Lévy statistics of light scattering in turbulent mixing layers

Experimental evidence and computational evidence suggest that the distribution of random segment lengths defined by the intersections of ray paths with the contorted and folded interface between two fluids mixed by turbulence follows a probability distribution with a Lévy law tail. Assuming that the two fluids have different optical properties, one finds that the statistics of light scattered by the mixing interface reflect the probability distribution for the random distances between the intersection points of straight ray paths with the interface. Examples of light-scattering calculations for limiting cases, including the planetary albedo problem and imaging through a transparent mixing layer, are discussed.     

Bridging the gulf between correlated random walks and Lévy walks: autocorrelation as a source of Lévy walk movement patterns

For many years, the dominant conceptual framework for describing non-oriented animal movement patterns has been the correlated random walk (CRW) model in which an individual''s trajectory through space is represented by a sequence of distinct, independent randomly oriented ‘moves’. It has long been recognized that the transformation of an animal''s continuous movement path into a broken line is necessarily arbitrary and that probability distributions of move lengths and turning angles are model artefacts. Continuous-time analogues of CRWs that overcome this inherent shortcoming have appeared in the literature and are gaining prominence. In these models, velocities evolve as a Markovian process and have exponential autocorrelation. Integration of the velocity process gives the position process. Here, through a simple scaling argument and through an exact analytical analysis, it is shown that autocorrelation inevitably leads to Lévy walk (LW) movement patterns on timescales less than the autocorrelation timescale. This is significant because over recent years there has been an accumulation of evidence from a variety of experimental and theoretical studies that many organisms have movement patterns that can be approximated by LWs, and there is now intense debate about the relative merits of CRWs and LWs as representations of non-orientated animal movement patterns.     

Assessing Lévy walks as models of animal foraging

The hypothesis that the optimal search strategy is a Lévy walk (LW) or Lévy flight, originally suggested in 1995, has generated an explosion of interest and controversy. Long-standing empirical evidence supporting the LW hypothesis has been overturned, while new models and data are constantly being published. Statistical methods have been criticized and new methods put forward. In parallel with the empirical studies, theoretical search models have been developed. Some theories have been disproved while others remain. Here, we gather together the current state of the art on the role of LWs in optimal foraging theory. We examine the body of theory underpinning the subject. Then we present new results showing that deviations from the idealized one-dimensional search model greatly reduce or remove the advantage of LWs. The search strategy of an LW with exponent μ = 2 is therefore not as robust as is widely thought. We also review the available techniques, and their potential pitfalls, for analysing field data. It is becoming increasingly recognized that there is a wide range of mechanisms that can lead to the apparent observation of power-law patterns. The consequence of this is that the detection of such patterns in field data implies neither that the foragers in question are performing an LW, nor that they have evolved to do so. We conclude that LWs are neither a universal optimal search strategy, nor are they as widespread in nature as was once thought.     

Lyapunov exponents of stochastic differential equations driven by Lévy processes

Lyapunov exponents are studied for stochastic differential equations driven by Lévy processes. The transformation from stochastic differential equations to random integral equations and a multiplicative ergodic theorem are used. Moreover, the result is applied to physical and economic problems.     

布朗单的Lévy马氏性适用范围

两参数过程的Lévy马氏性是相对于区域D而言的。对某些D,有Lévy马氏性,对另外一些则无Lévy马氏性.布朗单Lévy马氏性成立的D的全体叫做Lévy马氏性的适用范围。当D是单连通集并且边界(?)D是逐段光滑曲线时,适用范围是所有有限个正位矩形之并的集合L,这一结论是已知的。但当D是单连通区域而边界(?)D在黎曼可积意义     

Lévy statistics and beta model: A new solution of “wall” turbulence with a critical phenomenon

By applying statistical mechanics Brown's motion leads to the macroscopic diffusion equation, and similar Lévy flights and the β model are the specific flux origin to derive the Difference-Quotient Turbulence Model (DQTM). It is applied to a turbulent flow in the overlap region close to a wall and leads to a mean velocity deficit power law, which may serve as a substitute for the “logarithmic law” with the von Karman constant. Its exponent is slightly dependent on the stress parameter (e.g. the Reynolds number) and is confirmed by the Princeton super pipe data. The solution of this fluid dynamic standard example reveals a physical order concept, just as known in the theory of critical phenomena.     

Lévy稳定分布对住宅楼面活荷载的统计分析

该文基于Lévy稳定分布,给出了设计住宅楼面活荷载标准值的一种新的统计方法。该方法分为4个步骤,首先采用Lévy稳定分布,拟合楼面活荷载的累积分布函数;然后运用Lévy随机数,模拟设计基准期内活荷载的极大值分布;其次通过设定活荷载极大值分布的分位数,计算持久性活荷载和临时性活荷载的最大值;最终以赋权重的方式,确定楼面活荷载的标准值。我们的分析结果表明,这个新方法不需要大量的调查数据,避免了Turkstra荷载组合原则基于经验确定标准值的缺点。与极值I型分布和威布尔分布相比,Lévy稳定分布的模拟精度最高,并且其参数能够直接刻画活荷载分布的非对称性和拖尾性。此外,采用Lévy随机数模拟法,数学简单,方便工程技术人员的使用。     

Scalable attribute-based group key establishment: from passive to active and deniable

A protocol compiler is presented which transforms any unauthenticated (attribute-based) group key establishment protocol into an authenticated attribute-based group key establishment. If the protocol to which the compiler is applied does not make use of long-term secrets, then the resulting protocol is, in addition, deniable. In particular, applying our compiler to an unauthenticated 2-round protocol going back to Burmester and Desmedt results in a 3-round solution for attribute-based group key establishment, offering both forward secrecy and deniability.     

Fitness-maximizing foragers can use information about patch quality to decide how to search for and within patches: optimal Lévy walk searching patterns from optimal foraging theory

Optimal foraging theory shows how fitness-maximizing foragers can use information about patch quality to decide how to search within patches. It is amply supported by empirical studies. Nonetheless, the theory largely ignores the fact that foragers may need to search for patches as well as for the targets within them. Here, using an exact but simple mathematical argument, it is shown how foragers can use information about patch quality to facilitate the execution of Lévy walk movement patterns with μ = 2 at inter-patch scales. These movement patterns are advantageous when searching for patches that are not depleted or rejected once visited but instead remain profitable. The analytical results are verified by the results of numerical simulations. The findings bring forth an innovative theoretical synthesis of searching for and within patches and, suggest that foragers'' memories may be adaptive under spatially heterogeneous reward schedules.     

Memory-Reduction Method for Pricing American-Style Options under Exponential Lévy Processes

This paper concerns the Monte Carlo method in pricing American-style options under the general class of exponential Lévy models. Traditionally, one must store all the intermediate asset prices so that they can be used for the backward pricing in the least squares algorithm. Therefore the storage requirement grows like $O(mn)$, where m is the number of time steps and n is the number of simulated paths. In this paper, we propose a simulation method where the storage requirement is only $O(m+n)$. The total computational cost is less than twice that of the traditional method. For machines with limited memory, one can now enlarge m and n to improve the accuracy in pricing the options. In numerical experiments, we illustrate the efficiency and accuracy of our method by pricing American options where the log-prices of the underlying assets follow typical Lévy processes such as Brownian motion, lognormal jump-diffusion process, and variance gamma process.     

Four pioneers: Jany Thibault-Pénisson–Pierre Delavignette–Claude Goux–Jacques Lévy

In memory of four pioneers in the field of grain boundaries (GBs) and Interfaces, who recently expired: Jany Thibault-Pénisson (? 2011), Pierre Delavignette (? 2011), Claude Goux (? 2012) and Jacques Lévy (? 2012), I recall some of their major achievements. Jany Thibault-Pénisson was a pioneer in the use of High Resolution Transmission Electron Microscopy. She studied dislocations in Germanium with Alain Bourret in Grenoble, then GBs in Ge and Si, their structures and their interactions with extrinsic dislocations under plastic deformation, and finally interfaces in metallic multilayers. Pierre Delavignette also was a marvellous electron microscopist who worked on dislocations in non-metallic multilayered structures with Severin Amelinckx in Mol (Belgium), and on GBs in hexagonal metals, for which he contributed to the extension of Ranganathan’s approach, together with George Bleris, Theodoros Karakostas, Gérard Nouet, Serge Hagège and others, in parallel with Hans Grimmer, David Warrington, Walter Bollmann, and Roland Bonnet. Claude Goux developed a school of processing, observation, and atomistic simulation of GBs in very pure metals, bringing to the École des Mines of Saint-Étienne what he had learnt in Vitry under Georges Chaudron in the late 1950s. He also had a strong taste for modelling, and his impulse culminated with the 1975 Conference on GBs in Metals held in Saint-Étienne. Jacques Lévy has been one of Goux’s first Ph.D. students. They probably made the first electron microscopic observations of high-angle GBs around 1966. Jacques Lévy rapidly exerted to assume important positions and became, for instance, the Director of the École des Mines of Paris. Despite all his responsibilities, he still attended IIB conferences many times.     

A cuckoo search algorithm by Lévy flights for solving reliability redundancy allocation problems

A new metaheuristic optimization algorithm, called cuckoo search (CS), was recently developed by Yang and Deb (2009 Yang, X.S. and Deb, S., 2009. Cuckoo search via Lévy flights. In: Proceedings of world congress on nature &; biologically inspired computing (NaBIC 2009), 9–11 December 2009, Coimbatore, India. Piscataway, NJ: IEEE Press, 210–214.[Crossref] , [Google Scholar], 2010 Yang, X.S. and Deb, S., 2010. Engineering optimisation by cuckoo search. International Journal of Mathematical Modelling and Numerical Optimisation, 1, 330–343. doi: 10.1504/IJMMNO.2010.035430[Crossref] , [Google Scholar]). This article uses CS and Lévy flights to solve the reliability redundancy allocation problem. The redundancy allocation problem involves setting reliability objectives for components or subsystems in order to meet the resource consumption constraint, e.g. the total cost. The difficulties facing the redundancy allocation problem are to maintain feasibility with respect to three nonlinear constraints, namely, cost, weight and volume-related constraints. The redundancy allocation problems have been studied in the literature for decades, usually using mathematical programming or metaheuristic optimization algorithms. The performance of the algorithm is tested on five well-known reliability redundancy allocation problems and is compared with several well-known methods. Simulation results demonstrate that the optimal solutions obtained by CS are better than the best solutions obtained by other methods.     

Health monitoring of aerospace composite structures – Active and passive approach

Impact damage is one of the major concerns in maintenance of aircraft structures built from composite materials. Damage detection in composite materials can be divided into active and passive approaches. The active approach is usually based on various non-destructive techniques utilizing actuators and/or receivers. In contrast passive approaches do not involve any actuators; receivers are used to “sense and/or hear” any perturbations caused by possible hidden damage. Often strain data are used to localize impacts and estimate their energy. The assumption is that damage occurs above well-defined energy of impacts. The paper illustrates one active and one passive method recently developed for impact damage detection. The first method, based on guided ultrasonic waves, utilises 3-D laser vibrometry and does not require any signal processing. Simple laser scans, revealing the change in Lamb wave response amplitudes, have been used to locate delamination and estimate its severity in a composite plate. In contrast, the second method does not require any sophisticated instrumentation but relies on advanced signal processing. An array of piezoceramic sensors has been to detect strain waves transmitted from an impact applied to the composite aircraft structure. The modified multilateration procedure with Genetic Algorithms has been used to locate impact position.     

A hybrid cuckoo search via Lévy flights for the permutation flow shop scheduling problem

The permutation flow shop problem (PFSSP) is an NP-hard problem of wide engineering and theoretical background. In this paper, a cuckoo search (CS)-based memetic algorithm, called HCS, is proposed for the PFSSP. To make CS suitable for the PFSSP, a largest-ranked-value (LRV)-rule-based random key is used to convert the continuous position in CS into a discrete job permutation. The Nawaz-Enscore-Ham (NEH) heuristic is then combined with the random initialisation to initialise the population with a certain quality and diversity. After that, CS is employed to evolve nest vectors for exploration, and a fast local search is embedded to enhance the local exploitation ability. In addition, simulations and comparisons based on PFSSP benchmarks are carried out, which shows that our algorithm is both effective and efficient.     

Effects of Lévy-type distribution on transverse localization in one-dimensional disordered waveguide arrays

In this paper, we analyse a one-dimensional disordered waveguide array in which the refractive indices of the waveguides are selected to be random numbers drawn from a Lévy-type distribution characterized by the exponent α. We study the effect of changing the exponent α on the transverse localization effect. It is shown that the quality of transverse localization is improved by decreasing the exponent α. Furthermore, we study this effect for several refractive index ranges. The design of disordered waveguide arrays in this way permits control of the degree of transverse localization.     

Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion

Animals moving under the influence of spatio-temporal scaling and long-term memory generate a kind of space-use pattern that has proved difficult to model within a coherent theoretical framework. An extended kind of statistical mechanics is needed, accounting for both the effects of spatial memory and scale-free space use, and put into a context of ecological conditions. Simulations illustrating the distinction between scale-specific and scale-free locomotion are presented. The results show how observational scale (time lag between relocations of an individual) may critically influence the interpretation of the underlying process. In this respect, a novel protocol is proposed as a method to distinguish between some main movement classes. For example, the ‘power law in disguise’ paradox—from a composite Brownian motion consisting of a superposition of independent movement processes at different scales—may be resolved by shifting the focus from pattern analysis at one particular temporal resolution towards a more process-oriented approach involving several scales of observation. A more explicit consideration of system complexity within a statistical mechanical framework, supplementing the more traditional mechanistic modelling approach, is advocated.