Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations

The postcritical behavior of a generaln-dimensional system around a resonant double Hopf bifurcation isanalyzed. Both cases in which the critical eigenvalues are in ratios of1:2 and 1:3 are investigated. The Multiple Scale Method is employedto derive the bifurcation equations systematically in terms of thederivatives of the original vector field evaluated at the criticalstate. Expansions of the n-dimensional vector of state variables andof a three-dimensional vector of control parameters are performed interms of a unique perturbation parameter ε, of the order ofthe amplitude of motion. However, while resonant terms only appear atthe ε³-order in the 1:3 case, they already arise at theε²-order in the 1:2 case. Thus, by truncating theanalysis at the ε³-order in both cases, first orsecond-order bifurcation equations are respectively drawn, the latterrequiring resort to the reconstitution principle. A two-degrees-of-freedom system undergoing resonant double Hopf bifurcations isstudied. The complete postcritical scenario is analyzed in terms of thethree control parameters and the asymptotic results are compared withexact numerical integrations for both resonances. Branches of periodicas well as periodically modulated solutions are found and theirstability analyzed.     

Diastereoselective synthesis of 2-substituted-piperidin-4-ones as convenient precursors for an asymmetric approach to carbacephams

Optically active carbacephams can be efficiently prepared generating the β-lactam ring on 2-substituted-piperidin-4-ones. These can, in turn, be prepared diastereoselectively through a Michael-Michael reaction sequence initiated by benzylamine on precursors derived by Wittig reaction between serinals and 4-[(4-methylphenyl)sulfonyl]-1-(triphenylphosphoranylidene)-2-butanone.     

A new nitrile oxide based synthesis of the antitumor agent geiparvarin

Geiparvarin 1 has been synthesized utilizing the 3,5-disubstituted isoxazole 8 as protected synthon for the 3(2h)furanone system, the desired α′-hydroxy-1,3-diketone precursor being eventually revealed by Mo(CO)₆ promoted reductive cleavage.     

Highly reactive metals from potassium—graphite. Preparation and use of titanium—graphite and tin—graphite

The potassium—graphite route to active forms of metals has been extended to the preparation of titanium—graphite (Ti---Gr) and tin—graphite (Sn---Gr). The Ti---Gr is used to achieve the reductive coupling of ketones to give alkenes, and Sn---Gr is used in the preparation of diallyltin dibromide complexes which react with aldehydes to give homoallylic alcohols.     

Adsorption of succinonitrile at the mercury-aqueous solution interface: Unusual role of a fixed dipole in the adsorbed molecule

The adsorption of succinonitrile (SN) on a polarized Hg electrode has been studied from 0.25 M NaF aqueous solutions by means of both capacity and electrocapillary curves. Adsorption isotherms have been found to be strictly congruent with respect to both charge and potential. The charge of maximum adsorption is at ?4.6 μC cm^?2 and ΔG_ads⁰, the free energy of adsorption at zero coverage, decreases more rapidly as the charge is made positive with respect to O_max^M than on the other side. The adsorption obeys a Frumkin isotherm with a=?1.87 corresponding to attraction and saturation concentration of 4.8×10^?10 mol cm^?2 corresponding to an area per molecule of about 34.5 Ų. Experimental results suggest that SN adsorbs flat on the surface but, unlike acetonitrile and propionitrile, the CN groups are slightly turned towards the solution. This is responsible for the absence of electronic interaction of the CN groups with the metal at positive charges, which is apparent with the other two nitriles. Implications of the particular structure of the adsorbed molecule on the other adsorption parameters are discussed in detail.     

New catalytic approaches in the stereoselective Friedel-Crafts alkylation reaction

After more than 125 years, the Friedel–Crafts alkylation is still one of the most studied and most utilized reactions in organic synthesis. What is the secret of this astonishing success? Perhaps the great versatility in scope and applicability continues to justify its crucial role in the synthesis of more and more complex molecules. However, it has taken more than a century for asymmetric catalytic versions of this reaction to be developed and subsequently extended to a range of aromatic compounds and alkylating agents. Herein we review recent developments in the design and use of catalytic and stereoselective strategies for the alkylation of aromatic systems and synthesis of a wide range of polyfunctionalized enantiomerically enriched compounds.     

Enantioselective organocatalyzed Henry reaction with fluoromethyl ketones

Remarkable generality in scope of new C9-benzoylcupreines bearing electron-withdrawing substituents for the nitroaldol condensation with fluoromethyl ketones is presented. Both tri- and difluoromethyl ketones provided excellent levels of stereoinduction (ee 76-99%) under mild reaction conditions and low loading of catalyst (1-5 mol%).     

On a unifying approach to decomposition theorems of Yosida-Hewitt type

In this paper we deal with a very general form of the Yosida-Hewitt theorem on the decomposition of measures into countably additive («normal») and purely finitely additive («antinormal») parts. It expands a previous one by the authors with the aim of joining two different standpoints to the Yosida-Hewitt type theorems. The first goes back to the original publication defining the «antinormal» part as a certain disjoint complement to the «normal» one. The second approach goes deeper and characterizes this disjoint complement intrinsically i.e. as a measure, functional or operator which is equal to zero on a huge set. These two points of view are common for the publications connected, respectively, with measure theory and, theory of vector lattices; the second allows important applications. The unification of these approaches gives an opportunity to derive new information in the case of vector measures. We have taken the opportunity of this paper also to furnish a survey of the topic.     

Benzoin condensation in 1,3-dialkylimidazolium ionic liquids via electrochemical generation of N-heterocyclic carbene

N-heterocyclic carbenes generated by electrochemical reduction under galvanostatic control of 1,3-dialkylimidazolium-based ionic liquids were employed as organocatalysts in the benzoin condensation reaction. Benzoin was isolated in a good yield (84%) by a catalytic amount of carbene (20%). The use of toxic, volatile molecular solvents as well as any addition of bases has been avoided.     

Periastron shift in Weyl class spacetimes

The periastron position advance for geodesic motion in axially symmetric solutions of the Einstein field equations belonging to the Weyl class of vacuum solutions is investigated. Explicit examples corresponding to either static solutions (single Chazy-Curzon, Schwarzschild and a pair of them), or stationary solutions (single rotating Chazy-Curzon and Kerr black hole) are discussed. The results are then applied to the case of S2-SgrA^* binary system of which the periastron position advance will be soon measured with a great accuracy.