On the greatest common divisor of nn and the nnth Fibonacci number

Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. We prove that $\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x$ for all $x \geq 2$, and that $\mathcal{A}$ has zero asymptotic density. Our proofs rely on a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$

The Role of Research in Wine: the Emergence of a Regional Research Area in an Italian Wine Production System

The process of technological modernisation of the wine industry has increasingly become a global phenomenon involving both traditional wine producing countries – such as France, Italy and Portugal – as well as new emerging producers –as the US, Australia, South Africa, Chile and Argentina. At this respect, the literature (Aylward, 2003; Unwin, 1991) remarks that in the industry the process of technological renovation has been spurred by the consistent investment of new producer countries, as California, Australia, New Zealand and South Africa, in creating or strengthening research institutions. These efforts have led to the creation of a specialised wine research system. Conversely, it seems that producers in traditional wine producing areas have often been locked in old technologies and methods of production, due to path dependency. The aim of this paper is to shed some light on how old producing countries have reacted to the increased technological competition by emerging producers. In particular we focus on role played by research organisations (e.g. universities, public research centres) and intermediate institutions (e.g. sectoral associations, extension agencies, technological transfer centres) in generating and diffusing knowledge within the industry. The paper is based on first and second hand information collected in Piemonte, where it is located one of the most important Italian wine cluster. There, we have conducted an extensive fieldwork and interviewed several key informants working at research organisations; extension agencies, associations of producers and other associations acting as opinion/lobbying groups. The information collected are elaborated to reconstruct a detailed picture of actors, linkages and processes underpinning the regional innovation and production wine system.

Distribution of integral values for the ratio of two linear recurrences

Let $F$ and $G$ be linear recurrences over a number field $\mathbb{K}$, and let $\mathfrak{R}$ be a finitely generated subring of $\mathbb{K}$. Furthermore, let $\mathcal{N}$ be the set of positive integers $n$ such that $G(n) \neq 0$ and $F(n) / G(n) \in \mathfrak{R}$. Under mild hypothesis, Corvaja and Zannier proved that $\mathcal{N}$ has zero asymptotic density. We prove that $\#(\mathcal{N} \cap [1, x]) \ll x \cdot (\log\log x / \log x)^h$ for all $x \geq 3$, where $h$ is a positive integer that can be computed in terms of $F$ and $G$. Assuming the Hardy-Littlewood $k$-tuple conjecture, our result is optimal except for the term $\log \log x$

Is There an Italian Conception of International Law

In 1943, Angelo Piero Sereni wrote The Italian Conception of International Law, a book explicitly aimed at restoring Anglo-American respect for Italian international lawyers after the Fascist period. On the seventieth anniversary of the publication of this work, it is worth considering whether there is, in fact, such a thing as an 'Italian' conception of international law. Methodologically speaking, does thinking of international law in terms of national schools make sense? Although a comparative approach to international law is back in vogue, this article questions the validity of any attempt at finding any 'Italian distinctiveness' in the intellectual history of the Italian school(s) of international law. Sereni's enlisting of ancient masters to an 'Italian' conception between the 13th and 18th centuries is for the most part untenable. While a distinctively Italian conception of international law arguably came into existence in the 19th century with Mancini's theory of nationalities, Anzilotti successfully set out to dissolve this into the 20th century European mainstream of positivist international law. The ensuing absence of an 'Italian' conception may give pause for thought to contemporary proponents of 'comparative international law'

Measurement of ϒ(1S) Elliptic Flow at Forward Rapidity in Pb-Pb Collisions at sqrt[s_{NN}]=5.02 TeV.

The first measurement of the ϒ(1S) elliptic flow coefficient (v_{2}) is performed at forward rapidity (2.