The empirical process on Gaussian spherical harmonics

We establish weak convergence of the empirical process on the spherical harmonics of a Gaussian random field in the presence of an unknown angular power spectrum. This result suggests various Gaussianity tests with an asymptotic justification. The issue of testing for Gaussianity on isotropic spherical random fields has recently received strong empirical attention in the cosmological literature, in connection with the statistical analysis of cosmic microwave background radiation

Separable solutions of quasilinear Lane-Emden equations

For $0 < p-1 < q$ and $\ge=\pm 1$, we prove the existence of solutions of -\Gd_pu=\ge u^q in a cone $C_S$, with vertex 0 and opening $S$, vanishing on \prt C_S, under the form u(x)=|x|^\gb\gw(\frac{x}{|x|}). The problem reduces to a quasilinear elliptic equation on $S$ and existence is based upon degree theory and homotopy methods. We also obtain a non-existence result in some critical case by an integral type identity.Comment: To appear in Journal of the European Mathematical Societ

On the Voting Time of the Deterministic Majority Process

In the deterministic binary majority process we are given a simple graph where each node has one out of two initial opinions. In every round, every node adopts the majority opinion among its neighbors. By using a potential argument first discovered by Goles and Olivos (1980), it is known that this process always converges in $O(|E|)$ rounds to a two-periodic state in which every node either keeps its opinion or changes it in every round. It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the $O(|E|)$ bound on the convergence time of the deterministic binary majority process is indeed tight even for dense graphs. However, in many graphs such as the complete graph, from any initial opinion assignment, the process converges in just a constant number of rounds. By carefully exploiting the structure of the potential function by Goles and Olivos (1980), we derive a new upper bound on the convergence time of the deterministic binary majority process that accounts for such exceptional cases. We show that it is possible to identify certain modules of a graph $G$ in order to obtain a new graph $G^\Delta$ with the property that the worst-case convergence time of $G^\Delta$ is an upper bound on that of $G$. Moreover, even though our upper bound can be computed in linear time, we show that, given an integer $k$, it is NP-hard to decide whether there exists an initial opinion assignment for which it takes more than $k$ rounds to converge to the two-periodic state.Comment: full version of brief announcement accepted at DISC'1

Global Value Chains and the Great Recession: Evidence from Italian and German Firms

During the last two decades, profound changes in the international division of labour among firms have occurred, with impressive growth in outsourcing, off-shoring of some stages of production and the globalization of intermediates goods markets. This new model of the international division of labour has both initiated an increasing variety of relationships among producers and spurred the development of Global Value Chains. According to some recent research, Global Value Chains have been one of the main transmission mechanisms of the Great Trade Collapse that severely and simultaneously hit all OECD countries in 2009. Pervasive as it has been, it also appears that the impact of the crisis on firms involved in Global Value Chains has been highly heterogeneous. Our paper intends to contribute to this very recent and ongoing debate, providing a description of the effects of the crisis from a perspective that is both countrycomparative, Germany and Italy being the countries taken into consideration, and on firm level, as we pay particular attention to the positioning of the firms along Global Value Chains, i.e., whether intermediate or final firms- and to their strategies. Three are the main conclusions: i) intermediate firms were hit by the crisis more than final firms; ii) among intermediate firms, the ones that carried out innovation activities in the previous period (before 2008) were somewhat sheltered by the effect of crisis; iii) firms ’ positioning in GVCs and their strategies may help to explain the Italy-Germany performance gap

Simple Dynamics for Plurality Consensus

We study a \emph{Plurality-Consensus} process in which each of $n$ anonymous agents of a communication network initially supports an opinion (a color chosen from a finite set $[k]$). Then, in every (synchronous) round, each agent can revise his color according to the opinions currently held by a random sample of his neighbors. It is assumed that the initial color configuration exhibits a sufficiently large \emph{bias} $s$ towards a fixed plurality color, that is, the number of nodes supporting the plurality color exceeds the number of nodes supporting any other color by $s$ additional nodes. The goal is having the process to converge to the \emph{stable} configuration in which all nodes support the initial plurality. We consider a basic model in which the network is a clique and the update rule (called here the \emph{3-majority dynamics}) of the process is the following: each agent looks at the colors of three random neighbors and then applies the majority rule (breaking ties uniformly). We prove that the process converges in time $\mathcal{O}( \min\{ k, (n/\log n)^{1/3} \} \, \log n )$ with high probability, provided that $s \geqslant c \sqrt{ \min\{ 2k, (n/\log n)^{1/3} \}\, n \log n}$. We then prove that our upper bound above is tight as long as $k \leqslant (n/\log n)^{1/4}$. This fact implies an exponential time-gap between the plurality-consensus process and the \emph{median} process studied by Doerr et al. in [ACM SPAA'11]. A natural question is whether looking at more (than three) random neighbors can significantly speed up the process. We provide a negative answer to this question: In particular, we show that samples of polylogarithmic size can speed up the process by a polylogarithmic factor only.Comment: Preprint of journal versio

La stanza di Giorgio Manganelli

L'avversione di Giorgio Manganelli nei confronti del concetto d'autore era connessa a un ideale polimorfico e cangiante dell'individualità, a una critica dell'io, inteso come struttura stabile, riconoscibile, rappresentabile. Le sue concezioni filosofiche, lo spingevano a considerare l'arte come un processo impersonale. Lo psicanalista junghiano Ernst Bernhard gli aveva insegnato che lo spazio della memoria può dilatarsi oltre l'insieme dei ricordi del singolo individuo, per comprendere differenti regimi di oggetti come ad esempio il mondo animale, vegetale e minerale. Manganelli usava la scrittura per entrare in questo lungo repertorio di variazioni. Una delle sue immagini preferite era quella dell'uomo che si trasforma in un calamaio di inchiostro e che intinge il pennino in sé medesimo per poter scrivere delle cose, quasi trasformandosi in esse.Giorgio Manganelli's aversion to the concept of the author was linked to a polymorphic and changing ideal of individuality, to a critique of the self, conceived as a stable, recognisable and representable structure. His philosophical ideas drove him to consider art as an impersonal process. Ernst Bernhardh, the Jungian psychoanalyst, had taught him that memory space can expand beyond memories of every individual, to understand different patterns of objects, for example the animal world, the plant world and the mineral world. Manganelli used writing to enter into this long repertoire of variations. One of his favourite images was that of a man transforming into an inkwell who wets the nib in himself in order to write things, thereby becoming these things