Immobilized palladium on surface-modified Fe3O4/SiO2 nanoparticles: as a magnetically separable and stable recyclable high-performance catalyst for Suzuki and Heck cross-coupling reactions

A palladium-based catalyst (Fe₃O₄/SiO₂/HPG–OPPh₂–PNP) supported on chlorodiphenylphosphine-functionalized magnetic nanoparticles was successfully prepared from Fe₃O₄/SiO₂ with sequential attachment of glycerol and chlorodiphenylphosphine, followed by treatment of an ethanolic solution of palladium chloride with hydrazine. The as-prepared catalyst was characterized by ICP-AES, FTIR, XRD, SEM, and TEM. The Fe₃O₄/SiO₂/HPG–OPPh₂–PNP was found as a magnetically separable and highly active catalyst for Suzuki coupling reactions of aryl iodides, bromides, and chlorides as well as Heck reactions of aryl iodides and bromides. Under appropriate conditions, all reactions afforded the desired products in moderate to excellent yields. Moreover, this catalyst can be easily recovered by using a magnetic field and directly reused for at least six cycles without significant loss of its activity.     

钾原子中多种受激辐射参与的混频过程

当双原子共振激发钾原子于６Ｓ能级时,由光泵受激辐射和电离复合受激辐射参与的六波或四波混频,产生了位于紫外和紫区的诸多相干辐射、文中对多种混频机制进行了分析和讨论,并对所测得的相干辐射线进行了标识．     

有消费或投资的华氏宏观经济模型

本文对华罗庚教授的宏观经济模型进行了推广 ,在原来模型的基础上 ,分别增加了消费、投资 ,得到两种不同的模型 ,并对模型具有经济意义的解进行了研究     

Convergence rates of cascade algorithms

We consider solutions of a refinement equation of the form

where is a finitely supported sequence called the refinement mask. Associated with the mask is a linear operator defined on by . This paper is concerned with the convergence of the cascade algorithm associated with , i.e., the convergence of the sequence in the -norm.

Our main result gives estimates for the convergence rate of the cascade algorithm. Let be the normalized solution of the above refinement equation with the dilation matrix being isotropic. Suppose lies in the Lipschitz space , where 0$"> and . Under appropriate conditions on , the following estimate will be established:

where and is a constant. In particular, we confirm a conjecture of A. Ron on convergence of cascade algorithms.

     

Trapping neutral fermions with a PT-symmetric trigonometric potential in the presence of position-dependent mass

The relativistic problem of neutral fermions subject to PT-symmetric trigonometric potential (∼iαtanαx)$(\sim \mathrm{i}\alpha \tan \alpha x)$ in 1+1$1+1$ dimensions is investigated. By using the basic concepts of the supersymmetric quantum mechanics formalism and the functional analysis method, we solve exactly the position-dependent effective mass Dirac equation with the vector coupling scheme and obtain the bound state solutions in closed form. The behavior of the energy spectra is discussed in detail.