Metathesis-based synthesis of 3-methoxy α,β-unsaturated lactones: total synthesis of (R)-kavain and of the C1-C6 fragment of jerangolid D

The total synthesis of (R)-kavain and of the C1-C6 fragment of jerangolid D has been achieved in nine and seven steps, respectively, from commercially available dimethyl d-malate. A metathesis reaction of vinyl ethers and a sulfoxide-modified Julia olefination have been employed as the key steps.     

Total synthesis of peridinin and related C37-norcarotenoid butenolides

As an extension of our synthesis of symmetrical carotenoids, the preparation of the highly functionalized C37-norcarotenoid butenolide peridinin (1), its 6'-epi- and 11'Z stereoisomers has been completed. Featuring a central dihalogenated C8 linchpin unit 6, two synthetic routes, differing in the ordering of the last three steps were explored by using the C3,C3'-bisdehydroxylated target as the model system. The first route uses the combination of a modified Z-selective Julia reaction and two sequential Stille couplings, the last one producing the isomerisation of the polyene Z double bond. The second route inverts these steps and makes the isolation of the 11'Z stereoisomers as major products possible. An efficient Z to E isomerisation of the final carotenoid skeleton simply uses the Stille reaction conditions at ambient temperature. As the reaction of bromoallene 12 with alkenylstannane 11 occurs with inversion of configuration, 6'-epi-peridinin could also be prepared by route A. The advantages and limitations of the sequential Stille cross-coupling approach to carotenoids are highlighted.     

Limiting directions of julia sets of entire solutions to complex differential equations

In this paper, we mainly investigate the dynamical properties of entire solutions of complex differential equations. With some conditions on coefficients, we prove that the set of common limiting directions of Julia sets of solutions, their derivatives and their primitives must have a definite range of measure.     

Dynamics of Commuting Rational Maps on Berkovich Projective Space over C_p

We study the dynamics of commuting rational maps with coefficients in Cp. By lifting the dynamics from P1(Cp) to Berkovich projective space P1 Berk, we prove that two nonlinear commuting maps have the same Berkovich Julia set and the same canonical measure. As a consequence, two nonlinear commuting maps with coefficient in Cp have the same classical Julia set. We also prove that they have the same pre-periodic Berkovich Fatou components.     

Existence of herman rings for meromorphic functions

We apply the Shishikura surgery construction to transcendental maps in order to obtain examples of meromorphic functions with Herman rings, in a variety of possible arrangements. We give a sharp bound on the maximum possible number of such rings that a meromorphic function may have, in terms of the number of poles. Finally we discuss the possibility of having “unbounded” Herman rings (i.e., with an essential singularity in the boundary), and give some examples of maps with this property.     

Carathéodory–Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit disk

It is shown that the following conditions are equivalent for the generalized Schur class functions at a boundary point t₀ ∈ ??: 1) Carathéodory–Julia type condition of order n; 2) agreeing of asymptotics of the original function from inside and of its continuation by reflection from outside of the unit disk ?? up to order 2n + 1; 3) t₀‐isometry of the coefficients ofthe boundary asymptotics; 4) a certain structured matrix ? constructed from these coefficients being Hermitian. It is also shown that for an arbitrary analytic function, properties 2), 3), 4) are still equivalent to each other and imply 1) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two problems in the value distribution theory

This paper includes two parts. In the first part, we deal with the relations between the orders, the lower orders and the number of Julia directions of entire functions. In the second part, we deal with the existence of the zeros of a class of meromorphic functions. This class of meromorphic functions has an interesting physical background.