The convenient route to cd fragment for the synthesis of vitamin D3 relatives

The efficient degradation of $\text{1}$ to the α-methylene ketone $\text{4}$ is described. Compound $\text{4}$ was then converted to the allylic alcohol $\text{8a}$ - the precursor of vitamin D³ relatives.     

Some analytical semigroups occurring in probability theory

One-parameter semigroups occurring in operator-limit distributions are investigated. The topological-algebraic background of the relevant monoids is discussed and Lie semigroup theory is applied to the Urbanik Decomposability Semigroup.     

Correction procedure for the electron microprobe analysis of porous materials

A correction method is proposed for the quantitative determination of stoichiometric ratios in porous materials by electron microprobe analysis. Analysis of this kind of material is complicated by an imperfect surface that can only be improved to some extent. The correction is based on a linear dependence (different for each element) of the analyses on a sum of weight concentrations of oxides. With this correction it is possible to use analyses affected by the imperfection of the surface with much better confidence compared to common normalisation. An example is presented for a series of ceramic superconductors Bi_2.1(SrCa)_2.9Cu₂O₈₊.     

Strong Mixing and Operator-Selfdecomposability

For nonstationary, strongly mixing sequences of random variables taking their values in a finite-dimensional Euclidean space, with the partial sums being normalized via matrix multiplication, with certain standard conditions being met (an “infinitesimality” assumption, and a restriction to “full” distributions), the possible limit distributions are precisely the operator-self-decomposable laws.