Hypoellipticity in spaces of ultradistributions-Study of a model case

In this work we study C (a)-hypoellipticity in spaces of ultradistributions for analytic linear partial differential operators. Our main tool is a new a-priori inequality, which is stated in terms of the behaviour of holomorphic functions on appropriate wedges. In particular, for sum of squares operators satisfying Hormander's condition, we thus obtain a new method for studying analytic hypoellipticity for such a class. We also show how this method can be explicitly applied by studying a model operator, which is constructed as a perturbation of the so-called Baouendi-Goulaouic operator.

Numerical assessment of stability of interface discontinuous finite element pressure spaces

The stability of two recently developed pressure spaces has been assessed numerically: The space proposed by Ausas et al. [R.F. Ausas, F.S. Sousa, G.C. Buscaglia, An improved finite element space for discontinuous pressures, Comput. Methods Appl. Mech. Engrg. 199 (2010) 1019-1031], which is capable of representing discontinuous pressures, and the space proposed by Coppola-Owen and Codina [A.H. Coppola-Owen, R. Codina, Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions, Int. J. Numer. Methods Fluids, 49 (2005) 1287-1304], which can represent discontinuities in pressure gradients. We assess the stability of these spaces by numerically computing the inf-sup constants of several meshes. The inf-sup constant results as the solution of a generalized eigenvalue problems. Both spaces are in this way confirmed to be stable in their original form. An application of the same numerical assessment tool to the stabilized equal-order P-1/P-1 formulation is then reported. An interesting finding is that the stabilization coefficient can be safely set to zero in an arbitrary band of elements without compromising the formulation's stability. An analogous result is also reported for the mini-element P-1(+)/P-1 when the velocity bubbles are removed in an arbitrary band of elements. (C) 2012 Elsevier B.V. All rights reserved.

Reproducing kernel Hilbert spaces associated with kernels on topological spaces

We analyze reproducing kernel Hilbert spaces of positive definite kernels on a topological space X being either first countable or locally compact. The results include versions of Mercer's theorem and theorems on the embedding of these spaces into spaces of continuous and square integrable functions.

Conceptual problem for noncommutative inflation and the new approach for nonrelativistic inflationary equation of state

In a previous paper, we connected the phenomenological noncommutative inflation of Alexander, Brandenberger and Magueijo [ Phys. Rev. D 67 081301 (2003)] and Koh and Brandenberger [ J. Cosmol. Astropart Phys. 2007 21 ()] with the formal representation theory of groups and algebras and analyzed minimal conditions that the deformed dispersion relation should satisfy in order to lead to a successful inflation. In that paper, we showed that elementary tools of algebra allow a group-like procedure in which even Hopf algebras (roughly the symmetries of noncommutative spaces) could lead to the equation of state of inflationary radiation. Nevertheless, in this paper, we show that there exists a conceptual problem with the kind of representation that leads to the fundamental equations of the model. The problem comes from an incompatibility between one of the minimal conditions for successful inflation (the momentum of individual photons being bounded from above) and the Fock-space structure of the representation which leads to the fundamental inflationary equations of state. We show that the Fock structure, although mathematically allowed, would lead to problems with the overall consistency of physics, like leading to a problematic scattering theory, for example. We suggest replacing the Fock space by one of two possible structures that we propose. One of them relates to the general theory of Hopf algebras (here explained at an elementary level) while the other is based on a representation theorem of von Neumann algebras (a generalization of the Clebsch-Gordan coefficients), a proposal already suggested by us to take into account interactions in the inflationary equation of state.

Weighted Fourier–Laplace transforms in reproducing kernel Hilbert spaces on the sphere

We study the action of a weighted Fourier–Laplace transform on the functions in the reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel on the sphere. After defining a notion of smoothness implied by the transform, we show that smoothness of the kernel implies the same smoothness for the generating elements (spherical harmonics) in the Mercer expansion of the kernel. We prove a reproducing property for the weighted Fourier–Laplace transform of the functions in the RKHS and embed the RKHS into spaces of smooth functions. Some relevant properties of the embedding are considered, including compactness and boundedness. The approach taken in the paper includes two important notions of differentiability characterized by weighted Fourier–Laplace transforms: fractional derivatives and Laplace–Beltrami derivatives.

Pullback exponential attractors for evolution processes in Banach spaces: properties and applications

This article is a continuation of our previous work [5], where we formulated general existence theorems for pullback exponential attractors for asymptotically compact evolution processes in Banach spaces and discussed its implications in the autonomous case. We now study properties of the attractors and use our theoretical results to prove the existence of pullback exponential attractors in two examples, where previous results do not apply.

Intersections on moduli spaces of curves

We present a new approach to perform calculations with the certain standard classes in cohomology of the moduli spaces of curves. It is based on an important lemma of Ionel relating the intersection theoriy of the moduli space of curves and that of the space of admissible coverings. As particular results, we obtain expressions of Hurwitz numbers in terms of the intersections in the tautological ring, expressions of the simplest intersection numbers in terms of Hurwitz numbers, an algorithm of calculation of certain correlators which are the subject of the Witten conjecture, an improved algorithm for intersections related to the Boussinesq hierarchy, expressions for the Hodge integrals over two-pointed ramification cycles, cut-and-join type equations for a large class of intersection numbers, etc.

A fixed point theorem for Meir-Keeler contractions in ordered metric spaces

[EN] The purpose of this paper is to present some fixed point theorems for Meir-Keeler contractions in a complete metric space endowed with a partial order. MSC: 47H10.

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.

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. ..

Max Bill in Italia . Lo spazio logico dell'architettura

I Max Bill is an intense giornata of a big fresco. An analysis of the main social, artistic and cultural events throughout the twentieth century is needed in order to trace his career through his masterpieces and architectures. Some of the faces of this hypothetical mural painting are, among others, Le Corbusier, Walter Gropius, Ernesto Nathan Rogers, Kandinskij, Klee, Mondrian, Vatongerloo, Ignazio Silone, while the backcloth is given by artistic avant-gardes, Bauhaus, International Exhibitions, CIAM, war events, reconstruction, Milan Triennali, Venice Biennali, the School of Ulm. Architect, even though more known as painter, sculptor, designer and graphic artist, Max Bill attends the Bauhaus as a student in the years 1927-1929, and from this experience derives the main features of a rational, objective, constructive and non figurative art. His research is devoted to give his art a scientific methodology: each work proceeds from the analysis of a problem to the logical and always verifiable solution of the same problem. By means of composition elements (such as rhythm, seriality, theme and its variation, harmony and dissonance), he faces, with consistent results, themes apparently very distant from each other as the project for the H.f.G. or the design for a font. Mathematics are a constant reference frame as field of certainties, order, objectivity: ‘for Bill mathematics are never confined to a simple function: they represent a climate of spiritual certainties, and also the theme of non attempted in its purest state, objectivity of the sign and of the geometrical place, and at the same time restlessness of the infinity: Limited and Unlimited ’. In almost sixty years of activity, experiencing all artistic fields, Max Bill works, projects, designs, holds conferences and exhibitions in Europe, Asia and Americas, confronting himself with the most influencing personalities of the twentieth century. In such a vast scenery, the need to limit the investigation field combined with the necessity to address and analyse the unpublished and original aspect of Bill’s relations with Italy. The original contribution of the present research regards this particular ‘geographic delimitation’; in particular, beyond the deep cultural exchanges between Bill and a series of Milanese architects, most of all with Rogers, two main projects have been addressed: the realtà nuova at Milan Triennale in 1947, and the Contemporary Art Museum in Florence in 1980. It is important to note that these projects have not been previously investigated, and the former never appears in the sources either. These works, together with the most well-known ones, such as the projects for the VI and IX Triennale, and the Swiss pavilion for the Biennale, add important details to the reference frame of the relations which took place between Zurich and Milan. Most of the occasions for exchanges took part in between the Thirties and the Fifties, years during which Bill underwent a significant period of artistic growth. He meets the Swiss progressive architects and the Paris artists from the Abstraction-Création movement, enters the CIAM, collaborates with Le Corbusier to the third volume of his Complete Works, and in Milan he works and gets confronted with the events related to post-war reconstruction. In these years Bill defines his own working methodology, attaining an artistic maturity in his work. The present research investigates the mentioned time period, despite some necessary exceptions. II The official Max Bill bibliography is naturally wide, including spreading works along with ones more devoted to analytical investigation, mainly written in German and often translated into French and English (Max Bill himself published his works in three languages). Few works have been published in Italian and, excluding the catalogue of the Parma exhibition from 1977, they cannot be considered comprehensive. Many publications are exhibition catalogues, some of which include essays written by Max Bill himself, some others bring Bill’s comments in a educational-pedagogical approach, to accompany the observer towards a full understanding of the composition processes of his art works. Bill also left a great amount of theoretical speculations to encourage a critical reading of his works in the form of books edited or written by him, and essays published in ‘Werk’, magazine of the Swiss Werkbund, and other international reviews, among which Domus and Casabella. These three reviews have been important tools of analysis, since they include tracks of some of Max Bill’s architectural works. The architectural aspect is less investigated than the plastic and pictorial ones in all the main reference manuals on the subject: Benevolo, Tafuri and Dal Co, Frampton, Allenspach consider Max Bill as an artist proceeding in his work from Bauhaus in the Ulm experience . A first filing of his works was published in 2004 in the monographic issue of the Spanish magazine 2G, together with critical essays by Karin Gimmi, Stanislaus von Moos, Arthur Rüegg and Hans Frei, and in ‘Konkrete Architektur?’, again by Hans Frei. Moreover, the monographic essay on the Atelier Haus building by Arthur Rüegg from 1997, and the DPA 17 issue of the Catalonia Polytechnic with contributions of Carlos Martì, Bruno Reichlin and Ton Salvadò, the latter publication concentrating on a few Bill’s themes and architectures. An urge to studying and going in depth in Max Bill’s works was marked in 2008 by the centenary of his birth and by a recent rediscovery of Bill as initiator of the ‘minimalist’ tradition in Swiss architecture. Bill’s heirs are both very active in promoting exhibitions, researching and publishing. Jakob Bill, Max Bill’s son and painter himself, recently published a work on Bill’s experience in Bauhaus, and earlier on he had published an in-depth study on ‘Endless Ribbons’ sculptures. Angela Thomas Schmid, Bill’s wife and art historian, published in end 2008 the first volume of a biography on Max Bill and, together with the film maker Eric Schmid, produced a documentary film which was also presented at the last Locarno Film Festival. Both biography and documentary concentrate on Max Bill’s political involvement, from antifascism and 1968 protest movements to Bill experiences as Zurich Municipality councilman and member of the Swiss Confederation Parliament. In the present research, the bibliography includes also direct sources, such as interviews and original materials in the form of letters correspondence and graphic works together with related essays, kept in the max+binia+jakob bill stiftung archive in Zurich. III The results of the present research are organized into four main chapters, each of them subdivided into four parts. The first chapter concentrates on the research field, reasons, tools and methodologies employed, whereas the second one consists of a short biographical note organized by topics, introducing the subject of the research. The third chapter, which includes unpublished events, traces the historical and cultural frame with particular reference to the relations between Max Bill and the Italian scene, especially Milan and the architects Rogers and Baldessari around the Fifties, searching the themes and the keys for interpretation of Bill’s architectures and investigating the critical debate on the reviews and the plastic survey through sculpture. The fourth and last chapter examines four main architectures chosen on a geographical basis, all devoted to exhibition spaces, investigating Max Bill’s composition process related to the pictorial field. Paintings has surely been easier and faster to investigate and verify than the building field. A doctoral thesis discussed in Lausanne in 1977 investigating Max Bill’s plastic and pictorial works, provided a series of devices which were corrected and adapted for the definition of the interpretation grid for the composition structures of Bill’s main architectures. Four different tools are employed in the investigation of each work: a context analysis related to chapter three results; a specific theoretical essay by Max Bill briefly explaining his main theses, even though not directly linked to the very same work of art considered; the interpretation grid for the composition themes derived from a related pictorial work; the architecture drawing and digital three-dimensional model. The double analysis of the architectural and pictorial fields is functional to underlining the relation among the different elements of the composition process; the two fields, however, cannot be compared and they stay, in Max Bill’s works as in the present research, interdependent though self-sufficient. IV An important aspect of Max Bill production is self-referentiality: talking of Max Bill, also through Max Bill, as a need for coherence instead of a method limitation. Ernesto Nathan Rogers describes Bill as the last humanist, and his horizon is the known world but, as the ‘Concrete Art’ of which he is one of the main representatives, his production justifies itself: Max Bill not only found a method, but he autonomously re-wrote the ‘rules of the game’, derived timeless theoretical principles and verified them through a rich and interdisciplinary artistic production. The most recurrent words in the present research work are synthesis, unity, space and logic. These terms are part of Max Bill’s vocabulary and can be referred to his works. Similarly, graphic settings or analytical schemes in this research text referring to or commenting Bill’s architectural projects were drawn up keeping in mind the concise precision of his architectural design. As for Mies van der Rohe, it has been written that Max Bill took art to ‘zero degree’ reaching in this way a high complexity. His works are a synthesis of art: they conceptually encompass all previous and –considered their developments- most of contemporary pictures. Contents and message are generally explicitly declared in the title or in Bill’s essays on his artistic works and architectural projects: the beneficiary is invited to go through and re-build the process of synthesis generating the shape. In the course of the interview with the Milan artist Getulio Alviani, he tells how he would not write more than a page for an essay on Josef Albers: everything was already evident ‘on the surface’ and any additional sentence would be redundant. Two years after that interview, these pages attempt to decompose and single out the elements and processes connected with some of Max Bill’s works which, for their own origin, already contain all possible explanations and interpretations. The formal reduction in favour of contents maximization is, perhaps, Max Bill’s main lesson.