Catalytic asymmetric alkylation of α-cyanocarboxylates and acetoacetates using a phase-transfer catalyst

The catalytic asymmetric alkylation of α-cyanocarboxylates and acetoacetates with an alkyl halide was performed under phase-transfer conditions to afford compounds which have a chiral quaternary carbon with up to 97% and 94% ee, respectively. As applications of this method, chiral 2-oxindole derivatives and a β-lactam derivative were synthesized.     

Sharp bounds on Castelnuovo-Mumford regularity

The Castelnuovo-Mumford regularity is one of the most important invariants in studying the minimal free resolution of the defining ideals of the projective varieties. There are some bounds on the Castelnuovo-Mumford regularity of the projective variety in terms of the other basic measures such as dimension, codimension and degree. In this paper we consider an upper bound on the regularity of a nondegenerate projective variety , , provided is -Buchsbaum for , and investigate the projective variety with its Castelnuovo-Mumford regularity having such an upper bound.

     

Computation of multiple eigenvalues of infinite tridiagonal matrices

In this paper, it is first given as a necessary and sufficient condition that infinite matrices of a certain type have double eigenvalues. The computation of such double eigenvalues is enabled by the Newton method of two variables. The three-term recurrence relations obtained from its eigenvalue problem (EVP) subsume the well-known relations of (A) the zeros of ; (B) the zeros of ; (C) the EVP of the Mathieu differential equation; and (D) the EVP of the spheroidal wave equation. The results of experiments are shown for the three cases (A)-(C) for the computation of their ``double pairs'.

     

Mutual Kinetic Resolution in Aldol Condensation between 2-Phenylpropanal and 4,5-Dihydro-5-methyl-2-(trimethylsiloxy)-3-(trimethylsilyl)furan

The titanium(IV) chloride-mediated aldol condensation between title compounds proceeds with high diastereoselectivity (Cram) to give $\text{3a}$ exclusively (99% purity).