On soft capacities, quasi-stationary distributions and the pathwise approach to metastability

Motivated by the study of the metastable stochastic Ising model at subcritical temperature and in the limit of a vanishing magnetic field, we extend the notion of ($\kappa$, $\lambda$)-capacities between sets, as well as the associated notion of soft-measures, to the case of overlapping sets. We recover their essential properties, sometimes in a stronger form or in a simpler way, relying on weaker hypotheses. These properties allow to write the main quantities associated with reversible metastable dynamics, e.g. asymptotic transition and relaxation times, in terms of objects that are associated with two-sided variational principles. We also clarify the connection with the classical "pathwise approach" by referring to temporal means on the appropriate time scale.Comment: 29 pages, 1 figur

Thermodynamic Limit for Mean-Field Spin Models

If the Boltzmann-Gibbs state $\omega_N$ of a mean-field $N$-particle system with Hamiltonian $H_N$ verifies the condition $$\omega_N(H_N) \ge \omega_N(H_{N_1}+H_{N_2})$$ for every decomposition $N_1+N_2=N$, then its free energy density increases with $N$. We prove such a condition for a wide class of spin models which includes the Curie-Weiss model, its p-spin generalizations (for both even and odd p), its random field version and also the finite pattern Hopfield model. For all these cases the existence of the thermodynamic limit by subadditivity and boundedness follows.Comment: 15 pages, few improvements. To appear in MPE

Symmetrization in Geometry

The concept of an $i$-symmetrization is introduced, which provides a convenient framework for most of the familiar symmetrization processes on convex sets. Various properties of $i$-symmetrizations are introduced and the relations between them investigated. New expressions are provided for the Steiner and Minkowski symmetrals of a compact convex set which exhibit a dual relationship between them. Characterizations of Steiner, Minkowski and central symmetrization, in terms of natural properties that they enjoy, are given and examples are provided to show that none of the assumptions made can be dropped or significantly weakened. Other familiar symmetrizations, such as Schwarz symmetrization, are discussed and several new ones introduced.Comment: A chacterization of central symmetrization has been added and several typos have been corrected. This version has been accepted for publication on Advances in Mathematic

Il principio di imparzialità del giudice : dal Codice Teodosiano all'opera di Isidoro di Siviglia

Una costituzione della seconda metà del IV secolo, appartenente alla cancelleria di Graziano (CTh. 2.2.1, a. 376), vieta che si possa essere giudici in causa propria. Sancisce, in tal modo, uno dei principi cardine del diritto processuale, uno dei dogmi della giurisdizione. Rappresenta la base del principio dell’imparzialità del giudice espresso, ad es., nell’art. 111 della nostra Costituzione. Il divieto di Graziano non rappresenta un intervento innovativo, potendosi ravvisare dei precedenti a questa disposizione. Si fonda inoltre su un principio analogo in tema di testimonianza, risalente nel tempo, a cui lo stesso Graziano fa espresso rinvio, richiamando gli iura. Il precetto si trasmette attraverso la Consultatio, la Lex Romana Visigothorum (a cui dobbiamo la sua tradizione) la Lex Burgundionum e il Codice giustinianeo. Nel pensiero di Isidoro, influenzato, sotto il profilo giuridico, soprattutto dal diritto occidentale (Codice Teodosiano), troviamo echi di questo principio processualistico così fondante sul piano della iurisdictio? Esistono tracce di un’influenza della norma teodosiana? Nell’opera di Isidoro, sono le Sententiae, che tramandano una serie di precetti morali sulla figura dei giudici che, a mio giudizio, possono considerarsi attinenti al principio suddetto. Ho verificato infatti come il vescovo focalizzi la sua attenzione su criteri di equità e giustizia (da cui poi diramano altri parametri) che costituiscono il fondamento dei suoi precetti morali, tradotti in comportamenti giuridici. Gli stessi criteri fondanti delle norme romane sulla responsabilità del giudice. È presente anche uno specifico richiamo alla parzialità/imparzialità dell’organo giudicante. Inoltre illustrano le qualità del buon giudice che, solo attraverso alcuni comportamenti e caratteristiche determinate, può raggiungere nell’obiettività, la verità del giudizio. Il discorso sul potere, dei prìncipi e dei giudici, posto sul piano teologico, morale e cristiano, appare una trasposizione etica di principi giuridici. Le opinioni del vescovo di Siviglia traducono su un piano pedagogico-morale precetti normativi che, in tal modo, si ammantano di ulteriore autorità. La lettura delle Sententiae che intendo proporre (sotto il profilo indicato), può giovare ad una conoscenza ulteriore del suo pensiero e, allo stesso tempo, a verificare una tradizione, anche celata, di principi romanistici

Monte Carlo Predictions of Far-Infrared Emission from Spiral Galaxies

We present simulations of Far Infrared (FIR) emission by dust in spiral galaxies, based on the Monte Carlo radiative transfer code of Bianchi, Ferrara & Giovanardi (1996). The radiative transfer is carried out at several wavelength in the Ultraviolet, optical and Near Infrared, to cover the range of the stellar Spectral Energy Distribution (SED). Together with the images of the galactic model, a map of the energy absorbed by dust is produced. Using Galactic dust properties, the spatial distribution of dust temperature is derived under the assumption of thermal equilibrium. A correction is applied for non-equilibrium emission in the Mid Infrared. Images of dust emission can then be produced at any wavelength in the FIR. We show the application of the model to the spiral galaxy NGC 6946. The observed stellar SED is used as input and models are produced for different star-dust geometries. It is found that only optically thick dust disks can reproduce the observed amount of FIR radiation. However, it is not possible to reproduce the large FIR scalelength suggested by recent observation of spirals at 200 um, even when the scalelength of the dust disk is larger than that for stars. Optically thin models have ratios of optical/FIR scalelengths closer to the 200um observations, but with smaller absolute scalelengths than optically thick cases. The modelled temperature distributions are compatible with observations of the Galaxy and other spirals. We finally discuss the approximations of the model and the impact of a clumpy stellar and dust structure on the FIR simulations.Comment: 19 pages, 6 figures, accepted by A&

Cooperative Spectrum Sensing Using Random Matrix Theory

In this paper, using tools from asymptotic random matrix theory, a new cooperative scheme for frequency band sensing is introduced for both AWGN and fading channels. Unlike previous works in the field, the new scheme does not require the knowledge of the noise statistics or its variance and is related to the behavior of the largest and smallest eigenvalue of random matrices. Remarkably, simulations show that the asymptotic claims hold even for a small number of observations (which makes it convenient for time-varying topologies), outperforming classical energy detection techniques.Comment: Submitted to International Symposium on Wireless Pervasive Computing 200