Derivative Formula and Applications for Degenerate Diffusion Semigroups

By using the Malliavin calculus and solving a control problem, Bismut type derivative formulae are established for a class of degenerate diffusion semigroups with non-linear drifts. As applications, explicit gradient estimates and Harnack inequalities are derived.Comment: 18 page

Detection of gamma-ray emission from the Coma cluster with Fermi Large Area Telescope and tentative evidence for an extended spatial structure

Many galaxy clusters have giant halos of non-thermal radio emission, indicating the presence of relativistic electrons in the clusters. Relativistic protons may also be accelerated by merger and/or accretion shocks in galaxy clusters. These cosmic-ray (CR) electrons and/or protons are expected to produce gamma-rays through inverse-Compton scatterings or inelastic $pp$ collisions respectively. Despite of intense efforts in searching for high-energy gamma-ray emission from galaxy clusters, conclusive evidence is still missing so far. Here we report the discovery of $\ge 200$ MeV gamma-ray emission from the Coma cluster direction with an unbinned likelihood analysis of the 9 years of {\it Fermi}-LAT Pass 8 data. The gamma-ray emission shows a spatial morphology roughly coincident with the giant radio halo, with an apparent excess at the southwest of the cluster. Using the test statistic analysis, we further find tentative evidence that the gamma-ray emission at the Coma center is spatially extended. The extended component has an integral energy flux of $\sim 2\times 10^{-12}{\rm \ erg\ cm^{-2}\ s^{-1}}$ in the energy range of 0.2 - 300 GeV and the spectrum is soft with a photon index of $\simeq-2.7$. Interpreting the gamma-ray emission as arising from CR proton interaction, we find that the volume-averaged value of the CR to thermal pressure ratio in the Coma cluster is about $\sim 2\%$. Our results show that galaxy clusters are likely a new type of GeV gamma-ray sources, and they are probably also giant reservoirs of CR protons.Comment: 10 pages, 10 figures, Accepted by Physical Review D, more spatial models for the gamma-ray emission are used, systematic checks on the results are adde

Hermite–Hadamard-Type Integral Inequalities for Functions Whose First Derivatives are Convex

We establish some new Hermite–Hadamard-type inequalities for functions whose first derivatives are of convexity and apply these inequalities to construct inequalities for special means.Встановлено дєякі нові нєрівності типу Ерміта - Адамара для Функцій, похідні яких мають опуклість. Ці нєрівності застосовано при побудові нерівностей для спеціальних середніх