On three conjectures by K. E. Shuler

Some fifteen years ago, Shuler formulated three conjectures relating to the large-time asymptotic properties of a nearest-neighbor random walk on ² that is allowed to make horizontal steps everywhere but vertical steps only on a random fraction of the columns. We give a proof of his conjectures for the situation where the column distribution is stationary and satisfies a certain mixing codition. We also prove a strong form of scaling to anisotropic Brownian motion as well as a local limit theorem. The main ingredient of the proofs is a large-deviation estimate for the number of visits to a random set made by a simple random walk on . We briefly discuss extensions to higher dimension and to other types of random walk.Dedicated to Prof. K. E. Shuler on the occasion of his 70th birthday, celebrated at a Symposium in his honor on July 13, 1992, at the University of California at San Diego, La Jolla, California.     

The asymptotic behavior of a random walk on a dual-medium lattice

This paper considers the asymptotic distribution for the horizontal displacement of a random walk in a medium represented by a two-dimensional lattice, whose transitions are to nearest-neighbor sites, are symmetric in the horizontal and vertical directions, and depend on the column currently occupied. On either side of a change-point in the medium, the transition probabilities are assumed to obey an asymptotic density condition. The displacement, when suitably normalized, converges to a diffusion process of oscillating Brownian motion type. Various special cases are discussed.     

Random walks on periodic and random lattices. II. Random walk properties via generating function techniques

We investigate the random walk properties of a class of two-dimensional lattices with two different types of columns and discuss the dependence of the properties on the densities and detailed arrangements of the columns. We show that the row and column components of the mean square displacement are asymptotically independent of the details of the arrangement of columns. We reach the same conclusion for some other random walk properties (return to the origin and number of distinct sites visited) for various periodic arrangements of a given relative density of the two types of columns. We also derive exact asymptotic results for the occupation probabilities of the two types of distinct sites on our lattices which validate the basic conjecture on bond and step ratios made in the preceding paper in this series.Supported in part by a grant from Charles and Renée Taubman and by the National Science Foundation, Grant CHE 78-21460.     

Self-motile colloidal particles: from directed propulsion to random walk

The motion of an artificial microscale swimmer that uses a chemical reaction catalyzed on its own surface to achieve autonomous propulsion is fully characterized experimentally. It is shown that at short times it has a substantial component of directed motion, with a velocity that depends on the concentration of fuel molecules. At longer times, the motion reverts to a random walk with a substantially enhanced diffusion coefficient. Our results suggest strategies for designing artificial chemotactic systems.     

Nonmonotonic effects of perpendicular magnetic anisotropy on current-driven vortex wall motions in magnetic nanostripes

In a magnetic nanostripe, the effects of perpendicular magnetic anisotropy(PMA) on the current-driven horizontal motion of vortex wall along the stripe and the vertical motion of the vortex core are studied by micromagnetic simulations.The results show that the horizontal and vertical motion can generally be monotonously enhanced by PMA. However, when the current is small, a nonmonotonic phenomenon for the horizontal motion is found. Namely, the velocity of the horizontal motion firstly decreases and then increases with the increase of the PMA. We find that the reason for this is that the PMA can firstly increase and then decrease the confining force induced by the confining potential energy. In addition, the PMA always enhances the driving force induced by the current.     

Laplace operator and random walk on one-dimensional nonhomogeneous lattice

A classical result of probability theory states that under suitable space and time renormalization, a random walk converges to Brownian motion. We prove an analogous result in the case of nonhomogeneous random walk on onedimensional lattice. Under suitable conditions on the nonhomogeneous medium, we prove convergence to Brownian motion and explicitly compute the diffusion coefficient. The proofs are based on the study of the spectrum of random matrices of increasing dimension.     

Random walks on a lattice

This paper discusses the mean-square displacement for a random walk on a two-dimensional lattice, whose transitions to nearest-neighbor sites are symmetric in the horizontal and vertical directions and depend on the column currently occupied. Under a uniform density condition for the step probabilities it is shown that the horizontal mean-square displacement aftern steps is asymptotically proportional ton, and independent of the particular column configuration. The model generalizes that of Seshadri, Lindenberg, and Shuler and the arguments are essentially probabilistic.     

Neon Color Spreading Promoted by Motion Information

We conducted experiments to investigate the effects of head-motion and object-motion to produce a perception of neon color spreading. Our experimental results indicated that a two-dimensional motion of yellow patches promoted the color spreading. A horizontal head-motion with motion parallax and three-dimensional pictorial depth information together with changing the viewing point stimulated the color spreading even, more than the two-dimensional motion. Thus, motion signals appear to offer useful information for the formation of a subjective contour and/or to activate the filling-in effect needed to generate neon color spreading. We also tested whether there was a difference between the effects of vertical and horizontal motion parallaxes for neon color spreading and found that there was none. This suggests that vertical head motion provides information in depth and surface perception equivalent to horizontal head motion.     

Processive motor protein as an overdamped brownian stepper

The two headed motor protein kinesin appears to "walk" along the biopolymer microtubule in 8 nm steps. There is ample justification for a model where the motion of the detached head to the next docking site on the biopolymer is described as ratcheted diffusion. The forward reorientation of an attached head can be conceived of as a power stroke. A model that is based on these premises can accurately predict parameters of motor protein motion.     

Quantum Walks,Weyl Equation and the Lorentz Group

Quantum cellular automata and quantum walks provide a framework for the foundations of quantum field theory, since the equations of motion of free relativistic quantum fields can be derived as the small wave-vector limit of quantum automata and walks starting from very general principles. The intrinsic discreteness of this framework is reconciled with the continuous Lorentz symmetry by reformulating the notion of inertial reference frame in terms of the constants of motion of the quantum walk dynamics. In particular, among the symmetries of the quantum walk which recovers the Weyl equation—the so called Weyl walk—one finds a non linear realisation of the Poincaré group, which recovers the usual linear representation in the small wave-vector limit. In this paper we characterise the full symmetry group of the Weyl walk which is shown to be a non linear realization of a group which is the semidirect product of the Poincaré group and the group of dilations.     

Scaling Argument of Anisotropic Random Walk

In this paper, we analytically discuss the scaling properties of the average square end-to-end distance 〈R²〉for anisotropic random walk in D-dimensional space (D≥2), and the returning probability P_n( r₀) for the walker into a certain neighborhood of the origin. We will not only give the calculating formula for 〈R²〉and P_n(r₀), but also point out that if there is a symmetric axis for the distribution of the probability density of a single step displacement, we always obtain 〈R²_⊥n〉～n, where ⊥ refers to the projections of the displacement perpendicular to each symmetric axes of the walk; in D-dimensional space with D symmetric axes perpendicular to each other, we always have 〈R_n²〉～n and the random walk will be like a purely random motion; if the number of inter-perpendicular symmetric axis is smaller than the dimensions of the space, we must have 〈R_n²〉～n² for very large n and the walk will be like a ballistic motion. It is worth while to point out that unlike the isotropic random walk in one and two dimensions, which is certain to return into the neighborhood of the origin, generally there is only a nonzero probability for the anisotropic random walker in two dimensions to return to the neighborhood.     

Spatial instabilities in reaction random walks with direction-independent kinetics

We study spatial instabilities in reacting and diffusing systems, where diffusion is modeled by a persistent random walk instead of the usual Brownian motion. Perturbations in these reaction walk systems propagate with finite speed, whereas in reaction-diffusion systems localized disturbances affect every part instantly, albeit with heavy damping. We present evolution equations for reaction random walks whose kinetics do not depend on the particles' direction of motion. The homogeneous steady state of such systems can undergo two types of transport-driven instabilities. One type of bifurcation gives rise to stationary spatial patterns and corresponds to the Turing instability in reaction-diffusion systems. The other type occurs in the ballistic regime and leads to oscillatory spatial patterns; it has no analog in reaction-diffusion systems. The conditions for these bifurcations are derived and applied to two model systems. We also analyze the stability properties of one-variable systems and find that small wavelength perturbations decay in an oscillatory manner.     

一种基于布朗粒子的混合搜索模型

基于网络上的布朗粒子运动基本原理,提出了一种单粒子和多粒子相结合的混合搜索模型.该模型将一次搜索过程分成单粒子搜索与多粒子搜索两个阶段,既克服了单粒子搜索效率低下的缺点,又降低了多粒子搜索的硬件代价.在各种复杂网络拓扑上实施该模型,并与混合导航模型进行比较.结果表明,混合搜索模型的平均搜索时间收敛更快,硬件代价更小.将度大优先的目标选择策略与混合搜索模型相结合,能进一步提高搜索效率.此外通过仿真发现,在无标度网络上混合搜索模型的效率远高于单粒子随机行走,与多粒子随机行走的效率相当,但硬件代价远小于多粒子行走.最后针对该模型给出了一种能有效降低负载的"吸收"策略.     

Mott Law as Lower Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic stationary simple point process on ℝ^d, d≥2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.     

偏心类柱体的运动规律演示与分析

设计了偏心柱体的微摆动及其在圆弧面上的运动规律演示实验,得出了几何大小相同的不同材料偏心柱体的微摆动周期相同时,在同一圆弧面上的运动规律相同.理论分析结果与相关实验测量数据一致.     

Multiple support excitations of open-plane frames by a filtered white noise and soil-structure interaction

Seismic structural design is essentially the estimation of structural response to a forced motion, which may be deterministic or stochastic, imposed on the ground. The assumption that the same ground motion acts at every point of the base of the structure (or at every support) is not always justifiable; particularly in case of very large structures when considerable spatial variability in ground motion can exist over significant distances—example long span bridges. This variability is partly due to the delay in arrival of the excitation at different supports (which is called the wave passage effect) and due to heterogeneity in the ground medium which results in incoherency and local effects. The current study examines the influence of the wave passage effect (in terms of delay in arrival of horizontal ground excitation at different supports and neglecting transmission through the structure) on the response of a few open-plane frame building structures with soil-structure interaction. The ground acceleration has been modeled by a suitably filtered white noise. As a special case, the ground excitation at different supports has also been treated as statistically independent to model the extreme case of incoherence due to local effects and due to modifications to the ground motion resulting from wave reflections and refractions in heterogeneous soil media. The results indicate that, even for relatively short spanned building frames, wave passage effect can be significant. In the absence of soil-structure interaction, it can significantly increase the root mean square (rms) value of the shear in extreme end columns for the stiffer frames but has negligible effect on the flexible frames when total displacements are considered. It is seen that pseudo-static displacements increasingly contribute to the rms value of column shear as the time delay increases both for the stiffer and for the more flexible frames. When soil-structure interaction is considered, wave passage effect (in terms of total displacements) is significant only for low soil shear modulus, G_s, values (where soil-structure interaction significantly lowers the fundamental frequency) and for stiff frames. The contribution of pseudo-static displacement to these rms values is found to decrease with increase in G_s. In general, wave passage effect for most interactive frames is insignificant compared to the attenuating effect a decrease in G_s has on the response of the interactive structure to uniform support excitation. When the excitations at different supports are statistically independent, it is seen that for both the stiff and flexible frames, the rms value of the column shear in extreme end columns is several times larger (more for the stiffer frames) than the value corresponding to uniform base excitation with the pseudo-static displacements contributing over 99% of the rms value of column shear. Soil-structure interaction has an attenuating effect on the rms value of the column shear, the effect decreasing with increase in G_s. Here too, the pseudo-static displacements contribute very largely to the column shear. The influence of the wave passage effect on the response of three 2-bay frames with and without soil-structure interaction to a recorded horizontal accelerogram is also examined.     

A generalized Pearson random walk allowing for bias

We calculate asymptotic values of the first two moments of a planar walk in which the step lengths depend on the direction of motion. The model is suggested by experiments on the locomotion of biological cells. Internally induced persistence due to nonuniform turn angle distributions is also accounted for.     

Asymptotic results for a persistent diffusion model of Taylor dispersion of particles

We study Taylor diffusion for the case when the diffusion transverse to the bulk motion is a persistent random walk on a one-dimensional lattice. This is mapped onto a Markovian walk where each lattice site has two internal states. For such a model we find the effective diffusion coefficient which depends on the rate of transition among internal states of the lattice. The Markovian limit is recovered in the limit of infinite rate of transitions among internal states; the initial conditions have no role in the leading-order time-dependent term of the effective dispersion, but a strong effect on the constant term. We derive a continuum limit of the problem presented and study the asymptotic behavior of such limit.     

运动背景下的帧间稳像技术

针对运动背景下帧间稳像技术,提出了一种带运动矢量修正的灰度投影运动估计算法。采用该算法分别对当前帧和参考帧的行、列计算灰度投影序列;将当前帧投影序列局部分块,分别将每一分块与参考帧行、列投影曲线进行互相关计算,得到基于局部投影的行、列运动矢量集合;以分块区域的相关置信度为权系数衡量参数,计算每一分块的像素位移权值,从而计算某一方向帧间的加权运动矢量。实验结果表明：该方法可以使运动目标造成的影响只作用于其中若干个局部分块,而其他分块不受此影响,尽可能保证稳像的准确性。采用该方法稳像后的图像与参考帧图像的均方根误差（RMSE）值明显下降,与参考帧图像更加吻合。     

Diffusion of magnetotactic bacterium in rotating magnetic field

Swimming trajectory of a magnetotactic bacterium in a rotating magnetic field is a circle. Random reversals of the direction of the bacterium motion induces a random walk of the curvature center of the trajectory. In assumption of the distribution of the switching events according to the Poisson process the diffusion coefficient is calculated in dependence on the frequency of the rotating field and the characteristic time between the switching events. It is confirmed by the numerical simulation of the random walk of the bacterium in the rotating magnetic field.