Hierarchical structure in sharply divided phase space for the piecewise linear map.

We have studied a two-dimensional piecewise linear map to examine how the hierarchical structure of stable regions affects the slow dynamics in Hamiltonian systems. In the phase space there are infinitely many stable regions, each of which is polygonal-shaped, and the rest is occupied by chaotic orbits. By using symbolic representation of stable regions, a procedure to compute the edges of the polygons is presented. The stable regions are hierarchically distributed in phase space and the edges of the stable regions show the marginal instability. The cumulative distribution of the recurrence time obeys a power law as ∼t^{-2}, the same as the one for the system with phase space, which is composed of a single stable region and chaotic components. By studying the symbol sequence of recurrence trajectories, we show that the hierarchical structure of stable regions has no significant effect on the power-law exponent and that only the marginal instability on the boundary of stable regions is responsible for determining the exponent. We also discuss the relevance of the hierarchical structure to those in more generic chaotic systems.

Indium-Catalyzed Regioselective ß-Alkylation of Pyrroles with Carbonyl Compounds and Hydrosilanes and Its Application to Construction of a Quaternary Carbon Center with a ß-Pyrrolyl Group.

Treatment of N-substituted pyrroles with carbonyl compounds and nucleophiles under indium catalysis was found to be a promising method for preparing ß-alkylpyrroles without contamination by α-alkylpyrroles. With this methodology, a variety of alkyl groups, which are primary, secondary, and tertiary as well as cyclic and functionalized types, can be introduced in place onto the pyrrole ring. The simplicity performable as a catalytic one-step process is one of the important features of this reaction. The substituent on the nitrogen atom of the product ß-alkylpyrrole can be removed easily by literature procedures. Therefore, the indium-catalyzed ß-alkylation plus the N-deprotection is a powerful system for all six variations, which are N-substituted and N-unsubstituted ß-alkylpyrroles having primary, secondary, and tertiary alkyl groups. Our method is applicable to synthesizing, albeit in two steps, ß-pyrrolyl-group-connected unsymmetrical tetraarylmethanes that have not been addressed thus far. Mechanistic studies showed the following three aspects: (1) dipyrrolylalkanes produced in situ from the pyrrole and carbonyl compound are key intermediates, (2) the selective ß-alkylation is attributed to the selective elimination of an α-pyrrolyl group from the dipyrrolylalkane intermediates, and (3) the indium Lewis acid catalyst is indispensable for the progress of both stages.

An in situ quick XAFS spectroscopy study on the formation mechanism of small gold nanoparticles supported by porphyrin-cored tetradentate passivants.

We previously reported that a porphyrin-cored tetradentate passivant, which has two disulfide straps over one face of the porphyrin plane, can produce monolayer-protected gold nanoparticles, 2-4 nm in size, by the one-pot reduction of HAuCl(4) in DMF. The resulting nanoparticles are smaller than those prepared using the same S/Au molar ratio of a monodentate passivant. To examine the formation mechanism of small gold nanoparticles, the formation of gold nanoparticles in the presence of porphyrin-cored tetradentate passivants or a structurally related monodentate passivant was studied by time-resolved quick X-ray absorption fine structure spectroscopy. The results demonstrated that all of Au ions in solution are reduced to compose small Au clusters, i.e. nuclei, just after the NaBH(4) reduction of HAuCl(4) in both cases, but their size varied with the initial S/Au molar ratios and structure of the passivants. Thus, the size of Au nuclei was kinetically controlled by the passivants. Interestingly, the porphyrin-cored tetradentate passivant could stabilize smaller gold nanoparticles, 2-4 nm in size, but it was less efficient in trapping the Au nuclei formed at a very early stage, in comparison to the monodentate passivant.

Indium-catalyzed reductive alkylation of pyrroles with alkynes and hydrosilanes: selective synthesis of beta-alkylpyrroles.

Mixing readily available alkynes, pyrroles, and triethylsilane along with an indium catalyst was found to be an efficient procedure to introduce alkyl groups onto a beta-position of pyrroles in a complete regioselective manner. This is the first demonstration of catalytic beta-alkylation of pyrroles in a single step.