Carbon-carbon (C-C) bond formation is paramount in the synthesis of biologically relevant molecules, modern synthetic materials and commodity chemicals such as fuels and lubricants. Traditionally, the presence of a functional group is required at the site of C-C bond formation. Strategies that allow C-C bond formation at inert carbon-hydrogen (C-H) bonds allow scientists to access molecules which would otherwise be inaccessible and to develop more efficient syntheses of complex molecules.1,2 Herein we report a method for the formation of C-C bonds by directed cleavage of traditionally non-reactive C-H bonds and their subsequent coupling with readily available alkenes. Our methodology allows for the selective C-C bond formation at single C-H bonds in molecules that contain a multitude of seemingly indifferentiable such bonds. Selectivity arises through a relayed photoredox catalyzed oxidation of an N-H bond. We anticipate our findings to serve as a starting point for functionalization at inert C-H bonds through a hydrogen atom transfer strategy.

The functionalization of C(sp3)−H bonds streamlines chemical synthesis by allowing the use of simple molecules and providing novel synthetic disconnections. Intensive recent efforts in the development of new reactions based on C−H functionalization have led to its wider adoption across a range of research areas. This Review discusses the strengths and weaknesses of three main approaches: transition‐metal‐catalyzed C−H activation, 1,n‐hydrogen atom transfer, and transition‐metal‐catalyzed carbene/nitrene transfer, for the directed functionalization of unactivated C(sp3)−H bonds. For each strategy, the scope, the reactivity of different C−H bonds, the position of the reacting C−H bonds relative to the directing group, and stereochemical outcomes are illustrated with examples in the literature. The aim of this Review is to provide guidance for the use of C−H functionalization reactions and inspire future research in this area.

In this paper, we study positive periodic solutions to the repulsive singular perturbations of the Hill equations. It is proved that such a perturbation problem has at least two positive periodic solutions when the anti-maximum principle holds for the Hill operator and the perturbation is superlinear at infinity. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.

In this paper, we establish two different existence results of positive periodic solutions for second order non-autonomous singular dynamical systems. The first one is based on a nonlinear alternative principle of Leray-Schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity. The second one is based on Schauder's fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions. Recent results in the literature are generalized and significantly improved.