We have developed a direct amine/acid-catalyzed stereoselective hydrogenation of a variety of Wieland-Miescher (W-M) ketones, Hajos-Parrish (H-P) ketones and their analogs with organic hydrides (Hantzsch esters) as the hydrogen source. This astonishingly simple and biomimetic approach was used to construct highly functionalized chiral bicyclo[4.4.0]decane-1,6-diones in a diastereoselective fash...

Identity (1) is easy to prove: note that the left side is a convolution, so multiply it by xn , sum, and recognize the square of a binomial series. But a combinatorial interpretation of the same identity is not so easy to find. In [3], M. Sved recounts the story of the identity and its combinatorial proofs. She relates how, after she challenged her readers (in a previous article) to find a comb...

A hole is a chordless cycle of length at least four. A hole is even (resp. odd) if it contains an even (resp. odd) number of vertices. A cap is a graph induced by a hole with an additional vertex that is adjacent to exactly two adjacent vertices on the hole. In this note, we use a decomposition theorem by Conforti et al. (1999) to show that if a graph G does not contain any even hole or cap as ...

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We analyze three-body decays of and . Under the factorization approach, there are tree level diagrams for these decay modes and the transition matrix element of decay is factorized into a form factor multiplied by decay constant and form factor multiplied into weak vertices form factor. The transition matrix element of decay is also factorized into a form factor multiplied into weak vertic...