Contractive-like mapping principles in ordered metric spaces and application to ordinary differential equations

[EN] The purpose of this paper is to present a fixed point theorem for generalized contractions in partially ordered complete metric spaces. We also present an application to first-order ordinary differential equations.

Paisajes epigráficos en la Provincia Hispana Citerior en época de Augusto

[ES] Se estudia la diversidad de contextos de exhibición de las inscripciones romanas en la provincia Hispania citerior en época augustea, atendiendo a la importancia que adquirieron las ciudades en el proceso de romanización del territorio, así como a la configuración de nuevos espacios de exposición de las inscripciones, en algunos casos dentro de las estrategias de promoción y autorrepresentación de las elites locales.

Il contributo dell’Unione europea all’affermazione dello stato di diritto nella comunità internazionale

This thesis is aimed at analysing EU external relations from the perspective of the promotion of the rule of law in order to evaluate the effectiveness and consistency of its action within the international community. The research starts with an examination of the notion of the rule of law from a theoretical point of view. The first chapter initially describes the historical-political evolution of the establishment of the notion of the rule of law. Some of the most significant national experiences (France, the UK, Germany and Austria) are discussed. Then, the focus is put on the need to propose interpretations which explain the grounds of the rule of law, by highlighting the different formal and substantive interpretations. This philosophical-historical analysis is complemented by a reconstruction of how the notion of the rule of law was developed by the international community, with a view to searching a common notion at the international level by comparing theory and practice within the main international organisations such as the UN, OECD and the Council of Europe. Specific mention is made of the EU experience, whose configuration as a Community based on the rule of law is often debated, starting from the case law of the European Court of Justice. The second chapter deals with the conditionality policy and focuses on the development and scope of democratic conditionality according to the dominant approach of the doctrine. First, the birth of conditionality is analysed from an economic point of view, especially within international financial organisations and the different types of conditionality recreated in the scientific sector. Then an analysis is provided about the birth of democratic conditionality in the EC – in relation to its external relations – firstly as a mere political exercise to be then turned into a standardised system of clauses. Specific reference is made to the main scope of conditionality, that is to say enlargement policy and the development of the Copenhagen criteria. The third chapter provides further details about the legal questions connected to the use of democratic clauses: on the one hand, the power of the EC to include human rights clauses in international agreements, on the other, the variety and overlapping in the use of the legal basis. The chapter ends with an analysis of the measures of suspension of agreements with third countries in those rare but significant cases in which the suspension clause, included in the Lomè Convention first and in the Cotonou Agreement then, is applied. The last chapter is devoted to the analysis of democratic clauses in unilateral acts adopted by the European Union which affect third countries. The examination of this practice and the comparison with the approach analysed in the previous chapter entails a major theoretical question. It is the clear-cut distinction between conditionality and international sanction. This distinction is to be taken into account when considering the premises and consequences, in terms of legal relations, which are generated when democratic clauses are not complied with. The chapter ends with a brief analysis of what, according to the reconstruction suggested, can be rightly labelled as real democratic conditionality, that is to say the system of incentives, positive measures developed within the community GSP. The dissertation ends with a few general considerations about the difficulties experienced by the EU in promoting the rule of law. The contradictory aspects of the EU external actions are manifold, as well as its difficulties in choosing the most appropriate measures to be taken which, however, reflect all the repercussions and tension resulting from the balance of power within the international community. The thesis argues that it is difficult to grant full credibility to an entity like the EU which, although it proclaims itself as the guardian and promoter of the rule of law, in practice, is too often biased in managing its relations with third countries. However, she adds, we must acknowledge that the EU is committed and constantly strives towards identifying new spaces and strategies of action.

Fixed point theorems in partially ordered spaces and aplications

[ES]Recientemente, en la Teoría del punto fijo, han aparecido muchos resultados que obtienen condiciones suficientes para la existencia de un punto fijo si trabajamos con aplicaciones en un conjunto dotado de un orden parcial. Generalmente, estos resultados combinan dos teoremas del punto fijo fundamentales: el Teorema de la contracción de Banach y el Teorema de Knaster-Tarski.

Construction of polynomial spline spaces over quadtree and octree T-meshes

[EN]We present a new strategy for constructing tensor product spline spaces over quadtree and octree T-meshes. The proposed technique includes some simple rules for inferring local knot vectors to define spline blending functions. These rules allow to obtain for a given T-mesh a set of cubic spline functions that span a space with nice properties: it can reproduce cubic polynomials, the functions are C2-continuous, linearly independent, and spaces spanned by nested T-meshes are also nested. In order to span spaces with these properties applying the proposed rules, the T-mesh should fulfill the only requirement of being a 0-balanced quadtree or octree. ..

A simple strategy for defining polynomial spline spaces over hierarchical T-meses

[EN]We present a new strategy for constructing spline spaces over hierarchical T-meshes with quad- and octree subdivision scheme. The proposed technique includes some simple rules for inferring local knot vectors to define C 2 -continuous cubic tensor product spline blending functions. Our conjecture is that these rules allow to obtain, for a given T-mesh, a set of linearly independent spline functions with the property that spaces spanned by nested T-meshes are also nested, and therefore, the functions can reproduce cubic polynomials. In order to span spaces with these properties applying the proposed rules, the T-mesh should fulfill the only requirement of being a 0- balanced mesh...

Construction of polynomial spline spaces over quadtree and octree T-meshes

[EN]We present a new strategy for constructing tensor product spline spaces over quadtree and octree T-meshes. The proposed technique includes some simple rules for inferring local knot vectors to define spline blending functions. These rules allow to obtain for a given T-mesh a set of cubic spline functions that span a space with nice properties: it can reproduce cubic polynomials, the functions are C2-continuous, linearly independent, and spaces spanned by nested T-meshes are also nested. In order to span spaces with these properties applying the proposed rules, the T-mesh should fulfill the only requirement of being a 0-balanced quadtree or octree. ..

La via Emilia tra Rimini e Forlì. Ruolo e immagine della strada

Abstract Il tema delle infrastrutture, intese come parte dell’architettura dello spazio urbano e del territorio, assume un ruolo centrale in molti progetti contemporanei e costituisce la ragione di questa ricerca. E’ preso in esame, in particolare, il tracciato extraurbano della via Emilia, antica strada consolare romana la cui definizione risale al II sec. a.C., nel tratto compreso tra le città di Rimini e Forlì. Studiare la strada nel suo rapporto con il territorio locale ha significato in primo luogo prendere in considerazione la via Emilia in quanto manufatto, ma anche in quanto percorso che si compie nel tempo. Si è dunque cercato di mostrare come, in parallelo all’evoluzione della sua sezione e della geometria del suo tracciato, sia cambiata anche la sua fruizione, e come si sia evoluto il modo in cui la strada viene “misurata”, denominata e gestita. All’interno di una riflessione critica sulla forma e sul ruolo della strada nel corso dei secoli la Tesi rilegge il territorio nella sua dimensione di “palinsesto”, riconoscendo e isolando alcuni momenti in cui la via Emilia ha assunto un valore “simbolico” che rimanda alla Roma imperiale. La perdita del significato via Emilia, intesa come elemento di “costruzione” del territorio, ha origine con il processo di urbanizzazione diffusa che ha investito il territorio extraurbano a partire dalla fine della seconda guerra mondiale. La condizione attuale della strada, sempre più congestionata dal traffico veicolare, costituisce la premesse per una riflessione sul futuro della sua forma e degli insediamenti che attraversa. La strategia proposta dagli Enti locali che prevede il raddoppio della strada, con la costruzione della via Emilia Bis, non garantisce solo un potenziamento infrastrutturale ma rappresenta l’occasione per sottrarre al tracciato attuale la funzione di principale asse di comunicazione extraurbana. La via Emilia potrebbe così recuperare il ruolo di itinerario narrativo, attraverso la configurazione dei suoi spazi collettivi, l’architettura dei suoi edifici, il significato dei suoi monumenti, e diventare spazio privilegiato di relazione e di aggregazione. The theme of urban infrastructures, thought as part of the design of urban space and territory, has a central role in several contemporary projects and is the reason of this research. The object of the study is the extra urban route of the via Emilia, an ancient roman road which has been defined in the II century b. C., in its stretch between the cities of Rimini and Forlì. Studying the road in its relationship with the local environment has meant first of all considering the via Emilia as an “artefact” but also as a path that takes place over time. The aim of this research was also to demonstrate how its fruition has changed together with the evolution of the section and geometry of the route, and how the road itself is measured, named and managed. Within a critical approach on the shape and on the role played by the road through the centuries, this Essay reinterprets the territory in its dimension of “palimpsest”, identifying and isolating some periods of time when the via Emilia assumed a symbolic value which recalls the Imperial Rome. The loss of the meaning of the via Emilia, intended as an element that “constitutes” the territory originates from a process of diffused urbanization, which spread in the extra urban environment from the end of the second world war. The actual condition of the road, more and more congested by traffic, is the premise of a reflection about the future of its shape and of the settlements alongside. The strategy proposed by the local authorities, that foresees to double the size of the road, building the via Emilia Bis, not only guarantees an infrastructural enhancement but also it represents an opportunity to take off from the road itself the current function of being the principal axis of extra urban connection. In this way the via Emilia could regain its role as a narrative itinerary, through the configuration of its public spaces, the architecture of its buildings, the meaning of its monuments, and then become a privileged space of relationship and aggregation.

Semantic coordination through programmable Tuple spaces

Two of the main features of today complex software systems like pervasive computing systems and Internet-based applications are distribution and openness. Distribution revolves around three orthogonal dimensions: (i) distribution of control|systems are characterised by several independent computational entities and devices, each representing an autonomous and proactive locus of control; (ii) spatial distribution|entities and devices are physically distributed and connected in a global (such as the Internet) or local network; and (iii) temporal distribution|interacting system components come and go over time, and are not required to be available for interaction at the same time. Openness deals with the heterogeneity and dynamism of system components: complex computational systems are open to the integration of diverse components, heterogeneous in terms of architecture and technology, and are dynamic since they allow components to be updated, added, or removed while the system is running. The engineering of open and distributed computational systems mandates for the adoption of a software infrastructure whose underlying model and technology could provide the required level of uncoupling among system components. This is the main motivation behind current research trends in the area of coordination middleware to exploit tuple-based coordination models in the engineering of complex software systems, since they intrinsically provide coordinated components with communication uncoupling and further details in the references therein. An additional daunting challenge for tuple-based models comes from knowledge-intensive application scenarios, namely, scenarios where most of the activities are based on knowledge in some form|and where knowledge becomes the prominent means by which systems get coordinated. Handling knowledge in tuple-based systems induces problems in terms of syntax - e.g., two tuples containing the same data may not match due to differences in the tuple structure - and (mostly) of semantics|e.g., two tuples representing the same information may not match based on a dierent syntax adopted. Till now, the problem has been faced by exploiting tuple-based coordination within a middleware for knowledge intensive environments: e.g., experiments with tuple-based coordination within a Semantic Web middleware (surveys analogous approaches). However, they appear to be designed to tackle the design of coordination for specic application contexts like Semantic Web and Semantic Web Services, and they result in a rather involved extension of the tuple space model. The main goal of this thesis was to conceive a more general approach to semantic coordination. In particular, it was developed the model and technology of semantic tuple centres. It is adopted the tuple centre model as main coordination abstraction to manage system interactions. A tuple centre can be seen as a programmable tuple space, i.e. an extension of a Linda tuple space, where the behaviour of the tuple space can be programmed so as to react to interaction events. By encapsulating coordination laws within coordination media, tuple centres promote coordination uncoupling among coordinated components. Then, the tuple centre model was semantically enriched: a main design choice in this work was to try not to completely redesign the existing syntactic tuple space model, but rather provide a smooth extension that { although supporting semantic reasoning { keep the simplicity of tuple and tuple matching as easier as possible. By encapsulating the semantic representation of the domain of discourse within coordination media, semantic tuple centres promote semantic uncoupling among coordinated components. The main contributions of the thesis are: (i) the design of the semantic tuple centre model; (ii) the implementation and evaluation of the model based on an existent coordination infrastructure; (iii) a view of the application scenarios in which semantic tuple centres seem to be suitable as coordination media.

Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Pseudodifferential analysis in Psi*-algebras on transmission spaces, infinite solving ideal chains and K-theory for conformally compact spaces

The present thesis is a contribution to the theory of algebras of pseudodifferential operators on singular settings. In particular, we focus on the $b$-calculus and the calculus on conformally compact spaces in the sense of Mazzeo and Melrose in connection with the notion of spectral invariant transmission operator algebras. We summarize results given by Gramsch et. al. on the construction of $Psi_0$-and $Psi*$-algebras and the corresponding scales of generalized Sobolev spaces using commutators of certain closed operators and derivations. In the case of a manifold with corners $Z$ we construct a $Psi*$-completion $A_b(Z,{}^bOmega^{1/2})$ of the algebra of zero order $b$-pseudodifferential operators $Psi_{b,cl}(Z, {}^bOmega^{1/2})$ in the corresponding $C*$-closure $B(Z,{}^bOmega^{12})hookrightarrow L(L^2(Z,{}^bOmega^{1/2}))$. The construction will also provide that localised to the (smooth) interior of Z the operators in the $A_b(Z, {}^bOmega^{1/2})$ can be represented as ordinary pseudodifferential operators. In connection with the notion of solvable $C*$-algebras - introduced by Dynin - we calculate the length of the $C*$-closure of $Psi_{b,cl}^0(F,{}^bOmega^{1/2},R^{E(F)})$ in $B(F,{}^bOmega^{1/2}),R^{E(F)})$ by localizing $B(Z, {}^bOmega^{1/2})$ along the boundary face $F$ using the (extended) indical familiy $I^B_{FZ}$. Moreover, we discuss how one can localise a certain solving ideal chain of $B(Z, {}^bOmega^{1/2})$ in neighbourhoods $U_p$ of arbitrary points $pin Z$. This localisation process will recover the singular structure of $U_p$; further, the induced length function $l_p$ is shown to be upper semi-continuous. We give construction methods for $Psi*$- and $C*$-algebras admitting only infinite long solving ideal chains. These algebras will first be realized as unconnected direct sums of (solvable) $C*$-algebras and then refined such that the resulting algebras have arcwise connected spaces of one dimensional representations. In addition, we recall the notion of transmission algebras on manifolds with corners $(Z_i)_{iin N}$ following an idea of Ali Mehmeti, Gramsch et. al. Thereby, we connect the underlying $C^infty$-function spaces using point evaluations in the smooth parts of the $Z_i$ and use generalized Laplacians to generate an appropriate scale of Sobolev spaces. Moreover, it is possible to associate generalized (solving) ideal chains to these algebras, such that to every $ninN$ there exists an ideal chain of length $n$ within the algebra. Finally, we discuss the $K$-theory for algebras of pseudodifferential operators on conformally compact manifolds $X$ and give an index theorem for these operators. In addition, we prove that the Dirac-operator associated to the metric of a conformally compact manifold $X$ is not a Fredholm operator.

Weighted Inequalities and Lipschitz Spaces

In this thesis I have characterized the trace measures for particular potential spaces of functions defined on R^n, but "mollified" so that the potentials are de facto defined on the upper half-space of R^n. The potential functions are kind Riesz-Bessel. The characterization of trace measures for these spaces is a test condition on elementary sets of the upper half-space. To prove the test condition as sufficient condition for trace measures, I had give an extension to the case of upper half-space of the Muckenhoupt-Wheeden and Wolff inequalities. Finally I characterized the Carleson-trace measures for Besov spaces of discrete martingales. This is a simplified discrete model for harmonic extensions of Lipschitz-Besov spaces.

On moduli spaces of semistable sheaves on K3 surfaces

Ich untersuche die nicht bereits durch die Arbeit "Singular symplectic moduli spaces" von Kaledin, Lehn und Sorger (Invent. Math. 164 (2006), no. 3) abgedeckten Fälle von Modulräumen halbstabiler Garben auf projektiven K3-Flächen - die Fälle mit Mukai-Vektor (0,c,0) sowie die Modulräume zu nichtgenerischen amplen Divisoren - hinsichtlich der möglichen Konstruktion neuer Beispiele von kompakten irreduziblen symplektischen Mannigfaltigkeiten. Ich stelle einen Zusammenhang zu den bereits untersuchten Modulräumen und Verallgemeinerungen derselben her und erweitere bekannte Ergebnisse auf alle offenen Fälle von Garben vom Rang 0 und viele Fälle von Garben von positivem Rang. Insbesondere kann in diesen Fällen die Existenz neuer Beispiele von kompakten irreduziblen symplektischen Mannigfaltigkeiten, die birational über Komponenten des Modulraums liegen, ausgeschlossen werden.

Knots and links in lens spaces

The aim of this dissertation is to improve the knowledge of knots and links in lens spaces. If the lens space L(p,q) is defined as a 3-ball with suitable boundary identifications, then a link in L(p,q) can be represented by a disk diagram, i.e. a regular projection of the link on a disk. In this contest, we obtain a complete finite set of Reidemeister-type moves establishing equivalence, up to ambient isotopy. Moreover, the connections of this new diagram with both grid and band diagrams for links in lens spaces are shown. A Wirtinger-type presentation for the group of the link and a diagrammatic method giving the first homology group are described. A class of twisted Alexander polynomials for links in lens spaces is computed, showing its correlation with Reidemeister torsion. One of the most important geometric invariants of links in lens spaces is the lift in 3-sphere of a link L in L(p,q), that is the counterimage of L under the universal covering of L(p,q). Starting from the disk diagram of the link, we obtain a diagram of the lift in the 3-sphere. Using this construction it is possible to find different knots and links in L(p,q) having equivalent lifts, hence we cannot distinguish different links in lens spaces only from their lift. The two final chapters investigate whether several existing invariants for links in lens spaces are essential, i.e. whether they may assume different values on links with equivalent lift. Namely, we consider the fundamental quandle, the group of the link, the twisted Alexander polynomials, the Kauffman Bracket Skein Module and an HOMFLY-PT-type invariant.