Asymptotic Shapes of large Cells in Random Tessellations

We establish a close relationship between isoperimetric inequalities for convex bodies and asymptotic shapes of large random polytopes, which arise as cells in certain random mosaics in d-dimensional Euclidean space. These mosaics are generated by Poisson hyperplane processes satisfying a few natural assumptions (not necessarily stationarity or isotropy). The size of large cells is measured by a class of general functionals. The main result implies that the asymptotic shapes of large cells are completely determined by the extremal bodies of an inequality of isoperimetric type, which connects the size functional and the expected number of hyperplanes of the generating process hitting a given convex body. We obtain exponential estimates for the conditional probability of large deviations of zero cells from asymptotic or limit shapes, under the condition that the cells have large size. This work was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953. Received: May 2005 Accepted: September 2005     

非线性二阶三点边值问题系统正解的存在性

考虑二阶三点边值问题系统-u"=f(t,v),t∈(0,1),-v"=g(t,u),t∈(0,1),u(0)=αu(η),u(1)=βu(η),v(0)=αv(η),v(1)=βv(η),其中f,g∈C([0,1]×R+,R+),g(t,0)(=)0,η∈(0,1)且0＜β≤α＜1.首先给出了线性边值问题的Green函数;其次,给出了Green函数的一些很好的性质;最后,运用锥上拉伸与压缩不动点定理研究了上述边值问题系统至少一个或多个正解的存在性.     

随机二阶多值周期问题解的存在性

本文研究了随机二阶多值周期问题解的存在性,在非常一般的假定下建立了凸和非凸问题解的存在性定理,并给出了在随机周期反馈控制系统中的应用．     

The Edge Correlation of Random Forests

The conjecture was made by Kahn that a spanning forest F chosen uniformly at random from all forests of any finite graph G has the edge-negative association property. If true, the conjecture would mean that given any two edges ε₁ and ε₂ in G, the inequality \mathbbP(e₁ ? F, e₂ ? F) ￡ \mathbbP(e₁ ? F)\mathbbP(e₂ ? F){{\mathbb{P}(\varepsilon_{1} \in \mathbf{F}, \varepsilon_{2} \in \mathbf{F}) \leq \mathbb{P}(\varepsilon_{1} \in \mathbf{F})\mathbb{P}(\varepsilon_{2} \in \mathbf{F})}} would hold. We use enumerative methods to show that this conjecture is true for n large enough when G is a complete graph on n vertices. We derive explicit related results for random trees.     

Second-Order Term of Cover Time for Planar Simple Random Walk

Journal of Theoretical Probability - We consider the cover time for a simple random walk on the two-dimensional discrete torus of side length n. Dembo et al. (Ann Math 160:433–464, 2004)...     

一类二阶三点边值问题正解的存在性

The existence of positive solutions is established for a nonlinear second-order three-point boundary value problem. The result improves and extends the main result in Electron J. Differential Equations, 34(1999), 1-8.     

Existence and Multiplicity of Solutions of Second-Order Difference Boundary Value Problems

This paper concerns the existence and multiplicity of solutions of second-order difference boundary value problems. Under the assumptions which guarantee the existence of at least one nontrivial solution of the homogeneous problem, we obtain the existence of exactly three solutions of the nonhomogeneous problem with some other suitable conditions.     

二阶奇异积分边值问题的正解

By using the fixed point index in cone and the fixed theorem of cone expansion and compression, the existence of positive solutions to the singular second-order boundary value problem is considered.     

一类半正二阶三点边值问题的解和多解性

利用锥上的不动点定理,给出了非线性二阶三点边值问题解和多解的存在性定理.其中允许非线性项有一个负的下界.     

Estimating Second-Order Characteristics of Inhomogeneous Spatio-Temporal Point Processes

Non-parametric estimates of the K-function and the pair correlation function play a fundamental role for exploratory and explanatory analysis of spatial and spatio-temporal point patterns. These estimates usually require information from outside of the study region, resulting to the so-called edge effects which have to be corrected. They also depend on first-order characteristics, which have to be estimated in practice. In this paper, we extend classical edge correction methods to the spatio-temporal setting and compare the performance of the related estimators for stationary/non-stationary and/or isotropic/anisotropic point patterns. Further, we explore the influence of the estimated intensity function on these estimators.     

Model Based Estimation of Geometric Characteristics of Open Foams

Open cell foams are a class of modern materials which is interesting for a wide variety of applications and which is not accessible to classical materialography based on 2d images. 3d imaging by micro computed tomography is a practicable alternative. Analysis of the resulting volume images is either based on a simple binarisation of the image or on so-called cell reconstruction by image processing. The first approach allows to estimate mean characteristics like the mean cell volume using the typical cell of a random spatial tessellation as model for the cell shape. The cell reconstruction allows estimation of empirical distributions of cell characteristics. This paper summarises the theoretical background for the first method, in particular estimation of the intrinsic volumes and their densities from discretized data and models for random spatial tessellations. The accuracy of the estimation method is assessed using the dilated edge systems of simulated random spatial tessellations.     

P-集合的随机特征

在P-集合概念的基础上,利用元素迁移的随机性,蛤出P-集合的随机生成和P-集合的强随机生成,讨论了P-集合的随机特性.     

一类二阶拟线性边值问题的可解性

当非线性项奇异和无穷远处的极限增长函数存在时,考察了一类二阶拟线性边值问题.通过引入非线性项在有界集合上的高度函数,并且考察高度函数的积分,证明了一个解的存在定理.该定理表明当极限增长函数的积分具有适当值时此问题有一个解.     

Integral Analogue of the First Initial-Boundary Value Problem for Second-Order Hyperbolic and Parabolic Equations

Mathematical Notes - We study the solvability of initial-boundary value problems for second-order hyperbolic and parabolic equations with a boundary condition that integrally connects the values of...     

二阶m点奇异边值问题的多重正解

利用锥上的不动点定理,讨论了一类二阶m点奇异一维p-Laplacian算子的边值问题,得到该问题至少存在三个正解的充分条件.     

Real Options with Random Controls and the Value of Learning

In this paper we propose a conceptual framework for continuous-time valuation of real (investment) options in the presence of costly controls with random outcomes (learning), that affect the value of the underlying asset or a relevant state-variable. These controls represent optional efforts by management to add value to the underlying real investments over which it has monopoly power, albeit with uncertain results. Special cases of such controls include pure learning (but costly) actions, as in many research and development, marketing research or natural resource exploration projects. We demonstrate a discrete-time Markov-chain solution methodology implemented in a finite-difference scheme, and we discuss numerical results. The impact of such uncertain jumps is seen to be relatively more significant in the case of non-profitable options than in the case of very profitable real (investment) options. When the potential for information revelation is significant, we are even willing to pay for an action with a negative expected outcome. With numerical simulations we capture the value of embedded exploration (pure learning) options and we demonstrate the improvement over the traditional (sequential/compound) real options approach. We show that such exploration options enhance the value of investment opportunities in the most significant manner, and justify the (mostly unexplained) observed practice of overpaying for the purchase of rights to natural resources extraction.     

一类二阶边值问题解的存在性

讨论了一类椭圆问题:-u″+a(x)u=f(x,u),u(0)=u(1)=0,a∈C([0,1],R+),f∈C~1([0,1]×R~1,R~1)且对任意的x∈[0,1]有f(x,0)=0.我们首先给出了关于f的一些条件,然后运用强单调算子原理建立了此问题唯一解的存在性结果.     

二阶微分方程多点边值问题的三重正解

We consider the second-order differential equation u(t) + q(t)f(t,u(t),u (t)) = 0,0 t 1,subject to three-point boundry condition u(0) = 0,u(1) = a 0 u(ξ 0 ),or to m-point boundary conditionu (0) = m2 i=1 b i u (ξ i ),u(1) = m2 i=1 a i u(ξ i ).We show the existence of at least three positive solutions of the above multi-point boundary-value problem by applying a new fixed-point theorem introduced by Avery and Peterson.