Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ω). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result.

Working within the Zermelo-Frankel Axioms of set theory, we will introduce two important contradictory axioms: Axiom of Choice and Axiom of Determinacy. We will explore perfect polish spaces and games on these spaces to see that the Axiom of Determinacy is inconsistent with the Axiom of Choice. We will see some of the major consequences of accepting the Axiom of Determinacy and how some of these results change when accepting the Axiom of Choice. We will consider 2-player games of perfect information wherein we will see some powerful results having to do with properties of the real numbers. We will use a game to illustrate a weak proof of the continuum hypothesis.

Les graphes de données sont naturellement utilisés dans de nombreux contextes incluant par exemple les réseaux sociaux ou le Web sémantique. L'information contenue dans la base de données se trouve alors aussi bien dans les données mêmes que dans la topologie du graphe, c'est-à-dire dans la manière dont les données sont connectées. Cela implique donc de considérer les questions traditionnelles en théorie des bases de données pour des langages de requêtes capables de parler des chemins connectant les nœuds du graphe. Nous nous intéressons en particulier aux problèmes de la déterminabilité et de la réécriture d'une requête à l'aide de vues. Il s'agit alors de décider si une vue de la base de données contient suffisamment d'information pour répondre entièrement à une requête sans consulter la base de données directement, et dans ce cas, d'exprimer explicitement la réponse à la requête à partir de la vue. Ce cadre rencontre de nombreuses applications, notamment pour l'intégration de données et l'optimisation de requêtes. Nous commençons par comparer ces deux questions aux autres problèmes de décision classiques dans ce contexte : calcul des réponses certaines, test de cohérence et mise à jour d'une instance de vue. Nous améliorons ensuite ces résultats dans deux cas spécifiques. Tout d'abord, nous montrons que pour les requêtes régulières de chemin, l'existence d'une réécriture monotone coïncide avec l'existence d'une réécriture dans Datalog. Puis, nous montrons que pour des vues s'intéressant uniquement aux longueurs des chemins du graphe, une notion plus faible de déterminabilité, appelée déterminabilité asymptotique, est décidable et résulte en des réécritures du premier ordre. / Graph databases appear naturally in various scenarios, such as social networks and the semantic Web. In these cases, the information contained in the database lies as much in the data itself as in the topology of the graph, that is, in how the data points are linked together. This leads to considering traditional database theory questions for query languages that return data nodes based on the paths of the graph connecting them. We focus our attention on the view-based query determinacy and rewriting problems. They ask the question whether a view of the database contains enough information to fully answer a query without accessing the database directly. If so, we then want to express the answer to the query directly with regards to the view. This setting occurs in many applications, such as data integration and query optimization. We start by comparing these two tasks to other common task in this setting: computing certain answers, checking consistency of a view instance and updating it. We then build on these results in two specific cases. First, we show that for regular path queries, the existence of a monotone rewriting coincides with the existence of a rewriting expressible in Datalog. Then, we show that for views that only consider the lengths of the path in the graph, we can decide a weaker form of determinacy, called asymptotic determinacy, and produce first-order rewritings for the queries that are asymptotically determined.

abstract: In this thesis, I explore Cognate Object Constructions COCs (e.g. The clown "laughed" a creepy "laugh") through three research questions: (1) What verbs can accept Cognate Objects COs? (2) Why can these verbs accept COs and other verbs cannot? and (3) How are COCs derived? I demonstrate that Sorace's Hierarchy sheds light on which verbs can accept COs and which cannot by explaining the discrepancies in grammaticality judgments that exist in the literature. I then argue that Hale and Keyser's Conflation account of COCs is not minimalist because it relies on a phenomenon that can be reduced to Merge. After commenting and repairing their account, I provide an outline for a more minimalist framework, which I refer to as "Problems of Projection Extensions" PoP+, that focuses on MERGE, workspaces, labeling theory, phases, and determinacy. Inside this framework, I then develop my own account that depends on only Internal Merge and the constraint in English against stranded articles. With my account situated in this PoP+ framework, I am able to approach the research questions from a syntactic perspective, arguing that the Unergative Restriction on COCs is a result of a determinacy violation in the derivation of Unaccusative COCs. Finally, I point out that, being situated in the PoP+ framework, my account opens COCs up to further investigation not possible before. / Dissertation/Thesis / Masters Thesis Linguistics and Applied Linguistics 2019

In this paper we explore coloring theorems for the reals, its quotients, cardinals, and their combinations. This work is done under the scope of the axiom of determinacy. We also explore generalizations of Mycielski's theorem and show how these can be used to establish coloring theorems. To finish, we discuss the strange realm of long unions.

We characterize various types of σ-porosity via an infinite game in terms of winning strategies. We use a modification of the game to prove and reprove some new and older in- scribing theorems for σ-ideals of σ-porous type in locally compact metric spaces. We show that there exists a closed set which is σ-(1 − ε)-symmetrically porous for every 0 < ε < 1 but which is not σ-1-symmetrically porous. Next, we prove that the realizable by an action unitary representations of a finite abelian group Γ on an infinite-dimensional complex Hilbert space H form a comeager set in Rep(Γ, H). 1

[pt] Seja (Es)t o espaço de germes na origem de funções suaves entre os espaços euclidianos de dimensões e t. Nesta dissertação, apresentamos a parte da Teoria de Mather que descreve hipóteses suficientes para k-determinação em (Es)t sob duas ações diferentes, induzindo as chamadas R- e K-equivalências. Um germe é k-determinado se é equivalente a qualquer perturbação que deixa invariante seu k-jato, os termos de ordem até k de sua expansão de Taylor na origem. A R-equivalência consiste em compor germes com germes de difeomorfismos µa direita. A K- equivalência é mais difícil de descrever. / [en] Let (Es)t be the space of smooth map-germs at the origin between Euclidian spaces of dimensions s and t. In this dissertation, we present a section of Mather theory describing su±cient conditions for k- determinacy of this map-germs under two different actions, inducing the so called R- e K- equivalences. A map-germ is k-determined if it is equivalent to any perturbation that leaves invariant its k-jet, i.e., the terms up to order k of its Taylor expansion at the origin. The R-equivalence consists of compositions with germs of diffeomorphisms to the right. The K- equivalence is harder to describe.

La tesi è organizzata come segue: nella prima parte viene presentato il background matematico necessario per studiare il modello, introducendo la nozione di sistema dinamico (capitolo 2). Succesivamente viene introdotto il background economico, con il concetto di determinatezza sia dal punto di vista tecnico, con esempio applicativo, che dal punto di vista dei modelli presenti in letteratura in cui viene impiegato. Nella seconda parte viene presentato il framework analitico, basato sul modello di Woodford. In seguito, si passa studiare il modello con lo strumento della loglinearizzazione, fornendo le relazioni di equilibrio attorno allo stato stazionario. Poi si specifica il modello in un caso particolare e si prosegue per ottenere la versione non lineare del modello, introducendo nuove ipotesi compatibili con il framework analitico, in modo da poter studiare la determinatezza dell'equilibrio. / The thesis is organized as follow: we start presenting, in the first part, the mathematical background we need to study our model, introducing at first the notion of dynamical system, more specifically in the discrete time, as our model required (Chapter 2). Afterwards, for the economical background (Chapter 3-4), the concept of determinacy of the equilibrium is analyzed from the technical point of view for linear models, providing the analytical conditions that let us obtain a unique and determinate equilibrium. Once explored the techniques, we give an exhaustive example that allows us to better understand from the mathematical point of view the concept of determinacy and how it is linked to the concept of stability. After that, a brief survey of the models that involved the study of determinacy is exposed, showing the several fields of application. Then, since in our model of monetary policy we imply different interest-rate policy rules in order to study the stability of the macroeconomic system, we provide a short preamble of the rules for the operating target interest rate set by Central Bank. In the second part the analytical framework is presented. The starting point is a model for price level determination in a cashless economy, where nominal rigidities are introduced, based on Woodford's work and we give the equilibrium relations of the model in the implicit form (Chapter 5). Afterwards, we build the model in a particular case (Chapter 6), by specifying the functions involved in the model by using utility function of C.R.R.A. type and linear production function, compatible with the analytical characterization. In this way we get the components of a general economic equilibrium model, consistent with the optimizing behavior of households and firms. The results obtained after the specification of the functions are the same we can find in the analysis proposed by Walsh. At this point, we obtain the log-linearized version of the model that is the starting point for the study of the stability of the system in the linear case. This procedure let us find a two equations forward-looking and rational-expectations model for inflation and the output gap. Then we briefly present the different policy regimes used in the analysis according to our framework, providing the interest-rate relations that close the model Since our intention is to find a nonlinear version of the model, the step of using the log-linearization is essential in order to understand and to underline how this tool is useful not only for studying the equations around the steady-state but also to make these relations more treatable from a mathematical point of view: in fact it is important in order to figure out the obstacles we faced to build the nonlinear model and to find the solutions proposed in this work. In the third part (Chapter 7), to reach our purpose to go beyond the log-linearized and simplified version of the model, we try, under some assumption compatible with the behavior of the agents, to provide nonlinear conditions for this model. This is meaningful in order to avoid loss of informations due to the limited analysis around a neighborhood of the steady-state. Once illustrated the nonlinear model and the equilibrium relation, we study the determinacy of the equilibrium, using the techniques shown in the first part, under two different interest rate specifications that close the model. Chapter 8 concludes.

This thesis examines determinacy and E-stability of economic policy in monetary union models. Monetary policy takes the form of either a contemporaneous or a forecast based interest rate rule, while fiscal policy follows a contemporaneous government spending rule. In the absence of asymmetries, the results from the closed economy literature on learning are retained. However, when introducing asymmetries into monetary union frameworks, the determinacy and E-stability conditions for economic policy differ from both the closed and open economy cases. We find that a monetary union with heterogeneous price rigidities is more likely to be determinate and E-stable. Specifically, the Taylor principle, a key stability condition for the closed economy, is now relaxed. Furthermore, an interest rate rule that stabilizes the terms of trade in addition to output and inflation, is more likely to induce determinacy and local stability under RLS learning. If monetary policy is sufficiently aggressive in stabilizing the terms of trade, then determinacy and E-stability of the union economy can be achieved without direct stabilization of output and inflation. A fiscal policy rule that supports demand for domestic goods following a shock to competitiveness, can destabilize the union economy regardless of the interest rate rule employed by the union central bank. In this case, determinacy and E-stability conditions have to be simultaneously and independently met by both fiscal and monetary policy for the union economy to be stable. When fiscal policy instead stabilizes domestic output gaps while monetary policy stabilizes union output and inflation, fiscal policy directly affects the stability of monetary policy. A contemporaneous monetary policy rule has to be more aggressive to satisfy the Taylor principle, the more aggressive fiscal policy is. On the other hand, when monetary policy is forward looking, an aggressive fiscal policy rule can help induce determinacy.

Wie sollten Zentralbanken Geldpolitik gestalten und der Öffentlichkeit kommunizieren, um die Ökonomie bestmöglich zu stabilisieren? Diese Dissertation, bestehend aus drei selbständigen Essays in dynamischer Makroökonomik, widmet sich in erster Linie dem normativen Aspekt von Geldpolitik. Das Hauptresultat im ersten Essay ist, dass bei idiosynkratischen Risiko die öffentliche Bekanntgabe von Informationen zu aggregierten Risiko einen negativen Effekt auf die soziale Wohlfahrt haben kann: durch die Veröffentlichung von Informationen zu nicht-versicherbaren aggregierten Risiko werden die Versicherungsanreize der Individuen verzerrt und damit das individuelle Konsumrisiko erhöht. Als eine Anwendung, analysieren wir die Situation einer Zentralbank, die die Möglichkeit hat, Veränderungen in ihren Inflationszielen anzukündigen und dokumentieren, das der negative Effekt der verzerrten Versicherungsanreize konventionelle positive Aspekte der Ankündigung überwiegt. In zweiten Essay untersuchen wir optimale Geldpolitik in Falle von nominalen Rigiditäten und einer Transaktionsfriktion. In einem Standardmodell, Money-in-the-Utility function, zeigen wir, dass das langfristige Optimum durch die Friedmansche Regel gegeben ist. Daraus folgt für die kurze Frist, dass das Primat von Geldpolitik auf die Stabilisierung der Zinsen und nicht auf Inflationsstabilisierung ausgelegt sein sollte. Im dritten Essay untersuche ich, ob die Existenz und die Terminierung von Realkasseneffekten eine wichtige Rolle für die Determiniertheit des allgemeinen Preisniveaus spielen. Als wichtigstes neues Resultat zeige ich, dass auch bei Zinspolitik ein eindeutiges Preisniveau bestimmt werden kann, wenn die Geldmenge zu Beginn der Periode in Transaktionen verwendet wird. Unter diesen Umständen, hat prädeterminiertes reales Geld die Funktion einer Zustandsvariable und die Zinspolitik sollte passiv sein, um eindeutige, stabile und nicht-oszillierende Gleichgewichtssequenzen zu erreichen. / How should central banks conduct and communicate their policies to serve the goal of stabilizing the macroeconomy? This thesis -- consisting of three self-contained essays on dynamic macroeconomics -- is mainly intended as a progress report on exploring the normative aspect of monetary policy. The main result of the first essay is, that in the presence of idiosyncratic risk, the public revelation of information about uncertain aggregate outcomes can be detrimental. By announcing informative signals on non-insurable aggregate risk, the policy maker distorts agents'' insurance incentives and increases the riskiness of the optimal allocation that is feasible in self-enforceable arrangements. We consider a monetary authority that may reveal changes in the inflation target, and document that the negative effect of distorted insurance incentives can very well dominate conventional effects in favor for the release of better information. In the second essay, we study optimal monetary policy with the nominal interest rate as the single policy instrument. Firms set prices in a staggered way without indexation and real money balances contribute separately to households'' utility. The optimal deterministic steady state under commitment is the Friedman rule for a broad range of parameters. Optimal monetary policy in the short run is then characterized by stabilization of the nominal interest rate instead of inflation stabilization as the predominant principle. In the third essay, I examine whether the existence and the timing of real balance effects contribute to the determination of the absolute price level. As the main novel result, I show that there exists a unique price level sequence that is consistent with an equilibrium under interest rate policy, if beginning-of-period money yields transaction services. Predetermined real money balances can then serve as a state variable, implying that interest rate setting should be passive -- a violation of the Taylor-principle.