Advanced trajectory engineering of diffraction-resisting laser beams

We introduce an analytical technique for engineering the trajectory of diffraction-resisting laser beams. The generated beams have a Bessel-like transverse field distribution and can be navigated along rather arbitrary curved paths in free space, thus being an advanced hybrid between accelerating and non-accelerating diffraction-free optical waves. The method involves phase-modulating the wavefront of a Gaussian laser beam to create a continuum of conical ray bundles whose apexes define a prespecified focal curve, along which a nearly perfect circular intensity lobe propagates without diffracting. Through extensive numerical simulations, we demonstrate the great flexibility in the design of a gamut of different beam trajectories. Propagation around obstructions and self-healing scenarios are also investigated. The proposed wave entities can be used extensively for light trajectory control in applications such as laser microfabrication, optical tweezers and curved plasma filamentation spectroscopy

Limiting distributions for additive functionals on Catalan trees

Additive tree functionals represent the cost of many divide-and-conquer algorithms. We derive the limiting distribution of the additive functionals induced by toll functions of the form (a) n^\alpha when \alpha > 0 and (b) log n (the so-called shape functional) on uniformly distributed binary search trees, sometimes called Catalan trees. The Gaussian law obtained in the latter case complements the central limit theorem for the shape functional under the random permutation model. Our results give rise to an apparently new family of distributions containing the Airy distribution (\alpha = 1) and the normal distribution [case (b), and case (a) as $\alpha \downarrow 0$]. The main theoretical tools employed are recent results relating asymptotics of the generating functions of sequences to those of their Hadamard product, and the method of moments.Comment: 30 pages, 4 figures. Version 2 adds background information on singularity analysis and streamlines the presentatio

Proceedings for the ICASE Workshop on Heterogeneous Boundary Conditions

Domain Decomposition is a complex problem with many interesting aspects. The choice of decomposition can be made based on many different criteria, and the choice of interface of internal boundary conditions are numerous. The various regions under study may have different dynamical balances, indicating that different physical processes are dominating the flow in these regions. This conference was called in recognition of the need to more clearly define the nature of these complex problems. This proceedings is a collection of the presentations and the discussion groups

Software needs in special functions

AbstractCurrently available software for special functions exhibits gaps and defects in comparison to the needs of moderm high-performance scientific computing and also, surprisingly, in comparison to what could be constructed from current algorithms. In this paper we expose some of these deficiencies and identify the related need for user-oriented testing software