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).     

Efficient method for visualization and isolation of proteins resolved in polyacrylamide gels

Polyacrylamide gel electrophoresis is a popular method used to purify proteins for reconstitution experiments, amino acid composition and sequence determinations. In this report we describe methods that will be of general use in the isolation and characterization of proteins and the benefits of substituting boric acid for glycine in the electrophoresis tray buffers. We also describe how proteins resolved in a variety of gel systems (including those containing sodium dodecyl sulfate) may be rapidly visualized with 8-anilino-1-naphthalene sulfonic acid and efficiently transferred to a second gel for two-dimensional gel analysis, or isolated by electroelution for subsequent characterization.     

Period doubling forC 2+∈ mappings

The rigorous treatment of period-doubling cascades, developed by Lanford for analytic families, is extended to coverC ²⁺ families for any >0. This requires spectral analysis of the linearised doubling operator on spaces of differentiable mappings, and a version of stable manifold theory which takes account of the non-differentiability of the doubling operator on these spaces.     

Synthesis of the B-ring of FR182877. Investigation of the reactions of 6-fumaryl 1,3,8-nonatrienes

Investigation of the intramolecular Diels-Alder reactions of 6-fumaryl 1,3,8-nonatrienes, substituted at the C5 by a vinyl group, to afford the B-ring of FR182877 are reported. The synthesis of the required 1,3,8-nonatriene was achieved quickly and in high yield. 6-Fumaryl 1,3,8-nonatrienes substituted at the C5 by a vinyl group were found to undergo competing tandem sigmatropic rearrangement/Diels-Alder cyclisation when heated under standard Diels-Alder cyclisation conditions. This rearrangement became the exclusive pathway when the reaction was performed in the presence of a Lewis acid. As expected from modeling studies, the major intramolecular Diels-Alder cyclisation product was the desired exo-trans adduct, which was required for the synthesis of FR182877. Intrigued by the rearrangements, a number of alterations were made to the 1,3,8-nonatriene. Replacement of the fumaryl group by an acetyl group resulted in the diminished reactivity of the 1,3,8-nonatriene with neither rearrangements nor cycloadditions observed. Variation of the C5 substituent was found to be very important in determining the π-diastereoselectivity of the Diels-Alder cyclisation.     

Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations

We consider the numerical solution of the stochastic partial differential equation , where is space-time white noise, using finite differences. For this equation Gyöngy has obtained an estimate of the rate of convergence for a simple scheme, based on integrals of over a rectangular grid. We investigate the extent to which this order of convergence can be improved, and find that better approximations are possible for the case of additive noise ( ) if we wish to estimate space averages of the solution rather than pointwise estimates, or if we are permitted to generate other functionals of the noise. But for multiplicative noise ( ) we show that no such improvements are possible.

     

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.     

Kolmogorov Complexity and Noncomputability

We use a method suggested by Kolmogorov complexity to examine some relations between Kolmogorov complexity and noncomputability. In particular we show that the method consistently gives us more information than conventional ways of demonstrating noncomputability (e. g. by embedding in the halting problem). Also, many sets which are (at least) awkward to embed into the halting problem are easily shown noncomputable. We also prove a gap‐theorem for outputting consecutive integers and find, for a given length n, a statement of length n with maximal (shortest) proof length.     

Pointwise limits of analytic functions

This note is an addendum to the above-mentioned paper, whichwas recently published in this journal. Unfortunately, the authorwas unaware that the main results of Section 1 of this paper,Theorems 1.9 and 1.12, had already been proved in papers ofDanielyan and Mergelyan [1,2]. The author is grateful to ProfessorDanielyan for drawing his attention to these references, andapologises for the unintentional duplication of part of theirwork. Footnotes 2000 Mathematics Subject Classification 30E10 (primary), 46J10(secondary). Received May 6, 2008; published online August 22, 2008.