In this thesis we study the types of algebraically closed valued fields (ACVF). We prove the definable types of ACVF are residual and valuational and provide a defining schema for the definable types. We then conclude that all the types are invariant. / Thesis / Master of Science (MSc)

<div>We further the research in o-minimal topology by studying in full generality definable topological spaces in o-minimal structures. These are topological spaces $(X, \tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ has a basis that is (uniformly) definable. Examples include the canonical o-minimal "euclidean" topology, “definable spaces” in the sense of van den Dries [17], definable metric spaces [49], as well as generalizations of classical non-metrizable topological spaces such as the Split Interval and the Alexandrov Double Circle.</div><div><br></div><div>We develop a usable topological framework in our setting by introducing definable analogues of classical topological properties such as separability, compactness and metrizability. We characterize these notions, showing in particular that, whenever the underlying o-minimal structure expands $(\mathbb{R},<)$, definable separability and compactness are equivalent to their classical counterparts, and a similar weaker result for definable metrizability. We prove the equivalence of definable compactness and various other properties in terms of definable curves and types. We show that definable topological spaces in o-minimal expansions of ordered groups and fields have properties akin to first countability. Along the way we study o-minimal definable directed sets and types. We prove a density result for o-minimal types, and provide an elementary proof within o-minimality of a statement related to the known connection between dividing and definable types in o-minimal theories.</div><div><br></div><div>We prove classification and universality results for one-dimensional definable topological spaces, showing that these can be largely described in terms of a few canonical examples. We derive in particular that the three element basis conjecture of Gruenhage [25] holds for all infinite Hausdorff definable topological spaces in o-minimal structures expanding $(\mathbb{R},<)$, i.e. any such space has a definable copy of an interval with the euclidean, discrete or lower limit topology.</div><div><br></div><div>A definable topological space is affine if it is definably homeomorphic to a euclidean space. We prove affineness results in o-minimal expansions of ordered fields. This includes a result for Hausdorff one-dimensional definable topological spaces. We give two new proofs of an affineness theorem of Walsberg [49] for definable metric spaces. We also prove an affineness result for definable topological spaces of any dimension that are Tychonoff in a definable</div><div>sense, and derive that a large class of locally affine definable topological spaces are affine.</div>

An approach to aerodynamic configuration optimization is presented for the high-speed civil transport (HSCT). Methods to parameterize the wing shape, fuselage shape and nacelle placement are described. Variable-complexity design strategies are used to combine conceptual and preliminary-level design approaches, both to preserve interdisciplinary design influences and to reduce computational expense. The preliminary-design-level analysis methods used to estimate aircraft performance are described. Conceptual-design-level (approximate) methods are used to estimate aircraft weight, supersonic wave drag and drag due to lift, and landing angle of attack. The methodology is applied to the minimization of the gross weight of an HSCT that flies at Mach 2.4 with a range of 5500 n.mi. Results are presented for wing plan form shape optimization and for combined wing and fuselage optimization with nacelle placement. Case studies include both all-metal wings and advanced composite wings. The results indicate the beneficial effect of simultaneous design of an entire configuration over the optimization of the wing alone and illustrate the capability of the optimization procedure. / Ph. D.

Expressiveness, and more recently, succinctness, are two central concerns of finite model theory and descriptive complexity theory. Succinctness is particularly interesting because it is closely related to the complexity-theoretic trade-off between parallel time and the amount of hardware. We develop new bounds on the expressiveness and succinctness of first-order logic with two variables on finite words, present a related result about the complexity of the satisfiability problem for this logic, and explore a new approach to the generalized star-height problem from the perspective of logical expressiveness. We give a complete characterization of the expressive power of first-order logic with two variables on finite words. Our main tool for this investigation is the classical Ehrenfeucht-Fra¨ıss´e game. Using our new characterization, we prove that the quantifier alternation hierarchy for this logic is strict, settling the main remaining open question about the expressiveness of this logic. A second important question about first-order logic with two variables on finite words is about the complexity of the satisfiability problem for this logic. Previously it was only known that this problem is NP-hard and in NEXP. We prove a polynomialsize small-model property for this logic, leading to an NP algorithm and thus proving that the satisfiability problem for this logic is NP-complete. Finally, we investigate one of the most baffling open problems in formal language theory: the generalized star-height problem. As of today, we do not even know whether there exists a regular language that has generalized star-height larger than 1. This problem can be phrased as an expressiveness question for first-order logic with a restricted transitive closure operator, and thus allows us to use established tools from finite model theory to attack the generalized star-height problem. Besides our contribution to formalize this problem in a purely logical form, we have developed several example languages as candidates for languages of generalized star-height at least 2. While some of them still stand as promising candidates, for others we present new results that prove that they only have generalized star-height 1.

<p>In this thesis we prove a strong conceptual completeness result for first-order continuous logic. Strong conceptual completeness was proved in 1987 by Michael Makkai for classical first-order logic, and states that it is possible to recover a first-order theory T by looking at functors originating from the category Mod(T) of its models. </p> <p> We then give a brief account of simple theories in continuous logic, and give a proof that the characterization of simple theories using dividing holds in continuous structures. These results are a specialization of well established results for thick cats which appear in [Ben03b] and in [Ben03a].</p> <p> Finally, we turn to the study of non-archimedean Banach spaces over non-trivially valued fields. We give a natural language and axioms to describe them, and show that they admit quantifier elimination, and are N0-stable. We also show that the theory of non-archimedean Banach spaces has only one N 1-saturated model in any cardinality. </p> / Thesis / Doctor of Philosophy (PhD)

This thesis looks at characterising countably infinitely categorical theories. That is theories for which every countably infinite model is isomorphic to every other countably infinite model. The thesis looks at the Lindenbaum-Tarski algebra, Henkin theories, types and then ends with the Ryll-Nardzewski theorem which provides several equivalences to a theory being countably infinitely categorical.

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A definition of abstract logic is presented. This is used to explore and compare some abstract logics, such as logics with generalised quantifiers and infinitary logics, and their properties. Special focus is given to the properties of completeness, compactness, and the Löwenheim-Skolem property. A method of comparing different logics is presented and the concept of equivalent logics introduced. Lastly a proof is given for Lindström's theorem, which provides a characterization of elementary logic, also known as first-order logic, as the strongest logic for which both the compactness property and the Löwenheim-Skolem property, holds.

Le cadre général de cette thèse est celui de la propriété d’indépendance en théorie des modèles. Les théories sans cette propriété sont appelées NIP ou dépendantes. L’objectif principal est de trouver de nouveaux exemples de théories appartenant à cette classe. Nous montrons d’abord un résultat isolé qui répond une question de Pillay : dans un groupe NIP possédant une partie infinie de classe de nilpotence finie, on y trouve un sous-groupe définissable de même classe de nilpotence et contenant cette partie infinie. Le reste de la thèse est motivé par deux cadres extrêmement proches : les groupes abéliens munis d’une chaîne de sous-groupes uniformément définissables, et les groupes abéliens valués. Dans le premier cas nous identifions une certaine théorie et nous étudions plusieurs extensions de cette théorie. Nous prouvons une élimination des quantificateurs dans chacune des ses extensions, grâce à laquelle la NIP en découle facilement. Le dernier résultat est le plus substantiel. Nous montrons qu’une théorie naturelle de chaîne colorée munie quasi-automorphismes n’a pas la propriété d’indépendance. Nous appliquons ensuite ce résultat à une certaine théorie de groupes valués, étudiée par Simonetta dans le contexte des groupes C-minimaux, pour en conclure qu’elle est NIP. Nous montrons aussi d’une façon assez directe (en utilisant des résultats de Rubin et Poizat) qu’une chaîne colorée munie d’automorphismes est NIP. / This PhD thesis is in the general area of the independence property in model theory.Theories without this property are called NIP or dependent. The main objective of this thesis is to find new examples belonging to this class. Firstly, we prove an isolated result that answers a question stated by Pillay : if a NIP group contains an infinite set of finite nilpotency class, then there exists a definable subgroup of the same nilpotency class containing this set. The rest of this thesis is motivated by two extremely closed related contexts : abelian groups equipped with an uniformly definable chain of subgroups, and valued groups. In the first case we identify a theory and study several extensions of it. We prove quantifier elimination in each of these extensions, and use it to easily conclude that they are NIP. The last result is the most significant one. We prove that a natural theory of linear orderings equipped with quasi-automorphisms doesn’t have the independence property. Then we apply this result to a particular theory of valued abelian groups, which has been studied by Simonetta in the context of C-minimal groups, to conclude that it is NIP. We also prove in a rather straightforward way (using results by Rubin and Poizat) that a linear ordering equipped with automorphisms is NIP

Cette thèse est consacrée à l'étude d'un point de vue modèle théorique de corps valués algébriquement clos enrichis d'un prédicat qui représente soit un sous-groupe multiplicatif soit un sous-corps. Nous donnons un résultat d'élimination partielle des quantificateurs pour les structures du type (M , G), où M est un corps valué algébriquement clos et où G un sous-groupe multiplicatif sur lequel la valuation est injective... / This thesis is dedicated to the model theoretic study of algebraically closed valued fields equipped with a additional unary predicate for either a multiplicative subgroup or a subfield.We give a result of relative quantifier elimination for structures of the kind (M , G), where M is an algebraically closed valued field and G is a multiplicative subgroup on wich the valuation is injective...