Fields and Fusions: Hrushovski constructions and their definable groups

An overview is given of the various expansions of fields and fusions of strongly minimal sets obtained by means of Hrushovski's amalgamation method, as well as a characterization of the groups definable in these structures

Large sets in constructive set theory

This thesis presents an investigation into large sets and large set axioms in the context of the constructive set theory CZF. We determine the structure of large sets by classifying their von Neumann stages and use a new modified cumulative hierarchy to characterise their arrangement in the set theoretic universe. We prove that large set axioms have good metamathematical properties, including absoluteness for the common relative model constructions of CZF and a preservation of the witness existence properties CZF enjoys. Furthermore, we use realizability to establish new results about the relative consistency of a plurality of inaccessibles versus the existence of just one inaccessible. Developing a constructive theory of clubs, we present a characterisation theorem for Mahlo sets connecting classical and constructive approaches to Mahloness and determine the amount of induction contained in the assertion of a Mahlo set. We then present a characterisation theorem for 2-strong sets which proves them to be equivalent to a logically simpler concept. We also investigate several topics connected to elementary embeddings of the set theoretic universe into a transitive class model of CZF, where considering different equivalent classical formulations results in a rich and interconnected spectrum of measurability for the constructive case. We pay particular attention to the question of cofinality of elementary embeddings, achieving both very strong cofinality properties in the case of Reinhardt embeddings and constructing models of the failure of cofinality in the case of ordinary measurable embeddings, some of which require only surprisingly low conditions. We close with an investigation of constructive principles incompatible with elementary embeddings

2-categorical Brown representability and the relation between derivators and infinity-categories

In this thesis, we study the 2-category of infinity-categories, largelywith attention to its relationships with the 2-category of prederivators. We prove that the 2-category of infinity-categories admits a small set of objects detecting equivalences andsatisfies a Brown representability theorem, which we formulate using a new notion of compactly generated 2-category. We show that the canonical2-functor from the 2-category of infinity-categories into the 2-category of prederivators detects equivalences and, under appropriatesize conditions, induces an equivalence on hom-categories. We explain how to extend prederivators defined on the 2-category ofordinary categories to the domain of all infinity-categories using the delocalization theorem. We use theBrown representability theorem to give conditions under which a prederivator is representable by an infinity-category. We also show how to extend derivators defined on categoriesand satisfying a mild size condition to derivators on infinity-categories, using an extensionof Cisinski's theorem on the universality of derivators of spaces. This extension allows us to give conditions under which the small sub-prederivators of quite general derivators are all representable by infinity-categories

Alternative Finestructural and Computational Approaches to Constructibility

We consider attempts to simplify finestructural arguments concerning inner models of ZFC, in particular L; in addition, we exhibit different aspects of the computational strength of Infinite Time Register Machines.Die Arbeit befasst sich mit alternativen Methoden zur Analyse von Gödels konstruktiblem Universum L, dem ⊆-minimalen klassenmächtigen Modell von ZFC und anderer konstruktibler Strukturen. Im ersten Teil werden F-Strukturen eingeführt, ein Ansatz von Koepke zur Vereinfachung der Feinstrukturtheorie von Kernmodellen. Wir gewinnen einige Vorteile für die weitere Entwicklung aus der Einführung einer Namensfunktion N unter die Basisfunktionen und kleinerer Modifikationen des Hüllenoperators. Es wird demonstriert, dass die F-Hierarchie ein geeignetes Instrument zum Beweis wichtiger Eigenschaften von L ist, wie etwa der Hausdorffschen verallgemeinerten Kontinuumshypothese GCH oder des kombinatorische Prinzips ◊. Dann wird eine Methode zur Erweiterung strukturerhaltender Funktionen, sogenannter feiner Abbildungen, angegeben, Koepkes vereinfachter Beweis des Überdeckungssatzes für L erläutert und ein Approximationssatz für L gezeigt: Unter ¬0# ist jedes X ⊂ On von überabzählbarer Konfinalität, das unter den Basisfunktionen der F-Hierarchie abgeschlossen ist, Vereinigung von abzählbar vielen Elementen von L. Gegenüber dem Beweis des Approximationssatzes von Magidor, der die Abgeschlossenheit unter primitiv-rekursiven Mengenfunktionen voraussetzt, gewinnen wir deutlich an Kürze und Einfachheit. Weiter ergänzen wir die Basisfunktionen der F-Hierarchie durch Ansätze aus der Hyperfeinstrukturtheorie von Friedman und Koepke. Im Kontext der F-Hierarchie ergibt sich daraus das allgemeinere Konzept des Hyperings, das wir ausführen und benutzen, um die hyperfeinstrukturellen Beweise des Quadrat- und Morastprinzips in die F-Hierarchie zu übertragen. Das hierbei vornehmlich benutzte horizontale Hypering H2 sorgt dabei für eine Unabhängigkeit der konstruierten Objekte von der gewählten Aufzählung der Formeln. Anschließend betrachten wir Infinite Time Register Machines (ITRMs) sowohl als Anwendung von wie auch als weiteren Zugang zu konstruktiblen Methoden. ITRMs sind Registermaschinen, deren Laufzeiten beliebige Ordinalzahlen sein können. Wir beweisen das Lost-Melody-Theorem für ITRMs, d.h. die Existenz einer reellen Zahl, die durch eine ITRM als Orakelzahl erkannt, aber nicht berechnet werden kann. Wir führen getypte Maschinen ein, die Register mit verschiedenem Limesverhalten parallel verwenden und klassifizieren die Maschinentypen hinsichtlich ihrer Berechnungsstärke. Insbesondere zeigen wir, dass ITRMs mit n+1 überlaufenden Registern und einigen schwächeren Hilfsregistern den n-ten Hypersprung berechnen und das Halteproblem für ITRMs mit n überlaufenden Registern lösen können. Wir beweisen, dass die Menge der ITRM-erkennbaren reellen Zahlen in der Ordnung von L Lücken aufweist, und zwar mindestens von der Größe sup{ωiCK|i ∈ ω}, wobei ωiCK die i-te zulässige Ordinalzahl bezeichnet. Außerdem zeigen wir, dass die beweistheoretischen Analysen von Welch bezüglich der Existenz der Haltezahlen für verschiedene Maschinen im Kontext der getypten Maschinen zu präziseren Schranken führen. Im letzten Teil skizzieren wir Ansätze zu einer Übertragung alternativer Feinstrukturen auf allgemeinere konstruktible Strukturen, sogenannte Kernmodelle. Wir übertragen zentrale Konzepte, zeigen einige Erhaltungseigenschaften für die Ultrapotenzkonstruktion und ein Dodd-Jensen-Lemma für das entsprechende Iterationskonzept. Die Bewahrung der feinstrukturellen Information in Iterationen hingegen scheitert. Daran anschließend erläutern wir kurz die Gründe dieser Schwierigkeiten und diskutieren mögliche Auswege

Fuzzy Sets, Fuzzy Logic and Their Applications 2020

The present book contains the 24 total articles accepted and published in the Special Issue “Fuzzy Sets, Fuzzy Logic and Their Applications, 2020” of the MDPI Mathematics journal, which covers a wide range of topics connected to the theory and applications of fuzzy sets and systems of fuzzy logic and their extensions/generalizations. These topics include, among others, elements from fuzzy graphs; fuzzy numbers; fuzzy equations; fuzzy linear spaces; intuitionistic fuzzy sets; soft sets; type-2 fuzzy sets, bipolar fuzzy sets, plithogenic sets, fuzzy decision making, fuzzy governance, fuzzy models in mathematics of finance, a philosophical treatise on the connection of the scientific reasoning with fuzzy logic, etc. It is hoped that the book will be interesting and useful for those working in the area of fuzzy sets, fuzzy systems and fuzzy logic, as well as for those with the proper mathematical background and willing to become familiar with recent advances in fuzzy mathematics, which has become prevalent in almost all sectors of the human life and activity

Álgebras de funciones analíticas acotadas. Interpolación

RESUMEN Este trabajo resume, de forma parcial, la investigación realizada durante mi periodo predoctoral. Esta investigación pertenece, de forma general, a la teoría de álgebras de Banach conmutativas y álgebras uniformes y, en particular, se desarrolla principalmente en el ámbito de las álgebras de funciones analíticas acotadas en dominios de espacios de Banach ¯nito e in¯nito dimensionales. Las líneas centrales de este trabajo son las siguientes: ² Sucesiones de Interpolación para Álgebras Uniformes ² Operadores de Composición ² Propiedades Topológicas de Álgebras de Funciones Analíticas La investigación realizada sobre sucesiones de interpolación para álgebras uniformes se puede dividir en dos partes: una genérica en la que se propor- cionan algunos resultados de carácter general sobre sucesiones de interpo- lación para álgebras uniformes, y una parte más específica, en que se tratan sucesiones de interpolación para algunas álgebras de funciones analíticas acotadas. Estos puntos se tratan en los Capítulos 2 y 3. El estudio de oper- adores de composición, principalmente sobre H1(BE), centra el contenido del Capítulo 4. En este cap¶³tulo estudiaremos una descripci¶on del espectro de estos operadores y los llamados operadores de composición de Radon- Nikod¶ym. Para ello, se harí uso de algunos resultados de interpolación del capítulo anterior. Con respecto a la tercera línea que hemos citado, estu- diaremos los llamados operadores de tipo Hankel en el capítulo 5. ¶Estos nos permitirán tratar el concepto de álgebra tight y las álgebras de Bour- gain de un subespacio de C(K), que están estrechamente relacionadas con la propiedad de Dunford-Pettis. __________________________________________________________________________________________________The lines studied in this thesis are the following: ² Interpolating Sequences for Uniform Algebras ² Composition Operators ² Topological Properties in Algebras of Analytic Functions After the preliminaries, the second chapter is devoted to the study of interpolating sequences on uniform algebras A. We ¯rst deal with the con- nection between interpolating sequences and linear interpolating sequences. Next, we deal with dual uniform algebras A = X¤. In this context, we prove ¯rst that c0¡linear interpolating sequences are linear interpolating and then, we show that c0¡interpolating sequences are, indeed, c0¡linear interpolating, obtaining that c0¡interpolating sequences (xn) ½ MA X become linear interpolating. Finally, we provide a di®erent approach to prove that c0¡interpolating sequences are not c0¡linear interpolating via composition operators. We continue with the study of interpolating sequences for the algebras of analytic functions H1(BE) and A1(BE) in the third chapter. The study of interpolating sequences for H1 arises from the results of L. Carleson, W. K. Hayman and D. J. Newman. When we deal with general Banach spaces, we prove that the Hayman-Newman condition for the sequence of norms is su±cient for a sequence (xn) ½ BE¤¤ to be interpolating for H1(BE) if E is any ¯nite or in¯nite dimensional Banach space. This is a consequence of a stronger result : The Carleson condition for the sequence (kxnk) ½ D is su±cient for (xn) to be interpolating for H1(BE). Actually, the result holds for sequences in BE¤¤ thanks to the Davie- Gamelin extension. When we deal with A = A1(BE), the existence of interpolating se- quences for A was proved by J. Globevnik for a wide class of in¯nite- dimensional Banach spaces. We complete this study by proving the ex- istence of interpolating sequences for A1(BE) for any in¯nite-dimensional Banach space E, characterizing the separability of A1(BE) in terms of the ¯nite dimension of E. Finally, we study the metrizability of bounded subsets of MA when we deal with A = Au(BE). In chapter 4 we deal with composition operators on H1(BE). First we study the spectra of these operators. L. Zheng described the spectrum of some composition operators on H1. Her results where extended to H1(BE), E any complex Banach space, by P. Galindo, T. Gamelin and M. LindstrÄom for the power compact case. In this work, the authors also deal with the non power compact case for Hilbert spaces. Inspired by them and using some interpolating results, we provide a general theorem which describes the spectrum of H1(BE) for general Banach spaces. In partic- ular, we prove that conditions on this theorem are satis¯ed by the n¡fold product space Cn, completing the description of ¾(CÁ) in this case, which was an open question. Next, we study the class of Radon-Nikod¶ym composition operators from H1(BE) to H1(BF ). We characterize these operators in terms of the As- plund property. Chapter 5 deals with properties related to Hankel-type operators. The concept of tight algebra is related to these operators and was introduced by B. Cole and T. Gamelin. They proved that A(Dn) is not tight on its spectrum for n ¸ 2. We present a new approach to this result extending it to algebras Au(BE) for Banach spaces E = C £ F endowed with the supremum norm. In addition, we show that H1(BE) is never tight on its spectrum re- gardless the Banach space E. Hankel-type operators are also closely related to the Dunford-Pettis prop- erty through the so-called Bourgain algebras introduced by J. A. Cima and R. M. Timoney. We prove that the Bourgain algebras of A(Dn) as a sub- space of C( ¹D n) are themselves

Dirac Operators on Quantum Principal G-Bundles

In this thesis I discuss some results on the noncommutative (spin) geometry of quantum principal G-bundles. The first part of the thesis is devoted to the study of spectral triples over toral bundles; extending some recent results by L. Dabrowski and A. Sitarz, we introduce the notion of projectable spectral triple for T^n-bundles. Moreover, we work out twisted Dirac operators. We discuss, in particular, the application of these results to noncommutative tori. In the second part of the thesis, instead, we work out a method for constructing real spectral triples over cleft quantum principal G-bundles and we study the properties of these triples and their behaviour under gauge transformations. Some of the results discussed in this thesis can also be found in the following papers: arXiv:1305.6185 arXiv:1308.473