Hyperbolic constant mean curvature one surfaces: Spinor representation and trinoids in hypergeometric functions

We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors. This is a modification of Bryant's representation. It is used to derive explicit formulas in hypergeometric functions for CMC-1 surfaces of genus 0 with three regular ends which are asymptotic to catenoid cousins (CMC-1 trinoids).Comment: 29 pages, 9 figures. v2: figures of cmc1-surfaces correcte

Cooling down Levy flights

Let L(t) be a Levy flights process with a stability index \alpha\in(0,2), and U be an external multi-well potential. A jump-diffusion Z satisfying a stochastic differential equation dZ(t)=-U'(Z(t-))dt+\sigma(t)dL(t) describes an evolution of a Levy particle of an `instant temperature' \sigma(t) in an external force field. The temperature is supposed to decrease polynomially fast, i.e. \sigma(t)\approx t^{-\theta} for some \theta>0. We discover two different cooling regimes. If \theta<1/\alpha (slow cooling), the jump diffusion Z(t) has a non-trivial limiting distribution as t\to \infty, which is concentrated at the potential's local minima. If \theta>1/\alpha (fast cooling) the Levy particle gets trapped in one of the potential wells

One-dimensional space-discrete transport subject to Levy perturbations

In this paper we study a one-dimensional space-discrete transport equation subject to additive Levy forcing. The explicit form of the solutions allows their analytic study. In particular we discuss the invariance of the covariance structure of the stationary distribution for Levy perturbations with finite second moment. The situation of more general Levy perturbations lacking the second moment is considered as well. We moreover show that some of the properties of the solutions are pertinent to a discrete system and are not reproduced by its continuous analogue

The problem of analytical calculation of barrier crossing characteristics for Levy flights

By using the backward fractional Fokker-Planck equation we investigate the barrier crossing event in the presence of Levy noise. After shortly review recent results obtained with different approaches on the time characteristics of the barrier crossing, we derive a general differential equation useful to calculate the nonlinear relaxation time. We obtain analytically the nonlinear relaxation time for free Levy flights and a closed expression in quadrature of the same characteristics for cubic potential.Comment: 12 pages, 2 figures, presented at 5th International Conference on Unsolved Problems on Noise, Lyon, France, 2008, to appear in J. Stat. Mech.: Theory and Experimen

Attraction and diffusion in nature-inspired optimization algorithms

Nature-inspired algorithms usually use some form of attraction and diffusion as a mechanism for exploitation and exploration. In this paper, we investigate the role of attraction and diffusion in algorithms and their ways in controlling the behaviour and performance of nature-inspired algorithms. We highlight different ways of the implementations of attraction in algorithms such as the firefly algorithm, charged system search, and the gravitational search algorithm. We also analyze diffusion mechanisms such as random walks for exploration in algorithms. It is clear that attraction can be an effective way for enhancing exploitation, while diffusion is a common way for exploration. Furthermore, we also discuss the role of parameter tuning and parameter control in modern metaheuristic algorithms, and then point out some key topics for further research

Optimum design of reinforced concrete retaining walls with the flower pollination algorithm

The flower pollination algorithm (FPA) is anefficient metaheuristicoptimizationalgorithm mimickingthe pollinationprocessof flowering species. In this study, FPA is applied, for first time, to the optimum design of reinforced concrete (RC) cantilever retaining walls. It is foundthat FPA offers important savings with respect to conventional design approachesand that it outperformsgenetic algorithm (GA)andthe particle swarm optimization (PSO) algorithm in this designproblem.Furthermore, parameter tuning reveals that the best FPA performance is achieved for switch probability values ranging between 0.4 and 0.7, a population size of 20 individualsand aLévy flightstep sizescale factor of 0.5. Finally, parametric optimum designs show that theoptimumcost of RC retaining walls increases rapidly with the wallheight and smoothly with the magnitude of surcharge loadin

Social Algorithms

This article concerns the review of a special class of swarm intelligence based algorithms for solving optimization problems and these algorithms can be referred to as social algorithms. Social algorithms use multiple agents and the social interactions to design rules for algorithms so as to mimic certain successful characteristics of the social/biological systems such as ants, bees, bats, birds and animals.Comment: Encyclopedia of Complexity and Systems Science, 201