The Gauss-Green theorem in stratified groups

We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the $BV$ fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regularity on sets of finite perimeter. We show several properties of divergence-measure fields in stratified groups, ultimately achieving the related Gauss-Green theorem.Comment: 69 page

Optimizing momentum resolution with a new fitting method for silicon-strip detectors

A new fitting method is explored for momentum reconstruction of tracks in a constant magnetic field for a silicon-strip tracker. Substantial increases of momentum resolution respect to standard fit is obtained. The key point is the use of a realistic probability distribution for each hit (heteroscedasticity). Two different methods are used for the fits, the first method introduces an effective variance for each hit, the second method implements the maximum likelihood search. The tracker model is similar to the PAMELA tracker. Each side, of the two sided of the PAMELA detectors, is simulated as momentum reconstruction device. One of the two is similar to silicon micro-strip detectors of large use in running experiments. Two different position reconstructions are used for the standard fits, the $\eta$-algorithm (the best one) and the two-strip center of gravity. The gain obtained in momentum resolution is measured as the virtual magnetic field and the virtual signal-to-noise ratio required by the two standard fits to reach an overlap with the best of two new methods. For the best side, the virtual magnetic field must be increased 1.5 times respect to the real field to reach the overlap and 1.8 for the other. For the high noise side, the increases must be 1.8 and 2.0. The signal-to-noise ratio has similar increases but only for the $\eta$-algorithm. The signal-to-noise ratio has no effect on the fits with the center of gravity. Very important results are obtained if the number N of detecting layers is increased, our methods provide a momentum resolution growing linearly with N, much higher than standard fits that grow as the $\sqrt{N}$.Comment: This article supersedes arXiv:1606.03051, 22 pages and 10 figure

A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up

We introduce the new space $BV^{\alpha}(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $\alpha \in (0, 1)$ via a new distributional approach exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical $BV$ theory, we give a new notion of set $E$ of (locally) finite fractional Caccioppoli $\alpha$-perimeter and we define its fractional reduced boundary $\mathscr{F}^{\alpha} E$. We are able to show that $W^{\alpha,1}(\mathbb{R}^n)\subset BV^\alpha(\mathbb{R}^n)$ continuously and, similarly, that sets with (locally) finite standard fractional $\alpha$-perimeter have (locally) finite fractional Caccioppoli $\alpha$-perimeter, so that our theory provides a natural extension of the known fractional framework. Our main result partially extends De Giorgi's Blow-up Theorem to sets of locally finite fractional Caccioppoli $\alpha$-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets.Comment: 46 page

Global convergence of quorum-sensing networks

In many natural synchronization phenomena, communication between individual elements occurs not directly, but rather through the environment. One of these instances is bacterial quorum sensing, where bacteria release signaling molecules in the environment which in turn are sensed and used for population coordination. Extending this motivation to a general non- linear dynamical system context, this paper analyzes synchronization phenomena in networks where communication and coupling between nodes are mediated by shared dynamical quan- tities, typically provided by the nodes' environment. Our model includes the case when the dynamics of the shared variables themselves cannot be neglected or indeed play a central part. Applications to examples from systems biology illustrate the approach.Comment: Version 2: minor editions, added section on noise. Number of pages: 36

A comparative study on the reliability of open cluster parameters

Context. Open clusters are known as excellent tracers of the structure and chemical evolution of the Galactic disk, however, the accuracy and reliability of open cluster parameters is poorly known. Aims: In recent years, several studies aimed to present homogeneous open cluster parameter compilations, which are based on some different approaches and photometric data. These catalogues are excellent sources to facilitate testing of the actual accuracy of open cluster parameters. Methods: We compare seven cluster parameter compilations statistically and with an external sample, which comprises the mean results of individual studies. Furthermore, we selected the objects IC 4651, NGC 2158, NGC 2383, NGC 2489, NGC 2627, NGC 6603, and Trumpler 14, with the main aim to highlight differences in the fitting solutions. Results: We derived correction terms for each cluster parameter, using the external calibration sample. Most results by the compilations are reasonable scaled, but there are trends or constant offsets of different degree. We also identified one data set, which appears too erroneous to allow adjustments. After the correction, the mean intrinsic errors amount to about 0.2 dex for the age, 0.08 mag for the reddening, and 0.35 mag for the distance modulus. However, there is no study that characterises the cluster morphologies of all test cases in a correct and consistent manner. Furthermore, we found that the largest compilations probably include at least 20 percent of problematic objects, for which the parameters differ significantly. These could be among others doubtful or unlikely open clusters that do not facilitate an unambiguous fitting solution