Backward Clusters, Hierarchy and Wild Sums for a Hard Sphere System in a Low-Density Regime

We study the statistics of backward clusters in a gas of hard spheres at low density. A backward cluster is defined as the group of particles involved directly or indirectly in the backwards-in-time dynamics of a given tagged sphere. We derive upper and lower bounds on the average size of clusters by using the theory of the homogeneous Boltzmann equation combined with suitable hierarchical expansions. These representations are known in the easier context of Maxwellian molecules (Wild sums). We test our results with a numerical experiment based on molecular dynamics simulations

The nonlinear diffusion limit for generalized Carleman models: the initial-boundary value problem

Consider the initial-boundary value problem for the 2-speed Carleman model of the Boltzmann equation of the kinetic theory of gases set in some bounded interval with boundary conditions prescribing the density of particles entering the interval. Under the usual parabolic scaling, a nonlinear diffusion limit is established for this problem. In fact, the techniques presented here allow treating generalizations of the Carleman system where the collision frequency is proportional to some power of the macroscopic density, with exponent in [-1,1]

NEMICO: Mining network data through cloud-based data mining techniques

Thanks to the rapid advances in Internet-based applications, data acquisition and storage technologies, petabyte-sized network data collections are becoming more and more common, thus prompting the need for scalable data analysis solutions. By leveraging today’s ubiquitous many-core computer architectures and the increasingly popular cloud computing paradigm, the applicability of data mining algorithms to these large volumes of network data can be scaled up to gain interesting insights. This paper proposes NEMICO, a comprehensive Big Data mining system targeted to network traffic flow analyses (e.g., traffic flow characterization, anomaly detection, multiplelevel pattern mining). NEMICO comprises new approaches that contribute to a paradigm-shift in distributed data mining by addressing most challenging issues related to Big Data, such as data sparsity, horizontal scaling, and parallel computation

Semiclassical Propagation of Coherent States for the Hartree equation

In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in $L^2$ by C \sqrt {\var}, \var being the Planck constant. Finally we present a full formal asymptotic expansion

Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solutions and tail behavior

We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of so-called constant normal restitution coefficients. We prove the existence of self-similar solutions, and we give pointwise estimates on their tail. We also give general estimates on the tail and the regularity of generic solutions. In particular we prove Haff 's law on the rate of decay of temperature, as well as the algebraic decay of singularities. The proofs are based on the regularity study of a rescaled problem, with the help of the regularity properties of the gain part of the Boltzmann collision integral, well-known in the elastic case, and which are extended here in the context of granular gases.Comment: 41 page

The dissipative linear Boltzmann equation for hard spheres

We prove the existence and uniqueness of an equilibrium state with unit mass to the dissipative linear Boltzmann equation with hard--spheres collision kernel describing inelastic interactions of a gas particles with a fixed background. The equilibrium state is a universal Maxwellian distribution function with the same velocity as field particles and with a non--zero temperature lower than the background one, which depends on the details of the binary collision. Thanks to the H--theorem we then prove strong convergence of the solution to the Boltzmann equation towards the equilibrium.Comment: 17 pages, submitted to Journal of Statistical Physic

Diathermy of leaking sclerotomies after 23-gauge transconjunctival pars plana vitrectomy: a prospective study.

PURPOSE: To evaluate the efficacy of bipolar diathermy in ensuring closure of leaking sclerotomies after complete 23-gauge transconjunctival sutureless vitrectomy. METHODS: In this prospective, interventional case series, in 136 eyes of 136 patients with at least one leaking sclerotomy at the end of a complete 23-gauge transconjunctival sutureless vitrectomy, external bipolar wet-field diathermy was applied to leaking sclerotomies, including the conjunctiva and sclera. Intraoperative wound closure, and postoperatively, at 6 hours, 1 day and 3 days, sclerotomies leakage, intraocular pressure, hypotony, and hypotony-related complications were evaluated. RESULTS: Intraoperative closure was achieved in 231 of 238 leaking sclerotomies (97%) that received diathermy. One of these with postoperative leakage needed suture. Compared with baseline (14.4 ± 2.8 mmHg), mean intraocular pressure was lower at 6 hours (13.2 ± 3.8 mmHg, Tukey-Kramer P < 0.001) and not different at 24 hours or 72 hours. Hypotony (intraocular pressure <5 mmHg) was observed in 6 eyes (4.5%) at 6 hours, in 2 (1.5%) at 24 hours, and in none at 3 days. Logistic regression analysis showed that, 6 hours postoperatively, hypotony was related to younger age (≤50 years) at surgery (P = 0.031). No hypotony-related complications were recorded. CONCLUSION: Bipolar wet-field diathermy of sutureless sclerotomies is an effective method for ensuring a leaking sclerotomies closure

Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model

This paper deals with a one--dimensional model for granular materials, which boils down to an inelastic version of the Kac kinetic equation, with inelasticity parameter $p>0$. In particular, the paper provides bounds for certain distances -- such as specific weighted $\chi$--distances and the Kolmogorov distance -- between the solution of that equation and the limit. It is assumed that the even part of the initial datum (which determines the asymptotic properties of the solution) belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent \a=2/(1+p). With such initial data, it turns out that the limit exists and is just the aforementioned stable distribution. A necessary condition for the relaxation to equilibrium is also proved. Some bounds are obtained without introducing any extra--condition. Sharper bounds, of an exponential type, are exhibited in the presence of additional assumptions concerning either the behaviour, near to the origin, of the initial characteristic function, or the behaviour, at infinity, of the initial probability distribution function

Multispecies virial expansions

We study the virial expansion of mixtures of countably many different types of particles. The main tool is the Lagrange–Good inversion formula, which has other applications such as counting coloured trees or studying probability generating functions in multi-type branching processes. We prove that the virial expansion converges absolutely in a domain of small densities. In addition, we establish that the virial coefficients can be expressed in terms of two-connected graphs