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This article discusses the synthesis of ¿-aryl carbonyl compounds using palladium catalysis. The authors focus on the deprotonative ¿-arylation of Meyers's chiral bicyclic lactams (MCBLs) as a potential method for synthesizing these compounds. They explore different bases and catalytic systems to optimize the reaction conditions and achieve high yields. The authors also investigate the reaction scope by testing various aryl and heteroaryl bromides. They propose a mechanism for the formation of the arylated lactam products based on their experimental findings. Overall, this study provides valuable insights into the synthesis of ¿-aryl carbonyl compounds and highlights the potential of MCBLs as versatile substrates. [Extracted from the article]

The number of subtrees, or simply the subtree number, is one of the most studied counting-based graph invariants that has applications in many interdisciplinary fields such as phylogenetic reconstruction. Motivated from the study of graph surgeries on evolutionary dynamics, we consider the subtree problems of fan graphs, wheel graphs, and the class of graphs obtained from "partitioning" wheel graphs under dynamic evolution. The enumeration of these subtree numbers is done through the so-called subtree generation functions of graphs. With the enumerative result, we briefly explore the extremal problems in the corresponding class of graphs. Some interesting observations on the behavior of the subtree number are also presented. [ABSTRACT FROM AUTHOR]

A letter to the editor is presented concerning the effects of association in cohort studies of metformin and Vitamin B12 deficiency.