The projection operator approach to the Fokker-Planck equation. II. Dichotomic and nonlinear Gaussian noise

Two models for the Freedericksz transition in a fluctuating magnetic field are considered: one is based on a dichotomic and the other on a nonlinear Gaussian noise. Both noises are characterized by a finite correlation time. It is shown that the linear response assumption leading to the best Fokker-Planck approximation in the dichotomic and nonlinear Gaussian cases can be trusted only up to the order ¹ and ⁰, respectively. The role of the corrections to the linear response approximation is discussed and it is shown how to replace the non-Fokker-Planck terms stemming from these corrections with equivalent terms of standard type. This technique is shown to produce perfect agreement with the exact analytical results (dichotomic noise) and to satisfactorily fit the results of analog simulation (nonlinear Gaussian noise).     

Equilibrium states for a plane incompressible perfect fluid

We associate to the plane incompressible Euler equation with periodic conditions the corresponding Hopf equation, as an equation for measures on the space of solenoidal distributions. We define equilibrium states as the solutions of the stationary Hopf equation. We find a class of equilibrium states which corresponds to a class of infinitely divisible distributions, and investigate the properties of gaussian and poissonian states. Equilibrium dynamics for a class of poissonian states is constructed by means of the Onsager vortex equations.Research partially supported by C.N.R., G.N.F.M.     

Regularity in parabolic phase transition problems

We describe the main results obtained in a joint work with Athanasopoulos and Caffarelli on the regularity of viscosity solutions and of their free boundaries for a rather general class of parabolic phase transition problems. To a great unforgettable Master.     

On the mechanical modelling of masonry structures using the F.E. technique

A non-linear finite element for the mechanical modelling of masonry structures is presented. Both material and geometrical non-linearities are introduced, developing the expression of the potential strain energy through appropriate average functions. The characteristics of the numerical model are explained and the approximations introduced in the analysis are discussed.

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Sommario Viene sviluppata la formulazione di un nuovo elemento finito per la modellazione meccanica di strutture murarie in campo non lineare. Vengono prese in esame sia non-linearita' geometriche che materiali, sviluppando l'espressione dell'energia potenziale elastica del solido murario attraverso l'introduzione di opportune funzioni medie sulle deformazioni. Vengono discusse le caratteristiche del modello numerico e le approssimazioni introdotte nell'analisi.

     

Time behaviour of non-linear stochastic processes in the presence of multiplicative noise: From Kramers' to Suzuki's decay

The diffusional regime of a Brownian particle in a double-well potential in the presence of both additive and multiplicative noise is explored. As a relevant effect of the multiplicative noise, the escape rate from a well is shown to change from the small value of the Kramers theory into the large relaxation rate of the Suzuki regime. It is shown, furthermore, that the time required to get equilibrium in a well after sudden application of multiplicative noise (the activation time) is very much shorter than the Kramers relaxation time. We envisage therefore an operational scheme making available multiplicative noise for a short interval of time (for example using a light pulse) as an efficient tool to get a fast process of escape from a well. These results are obtained by using a continued-fraction algorithm which makes it possible even to successfully deal with the decay of an unstable state at the critical point.     

The palladium-catalyzed reaction of aryl iodides with mono and disubstituted acetylenes: a new synthesis of trisubstituted alkenes.

Mono and disubstituted acetylenes react with aryl iodides containing a variety of functional groups in the presence of palladium catalyst, formic acid and a tertiary amine to give trisubstituted alkenes.     

Palladium-catalyzed carbonylation of enol triflates. A novel method for one-carbon homologation of ketones to α,β-unsaturated carboxylic acid derivatives

Ketones can be homologated to α,gb-unsaturated esters or amides via their enol triflates by a palladium-catalyzed reaction with carbon monoxide and alcohols or amines.     

Ru-based oxidation catalysis

Ranging from the oxidative conversion of water to O(2) to the elegant hydroxylation of olefins and to oxidative dehydrogenation of alcohols Ru-mediated oxidations are finding increasing application due to the unique properties of this extremely versatile transition metal, whose oxidation state can vary from -II to +VIII. Covering recent developments in both homogeneously and heterogeneously catalysed oxidations (in liquid-phase as well as in novel reaction media), this tutorial review aims to provide investigators with a general picture of the chemical and structural origins of the excellent performance of many ruthenium catalysts and to promote further advancement that, it is envisaged, will soon benefit society at large.     

Origin of the sphere-to-rod transition in cationic micelles with aromatic counterions: specific ion hydration in the interfacial region matters

Sphere-to-rod transitions of cetyltrimethylammonium (CTA+) micelles with dichlorobenzoate counterions are remarkably substituent dependent. Simultaneous estimates of the interfacial molarities of H2O, MeOH, and Cl- and 2,6- and 3,5-dichlorobenzoate (2,6OBz and 3,5OBz) counterions were obtained by the chemical trapping method in mixed micelles of CTACl/CTA3,5OBz and CTACl/CTA2,6OBz without added salt. Increasing the CTA3,5OBz mole fraction produces a marked concurrent increase in interfacial 3,5OBz- and a decrease in interfacial H2O concentrations through the sphere-to-rod transition. No abrupt concentration changes are observed with increasing CTA2,6OBz mole fraction. Counterion-specific changes in the interfacial water concentration may be a major contributor to the delicate balance of forces governing micellar morphology.