三维自助式类渗流细胞自动机的临界值

本文研究三维自助式类渗流细胞自动机的临界值．得出临界值pc＝0的一些充分条件和必要条件．当pc＞0时，利用定向点渗流的临界值，给出了pc上、下界的显式估计．     

Planck尺度的Hopf代数的Poisson括符

本文讨论拟微分算子象证的Witt乘积与C[x]▲△A.G[p]的乘积之间的关系,得到以下结果：具有Witt乘积的象证类S1.0^m。是具有普通乘积的函数类S1.0^m经过对Hopf代数的乘积量子化得到的,进一步，给出了C[x]▲△h.cC[p]上标准的辫导数的显示表示，并且证明了其上的Poisson括号与经典的具有同一形式．     

二维双重定向渗流及其临界概率函数

本文研究二维双重定向渗流模型，我们给出模型临界概率函数的一些基本性质，包括严格单调性、对称性和连续性，另外，我们指出，该临界概率函数的严格凹性是Grimmett相关猜想的充分条件。     

Poisson过程分解问题的一个注记

在研究Poisson过程分解问题时,现有文献的证明往往令人费解,本文主要运用极限理论,给出了一个简明易懂的证明.     

两个轮换对称不等式猜想的证明

文[1]提出了四个不等式猜想,其中的猜想1和猜想2已分别在文[2]和[3]中解决．在本文中,笔者将给出猜想3和猜想4的证明．     

关于一个不等式猜想的新证明

文[1]提出了如下猜想:若x,y为满足x y=1的正数,n为不小于3的整数,则ynx2 y3 xnx3 y2≤2(xn 1x2 y3 yn 1x3 y2)≤2(yn 1x2 y3 xn 1x3 y2).文[2]给出了这个猜想的严谨、详尽的证明.笔者现给出这个猜想的一个新证明,证明所用的方法是证不等式最基本的方法——比较法,相对于文[2]而     

广义的张量积Poisson函数的升阶问题

1 引言 文[2]讨论了Poisson函数的若干性质,及以Poisson函数表示的曲线的一种细分格式。而文[1]则对Poisson函数,Bézier函数作了一般的推广,引进了广义的Poisson函数。受文[1],[2]的启发,本文将讨论张量积形式下的相关结论。我们将会看到广义的张量积Poisson函数将不再局限于张量积形式。     

高维 Hadamard 猜想的新证明

众所周知,2维 Hadamard 矩阵的阶数必须是1或2或4t(此处 t 是某个正整数).反过来,著名的 Hadamard 猜想则说:“对任意正整数 t,至少存在一个2维4t 阶的Hadamard 矩阵.”此猜想至今已有近百年的历史了,虽然许多数学家都曾经或正在为此猜想而绞尽脑汁,但是仍然没人能证明或否定它.1979年美国学者 P.J.Shlichta 将Hadamard 矩阵的理论从2维推广到高维情形,并提出了这样一个高维 Hadamard 猜想:“高维 Hadamard 矩阵的阶数不受4t 的限制,即有可能存在阶数为2s(?)4t(s 是奇数)的高维 Hadamard 矩阵.”最近杨义先已在[2]中证明了上述高维 Hadamard 猜想是正确的.在本文中我们将再给出一个更简单、更有力的新证明.最后我们还得出了如下     

首达渗流中关于lim{(N_(on))/n}(n→∞)的一个结果

在二维首达渗流中,设边上通过时间的分布为F（x）,首达时a_（on）的轨道（route）的最短长度为N_（on）,人们猜测存在.本文对F（0）<1/2的情形,就一类特殊的分布证明此猜想成立.     

首达渗流中关于lin n→∞ Non—n的一个结果

在二维首达渗流中,设边上通过时间的分布为F(x),首达时aon的轨道(route)的最短长度为Non,人们猜测lim n→∞ Non/n 存在.本文对F(0)＜1/2的情形,就一类特殊的分布证明此猜想成立.     

Continuum Percolation at and above the Uniqueness Threshold on Homogeneous Spaces

We consider the Poisson Boolean model of continuum percolation on a homogeneous space M. Let λ be the intensity of the underlying Poisson process. Let λ _u be the infimum of the set of intensities that a.s. produce a unique unbounded component. First we show that if λ>λ _u , then there is a.s. a unique unbounded component at λ. Then we let M=?²×? and show that at λ _u there is a.s. not a unique unbounded component. These results are continuum analogs of theorems by Häggström, Peres and Schonmann.     

Poisson Broken Lines Process and its Application to Bernoulli First Passage Percolation

In this note we introduce a process, which we call 'the Poisson broken lines process", and we compute the intensity of a point process which is obtained by intersecting the Poisson broken lines process with an abscissa axis. In the second part we apply this result to compute an explicit lower bound for the time constant of a planar Bernoulli first passage percolation model with the parameter p < p_c.     

On a multiscale continuous percolation model with unbounded defects

We study the multiscale (fractal) percolation in dimension greater than or equal to 2, where the model at each level is the Poisson Boolean model [[, ]]. Also, the random radius is supposed to be unbounded. We prove that if the rate of Poisson field is less than some critical value, then by choosing the scaling parameter large enough one can assure that there is no multiscale percolation. Another result of this paper is that if the expectation of ^2d is finite, then the expectation of the size of the cluster raised to the power is also finite for small , which is a generalization of one of the results of [8].1 Partially supported by FAPESP (97/12826–6, 02/02984–3) and CNPq (300676/00–0)2 Partially supported by FAPESP (00/11462–5, 02/03012–5)     

On the visibility in well-behaved random sets in Euclidean space

In Calka et al. (2009), the decay of the probability of reaching distance at least r in some direction from a given point without hitting any ball, in the Poisson Boolean model of continuum percolation, was studied. The methods used in Calka et al. (2009) include coverage techniques, and the most precise results were obtained in dimension 2. In this note, we strengthen some of the results obtained in Calka et al. (2009) to dimension 3 and higher and at the same time extend them to more general random sets.     

Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces

In this paper we prove that the moduli spaces of framed vector bundles over a surface X, satisfying certain conditions, admit a family of Poisson structures parametrized by the global sections of a certain line bundle on X. This generalizes to the case of framed vector bundles previous results obtained in [B2] for the moduli space of vector bundles over a Poisson surface. As a corollary of this result we prove that the moduli spaces of framed SU(r) – instantons on S⁴ = ℝ⁴ ∪ {∞} admit a natural holomorphic symplectic structure.     

Two-dimensional scaling limits via marked nonsimple loops

We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE₆ and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane filling curves, and invasion percolation. We showthat this yields for some of the continuum objects a conformal covariance property that generalizes the conformal invariance of critical systems. It is an open problem to rigorously construct the continuum objects and to prove that they are indeed the scaling limits of the corresponding lattice objects.     

The critical probability for Voronoi percolation in the hyperbolic plane tends to 1/2

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster tends to 1/2 as the intensity of the Poisson process tends to infinity. This confirms a conjecture of Benjamini and Schramm [5].     

Stochastic representation for solutions of Isaacs’ type integral–partial differential equations

In this paper we study the integral–partial differential equations of Isaacs’ type by zero-sum two-player stochastic differential games (SDGs) with jump-diffusion. The results of Fleming and Souganidis (1989) [9] and those of Biswas (2009) [3] are extended, we investigate a controlled stochastic system with a Brownian motion and a Poisson random measure, and with nonlinear cost functionals defined by controlled backward stochastic differential equations (BSDEs). Furthermore, unlike the two papers cited above the admissible control processes of the two players are allowed to rely on all events from the past. This quite natural generalization permits the players to consider those earlier information, and it makes more convenient to get the dynamic programming principle (DPP). However, the cost functionals are not deterministic anymore and hence also the upper and the lower value functions become a priori random fields. We use a new method to prove that, indeed, the upper and the lower value functions are deterministic. On the other hand, thanks to BSDE methods (Peng, 1997) [18] we can directly prove a DPP for the upper and the lower value functions, and also that both these functions are the unique viscosity solutions of the upper and the lower integral–partial differential equations of Hamilton–Jacobi–Bellman–Isaacs’ type, respectively. Moreover, the existence of the value of the game is got in this more general setting under Isaacs’ condition.     

Spread‐out percolation in ℝd

Fix d ≥ 2, and let X be either ℤ^d or the points of a Poisson process in ℝ^d of intensity 1. Given parameters r and p, join each pair of points of X within distance r independently with probability p. This is the simplest case of a “spread‐out” percolation model studied by Penrose [Ann Appl Probab 3 (1993) 253–276], who showed that, as r → ∞, the average degree of the corresponding random graph at the percolation threshold tends to 1, i.e., the percolation threshold and the threshold for criticality of the naturally associated branching process approach one another. Here we show that this result follows immediately from of a general result of [3] on inhomogeneous random graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007