Recent results of the searches for Supersymmetry in final states with one or two leptons at CMS are presented. Many Supersymmetry scenarios, including the Constrained Minimal Supersymmetric extension of the Standard Model (CMSSM), predict a substantial amount of events containing leptons, while the largest fraction of Standard Model background events -which are QCD interactions -gets strongly reduced by requiring isolated leptons. The analyzed data was taken in 2011 and corresponds to an integrated luminosity of approximately L = 1 fb −1 . The center-of-mass energy of the pp collisions was √ s = 7 TeV.

We study the dynamics of the restricted planar three-body problem near mean motion resonances, i.e. a resonance involving the Keplerian periods of the two lighter bodies revolving around the most massive one. This problem is often used to model Sun-Jupiter-asteroid systems. For the primaries (Sun and Jupiter), we pick a realistic mass ratio µ = 10 −3 and a small eccentricity e0 > 0. The main result is a construction of a variety of non local diffusing orbits which show a drastic change of the osculating (instant) eccentricity of the asteroid, while the osculating semi major axis is kept almost constant. The proof relies on the careful analysis of the circular problem, which has a hyperbolic structure, but for which diffusion is prevented by KAM tori. In the proof we verify certain nondegeneracy conditions numerically.Based on the work of Treschev, it is natural to conjecture that the time of diffusion for this problem is ∼ − ln(µe 0 ) µ 3/2 e 0 . We expect our instability mechanism to apply to realistic values of e0 and we give heuristic arguments in its favor. If so, the applicability of Nekhoroshev theory to the three-body problem as well as the long time stability become questionable.It is well known that, in the Asteroid Belt, located between the orbits of Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly at mean motion resonances of low order. Our mechanism gives a possible explanation of their existence. To relate the existence of Kirkwood gaps with Arnol'd diffusion, we also state a conjecture on its existence for a typical ε-perturbation of the product of the pendulum and the rotator. Namely, we predict that a positive conditional measure of initial conditions concentrated in the main resonance exhibits Arnol'd diffusion on time scales − ln ε ε 2 .

Preliminary molecular data obtained from one mitochondrial (COI) and one nuclear marker (H3) from the available species of the genus Okenia have revealed a new cryptic species of this genus. The new species (Okenia harastii sp. nov.), from New South Wales (NSW) eastern Australia, has a light brown body with scattered dark brown and white spots of different sizes, and six pairs of translucent mantle processes. Okenia harastii sp. nov. is compared with other Okenia species from NSW and with other morphologically similar species.

We give a proof of existence of centre manifolds within large domains for systems with an integral of motion. The proof is based on a combination of topological tools, normal forms and rigorous-computer-assisted computations. We apply our method to obtain an explicit region in which we prove existence of a center manifold in the planar Restricted Three Body Problem.

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