Random rotations of the Wiener path

Summary Let (W, H, ) be an abstract Wiener space and letR(w) be a strongly measurable random variable with values in the set of isometries onH. Suppose that Rh is smooth in the Sobolev sense and that it is a quasi-nilpotent operator onH for everyhH. It is shown that (R(w)h) is again a Gaussian (0, |h| _H ² )-random variable. Consequently, if (e _i ,i)W ^* is a complete, orthonormal basis ofH, then defines a measure preserving transformation, a rotation, onW. It is also shown that if for some strongly measurable, operator valued (onH) random variableR, (R(w+k)h) is (0, |h| _H ² )-Gaussian for allk, hH, thenR is an isometry and Rh is quasi-nilpotent for allHH. The relation between the stochastic calculi for these Wiener pathsw and , as well as the conditions of the inverbibility of the map are discussed and the problem of the absolute continuity of the image of the Wiener measure under Euclidean motion on the Wiener space (i.e. composed with a shift) is studied.The research of the second author was supported by the Fund for the Promotion of Research at the TechnionDedicated to the memory of Albert Badrikian     

Stereoselectivity of Dienamine [4 + 2] Cycloadditions Synthesis of Functionalised Decalins and Drimanes

The [4 + 2] cycloaddition stereoselectivity of dienamines 1–4 with dimethyl fumarate and fumaronitrile has been investigated, and functionalised decalins 21–40 have been prepared by elimination of Me₂NH from cycloadducts 7–11 and 15–20 ; in the context of the synthesis of drimane sesquiterpenes, the reduction of dienediesters 29 and 30 is also described.     

Theoretical analysis of frontier orbitals,electronic transitions,and global reactivity descriptors of M(CO)4L2 type metal carbonyl complexes: a DFT/TDDFT study

Structural Chemistry - Metal carbonyl complexes, which have been known as effective catalysts since early days, find use in many fields both directly and indirectly. Although the use of metal...