A non-uniform Berry–Esseen bound via Stein's method

This paper is part of our efforts to develop Stein's method beyond uniform bounds in normal approximation. Our main result is a proof for a non-uniform Berry–Esseen bound for independent and not necessarily identically distributed random variables without assuming the existence of third moments. It is proved by combining truncation with Stein's method and by taking the concentration inequality approach, improved and adapted for non-uniform bounds. To illustrate the technique, we give a proof for a uniform Berry–Esseen bound without assuming the existence of third moments. Received: 2 March 2000 / Revised version: 20 July 2000 / Published online: 26 April 2001     

Phase transition of the principal Dirichlet eigenvalue in a scaled Poissonian potential

We consider d-dimensional Brownian motion in a scaled Poissonian potential and the principal Dirichlet eigenvalue (ground state energy) of the corresponding Schr?dinger operator. The scaling is chosen to be of critical order, i.e. it is determined by the typical size of large holes in the Poissonian cloud. We prove existence of a phase transition in dimensions d≥ 4: There exists a critical scaling constant for the potential. Below this constant the scaled infinite volume limit of the corresponding principal Dirichlet eigenvalue is linear in the scale. On the other hand, for large values of the scaling constant this limit is strictly smaller than the linear bound. For d > 4 we prove that this phase transition does not take place on that scale. Further we show that the analogous picture holds true for the partition sum of the underlying motion process. Received: 10 December 1999 / Revised version: 14 July 2000/?Published online: 15 February 2001     

Stochastic functional differential equations on manifolds

In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters. Received: 6 July 1999 / Revised version: 19 April 2000 /?Published online: 14 June 2001     

Elliptic equations for measures on infinite dimensional spaces and applications

We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models. Received: 23 January 2000 / Revised version: 4 October 2000 / Published online: 5 June 2001     

富勒烯与β淀粉样肽低聚体结合的分子动力学模拟

本文通过分子动力学模拟研究富勒烯在Aβ42 低聚体表面的结合过程. 在结合过程中,C₆₀在Aβ表面经历一系列尝试过程,最终找到某个稳定的结合位点. 根据结合的残基不同,这些结合位点可以分为六类,其中核心疏水区域（CHC）位点（¹⁷LVFFA²¹）及Turn 27-31 位点（²⁷NKGAI³¹）具有最强的结合稳定性. 二者的结合主要通过范德华作用稳定,而溶剂化效应则起相反作用. 在这六类位点中的两个位点,观察到C₆₀会对Aβ二级结构起破坏作用. 其一位于核心疏水区域,C₆₀有挤入多肽β片层中间的趋势;另外一位点位于N端,C₆₀能够破坏外侧Aβ的3-5 号残基的主链氢键,瓦解其末端的β片结构. 这两个过程对理解富勒烯抑制Aβ聚集的微观机制提供了帮助. 此外,在Turn 27-31 位点以及Y10-H14 位点,发现了富勒烯与Aβ纤维样聚集体结合的沟槽滚动机制,即富勒烯能够在淀粉样低聚体表面形成的特定沟槽内滚动. 这一特征有助于预测富勒烯在其它淀粉样多肽表面的结合位点及结合行为.     

The Willmore conjecture for immersed tori with small curvature integral

The Willmore conjecture states that any immersion of a 2-torus into euclidean space satisfies . We prove it under the condition that the L ^p -norm of the Gaussian curvature is sufficiently small. Received: 10 June 1999     

Motivic equivalence of quadratic forms. II

Let F be a field of characteristic ≠2 and φ be a quadratic form over F. By X _φ we denote the projective variety given by the equation φ=0. For each positive even integer d≥8 (except for d=12) we construct a field F and a pair φ, ψ of anisotropic d-dimensional forms over F such that the Chow motives of X _φ and X _ψ coincide but . For a pair of anisotropic (2 ^{n -1})-dimensional quadrics X and Y, we prove that existence of a rational morphism Y→X is equivalent to existence of a rational morphism Y→X. Received: 27 September 1999 / Revised version: 27 December 1999