Multifold factorizations of cyclic groups into subsets

Let $(G,+)$ be an abelian group. A finite multiset A over G is said to give a λ-fold factorization of G if there exists a multiset B over G such that each element of G occurs λ times in the multiset $A+B:=\{a+b:a\in A,b\in B\}$. In this article, restricting G to a cyclic group, we will provide sufficient conditions on a given multiset A under which the exact value or an upper bound of the minimum multiplicity λ of a factorization of G can be given by introducing a concept of ‘lcm-closure’. Furthermore, a couple of properties on a given factor A will be shown when A has a prime or prime power order (cardinality). A relation to multifold factorizations of the set of integers will be also glanced at a general perspective.     

Carboxylic Acids as Directing Groups for C−H Bond Functionalization

The selective transformation of C?H bonds is one of the most desirable approaches to creating complexity from simple building blocks. Several directing groups are efficient in controlling the regioselectivity of catalytic C?H bond functionalizations. Among them, carboxylic acids are particularly advantageous, since they are widely available in great structural diversity and at low cost. The carboxylate directing groups can be tracelessly cleaved or may serve as the anchor point for further functionalization through decarboxylative couplings. This Minireview summarizes the substantial progress made in the last few years in the development of reactions in which carboxylate groups direct C?H bond functionalizations with formation of C?C, C?O, C?N, or C?halogen bonds at specific positions. It is divided into sections on C?C, C?O, C?N, and C?halogen bond formation, each of which is subdivided by reactions and product classes. Particular emphasis is placed on methods that enable multiple derivatizations by combining carboxylate‐directed C?H functionalization with decarboxylative couplings.     

Solubility and Crystallizability: Facile Access to Functionalized π‐Conjugated Compounds with Chlorendylimide Protecting Groups

Functional π‐conjugated molecules are relevant for the preparation of new organic electronic materials with improved performance. However, their synthesis is often rendered difficult by their inherently low solubility, and the permanent attachment of solubilizing groups may change the properties of the material. Here, we introduced the chlorendylimidyl moiety as a new temporary protecting group for the straightforward large‐scale synthesis of protected quarter‐, sexi‐, octathiophene, and perylene bisimide diamine and dicarboxylic acid derivatives. The obtained chlorendylimides and chlorendylimidyl active esters were highly soluble in organic solvents, and optical spectroscopy confirmed the low tendency of the compounds to aggregate in solution. At the same time, they could be conveniently purified by recrystallization or precipitation. Single‐crystal X‐ray structures obtained for most compounds showed supramolecular motifs highlighting the role of the rigid, polychlorinated chlorendyl moieties in their crystallization. The obtained protected diamine and dicarboxylic acid derivatives were easily deprotected and converted into various amide‐substituted oligothiophenes and perylene bisimides that are of interest as new functional materials for organic electronic thin film or nanowire devices.     

m‐Methoxy Substituents in a Tetraphenylethylene‐Based Hole‐Transport Material for Efficient Perovskite Solar Cells

Three tetrapheynlethylene derivatives (N,N‐di(4‐methoxyphenyl)aminophenyl‐substituted tetraphenylethylene; TPE‐4DPA) with different methoxy positions (pp‐, pm‐, and po‐) have been synthesized and characterized. The methoxy groups can control the oxidation potential of the materials, and the electronic properties of the derivatives were affected by the position of the methoxy substituents. These compounds were synthesized in a facile and cost‐effective way, and were applied as hole‐transport materials in perovskite solar cells. The corresponding cell performances were compared with respect to their structure modifications, and it was found that the derivative with m‐OMe substituents showed the highest power conversion efficiency (PCE) of 15.4 %, with a J_sc value of 20.04 mA cm^?2, a V_oc value of 1.07 V, and a fill factor (FF) value of 0.72, which is higher than the p‐OMe and o‐OMe substituents. Moreover, the PCE of pm‐TPE‐4DPA is comparable with that of the state‐of‐the‐art 2,2′,7,7′‐tetrakis(N,N′‐di‐p‐methoxyphenylamine)‐9,9′‐spirobifluorene under identical conditions.