Prime numbers in logarithmic intervals

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and h=\odi{X}. Then we will apply this to prove that for every $\lambda>1/2$ there exists a positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains at least a prime number. As a consequence we improve Cheer and Goldston's result on the size of real numbers $\lambda>1$ with the property that there is a positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains at least a prime number. The last application of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains no primes.Comment: 17 page

The exceptional set for the number of primes in short intervals

We give upper bounds for the number of x up to X such that the interval (x, x+h) does not contain the expected quantity of primes. Here h is small with respect to x

High quality factor 1-D Er 3+ -activated dielectric microcavity fabricated by RF-sputtering

Rare earth-activated 1-D photonic crystals were fabricated by RF-sputtering technique. The cavity is constituted by an Er3+-doped SiO2 active layer inserted between two Bragg reflectors consisting of ten pairs of SiO2/TiO2 layers. Scanning electron microscopy is employed to put in evidence the quality of the sample, the homogeneities of the layers thickness and the good adhesion among them. Near infrared transmittance and variable angle reflectance spectra confirm the presence of a stop band from 1500 nm to 2000 nm with a cavity resonance centered at 1749 nm at 0° and a quality factor of 890. The influence of the cavity on the 4I13/2 -> 4I15/2 emission band of Er3+ ion is also demonstrated

On the asymptotic formula for Goldbach numbers in short intervals

Let $R(k)=\sum\limits_{l+m=k}\Lambda(l)\Lambda(m)$, \Sing(k) = 2 \prod\limits_{p>2}\left(1-\frac{1}{(p-1)^2}\right) \prod\limits_{\substack{ p\mid k\\ p>2 }} \left(\frac{p-1}{p-2}\right) if $k$ is even and \Sing(k) =0 if $k$ is odd. It is known that R(k) \sim k\Sing(k) as $N\to \infty$ for almost all $k\in [N,2N]$ and that \sum_{k\in [n,n+H)}R(k) \sim \sum_{k\in [n,n+H)} k\Sing(k) \quad\hbox{for} \quad n\to \infty \eqno{(1)} uniformly for $H\geq n^{1/6+\epsilon}$. Here we prove, assuming $N^\epsilon\leq H\leq N^{1/6+\epsilon}$ and $N\to\infty$, that (1) holds for almost all $n\in [N,2N]$

The short term debt vs. long term debt puzzle: a model for the optimal mix

This paper argues that the existing finance literature is inadequate with respect to its coverage of capital structure of small and medium sized enterprises (SMEs). In particular it is argued that the cost of equity (being both conceptually ill defined and empirically non quantifiable) is not applicable to the capital structure decisions for a large proportion of SMEs and the optimal capital structure depends only on the mix of short and long term debt. The paper then presents a model, developed by practitioners for optimising the debt mix and demonstrates its practical application using an Italian firm's debt structure as a case study

Proliferative activity in human breast cancer: Ki-67 automated evaluation and the influence of different Ki-67 equivalent antibodies

<p>Abstract</p> <p>Background</p> <p>Ki67 labeling index (Ki67 LI), the percentage Ki67 immunoreactive cells, is a measure of tumor proliferation, with important clinical relevance in breast cancer, and it is extremely important to standardize its evaluation.</p> <p>Aim</p> <p>To test the efficacy of computer assisted image analysis (CAIA) applied to completely digitized slides and to assess its feasibility in routine practice and compare the results obtained using two different Ki67 monoclonal antibodies.</p> <p>Materials and methods</p> <p>315 consecutive breast cancer routinely immunostained for Ki-67 (223 with SP6 and 92 with MM1 antibodies previously examined by an experienced pathologist, have been re-evaluated using Aperio Scanscope Xs.</p> <p>Results</p> <p>Mean human Ki67 LI values were 36%± 14.% and 28% ± 18% respectively for SP6 and MM1 antibodies; mean CAM Ki67 LI values were 31%± 19% and 22% ± 18% respectively for SP6 and MM1. Human and CAIA evaluation are statistically highly correlated (Pearson: 0.859, p<0.0001), although human LI are systematically higher. An interobserver variation study on CAIA performed on 84 cases showed that the correlation between the two evaluations was linear to an excellent degree.</p> <p>Discussion</p> <p>Our study shows that a) CAIA can be easily adopted in routine practice, b) human and CAIA Ki67 LI are highly correlated, although human LI are systematically higher, c) Ki67 LI using different evaluation methods and different antibodies shows important differences in cut-off values.</p