A note on primes in certain residue classes

Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$ such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic density relative to the set of all primes which is at least $\prod_{i=1}^k \left(1-\frac{1}{\varphi(a_i)}\right)$, where $\varphi$ is the Euler's totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer $n$ such that $n \not\equiv 0 \bmod a_i$ for $i=1,\ldots,k$ admits asymptotic density which is at least $\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)$

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 $F_k$

Principio di precauzione e partecipazione del pubblico in materia ambientale

La tesi analizza il quadro normativo relativo alla partecipazione del pubblico nei procedimenti amministrativi applicativi del principio di precauzione. Il lavoro si sviluppa in tre Capitoli: il primo inquadra il tema della partecipazione in materia ambientale nel quadro più generale dei modelli di partecipazione all’attività amministrativa generale presenti in altri settori, come quello dell’urbanistica e della regolazione condizionale dell’attività economica. In questo si precisa, inoltre, la ratio e il contenuto del principio di precauzione e si illustrano i relativi nessi con quello di partecipazione. Il secondo Capitolo affronta l’analisi della disciplina europea ed internazionale in materia di partecipazione ambientale laddove la qualità delle forme partecipative è molto più sofisticata e incisiva rispetto alla disciplina domestica; in esso è analizzato anche il procedimento di autorizzazione per l’immissione in commercio di cibi contenenti OGM. Il terzo ed ultimo Capitolo analizza le procedure di valutazione di impatto ambientale e il ruolo significativo rivestito, in esse, dai modelli di partecipazione. Nella fase conclusiva si mettono in luce gli aspetti critici, sotto il profilo procedurale, dei modelli di partecipazione analizzati e si offrono alcuni suggerimenti diretti al loro superamento.The thesis analyzes the legal framework provided for the public participation in administrative proceedings connected to environmental matters, especially when the precautionary principle is applied. The work consists of three chapters: the first situates participation in in the Italian general framework for public participation, especially regarding planning and regulatory proceedings. The meaning of precautionary principle and its connections with the public participation are also analyzed. The second chapter analyses the main features of participation in the international and European legal systems and their influence on the national regulations. This chapter deals in particular with the regulation provided by the EU law for the authorization of GM food. The last chapter discuss the role of public participation in the environmental impact assessments. Finally, the thesis highlights the weak points of environmental participation and propose some measures to improve its legal regulation

The History and geography of the Y chromosome SNPs in Europe: an update

The knowledge of the evolution of the human genome is strictly dependent on the availability of appropriate genetic markers and their relative coverage of genetic variation which refine the phylogenetic reconstruction. While autosomal markers are particularly valuable for recognizing correspondence between genetic and geographic distances, markers on mitochondrial DNA (mtDNA) or Non Recombining Portion of Y Chromosome (NRY), because of their unilinear transmission, can effectively trace diachronical patterns of the human peopling. The maximum extent of polymorphism coverage has already been reached for the very small mitochondrial genome (about 16,5 Kbp), whereas the first studies based on RFLPs (Restriction Fragment Length Polymorphisms) (Cann et al., 1987) and on sequencing of the hypervariable regions (Vigliant et al., 1991), were then combined to get higher resolution (Torroni et al., 1996), and finally the complete genome sequencing is now routinely performed (Achilli et al., 2004, Pala et al., 2009), in order to detect the whole mtDNA variation. A similar approach cannot be used yet at population level for the by far larger nuclear genome. However, advances in genotyping technology have dramatically enhanced the resolution of the analysis at genome-wide level, and recent papers significantly improved the knowledge of the relationships among European populations, using 300 to 500 K SNPs (Single Nucleotide Polymorphisms) on microarrays chips (Tian et al., 2008; Novembre et al., 2008). As to the NRY, most of the studies before the year 2000 were performed using Alu insertion (Hammer, 1995) or STRs (Short Tandem Repeats) (De Knijff et al., 1997; Pritchard et al., 1999) with the known limitations due to recurrence and reversion of this kind of polymorphisms. Using D-HPLC (Denaturing High Performance Liquid Chromatography) technology, Underhill and coworkers (1997) discovered 22 new SNP biallelic markers, rapidly raising in number to 167 (Underhill et al., 2000), 242 (YCC, 2002), about 600 (Karafet et al., 2008), up to more than 725 presently listed in the Y-DNA SNP Index 2009, (www.isogg. org), and the knowledge of Y chromosome phylogeny and of the spread worldwide of human populations raised proportionally. The next goal of the research on Y chromosome will be the use of specific microarrays that can genotype a much higher number of SNPs than nowadays routinely performed, and, ultimately, the complete Y chromosome sequencing. Waiting for future developments, this short note reports the state of the art of the phylogenetic (“history”) and phylogeographic (“geography”) research on Y chromosome SNP analyses in Europe, updating the review published in this Journal by Francalacci & Sanna at the beginning of 2008

Practical numbers among the binomial coefficients

A practical number is a positive integer n such that every positive integer less than n can be written as a sum of distinct divisors of n. We prove that most of the binomial coefficients are practical numbers. Precisely, letting f(n) denote the number of binomial coefficients (nk), with 0≤k≤n, that are not practical numbers, we show that f(n

The Naica Project - A Multidisciplinary Study of the Largest Gypsum Crystals of the World

The caves of Naica (Chihuahua, Mexico) are perhaps the most famous mine caves of the world due to the presence of gigantic gypsum crystals. Nevertheless, very little research has been carried out on these crystals until now. An international multidisciplinary investigation started in 2006 with the aim not only to define the genesis and the age of the Naica gypsum crystals, but also to focus other important scientific aspects of these caves and to ensure a complete documentation and knowledge of these natural wonders which will not be accessible anymore in a few couple of years. The preliminary results of this, still in progress, research allow to date the giant crystals and to define the boundary conditions and the mechanisms which induced their development. For the first time pollens have been extracted from gypsum crystals and their analyses evidenced that some 35 Ky BP the Naica climate was cooler and more humid than today

On the p-adic valuation of Stirling numbers of the first kind

For all integers n≥k≥1, define H(n,k):=∑1/(i1⋯ik), where the sum is extended over all positive integers i1<⋯−klogpn+Ok(1) and we conjecture that there exists a positive constant c = c(p, k) such that νp(H(n,k))<−clogn for all large n. In this respect, we prove the conjecture in the affirmative for all n≤x whose base p representations start with the base p representation of k − 1, but at most 3x0.835 exceptions. We also generalize a result of Lengyel by giving a description of ν2(H(n,2)) in terms of an infinite binary sequence

Design and evaluation of dry-coated tablets for colonic delivery of diclofenac sodium

A colonic drug delivery system is required to protect a drug during its transit through the upper gastro-intestinal tract and allow its release in the colon. The aim of this study was the preparation of dry-coated tablets designed for colonic release of the model drug Diclofenac sodium (DS). The system consists of a drug-pectine (PC) mixture as the core and hydroxypropylmethylcellulose (HPMC) alone or mixed in different ratios with poly(ε-caprolactone) (CL) as the coat layer

A Multivariate measure of lactation persistency for dairy sheep

The persistency of lactation, i.e. the ability of animals to maintain a rconstant level of production after the lactation peak, represents an interesting trait for animal breeding strategies, allowing for the increase of profitability of animal husbandry via the reduction of production costs. Dairy cattle with flatter curves show a higher reproductive efficiency, a better metabolic status and have their nutritional requirements more constantly spread throughout lactation, allowing for the use of cheaper feeds (Dekkers et al., 1998; Solkner and Fucks, 1987). Also in dairy sheep the persistency could represent an interesting trait for breeding purposes. A main problem for the introduction of this trait in an aggregate genotype is represented by the difficulty in finding an objective measure: several measurements of lactation persistency have been proposed but none of them is widely accepted (Gengler, 1996). Recently a new index of the persistency based on multivariate Factor analysis, has been proposed for dairy cattle (Macciotta et al., 2002). Aim of the present work is to check the suitability of this index to discriminate lactation curves with different persistency and to analyse the effect of some environmental factor on this trait