In 2002, Faugere presented the famous F5 algorithm for computing GrSbner basis where two cri- teria, syzygy criterion and rewritten criterion, were proposed to avoid redundant computations. He proved the correctness o...In 2002, Faugere presented the famous F5 algorithm for computing GrSbner basis where two cri- teria, syzygy criterion and rewritten criterion, were proposed to avoid redundant computations. He proved the correctness of the syzygy criterion, but the proof for the correctness of the rewritten criterion was left. Since then, F5 has been studied extensively. Some proofs for the correctness of F5 were proposed, but these proofs are valid only under some extra assumptions. In this paper, we give a proof for the correctness of F5B, an equivalent version of F5 in Buchberger＇s style. The proof is valid for both homogeneous and non-homogeneous polynomial systems. Since this proof does not depend on the computing order of the S-pairs, any strategy of selecting S-pairs could be used in F5B or F5. Furthermore, we propose a natural and non-incremental variant of F5 where two revised criteria can be used to remove almost all redundant S-pairs.展开更多
The phosphoryl transfer reactions of diphenyl chlorophosphate with a wide range of alcohols were catalyzed by tris（pentafluorophenyl）boron [B（C6F5）3], giving the corresponding phosphate esters in excellent yields....The phosphoryl transfer reactions of diphenyl chlorophosphate with a wide range of alcohols were catalyzed by tris（pentafluorophenyl）boron [B（C6F5）3], giving the corresponding phosphate esters in excellent yields. This methodology has advantages over older ones because it employs the easily handled B（C6F5）3, avoids the side reaction, and has shorter reaction time.展开更多
基金supported by National Key Basic Research Project of China (Grant No.2011CB302400)National Natural Science Foundation of China (Grant Nos. 10971217 and 61121062)
文摘In 2002, Faugere presented the famous F5 algorithm for computing GrSbner basis where two cri- teria, syzygy criterion and rewritten criterion, were proposed to avoid redundant computations. He proved the correctness of the syzygy criterion, but the proof for the correctness of the rewritten criterion was left. Since then, F5 has been studied extensively. Some proofs for the correctness of F5 were proposed, but these proofs are valid only under some extra assumptions. In this paper, we give a proof for the correctness of F5B, an equivalent version of F5 in Buchberger＇s style. The proof is valid for both homogeneous and non-homogeneous polynomial systems. Since this proof does not depend on the computing order of the S-pairs, any strategy of selecting S-pairs could be used in F5B or F5. Furthermore, we propose a natural and non-incremental variant of F5 where two revised criteria can be used to remove almost all redundant S-pairs.
文摘The phosphoryl transfer reactions of diphenyl chlorophosphate with a wide range of alcohols were catalyzed by tris（pentafluorophenyl）boron [B（C6F5）3], giving the corresponding phosphate esters in excellent yields. This methodology has advantages over older ones because it employs the easily handled B（C6F5）3, avoids the side reaction, and has shorter reaction time.
