Pseudo-Index of Fano Manifolds and Smooth Blow-Ups

Suppose π: X → Y is a smooth blow-up along a submanifold Z of Y between complex Fano manifolds X and Y of pseudo-indices i_X and i_Y respectively (recall that i_X is defined by i_X :=min {−K_X·C | C is a rational curve of X}). We prove that if 2 dim (Z) < dim (Y)+i_Y −1 and show that this result is optimal by classifying the ‘boundary’ cases. As expected, these results are obtained by studying rational curves on X and Y.     

Modified logarithmic Sobolev inequalities and transportation inequalities

We present a class of modified logarithmic Sobolev inequality, interpolating between Poincaré and logarithmic Sobolev inequalities, suitable for measures of the type exp (−|x|^α) or exp (−|x|^α log ^β(2+|x|)) (α ∈]1,2[ and β ∈ ℝ) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincaré inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities. Send offprint requests to: Ivan Gentil     

Equilateral Dimension of the Rectilinear Space

Itis conjectured that there exist at most 2k equidistantpoints in the k-dimensional rectilinear space.This conjecture has been verified for k ≤ 3; we show here its validity in dimension k = 4. We alsodiscuss a number of related questions. For instance, what isthe maximum number of equidistant points lying in the hyperplane: $\sum\nolimits_{i = 1}^k {x_i } = 0?$ If this number would be equal to k, then the above conjecture would follow. Weshow, however, that this number is ≥ k + 1 for k ≥ 4.     

Selective alkylation of a 6,7-dihydroxyquinazoline

A convenient 3-step multi-parallel process for the preparation of 4-(3-chloro-2-fluoroanilino)-6,7-bisalkoxyquinazolines is highlighted.     

Structure of the bromination product of 2-ethyl-1,4-benzothiazin-3(4H)-one

2-Ethyl-1,4-benzothiazin-3(4H)-one and bromine react smoothly to give Z-2-(1-bromoethylidine)-2(H)-1,4-benzothiazin-3(4H)-one, which results from a complex bromination-oxidation sequence. The structure of this product was determined by an X-ray study.     

Novel transition metal phospha-alkyne complexes: tBuCP acting as a 6 electron donor ligand. Synthesis,crystal and molecular structure of [Co2(CO)6(μ-tBuCP)W(CO)5]

A single crystal X-ray diffraction study of [Co₂(CO)₆(μ-^tBuCP)W(CO)₅] establishes that phospha-alkyne behaves as a 6e donor. Synthesis of the related [(η⁵-C₅H₅)₂Mo₂(CO)₄(μ-^tBuCP)W(CO)₅] complex is reported.