Projectively related complex Finsler metrics

In this paper we introduce in study the projectively related complex Finsler metrics. We prove the complex versions of the Rapcsák’s theorem and characterize the weakly Kähler and generalized Berwald projectively related complex Finsler metrics. The complex version of Hilbert’s Fourth Problem is also pointed out. As an application, the projectiveness of a complex Randers metric is described.     

The Adjoints of Dilation Operators on H-Cones

If S is an H-cone and P:SS is a localizable dilation operator on S (i.e., P is additive increasing, contractive, continuous in order from below and s(Ps+t–Pt+Pf)S, s,tS,f(S–S)₊), then it is proved that its adjoint P ^*:S ^*S ^* (i.e., P ^*=P) is also a localizable dilation operator. This is an improvement of a result obtained by G. Mokobodzki in the frame of excessive functions.     

A contraction proximal point algorithm with two monotone operators

It is a known fact that the method of alternating projections introduced long ago by von Neumann fails to converge strongly for two arbitrary nonempty, closed and convex subsets of a real Hilbert space. In this paper, a new iterative process for finding common zeros of two maximal monotone operators is introduced and strong convergence results associated with it are proved. If the two operators are subdifferentials of indicator functions, this new algorithm coincides with the old method of alternating projections. Several other important algorithms, such as the contraction proximal point algorithm, occur as special cases of our algorithm. Hence our main results generalize and unify many results that occur in the literature.     

Cytochrome P450 Catalyzed Oxidative Hydroxylation of Achiral Organic Compounds with Simultaneous Creation of Two Chirality Centers in a Single C?H Activation Step

Regio‐ and stereoselective oxidative hydroxylation of achiral or chiral organic compounds mediated by synthetic reagents, catalysts, or enzymes generally leads to the formation of one new chiral center that appears in the respective enantiomeric or diastereomeric alcohols. By contrast, when subjecting appropriate achiral compounds to this type of C H activation, the simultaneous creation of two chiral centers with a defined relative and absolute configuration may result, provided that control of the regio‐, diastereo‐, and enantioselectivity is ensured. The present study demonstrates that such control is possible by using wild type or mutant forms of the monooxygenase cytochrome P450 BM3 as catalysts in the oxidative hydroxylation of methylcyclohexane and seven other monosubstituted cyclohexane derivatives.