A note on differentiability of the cluster density for independent percolation in high dimensions

The cluster density function of independent percolation in ad-dimensional lattice is considered. For eachn, it is shown that(p) has finitenth leftderivative at critical probabilityp _c ifd is sufficiently large. This result agrees with the Bethe lattice approximation, where thenth one-sided derivative of(p) is bounded atp _c for alln.     

On an Open Question and a Fixed Point Theorem for a Class of Contractive Mappings

Let X be a complete metric space with, distance fcmction d, and T a mapping from X into X. T is said to be a contractive mapping (25), if it satisfies tbe following condition d(Tx, Ty) 相似文献   

Rapid chiral analysis based on liquid-phase cyclic chemiluminescence

Rapid chiral analysis has become one of the important aspects of academic and industrial research. Here we describe a new strategy based on liquid-phase cyclic chemiluminescence (CCL) for rapid resolution of enantiomers and determination of enantiomeric excess (ee). A single CCL measurement can acquire multistage signals that provide a unique way to examine the intermolecular interactions between chiral hosts and chiral guests, because the lifetime (τ) of the multistage signals is a concentration-independent and distinguishable constant for a given chiral host–guest system. According to the τ values, CCL allows discrimination between a wide range of enantiomeric pairs including chiral alcohols, amines and acids by using only one chiral host. Even the chiral systems hardly distinguished by nuclear magnetic resonance and fluorescence methods can be distinguished easily by CCL. Additionally, the τ value of a mixture of two enantiomers is equal to the weighted average of each enantiomer, which can be used for the direct determination of ee without the need to separate the chiral mixture and create calibration curves. This is extremely crucial for the cases without readily available enantiomerically pure samples. This strategy was successfully applied to monitoring of the Walden inversion reaction and analysis of chiral drugs. The results were in good agreement with those obtained by high-performance liquid chromatography, indicating the utility of CCL for routine quick ee analysis. Mechanism study revealed that the τ value is possibly related to the activity of the chiral substance to catalyze a luminol–H₂O₂ reaction. Our research provides an unprecedented and general protocol for chirality differentiation and ee determination, which is anticipated to be a useful technology that will find wide application in chirality-related fields, particularly in asymmetric synthesis and the pharmaceutical industry.

Rapid chiral analysis has become one of the important aspects of academic and industrial research.