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Allylic functionalization reactions are an important tool for constructing many organic compounds, as they allow for the insertion of the versatile allyl group. Numerous catalytic processes have been developed, primarily utilizing noble metals such as palladium and pre‐functionalized substrates. Conversely, the copper‐catalyzed oxidative C−H derivatization route, whether thermal or photocatalyzed, remains relatively underdeveloped. This is mostly due to the lack of a general concept and corresponding mechanistic understanding. Here we present a comprehensive analysis of the mechanism of this transformation. We utilized a range of spectroscopic techniques, including UV‐Vis, NMR, and EPR, single crystal x‐ray diffraction, as well as cyclovoltammetry, and DFT calculations. We propose that there are no free organic radicals involved in this cross‐dehydrogenative coupling, and that the reaction pathway is analogous to the well‐known Karash‐Sosnovsky reaction. [ABSTRACT FROM AUTHOR]

Carboxy to methyl reduction is an important transformation in organic synthesis, yet existing methodologies often require multi‐step procedures or use hazardous metal hydrides. Herein, a metal‐free catalytic system is reported for the one‐step reduction of esters, carboxylic acids, and carbamates to a methyl group, in the presence of catalytic amounts of boronic acids. By using ammonia borane as a hydrogen donor, a wide range of products bearing different functional groups can be obtained in high yields under relatively mild conditions. Mechanistic studies and control experiments elucidate the complexity of the mechanism and provide an explanation for the observed selectivity. [ABSTRACT FROM AUTHOR]

The Analytic Hierarchy Process (AHP) is a decision making method, which has as its greatest criticism the rank reversal effect. Here a new mathematical analysis of this method is performed, and three new results are highlighted. First, the method is formulated as a linear system of equations, where it is possible to assign a geometric interpretation, determine the number of possible solutions, and perform an sensitivity analysis based on the condition number of the matrix. Second, the causes of rank reversal can be encompassed by two mathematical aspects related to the properties of the matrix of AHP: high condition number and deficient rank. When the matrix is deficient rank, it is possible to obtain a condensed formulation of the AHP with a new full rank matrix. This guarantees greater stability to the method. Third, some mathematical results can be used as a robustness test for the matrix of AHP. [ABSTRACT FROM AUTHOR]