Runge–Kutta Methods for Itô Stochastic Differential Equations with Scalar Noise

A general class of stochastic Runge–Kutta methods for Itô stochastic differential equation systems w.r.t. a one-dimensional Wiener process is introduced. The colored rooted tree analysis is applied to derive conditions for the coefficients of the stochastic Runge–Kutta method assuring convergence in the weak sense with a prescribed order. Some coefficients for new stochastic Runge–Kutta schemes of order two are calculated explicitly and a simulation study reveals their good performance.     

Second order Runge–Kutta methods for Stratonovich stochastic differential equations

The weak approximation of the solution of a system of Stratonovich stochastic differential equations with a m–dimensional Wiener process is studied. Therefore, a new class of stochastic Runge–Kutta methods is introduced. As the main novelty, the number of stages does not depend on the dimension m of the driving Wiener process which reduces the computational effort significantly. The colored rooted tree analysis due to the author is applied to determine order conditions for the new stochastic Runge–Kutta methods assuring convergence with order two in the weak sense. Further, some coefficients for second order stochastic Runge–Kutta schemes are calculated explicitly. AMS subject classification (2000)  65C30, 65L06, 60H35, 60H10     

Growth of Lévy trees

We construct random locally compact real trees called Lévy trees that are the genealogical trees associated with continuous-state branching processes. More precisely, we define a growing family of discrete Galton–Watson trees with i.i.d. exponential branch lengths that is consistent under Bernoulli percolation on leaves; we define the Lévy tree as the limit of this growing family with respect to the Gromov–Hausdorff topology on metric spaces. This elementary approach notably includes supercritical trees and does not make use of the height process introduced by Le Gall and Le Jan to code the genealogy of (sub)critical continuous-state branching processes. We construct the mass measure of Lévy trees and we give a decomposition along the ancestral subtree of a Poisson sampling directed by the mass measure. T. Duquesne is supported by NSF Grants DMS-0203066 and DMS-0405779. M. Winkel is supported by Aon and the Institute of Actuaries, EPSRC Grant GR/T26368/01, le département de mathématique de l’Université d’Orsay and NSF Grant DMS-0405779.     

小球与均质自由杆的碰撞

讨论了小球与自由杆碰撞过程中所遵从的物理规律,明确了完全非弹性碰撞过程的能量特征,指出完全非弹性碰撞后两物体不一定粘在一起共同运动的事实.     

晶体学与准晶体学点群的母子群关系

依据群论，得出了准晶体学点群的直积或半直积推导算式；依据结晶学理论，绘出了五角、八角、十角和十二角晶系各点群的极赤投影图.据此推导出了每一个准晶体学点群的全部最大子群，从而推导并绘制出了三维晶体学和准晶体学点群之间的母子群关系(60个点群的“家谱”).该“家谱”以最大子群链的图解形式直观地给出了每个点群的最小母群和最大子群. 关键词： 准晶体 晶体 点群 最大子群 最小母群 群链     

Synthesis of new 7-aminosterol squalamine analogues with high antimicrobial activities through a stereoselective titanium reductive amination reaction

A series of 7-amino- and polyaminosterol analogues of squalamine and trodusquemine were synthesized involving a new stereoselective titanium reductive amination reaction in high chemical yields of up to 95% in numerous cases. These derivatives were evaluated for their in vitro antimicrobial properties against human pathogens. All the compounds present excellent activities against Gram-positive bacteria exhibiting similar results against Staphylococcus aureus and Streptococcus faecalis with minimum inhibitory concentrations (MICs) varying from 2.5 to 10 μg/mL. Numerous derivatives possess also MICs against Gram-negative Escherichia coli bacteria (MICs varying from 2.5 to 10 μg/mL) suggesting that nature of the amino group attached to the sterol moiety plays an important role on the activities of such products.     

Corrections of proofs for Hansen and Mélot's two theorems

The Randi? index R(G) of a (chemical) graph G is also called connectivity index. Hansen and Mélot [Variable neighborhood search for extremal graphs 6: analyzing bounds for the connectivity index, J. Chem. Inf. Comput. Sci. 43 (2003) 1-14] in their paper, characterized the chemical trees of given order and number of pendent vertices which have the minimum and maximum Randi? index, respectively. In this note, we point out the mistakes in the proofs of their results Theorems 8 and 10, while we still believe that the two theorems are true, and then we give their corrected proofs.     

图的k-单圈划分中的优化问题

图的划分问题曾引起图论界的广泛关注，在文献[4]中讨论了k-单圈划分，本文进一步研究基于k-单圈划分的优化问题，即在一个赋权图中求一个最小权可k-单圈划分的支撑子图，以及对一个不存在k-单圈划分支撑子图的图，如何添最少的边使得它有k-单圈划分的支撑子图。