Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures

This arXived paper has two independant parts, that are improved and corrected versions of different parts of a single paper once named "On equations in relatively hyperbolic groups". The first part is entitled "Existential questions in (relatively) hyperbolic groups". We study there the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. In the second part, called "Finding relative hyperbolic structures", we provide a general algorithm that recognizes the class of groups that are hyperbolic relative to abelian subgroups.Comment: Two independant parts 23p + 9p, revised. To appear separately in Israel J. Math, and Bull. London Math. Soc. respectivel

Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups, $Out(F_n)$, and the Cremona group. Other examples can be found among groups acting geometrically on $CAT(0)$ spaces, fundamental groups of graphs of groups, etc. We obtain a number of general results about rotating families and hyperbolically embedded subgroups; although our technique applies to a wide class of groups, it is capable of producing new results even for well-studied particular classes. For instance, we solve two open problems about mapping class groups, and obtain some results which are new even for relatively hyperbolic groups.Comment: Revision, corrections and improvement of the expositio

Accidental parabolics and relatively hyperbolic groups

By constructing, in the relative case, objects analoguous to Rips and Sela's canonical representatives, we prove that the set of images by morphisms without accidental parabolic, of a finitely presented group in a relatively hyperbolic group, is finite, up to conjugacy.Comment: Revision, 24 pages, 4 figure

Some integral inequalities on time scales

In this paper, some new integral inequalities on time scales are presented by using elementarily analytic methods in calculus of time scales.Comment: 8 page

Non Insulin Dependent Diabetes in Sand Rat (Psammomys obesus) and Production of Collagen in Cultured Aortic Smooth Muscle Cells. Influence of Insulin

In this report, we have shown that the standard laboratory diet administered to Psammomys obesus (sand rat) from Beni Abbes in Algeria, induced a non-insulin dependant diabetes, characterised by increase of body weight (p<0.001) as well as hyperinsulinemia, hyperglycemia and hypercholesterolemia. In cultured aortic smooth muscle cells (SMC) of sand rats, type I and type III collagen biosynthesis and insulin effects, at low dose, on these parameters were investigated. In all experimental conditions of cultured SMC study, The α chains of type I collagen were analysed by immunoblotting in media and cells

6-Bromo-1-(1,2-propadien­yl)-3-(2-propyn­yl)-1H-imidazo[4,5-b]pyridin-2(3H)-one

The reaction of propargyl bromide and 6-bromo-1,3-dihydro­imidazo[4,5-b]pyridin-2-one in refluxing dimethyl­formamide yields the title compound, C12H8BrN3O, which features nitro­gen-bound propadienyl and propynyl substituents. The imidazolopyridine fused ring is planar (r.m.s. deviation = 0.012 Å); the propadienyl chain is coplanar with the fused ring as it is conjugated with it, whereas the propynyl chain is not as the nitro­gen-bound C atom is a methyl­ene linkage. The acetyl­enic H atom is hydrogen bonded to the carbonyl O atom of an adjacent mol­ecule, forming a helical chain runnning along the b axis

1,3-Dibenzyl-6-bromo-1H-imidazo[4,5-b]pyridin-2(3H)-one

The imidazopyridine fused-ring in the title compound, C20H16BrN3O, is planar (r.m.s. deviation = 0.011 Å). The phenyl rings of the benzyl substitutents twist away from the central five-membered ring in opposite directions; the rings are aligned at 61.3 (1) and 71.2 (1)° with respect to this ring

Observation of Collective-Emission-Induced Cooling inside an Optical Cavity

We report the observation of collective-emission-induced, velocity-dependent light forces. One third of a falling sample containing 3 x 10^6 cesium atoms illuminated by a horizontal standing wave is stopped by cooperatively emitting light into a vertically oriented confocal resonator. We observe decelerations up to 1500 m/s^2 and cooling to temperatures as low as 7 uK, well below the free space Doppler limit. The measured forces substantially exceed those predicted for a single two-level atom.Comment: 10 pages, 5 figure