Proofs Versus Experiments: Wittgensteinian Themes Surrounding the Four-Color Theorem

The Four-Colour Theorem (4CT) proof, presented to the mathematical community in a pair of papers by Appel and Haken in the late 1970's, provoked a series of philosophical debates. Many conceptual points of these disputes still require some elucidation. After a brief presentation of the main ideas of Appel and Haken’s procedure for the proof and a reconstruction of Thomas Tymoczko’s argument for the novelty of 4CT’s proof, we shall formulate some questions regarding the connections between the points raised by Tymoczko and some Wittgensteinian topics in the philosophy of mathematics such as the importance of the surveyability as a criterion for distinguishing mathematical proofs from empirical experiments. Our aim is to show that the “characteristic Wittgensteinian invention” (Mühlhölzer 2006) – the strong distinction between proofs and experiments – can shed some light in the conceptual confusions surrounding the Four-Colour Theorem

Clausius' Virial vs. Total Potential Energy in the dynamics of a two-component system

In a gravitational virialized bound system built up of two components, one of which is embedded in the other, the Clausius' virial energy of one subcomponent is not, in general, equal to its total potential energy, as occurs in a single system without external forces. This is the main reason for the presence, in the case of two non-coinciding concentric spheroidal subsystems, of a minimum (in absolute value) in the Clausius' virial of the inner component B, when it assumes a special configuration characterized by a value of its semi-major axis we have named "tidal radius". The physical meaning, connected with its appearance, is to introduce a scale length on the gravity field of the inner subsystem, which is induced from the outer one. Its relevance in the galaxy dynamics has been stressed by demonstrating that some of the main features of the Fundamental Plane may follow as consequence of its existence. More physical insight into the dynamics of a two component system may be got by looking at the location of this scale length inside the plots of the potential energies of each subsystem and of the whole system and by also taking into account the trend of the anti-symmetric residual-energy, that is the difference between the tidal and the interaction-energy of each component. Some thermodynamical arguments related to the inner component are also added to prove as special is the "tidal radius configuration". Moreover the role of the divergency at the center of the two subsystems in obtaining this scale length is considered. For the sake of simplicity the analysis has been performed in the case of a frozen external component even if this constraint does not appear to be too relevant in order to preserve the main results.Comment: New Astronomy, accepte

Effects of the Planar Galactic Tides and Stellar Mass on Comet Cloud Dynamics

We report the first results of a research program to explore the sensitivity of the orbits of Oort cloud comets to changes in the strength of the Galactic tides in the plane of the disk and also to changes in the mass of the host star. We performed 2D simulations that confirm that the effects of the tides on comet orbits are sensitive to a star's distance from the Galactic center. A comet cloud closer to the Galactic center than the Sun will have comet perihelia reduced to the region of the inner planets more effectively by the planar tides alone. Similar results are found for a star of smaller mass. We also show how this phenomenon of comet injection persists for a set of alternative Galactic potential models. These preliminary results suggest a fruitful line of research, one that aims to generalize the study of comet cloud dynamics to systems different from the Solar System. In particular, it will allow us to study the roles played by comet clouds in defining the boundaries of the Galactic Habitable Zone.Comment: Published by The Open Astronomy Journal; some slight differences from published versio

From failure to value: Economic valuation for a selected set of products and services from Mediterranean forests

Aim of study: the paper estimates the economic value of a selected range of forest products and services, i.e. roundwood, nonwood forest products (NWFPs), and carbon-related services. Area of study: the research covers 21 Mediterranean countries, distinguished into four sub-regions. Material and methods: data have been gathered from official statistical sources (e.g. FAOSTAT), scientific literature and technical reports. Different estimation approaches based on market-price have been used for different products/services. Main results: the estimated value ranges between \u20ac10,512 and \u20ac11,158 million (M). Wood products represent more than 85% of the total value. Within them, industrial timber is the most relevant component (65%). Figures for NWFPs are likely to be underestimated because data are available only for some products and countries. When using alternative estimates for pine nuts, pine resin and cork, figures show a \u20ac36.8-572 M increase. In geographical terms, the economic value of Mediterranean forests is highly concentrated: North-West Mediterranean countries account for 70%, and nearly 90% is in just four countries (France, Spain, Turkey and Italy). Research highlights: enhancing the offer of Mediterranean forest products and increasing their role in the rural economy could help to reduce the costs of forest protection: a well-structured forest economy ensuring stable flows of incomes can provide a fundamental set of public non-market services and social values to both local people and the whole community. Understanding the true value of natural resources, then, is an essential step for promoting their protection and sustainable use

On Mechanics and Thermodynamics of a stellar galaxy in a two-component virial system and the Fundamental Plane

The paper confirms the existence of a special configuration (among the infinite number of virial states) which a B stellar(Baryonic) component may assume inside a given D dark halo potential well.This satisfies the d'Alembert Principle of virtual works and its typical dimension works as a scale length (tidal radius)induced on the gravitational field of the bright component by the dark one.Its dynamic and thermodynamic properties are here analyzed in connection with the physical reason for the existence of the Fundamental Plane (FP) for ellipticals.The analysis is performed by using 2-component models with two power-law density profiles and two homogeneous cores and compared with some observable scaling relations for pressure supported ellipticals.The virial equilibrium stages of the 2-component system have to occur after a previous violent relaxation phase. If the stellar B component is allowed to cool slowly its virial evolution consists of a sequence of contractions with enough time to rearrange the virial equilibrium after any step. The thermodynamic process during the dynamical evolution is so divided into a sequence of transformations which are irreversible but occur between two quasi-equilibrium stages.The analysis allows the conclusion that the induced scale length is a real confinement for the stellar system. The presence of this specific border on the space of the baryonic luminous component has to be regarded as the physical reason why a stellar galaxy belongs to the FP and why astrophysical objects, with a completely different history and formation, but characterized by a tidal radius (as the globular clusters are) lie on the same FP. An other problem addressed is how this special configuration may be reached and why an elliptical is not completely relaxed in respect to its dark halo.Comment: 42 pages, 7 figures, in press in New Astronomy 200

Municipal transitions: The social, energy, and spatial dynamics of sociotechnical change in South Tyrol, Italy

With the aim of proposing recommendations on how to use social and territorial specificities as levers for wider achievement of climate and energy targets at local level, this research analyses territories as sociotechnical systems. Defining the territory as a sociotechnical system allows us to underline the interrelations between space, energy and society. Groups of municipalities in a region can be identified with respect to their potential production of renewable energy by means of well-known data-mining approaches. Similar municipalities linking together can share ideas and promote collaborations, supporting clever social planning in the transition towards a new energy system. The methodology is applied to the South Tyrol case study (Italy). Results show eight different spatially-based sociotechnical systems within the coherent cultural and institutional context of South Tyrol. In particular, this paper observes eight different systems in terms of (1) different renewable energy source preferences in semi-urban and rural contexts; (2) different links with other local planning, management, and policy needs; (3) different socio-demographic specificities of individuals and families; (4) presence of different kinds of stakeholders or of (5) different socio-spatial organizations based on land cover. Each energy system has its own specificities and potentialities, including social and spatial dimensions, that can address a more balanced, inclusive, equal, and accelerated energy transition at the local and translocal scale

Bits et Rarezas : deux compositions assistées par ordinateur

La version intégrale de ce mémoire est disponible uniquement pour consultation individuelle à la Bibliothèque de musique de l'Université de Montréal (http://www.bib.umontreal.ca/MU)Ce projet a consisté en la composition de deux pièces électroacoustiques sur support, Bits et Rarezas, à partir de sons conçus dans l'environnement de programmation Max. Cet outil nous a permis de mettre en place une démarche portant principalement sur la composition du son à partir de la manipulation et de l'organisation d'éléments sonores très simples (comme les micro-sons ou les ondes sinusoïdales). Les algorithmes sous forme de patche Max, nous ont permis de contrôler ces matériaux de façon efficace, pour construire les textures sonores complexes utilisées dans les pièces. Dans le premier chapitre de ce mémoire, nous avons abordé les idées de base nous ayant guidé sur les plans techniques (surtout pour l'élaboration des patches Max) et esthétiques; les sujets abordés dans cette partie incluent le micro-son, la synthèse granulaire et le minimalisme électronique. Le deuxième chapitre présente premièrement des généralités sur les patches Max, considérant tant les concepts de base des algorithmes que les stratégies utilisées pour interagir avec les patches et créer du son. Le reste du chapitre II aborde en détail les patches utilisés dans notre projet; nous y présentons la description en détail des interfaces-utilisateur, ainsi que les démarches typiques pour produire des textures sonores diverses; des extraits sonores (inclus dans le CD fourni) servent à illustrer les différentes possibilités. Enfin le dernier chapitre, aborde l'analyse des pièces Bits et Rarezas; nous avons choisi la terminologie proposée par Pierre Schaeffer dans son célèbre Traité des objets musicaux pour décrire les différents sons et textures qui constituent les pièces.This project consists in the composition of two electroacoustic pieces for fixed media entitled Bits and Rarezas. Both works are based on sounds created in Max/MSP programming environment, a tool that has enabled an approach based mainly on the composition of sound by the manipulation and organization of very simple sound materials, such as microsounds and sinusoidal waves. The algorithms implemented in Max were used to efficiently control these sounds, building the complex sound textures used in the pieces. The first chapter is centered on the basic technical and aesthetic ideas that have guided me. These concepts include microsound, granular synthesis and electronic minimalism. For the second chapter I begin with an overview on the Max patches used, focusing on the concepts behind the algorithms as well as on the interaction strategies used to model the sound. The remaining part of the second chapter discusses in more detail the five patches Max used in the project, presenting a description of the user interfaces, as well as a discussion of the typical procedures to compose different types of sounds. A few sound excerpts have been included in the accompanying CD to illustrate the possibilities shown. Finally, the last chapter analyses in detail the pieces Bits and Rarezas. In order to describe the sounds and textures included in the pieces, I have used as a reference the terms proposed by Pierre Schaeffer in his seminal work: Traité des objets musicaux