado que Dedekind foi o primeiro a tratar o ato de contar como sendo, em sua essência, uma forma de bem ordenar o mundo espáciotemporal pelos números naturais. Nesta tese, o conceito de número ordinal transfinito, de Cantor, é entendido como uma extensão do conceito dedekindiano de número natural. / [en] Underlying Cantor s transfinite ordinal numbers theory, there is a finistic perspective. Accordingly that perspective, God can well order the infinite using, for that, similar procedures to the act of counting, understood as the act of well order the finite. That s why a natural dialog between Cantor and Dedekind becomes possible, since Dedekind was the first to consider the act of counting as being, in its essence, a way of well order the spatial-temporal world by natural numbers. In this thesis, the concept of Cantor´s transfinite ordinal number is understood as an extension of dedekindian concept of natural number.

An analogue of the Riemannian structure of a manifold is created for an ultrametric Cantor set using the techniques of Noncommutative Geometry. In particular, a spectral triple is created that can recover much of the fractal geometry of the original Cantor set. It is shown that this spectral triple can recover the metric, the upper box dimension, and in certain cases the Hausdorff measure. The analogy with Riemannian geometry is then taken further and an analogue of the Laplace-Beltrami operator is created for an ultrametric Cantor set. The Laplacian then allows to create an analogue of Brownian motion generated by this Laplacian. All these tools are then applied to the triadic Cantor set. Other examples of ultrametric Cantor sets are then presented: attractors of self-similar iterated function systems, attractors of cookie cutter systems, and the transversal of an aperiodic, repetitive Delone set of finite type. In particular, the example of the transversal of the Fibonacci tiling is studied.

Στο κεφάλαιο 1 γίνεται μελέτη διατεταγμένων αλγεβρικών δομών. Δίνονται ορισμοί, αποδείξεις και στοιχειώδη αποτελέσματα, απαραίτητα σε όλη την πορεία της εργασίας. Ορίζουμε μερικώς διατεταγμένα σύνολα και μερική διάταξη σε αλγεβρικά συστήματα, βλέπουμε υπό ποίες προϋποθέσεις η μερική διάταξη επεκτείνεται σε ολική και άρα το σύνολο γίνεται ολικώς διατεταγμένο και στη συνέχεια τα διατεταγμένα σύνολα με μία εσωτερική πράξη ορίζουν μερικώς ή ολικώς διατεταγμένες ομάδες. Στο κεφάλαιο 2 παρουσιάζουμε συμπληρώσεις διατεταγμένων συνόλων και συγκεκριμένα, τα συμπληρώματα Dedekind, Kurepa και Krasner καθώς και ορισμένες ιδιότητες αυτών. Ο Dedekind (1831-1916) όρισε τις τομές Dedekind με τη βοήθεια των οποίων επέκτεινε τη διάταξη των φυσικών στο σύνολο των πραγματικών και θεμελίωσε με αυτόν τον τρόπο το σύνολο αυτό ως ένα διατεταγμένο σώμα. Η κατασκευή της δομής των πραγματικών εφοδιασμένη με τις πράξεις της πρόσθεσης και του πολλαπλασιασμού και τη δίαταξη, καθώς και η κατασκευή της δομής του επιπέδου με τις ίδιες πραξεις και διάταξη κατά Dedekind παρουσιάζεται εκτενέστερα στο κεφάλαιο 3. Η γενίκευση της έννοιας του συμπληρώματος Kurepa και η εισαγωγή του συμπληρώματος Krasner, οφείλονται στον καθηγητή Λ. Ντόκα (1963). Η μέθοδος του Dedekind της συμπλήρωσης με τομές δεν είναι η μόνη μέθοδος κατασκευής των πραγματικών αριθμών. Η μέθοδος του G. Cantor (1845-1918) της συμπλήρωσης με ακολουθίες, είναι η δεύτερη εξίσου σημαντική μέθοδος, την οποία θα παρουσιάσουμε στο κεφάλαιο 4. Η μελέτη μας ολοκληρώνεται στο κεφάλαιο 5, όπου παρουσιάζεται ένα ενδιαφέρον αποτέλεσμα για τις μερικώς διατεταγμένες ομάδες και τις συνθήκες κάτω από τις οποίες αυτές επεκτείνονται σε ολικώς διατεταγμένες ομάδες, στηριζόμενοι στην εργασία “embedding groups into linear or lattice structures” των Κοντολάτου-Σταμπάκη (1987), όπου πραγματοποιούν επέκταση μίας μερικώς διατεταγμένης ομάδας, χρησιμοποιώντας τα αποτελέσματα του Fuchs για ύπαρξη επέκτασης ενός μερικώς διατεταγμένου συνόλου σε ολικώς διατεταγμένο. / In chapter 1 ordered algebraic structures are considered and we present certain definitions, proofs and elementary results which are necessary in the whole project. Partially ordered sets and partial order in algebraic systems is defined. Then we analyze under which conditions partial order can be extended to full order. This leads to fully ordered sets and those sets, along with an internal operation, define partially or fully ordered groups. In chapter 2 we present specific ordered set complements and in particular those of Dedekind, Kurepa and Krasner and furthermore we mention some of their properties. Dedekind sections where introduced by Dedekind (1831-1916), who used them in order to extend the order of natural numbers to the set of real numbers, making this set an ordered field. The construction of the real numbers structure along with the internal operations of addition and multiplication and order and the construction of the plane structure with the same operations and order, using Dedekind theory, is analytically presented in chapter 3. Due to L. Docas (1963), Kurepa complement was generalized and Krasner complement was introduced. Dedekind’s sections is not the only way to construct the set of real numbers. Another important method is that of G. Cantor (1845-1918), who used sequences for completion. We present this method in chapter 4. Finally, in chapter 5, we consider a paper published by A. Kontolatou and J. Stabakis (1987) entitled “Embedding groups into linear or lattice structures”. Fuchs’s results on the extend existence of a partially ordered set to fully ordered set is used. Based on the Kontolatou-Stabakis paper, we present an interesting result for partially ordered groups and certain conditions of how to extend those groups to fully ordered ones.

The purpose of this thesis is to examine certain questions concerning the Cantor ternary set. The second chapter deals with proving that the Cantor ternary set is equivalent to the middle thirds set of [0,1], closed, compact, and has Lebesgue measure zero. Further a proof that the Cantor ternary set is a locally compact, Hausdorff topological group is given. The third chapter is concerned with establishing the existence of a Haar integral on certain topological groups. In particular if G is a locally compact and Hausdorff topological group, then there is a non-zero translation invariant positive linear form on G. The fourth chapter deals with proving that for any Haar integral I on G there exists a unique Haar measure on G that represents I.

We give conditions which, given two Bernoulli trial measures, determine whether there exists a homeomorphism of Cantor space which sends one measure to the other, answering a question of Oxtoby. We then provide examples, relating these results to the notions of good and refinable measures on Cantor space.

Master of Science / Department of Mathematics / Hrant Hakobyan / When studying geometrical objects less regular than ordinary ones, fractal analysis becomes a valuable tool. Over the last 30 years, this small branch of mathematics has developed extensively. Fractals can be de fined as those sets which have non-integral Hausdor ff dimension. In this thesis, we take a look at some basic measure theory needed to introduce certain de finitions of fractal dimensions, which can be used to measure a set's fractal degree. We introduce Minkowski dimension and Hausdor ff dimension as well as explore some examples where they coincide. Then we look at the dimension of a measure and some very useful applications. We conclude with a well known result of Bedford and McMullen about the Hausdor ff dimension of self-a ffine sets.

Thesis advisor: Patrick Byrne / In the late 19th century, Georg Cantor opened up the mathematical field of set theory with his development of transfinite numbers. In his radical departure from previous notions of infinity espoused by both mathematicians and philosophers, Cantor created new notions of transcendence in order to clearly described infinities of different sizes. Leading the opposition against Cantor's theory was Leopold Kronecker, Cantor's former mentor and the leading contemporary German mathematician. In their lifelong dispute over the transfinite numbers emerge philosophical disagreements over mathematical existence, consistency, and freedom. This thesis presents a short summary of Cantor's controversial theories, describes Cantor and Kronecker's philosophical ideas, and attempts to state clearly their differences of opinion. In the end, the author hopes to present the shock caused by Cantor's work and an appreciation of the two very different philosophies of mathematics represented by Cantor and Kronecker. / Thesis (BA) — Boston College, 2005. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Philosophy. / Discipline: College Honors Program.

Lie, Nga Sze. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 103-111). / Abstracts in English and Chinese. / Abstract --- p.i / Chapter 1 --- Introduction and Preliminary Discussions --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.1.1 --- Overview of the Thesis --- p.2 / Chapter 1.1.2 --- Background --- p.3 / Chapter 1.1.3 --- About Chapter 3: Details of the Theory --- p.4 / Chapter 1.1.4 --- About Chapter 4: Defects of the Theory --- p.7 / Chapter 1.2 --- Preliminary Discussions --- p.12 / Chapter 1.2.1 --- number --- p.12 / Chapter 1.2.2 --- mathematical existence and abstract reality --- p.12 / Chapter 1.2.3 --- finite/infinite --- p.12 / Chapter 1.2.4 --- actually/potentially infinite --- p.13 / Chapter 1.2.5 --- denumerability --- p.13 / Chapter 1.3 --- Concluding Remarks --- p.14 / Chapter 2 --- Mapping Mathematical Philosophies --- p.15 / Chapter 2.1 --- Preview --- p.15 / Chapter 2.1.1 --- Nominalism --- p.16 / Chapter 2.1.2 --- Conceptualism --- p.16 / Chapter 2.1.3 --- Intuitionism --- p.17 / Chapter 2.1.4 --- Realism --- p.18 / Chapter 2.1.5 --- Empiricism --- p.19 / Chapter 2.1.6 --- Logicism --- p.19 / Chapter 2.1.7 --- Neo-logicism --- p.21 / Chapter 2.1.8 --- Formalism --- p.21 / Chapter 2.1.9 --- Practicism --- p.23 / Chapter 2.2 --- Central Problem of Philosophy of Mathematics --- p.23 / Chapter 2.3 --- Metaphysics --- p.24 / Chapter 2.3.1 --- Abstractism --- p.24 / Chapter 2.3.2 --- Abstractist Schools --- p.25 / Chapter 2.3.3 --- Non-abstractism --- p.25 / Chapter 2.3.4 --- Non-abstractist Schools --- p.26 / Chapter 2.4 --- Semantics --- p.26 / Chapter 2.4.1 --- Literalism --- p.26 / Chapter 2.4.2 --- Literalistic schools --- p.27 / Chapter 2.4.3 --- Non-literalism --- p.27 / Chapter 2.4.4 --- Non-literalistic schools --- p.27 / Chapter 2.5 --- Epistemology --- p.28 / Chapter 2.5.1 --- Scepticism --- p.28 / Chapter 2.5.2 --- Scepticist Schools --- p.28 / Chapter 2.5.3 --- Non-scepticism --- p.29 / Chapter 2.5.4 --- Non-scepticist Schools --- p.29 / Chapter 2.6 --- Foundations of Mathematics --- p.30 / Chapter 2.6.1 --- Foundationalism --- p.31 / Chapter 2.6.2 --- Foundationalist Schools --- p.32 / Chapter 2.6.3 --- N on-foundationalism --- p.33 / Chapter 2.6.4 --- Non-foundationalist schools --- p.33 / Chapter 2.7 --- Finitistic Considerations --- p.33 / Chapter 2.7.1 --- Finitism --- p.41 / Chapter 2.7.2 --- Finitist Schools --- p.42 / Chapter 2.7.3 --- Non-finitism --- p.44 / Chapter 2.7.4 --- Non-finitist Schools --- p.44 / Chapter 2.8 --- Finitistic Reconsiderations --- p.44 / Chapter 2.8.1 --- C-finitism --- p.45 / Chapter 2.8.2 --- C-finitist Schools --- p.45 / Chapter 2.8.3 --- Non-C-finitism --- p.46 / Chapter 2.8.4 --- Non-C-finitist Schools --- p.46 / Chapter 2.9 --- Concluding Remarks --- p.47 / Chapter 3 --- Principles of Transfinite Theory --- p.48 / Chapter 3.0.1 --- Historical Notes on Infinity --- p.48 / Chapter 3.0.2 --- Cantor´ةs Proof --- p.49 / Chapter 3.1 --- The Domain Principle --- p.51 / Chapter 3.1.1 --- Variables and Domain --- p.53 / Chapter 3.1.2 --- Attack and Defense --- p.54 / Chapter 3.2 --- The Enumeral Principle --- p.56 / Chapter 3.2.1 --- Cantor´ةs Ordinal Theory of Numbers --- p.58 / Chapter 3.2.2 --- A Well-ordered Set --- p.59 / Chapter 3.2.3 --- An Enumeral --- p.59 / Chapter 3.2.4 --- An Ordinal Number --- p.60 / Chapter 3.2.5 --- Attack and Defense --- p.60 / Chapter 3.3 --- The Abstraction Principle --- p.63 / Chapter 3.3.1 --- Cantor´ةs Cardinal Theory of Numbers --- p.64 / Chapter 3.3.2 --- An Abstract One --- p.65 / Chapter 3.3.3 --- One-one Correspondence --- p.65 / Chapter 3.3.4 --- A Cardinal Number --- p.65 / Chapter 3.3.5 --- Attack and Defense --- p.65 / Chapter 3.4 --- Concluding Remarks --- p.68 / Chapter 4 --- Problems in Transfinite Theory --- p.70 / Chapter 4.1 --- Structure and Procedure --- p.70 / Chapter 4.1.1 --- Free Mathematics --- p.72 / Chapter 4.1.2 --- Non-constructive Proof --- p.75 / Chapter 4.2 --- Number and Numerosity --- p.85 / Chapter 4.2.1 --- Weak Reductionism --- p.85 / Chapter 4.2.2 --- Non-Cantorian Sets --- p.87 / Chapter 4.2.3 --- Intension in an Extensional Theory --- p.89 / Chapter 4.3 --- Conceivability and Comparability --- p.95 / Chapter 4.3.1 --- Tension with Absolute Infinity --- p.95 / Chapter 4.4 --- Conclusion --- p.100 / Bibliography --- p.103

Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Bellissard, Jean; Committee Member: Baker, Matt; Committee Member: Bakhtin, Yuri; Committee Member: Garoufalidis, Stavros; Committee Member: Putnam, Ian

Dissertação (mestrado)—Universidade de Brasília, Instituto de Artes, Departamento de Música, 2015. / Submitted by Fernanda Percia França (fernandafranca@bce.unb.br) on 2016-04-28T19:26:51Z No. of bitstreams: 1 2015_MárciaLyraFerreira_Parcial.pdf: 688617 bytes, checksum: 3af53fdd0f44e17f45a048b92ff24177 (MD5) / Approved for entry into archive by Patrícia Nunes da Silva(patricia@bce.unb.br) on 2016-05-16T17:01:00Z (GMT) No. of bitstreams: 1 2015_MárciaLyraFerreira_Parcial.pdf: 688617 bytes, checksum: 3af53fdd0f44e17f45a048b92ff24177 (MD5) / Made available in DSpace on 2016-05-16T17:01:00Z (GMT). No. of bitstreams: 1 2015_MárciaLyraFerreira_Parcial.pdf: 688617 bytes, checksum: 3af53fdd0f44e17f45a048b92ff24177 (MD5) / Este estudo de caráter qualitativo e exploratório consiste em uma investigação a respeito da maneira que cantores líricos integram os elementos músico e vocais aos elementos cênicos ao elaborar personagens para obras que demandam encenação. Como forma de delimitação para este trabalho os cantores a serem investigados foram os cantores de óperas do Distrito Federal. Objetivou compreender as formas como os cantores líricos desenvolvem cenicamente suas personagens levando em consideração a integração entre os elementos da técnica vocal (tais como estilo dos períodos, textos cantados) e os elementos cênicos, ou seja, entre a voz e o corpo. A partir da revisão da literatura foi possível perceber que a construção cênica do cantor lírico representa um ciclo em que várias etapas devem ser cumpridas para o alcance da performance ideal. Estas etapas são interdependentes e perpassam por sub-temas como expressividade, habilidades cênicas do cantor lírico, gestos e emoções como recurso expressivo e a preparação corporal do cantor lírico. Como forma de averiguar a teoria com a prática, entrevistas semi-estruturadas foram realizadas envolvendo sete cantores líricos do Distrito Federal. Duas dessas entrevistas funcionaram como pré-teste (entrevista piloto). O resultado almejado para esta pesquisa foi colaborar com outros cantores que passam por dificuldades semelhantes acerca da integração dos elementos cênicos aos elementos músico-vocais, de uma maneira mais natural, orgânica, como também, contribuir para auto-reflexões sobre a elaboração de personagens bem como sobre os conceitos e atitudes práticas necessárias para a atuação cênica do cantor de óperas, além de contribuições para a área da performance em óperas. A partir dos resultados obtidos foi possível compreender que a atuação em ópera demanda uma preparação global do cantor e que consequentemente, lhe exige constantes estudos e capacidade de adaptar-se e readaptar-se aos vários contextos a que estão inseridos. / This qualitative and exploratory study consists of an investigation into the way that opera singers are part of the musician and vocal elements to the scenic elements when designing characters for works that demand scenario. As a way of delimitation for this job singers to be investigated were the opera singers of the Federal District. Aimed at understanding the ways in which singers scenically develop their characters taking into account the integration of the elements of vocal technique (such as style periods, sung texts) and the scenic elements, that is, between the voice and the body. From the literature review it was possible to realize the scenic construction of the opera singer is a cycle in which several steps must be completed to achieve the optimal performance. These steps are interdependent and run through by sub-themes such as expressiveness, performing skills opera singer, gestures and emotions as expressive resource and body preparation of the opera singer. In order to ascertain the theory with practice, semi-structured interviews were conducted involving seven singers of the Federal District. Two of these interviews acted as pretest (pilot interview). The expected result for this research was to collaborate with other singers who go through similar difficulties concerning the integration of scenic elements to the music-vocal elements in a more natural, organic way, but also contribute to self-reflections on the development of characters and how about the concepts and practical steps necessary for the scenic performance of the opera singer, and contributions to the field of performance in operas. From the results it was possible to understand that the opera performance requires an overall preparation of the singer and that consequently, it requires constant study and ability to adapt and re-adapt yourself to the various contexts in which they live.