Ergodicity for a weakly damped stochastic non-linear Schrödinger equation

We study a damped stochastic non-linear Schr?dinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markov transition semi-group toward a unique invariant probability measure. This kind of method was originally developed to prove exponential mixing for strongly dissipative equations such as the Navier-Stokes equations. We consider here a weakly dissipative equation, the damped nonlinear Schr?dinger equation in the one-dimensional cubic case. We prove that the mixing property holds and that the rate of convergence to equilibrium is at least polynomial of any power.     

A Reexamination of Inductive Substituent Constants Derived from pKa Values of 4-Substituted Quinuclidines. Polar effects. Part VIII

pK_a Values of 42 quinuclidinium perchlorates I have been measured in 0.1 M aqueous KCl-solution. In a few cases small corrections of earlier thermodynamic pK_a values are indicated. These measurements, in conjunction with recent X-ray structure determinations, confirm the reliability of the inductive substituent constants σ_I^q derived from them.     

Nucleophilic Reactions at Tertiary Carbon. Part 2. σ- and π-routes to the 8-hydrindanyl cation

Stereoisomeric ion pairs are implicated as intermediates in the solvolysis of cis and trans-8-hydrindanyl chloride 3 , whereas 4-(cyclopenten-1-yl)butyl tosylate 5 appears to cyclize by way of an unsymmetrically solvated 8-hydrindanyl cation. This follows from the solvolysis products and rates of these compounds in aqueous solvents. The rate and equilibrium constants of the chlorides 3 show that the transition state for the trans-isomer is more stable by 0.5 kcal than the one for the cis-isomer. By inference the intermediates differ by a similar amount of energy. Experimental results are not explained satisfactorily by conformationally isomeric 8-hydrindanyl cations, as suggested in the literature.     

Synthese und Basizität 4-substituierter Chinuclidine. Chinuclidinreihe 12. Mitteilung

The preparation of a number of 4-substituted quinuclidines and the PK_a values of their hydroperchlorates in water are reported.     

Artificial glycosyl phosphorylases

alpha- and beta-Cyclodextrin 6(A),6(D)-diacids (1 and 2), beta-cyclodextrin-6-monoacid (14), beta-cyclodextrin 6(A),6(D)-di-O-sulfate (16) and beta-cyclodextrin-6-heptasulfate (19) were synthesised. Acids 1, 2 and 14 were made from perbenzylated alpha- or beta-cyclodextrin, by diisobutylaluminum hydride (DIBAL)-promoted debenzylation, oxidation and deprotection. Addition of molecular sieves was found to improve the debenzylation reaction. Sulfates 16 and 19 were made by sulfation of the appropriately partially protected derivatives and deprotection. Catalysis of 4-nitrophenyl glycoside cleavage by these cyclodextrin derivatives was studied. Compounds 1, 2 and 16 were found to catalyse the reaction, with the catalysis following Michaelis-Menten kinetics and depending first order on the phosphate concentration. In a phosphate buffer (0.5 M, 59 degrees C, pH 8.0), K(M) varied from 2-10 mM and the k(cat)/k(uncat) ratio from 80-1000 depending on the stereochemistry of the substrate and the catalyst, with 2 being the best catalyst and with the sulfated 16 also displaying catalytic ability. The monoacid 14 and the heptasulfate 19 were not catalytic.     

Amine capture strategy for peptide bond formation by means of quinolinium thioester salts

A new and unprecedented exploitation of quinolinium thioester salts 2 in peptide bond formation is reported. These synthetic tools were assessed during the preparation of a number of dipeptides 3a-f obtained in good yields with complete stereochemical integrity. A sequential mechanism related to a prior amine capture strategy is well-established. Additionally, a tripeptide 3g was prepared according to a "safety-catch" approach, thus demonstrating the important potential of these new synthetic tools in the design of new safety-catch linkers exploitable in Solid-Phase Peptide Synthesis (SPPS).     

On L1 and Chebyshev estimation

The problem considered here is that of fitting a linear function to a set of points. The criterion normally used for this is least squares. We consider two alternatives, viz., least sum of absolute deviations (called the L₁ criterion) and the least maximum absolute deviation (called the Chebyshev criterion). Each of these criteria give rise to a linear program. We develop some theoretical properties of the solutions and in the light of these, examine the suitability of these criteria for linear estimation. Some of the estimates obtained by using them are shown to be counter-intuitive.     

An Efficient Synthesis of 2,3,5,6-Tetramethylidene-7-oxanorbornane

Palladium-catalyzed double carbomethoxylation of the Diels-Alder adduct of furan and maleic anhydride yielded the methyl all-exo-7-oxanorbornane-2,3,5,6-tetracarboxylate (7) which was transformed in three steps into 2,3,5,6-tetramethylidene-7-oxanorbornane (1) , a useful synthon. Six isomeric methyl 7-oxanor-bornane-2,3,5,6-tetracarboxylates (7–12) have been isolated and their ¹H- and ¹³C-NMR. data are compared.