Exposing the Delocalized Cu−S π Bonds on the Au24Cu6(SPhtBu)22 Nanocluster and Its Application in Ring‐Opening Reactions

Bimetallic nanomaterials are of major importance in catalysis. A Au‐Cu bimetallic nanocluster was synthesized that is effective in catalyzing the epoxide ring‐opening reaction. The catalyst was analyzed by SCXRD and ESI‐MS and found to be Au₂₄Cu₆(SPhtBu)₂₂ (Au₂₄Cu₆ for short). Six copper atoms exclusively occupy the surface positions in two groups with three atoms for each, and each group was bonded with three thiolate ligands to give a planar motif reminiscent of a benzene ring. In the epoxide‐ring opening reaction, Au₂₄Cu₆ exhibited superior catalytic activity compared to other homometallic and Au‐Cu alloy NCs, such as Au₂₅ and Au_38?xCu_x. Control experiments and DFT calculations revealed that the π conjugation among the Cu?S bonds played a pivotal role. This study demonstrates a unique π conjugation established among the Cu?S bonds as a critical structural motif in the nanocluster, which facilitates the catalysis of a ring‐opening reaction.     

Synthesis of (7-polyfluoroalkyl)pyrazolo[1,5-a]pyrimidines based on lithium fluorine-containing β-diketonates

The reactions of Li enolates of fluorine-containing -diketones with 3-aminopyrazoles afforded (7-polyfluoroalkyl)pyrazolo[1,5-a]pyrimidines. The structure of 3-bromo-2-methyl-5-phenyl-7-trifluoromethylpyrazolo[1,5-a]pyrimidine was established by X-ray diffraction analysis.     

Applications of Hilbert’s projective metric to a class of positive nonlinear operators

The purpose of this paper is to study the eigenvalue problems for a class of positive nonlinear operators. Using projective metric techniques and the contraction mapping principle, we establish existence, uniqueness and continuity results for positive eigensolutions of a particular type of positive nonlinear operator. In addition, we prove the existence of a unique fixed point of the operator with explicit norm-estimates. Applications to nonlinear systems of equations and to matrix equations are considered.     

Central extensions of quantum current groups

We describe Hopf algebras which are central extensions of quantum current groups. For a special value of the central charge, we describe Casimir elements in these algebras. New types of generators for quantum current algebra and its central extension for quantum simple Lie groups, are obtained. The application of our construction to the elliptic case is also discussed.     

On the theory of quantum groups

By using the results of S. L. Woronowicz, we show that for the twisted version of the classical compact matrix groups, the Hopf algebraA _h of representative elements is isomorphic as a co-algebra to the Hopf algebraA _O of representative functions on the classical group. As a consequence,A _h can be identified withA _O as a co-algebra but with an associative product, called the star-product, which is a deformation of the original commutative product ofA _O . Furthermore, the construction of this star product from the original product is connected to the Fourier transformation in a manner which is similar to the construction of quantum mechanics from classical mechanics on phase space. In fact, we shall describe the analog of the Weyl correspondence.     

Filling in solvable groups and in lattices in semisimple groups

We prove that the filling order is quadratic for a large class of solvable groups and asymptotically quadratic for all -rank one lattices in semisimple groups of -rank at least 3. As a byproduct of auxiliary results we give a shorter proof of the theorem on the nondistorsion of horospheres providing also an estimate of a nondistorsion constant.     

Unitary representations of the quantum group SU q (1, 1): II — Matrix elements of unitary representations and the basic hypergoemetric functions

Some series of unitary representations of the quantum group SU _q (1, 1) are introduced. Their matrix elements are expressed in terms of the basic hypergeometric functions. Operator realization of the coordinate elements of SU _q (1, 1) and aq-analogue of some classical identities are discussed.     

A Short Note on the Lindelöf Hypothesis

We obtain a quantitative estimate for the measure of the set of large values of the zeta-function on the critical line which is related to the Lindelöf hypothesis and give its application to extreme values (-results). We also investigate the problem of small values.     

Superdiffusivity in first-passage percolation

Summary. In standard first-passage percolation on (with ), the time-minimizing paths from a point to a plane at distance are expected to have transverse fluctuations of order . It has been conjectured that with the inequality strict (superdiffusivity) at least for low and with . We prove (versions of) for all and . Received: 30 August 1995 / In revised form: 28 March 1996