Self-intersection local times and collision local times of bifractional Brownian motions

In this paper, we consider the local time and the self-intersection local time for a bifractional Brownian motion, and the collision local time for two independent bifractional Brownian motions. We mainly prove the existence and smoothness of the self-intersection local time and the collision local time, through the strong local nondeterminism of bifractional Brownian motion, L2 convergence and Chaos expansion.     

多参数双分数布朗运动相遇局部时的存在性和联合连续性

本文研究了两个相互独立的(N,d)双分数布朗运动BH1,K1和BH2,K2的相遇局部时的问题.利用Fourier分析,获得了相遇局部时的存在性和联合连续性的结果,推广了分数布朗运动相遇局部时的相关结果.     

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Let, be two independent,
-dimensional sub-fractional Brownian motions with respective indices.
Assume. Our principal results are the necessary and sufficient condition for the
existence and smoothness of the collision local time and the intersection local time of
and through chaos expansion and elementary inequalities.     

Collision Measure and Collision Local Time for (α, d, β) Superprocesses

Existence and uniqueness of the solutions for some nonlinear evolution equations with measure-valued boundary conditions is established. This gives the existence of the collision local time and the collision measure for two independent (₁, d, ₁) and (₂, d, ₂) superprocesses without using any moment conditions on the mass processes. We obtain expressions for the Laplace transforms of the collision local time and the collision measure.     

On collision local time of two independent fractional ornstein-uhlenbeck processes

In this article, we study the existence of collision local time of two independent d-dimensional fractional Ornstein-Uhlenbeck processes X_t~(H_1)and _t~(H_2),with different parameters H_i∈(0, 1), i = 1, 2. Under the canonical framework of white noise analysis,we characterize the collision local time as a Hida distribution and obtain its' chaos expansion.     

A construction of catalytic super-Brownian motion via collision local time

We give a direct construction of a random measure which is equal in law to the collision local time between a catalytic super-Brownian motion and its catalytic measure. Under a regularity assumption on the catalytic measure, we show that the catalytic super-Brownian motion can be constructed deterministically from this measure.     

Collision local times of two independent fractional Brownian motions

In this paper, the collision local times for two independent fractional Brownian motions are considered as generalized white noise functionals. Moreover, the collision local times exist in L ² under mild conditions and chaos expansions are also given.     

基于耦合纯无网格方法时间分数阶下孤立子波碰撞过程的数值模拟研究北大核心CSCD

为数值预测时间分数阶耦合非线性Schrödinger(TF-CNLS)方程描述的孤立子波非弹性碰撞过程,首次发展了一种耦合纯无网格有限点集法(coupled finite pointset method,CFPM).其构造过程为:1)对时间分数阶Caputo导数项采用一种高精度的差分格式;2)对空间导数采用基于Taylor展开和加权最小二乘法的有限粒子法(FPM)离散格式;3)对区域进行局部加密和采用稳定性好的双曲余弦核函数以提高数值精度.数值研究中,首先,运用CFPM对有解析解的一维TF-CNLS方程进行求解,分析了节点均匀分布或局部加密情况下的误差和收敛阶,表明给出的耦合无网格法具有近似二阶精度和易局部加密求解的灵活性;其次,运用CFPM对无解析解一维TF-CNLS方程描述的孤立子波非弹性碰撞过程进行了数值预测,其出现的波塌缩现象与整数阶下出现的多波现象截然不同;最后,与有限差分结果作对比,表明CFPM数值预测时间分数阶下孤立子波非弹性碰撞过程的复杂传播现象是可靠的.     

Doubly localized two-dimensional rogue waves generated by resonant collision in Maccari system

In this paper, the Maccari system is investigated, which is viewed as a two-dimensional extension of nonlinear Schrödinger equation. We derive doubly localized two-dimensional rogue waves on the dark solitons of the Maccari system with Kadomtsev–Petviashvili hierarchy reduction method. The two-dimensional rogue waves include line segment rogue waves and rogue-lump waves, which are localized in two-dimensional space and time. These rogue waves are generated by the resonant collision of rational solitary waves and dark solitons, the whole process of transforming elastic collision into resonant collision is analytically studied. Furthermore, we also discuss the local characteristics and asymptotic properties of these rogue waves. Simultaneously, the generating conditions of the line segment rogue wave and rogue-lump wave are also given, which provides the possibility to predict rogue wave. Finally, a new way to obtain the high-order rogue waves of the nonlinear Schrödinger equation are given by proper reduction from the semi-rational solutions of the Maccari system.     

Two Brownian particles with rank-based characteristics and skew-elastic collisions

We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coëfficients, and with a full range of patterns of behavior upon collision that range from totally frictionless interaction, to elastic collision, to perfect reflection of one particle on the other. These interactions are governed by the left- and right-local times at the origin for the distance between the two particles. We realize this diffusion in terms of appropriate, apparently novel systems of stochastic differential equations involving local times, which we show are well posed. Questions of pathwise uniqueness and strength are also discussed for these systems.     

Smoothness of local times and self-intersection local times of Gaussian random fields

This paper is concerned with the smoothness (in the sense of Meyer- Watanabe) of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.     

Collision computation of moving bodies

In this paper, an explicit mathematical representation of n-dimensional bodies moving in translation along general trajectories is derived. This representation is used to find out if two moving bodies are going to collide. An optimization problem is developed for finding the time and location of collision. We consider the special cases of linear and piece-wise linear trajectories. The collision in this case can be obtained by solving a linear program or a sequence of linear programs, respectively. The problem of finding the collision time and location of several moving bodies is cast as an integer programming problem. A comprehensive simulation study shows that this approach requires much lesser computation time when compared with the current approach of finding the collision between all pairs of bodies.     

An estimate of the time of a direct-axis collision of a body of finite mass and a semiinfinite cylindrical shell with fluid

The problem of a direct-axis collision of a body affinite mass and a semi-infinite cylindrical shell with fluid is considered, and is solved using an analytic-numerical method, which is based on an application of the Laplace-Carson transform with its subsequent numerical inverse. An expression is found for the interaction force for the body and the hydro-elastic system, which allows us to determine the time of collision. Numerical results of calculations of the interaction force and the time of collision are given.Translated from Dinamicheskie Sistemy, No. 7, pp. 29–32, 1988.     

液滴碰撞和聚合模型研究

建立了描述液滴碰撞和聚合过程的数学模型,设计了一种高效的计算液滴碰撞对搜索算法,并在已有研究成果的基础上对液滴碰撞的结果做了进一步的发展．借助平滑粒子流体动力学方法两种实现方式各自的优点,将液滴间的相互作用局限在其周围一定数目的液滴之间,并采用积分核函数定义了液滴间碰撞的概率,通过数值模拟探讨了模型的特性．结果表明,所建模型对所采用的计算网格没有明显的依赖性,具有较高的计算精度和计算效率,不但能很好地维持系统动量的守恒性,而且对液滴初始尺寸分布没有明显的依赖性．     

A scenario for torus T destruction via a global bifurcation

We show a scenario of a two-frequency torus breakdown, in which a global bifurcation occurs due to the collision of a quasi-periodic torus T² with saddle points, creating a heteroclinic saddle connection. We analyze the geometry of this torus-saddle collision by showing the local dynamics and the invariant manifolds (global dynamics) of the saddle points. Moreover, we present detailed evidences of a heteroclinic saddle-focus orbit responsible for the type-II intermittency induced by this global bifurcation. We also characterize this transition to chaos by measuring the Lyapunov exponents and the scaling laws.     

Stability of rarefaction waves for the non-cutoff Vlasov–Poisson–Boltzmann system with physical boundary

In this paper, we are concerned with the Vlasov–Poisson–Boltzmann (VPB) system in three-dimensional spatial space without angular cutoff in a rectangular duct with or without physical boundary conditions. Near a local Maxwellian with macroscopic quantities given by rarefaction wave solution of one-dimensional compressible Euler equations, we establish the time-asymptotic stability of planar rarefaction wave solutions for the Cauchy problem to VPB system with periodic or specular-reflection boundary condition. In particular, we successfully introduce physical boundaries, namely, specular-reflection boundary, to the models describing wave patterns of kinetic equations. Moreover, we treat the non-cutoff collision kernel instead of the cutoff one. As a simplified model, we also consider the stability and large time behavior of the rarefaction wave solution for the Boltzmann equation.     

Existence and conservation laws for the Boltzmann–Fermi–Dirac equation in a general domain

We prove an existence theorem for the Boltzmann–Fermi–Dirac equation for integrable collision kernels in possibly bounded domains with specular reflection at the boundaries, using the characteristic lines of the free transport. We then obtain that the solution satisfies the local conservations of mass, momentum and kinetic energy thanks to a dispersion technique.     

Landau-Fermi-Dirac方程的整体经典解

研究了周期区域上平衡态附近Landau-Fermi-Dirac方程的Cauchy问题.利用宏观-微观分解以及局部的守恒律得到一致空间能量估计.接着结合对非线性碰撞算子的细致估计,推导了包含随时间演化的等价瞬时能量的非线性能量估计,进而得到一致的先验估计.最后通过局部存在性、一致的先验估计以及连续性技巧,得到了Landau-Fermi-Dirac方程平衡态附近整体光滑解的存在性.     

Collision Detection Using Interpolation Schemes

An algorithm using interpolation methods for the efficient search of the collision time and state of planar bodies is presented. Using interpolation and directed distances, the algorithm can efficiently obtain information about the collision. Further, a simulation system for multiple bodies is investigated and for some simple examples comparisons are shown of the proposed method and a traditional approach.     

On the modeling of pedestrian motion

A model for the simulation of pedestrian flows and crowd dynamics has been developed. The model is based on a series of forces, such as: will forces (the desire to reach a place at a certain time), pedestrian collision avoidance forces, obstacle/wall avoidance forces; pedestrian contact forces, and obstacle/wall contact forces. Except for the will force, it is assumed that for any given pedestrian these forces are the result of only local (nearest neighbour) situations. The near-neighbour search problem is solved by an efficient incremental Delaunay triangulation that is updated at every timestep. In order to allow for general geometries a so-called background triangulation is used to carry all geographic information. At any given time the location of any given pedestrian is updated on this mesh. The results obtained to date show that the model performs well for standard benchmarks, and allows for typical crowd dynamics, such as lane forming, overtaking, avoidance of obstacles and panic behaviour.