When an integrable Hamiltonian system, possessing an m-resonant lower dimensional normally parabolic torus is perturbed, a parabolic m-resonance occurs. If, in addition, the iso-energetic non-degeneracy condition for the integrable system fails, the near integrable Hamiltonian exhibits a at parabolic m-resonance. It is established that most kinds of parabolic resonances are persistent in n (n 3...

This is an expanded version of a series of lectures delivered at the 25th Winter School “Geometry and Physics” in Srni. After a short introduction to Cartan geometries and parabolic geometries, we give a detailed description of the equivalence between parabolic geometries and underlying geometric structures. The second part of the paper is devoted to constructions which relate parabolic geometr...

We define the notion of preGarside group slightly lightening the definition of Garside group so that all Artin-Tits groups are preGarside groups. This paper intends to give a first basic study on these groups. Firstly, we introduce the notion of parabolic subgroup, we prove that any preGarside group has a (partial) complemented presentation, and we characterize the parbolic subgroups in terms o...

We introduce and study parallel space-time domain decomposition methods for solving deterministic and stochastic parabolic equations. Traditional parallel algorithms solve parabolic problems time step by time step. The parallelism is restricted to each time step, and the algorithms are purely sequential in time. In this paper, we develop some overlapping Schwarz methods whose subdomains cover b...

Parabolic concentrator mirrors are an important component of many solar energy systems, particularly solar mirror collectors. Precision parabolic mirrors are expensive to fabricate and to transport. Here, a new concept for designing and fabricating precision parabolic mirrors is presented. The mirror is formed from a thin flat very flexible metal sheet with a highly reflective surface. Attached...

Conditional and Lie symmetries of semi-linear 1D Schrödinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrödinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra (conf3)C. We consider non-hermitian re...

The twistor transform of a parabolic geometry has two steps: lift up to a geometry of higher dimension, and then descend to a geometry of lower dimension. The first step is a functor, but the second requires some compatibility conditions. Local necessary conditions were uncovered by Andreas Čap [12]. We uncover necessary and sufficient global conditions for complex analytic geometries: rational...

Let Cg be a smooth projective irreducible curve over C of genus g ≥ 1 and let {p1, p2, . . . , ps} be a set of distinct points on Cg. We fix a nonnegative integer l and denote by Mg(p, λ) the moduli space of parabolic semistable vector bundles of rank r with trivial determinant and fixed parabolic structure of type λ = (λ1, λ2, . . . , λs) at p = (p1, p2, . . . , ps), where each weight λi is in...