The cyclodimerisation of 3-methyl-2-butenenitrile

3-Methyl-2-butenenitrile (1) cyclodimerised on treatment with lithium diisopropylamide in dimethoxyethane at temperatures between ?78°C and 0°C to 3-amino-4-cyano-1,5,5-trimethyl-1,3-cyclohexadiene (2) the structure of which was established by acid hydrolysis to the known 4-cyano-1,5,5-trimethyl-1-cyclohexene-3-one (3).     

Diastereoselective preparation and characterization of ruthenium bis(bipyridine) sulfoxide complexes

A new concept in the synthesis of optically active octahedral ruthenium complexes was realized for the first time when cis- or trans-Ru(bpy)2Cl, (cis- or trans-1) was reacted with either (R)-(+)- or (S)-(-)-methyl p-tolyl sulfoxide (2 or 3); this novel asymmetric synthesis leads to the diastereoselective formation of the ruthenium bis(bipyridine) complex cis-delta-[Ru(bpy)2(2)Cl]Cl (4) (49.6% de) or cis-lambda-[Ru(bpy)2(3)Cl]Cl (5) (48.4% de), respectively. cis- or trans-Ru(dmbpy)2Cl2 (cis- or trans-6) (dmbpy = 4,4'-dimethyl-2,2'-bipyridine) also reacts with 2 or 3, leading to the diastereoselective formation of cis-delta-[Ru(dmbpy)2(2)Cl]Cl (7) (59.5% de) or cis-lambda-[Ru(dmbpy)2(3)Cl]Cl (8) (57.2% de), respectively. The diastereoselectivity of these reactions is governed solely by the chirality of the sulfoxide nucleophile. This represents the first process by which a sigma-bonded ligand occupying only a single coordination site has had such an important influence on the stereochemical outcome of a ruthenium bis(bipyridine) complex formation. These novel complexes were fully characterized by elemental analysis and IR, UV/vis, and 1H, 13C, and 2D NMR spectroscopy. An investigation into the chiroptical properties of these novel ruthenium bis(bipyridine) sulfoxide complexes has been carried out, and circular dichroism spectra are used to assign absolute stereochemistry.     

The left-definite Legendre type boundary problem

The left-definite Legendre type boundary problem concerns the study of a fourth-order singular differential expressionM _k [–] in a weighted Sobolev spaceH generated by a Dirichlet inner product. The fourth-order differential equation

Right-angled Coxeter polytopes, hyperbolic six-manifolds, and a problem of Siegel

By gluing together the sides of eight copies of an all-right angled hyperbolic six-dimensional polytope, two orientable hyperbolic six-manifolds with Euler characteristic ?1 are constructed. They are the first known examples of orientable hyperbolic six-manifolds having the smallest possible volume.     

Conical parametric scattering of a single extraordinary beam in BaTiO3 crystal

In this paper, the parametric scattering of a single extraordinary polarized beam of laser in BaTiO₃ photorefractive crystal has been investigated experimentally and theoretically. The resulting pattern consists of beam fanning, isotropic ring, and anisotropic one. Among all parts of scattering pattern, isotropic ring has not been studied as much as beam fanning and anisotropic ring, and there still are some differences in reports about it. Therefore, the study has mainly focused on this part. In this experimental configuration, isotropic ring is just visible in positive angles although the other parts of parametric scattering pattern can be visible from behind and in front of the crystal. In addition to steady state pattern in forward and backward directions, its transient behavior with the rotation of crystal has been studied. The results of experiments have been analyzed carefully, and their theoretical explanations have been presented based on the standard theory of parametric scattering in photorefractive crystals. It has been shown that this configuration corresponds to the so called parametric B-process scattering.     

A note on the analytic form of the Kramer sampling theorem

This paper discusses the extension of the Kramer sampling theorem to the case when the kernel is Cauchy analytic in the sampling parameter.