Recent advance in direct sp3 carbon-hydrogen bond functionalizations

Direct and selective carbon-hydrogen bond functionalization has attracted enormous attention because it provides more efficient strategies for preparing valuable functional molecules from easily available substrates. Significant and exciting developments in functionalization of un-activation Csp3-H bonds have continuously been made over the past few decades. This review mainly summarizes recent advances on direct Csp3-H bond functionalization for the formation of C-C and C-heteroatom bond under transition-metal-catalysed and metal-free conditions

Existence of positive solutions for singular boundary value problem on time scales

AbstractThis paper deals with a class of boundary value problem of singular differential equations on time scales. The conditions we used here differ from those in the majority of papers as we know. An existence theorem of positive solutions is established by using the Krasnosel'skii fixed point theorem and an example is given to illustrate it

Optimal Pole Assignment of Linear Systems by the Sylvester Matrix Equations

The problem of state feedback optimal pole assignment is to design a feedback gain such that the closed-loop system has desired eigenvalues and such that certain quadratic performance index is minimized. Optimal pole assignment controller can guarantee both good dynamic response and well robustness properties of the closed-loop system. With the help of a class of linear matrix equations, necessary and sufficient conditions for the existence of a solution to the optimal pole assignment problem are proposed in this paper. By properly choosing the free parameters in the parametric solutions to this class of linear matrix equations, complete solutions to the optimal pole assignment problem can be obtained. A numerical example is used to illustrate the effectiveness of the proposed approach