Fractional motions

Brownian motion is the archetypal model for random transport processes in science and engineering. Brownian motion displays neither wild fluctuations (the “Noah effect”), nor long-range correlations (the “Joseph effect”). The quintessential model for processes displaying the Noah effect is Lévy motion, the quintessential model for processes displaying the Joseph effect is fractional Brownian motion, and the prototypical model for processes displaying both the Noah and Joseph effects is fractional Lévy motion. In this paper we review these four random-motion models–henceforth termed “fractional motions” –via a unified physical setting that is based on Langevin’s equation, the Einstein–Smoluchowski paradigm, and stochastic scaling limits. The unified setting explains the universal macroscopic emergence of fractional motions, and predicts–according to microscopic-level details–which of the four fractional motions will emerge on the macroscopic level. The statistical properties of fractional motions are classified and parametrized by two exponents—a “Noah exponent” governing their fluctuations, and a “Joseph exponent” governing their dispersions and correlations. This self-contained review provides a concise and cohesive introduction to fractional motions.     

The Weierstrass–Mandelbrot Process Revisited

We derive a functional central limit theorem for quasi-Gaussian processes. In particular, we prove that the limit of the Mandelbrot–Weierstrass process is a complex fractional Brownian motion.     

Frequency dependence of the photonic noise spectrum in an absorbing or amplifying diffusive medium

A theory is presented for the frequency dependence of the power spectrum of photon current fluctuations originating from a disordered medium. Both the cases of an absorbing medium (“grey body”) and of an amplifying medium (“random laser”) are considered in a waveguide geometry. The semiclassical approach (based on a Boltzmann-Langevin equation) is shown to be in complete agreement with a fully quantum mechanical theory, provided that the effects of wave localization can be neglected. The width of the peak in the power spectrum around zero frequency is much smaller than the inverse coherence time, characteristic for black-body radiation. Simple expressions for the shape of this peak are obtained, in the absorbing case, for waveguide lengths large compared to the absorption length, and, in the amplifying case, close to the laser threshold. Received 8 August 2000     

Nonholonomic Mapping Principle for Classical and Quantum Mechanics in Spaces with Curvature and Torsion

I explain the geometric basis for the recently-discovered nonholonomic mapping principle which permits deriving laws of nature in spacetimes with curvature and torsion from those in flat spacetime, thus replacing and extending Einstein's equivalence principle. As an important consequence, it yields a new action principle for determining the equation of motion of a free spinless point particle in such spacetimes. Surprisingly, this equation contains a torsion force, although the action involves only the metric. This force makes trajectories autoparallel rather than geodesic, as a manifestation of inertia. A generalization of the mapping principle transforms path integrals from flat spacetimes to those with curvature and torsion, thus playing the role of a quantum equivalence principle. This generalization yields consistent results only for completely antisymmetric or for gradient torsion.     

三维相位展开算法在精密运动测试中的应用

三维相位展开算法将基于空间的相位展开算法拓展到时间轴上,包含了一维时间和二维空间信息,适合于运动测试。通过建立一维时间相位展开算法的数学模型,分析了该算法运动测试的基本原理,讨论了在精密测试中的使用方法和适用范围,并对微谐振器的微结构进行运动测试。实验表明,算法在相位展开中引入时间信息,不仅消除了形变测量中需选取静止点为相位参考点或参考平面的限制,还可以同时测量物体表面某点纳米级的离面运动曲线和表面形变。     

滚摆运动的研究

从实验和理论上研究了滚摆的运动,得到了滚摆滚动最大高度和滚动时间随滚动次数的变化关系。揭示了空气阻力及摆线间摩擦力等因素对滚摆运动的影响及其引起的系统机械能损失。     

多帧融合处理技术在运动模糊图像复原中的应用

运动模糊图像是电视监控中常见的退化图像 ,也是图像处理和检验工作中的难题。针对电视摄像的特点 ,多帧融合处理技术从图像时间序列中提取景物的运动信息 ,构造复原滤波器 ,通过频率域反转滤波方式复原模糊图像。这项技术能够有效地增强电视监控中存在的运动模糊图像 ,并且适用于数字视频图像处理