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     

Aufschlu? von Zinn(IV)-oxid

Ohne Zusammenfassung

---

Decomposition of tin(IV) oxide

     

Auxiliary controlled singlet-oxygen ene reactions of cyclohexenes

The photooxygenation of homochiral cyclohexene ketals, which are easily available from 2-cyclohexenone and l-tartrates, affords hydroperoxides and after reduction the corresponding allylic alcohols in good yields and high regioselectivities. This can be rationalized by electronic repulsions in a perepoxide intermediate and provides evidence for unfavorable 1,3 diaxial interactions with a dioxolane oxygen atom. Only low stereoselectivities were observed, due to the flexibility of the cyclohexene ring. However, the diastereomers could be separated and after cleavage of the auxiliary, 4-hydroxy-2-cyclohexen-1-one was isolated in enantiomerically pure form, which can serve as a building block for natural product synthesis.     

Minimum boundary touching tilings of polyominoes

We study the problem of tiling a polyomino P with as few squares as possible such that every square in the tiling has a non‐empty intersection with the boundary of P . Our main result is an algorithm which given a simply connected polyomino P computes such a tiling of P . We indicate how one can improve the running time of this algorithm for the more restricted row‐column‐convex polyominoes. Finally we show that a related decision problem is in NP for rectangular polyominoes. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)