Error estimates for approximate approximations on compact intervals

The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact in- tervals, only. This method, which is based on an approximate partition of unity, was introduced by V. Mazya in 1991 and has mainly been used for functions defied on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed. In the present paper we apply the method of approximate approximations to functions which are defined on compact intervals. In contrast to the whole space case here a truncation error has to be controlled in addition. For the resulting total error pointwise estimates and L1-estimates are given, where all the constants are determined explicitly.

Approximate solutions and error estimates for a Stokes boundary value problem

The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes' equations. For this we extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the system of Stokes' equations in two dimensions. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.

Microsatellites from Fosterella christophii (Bromeliaceae) by de novo transcriptome sequencing on the Pacific Biosciences RS platform

• Premise of the study: Microsatellite markers were developed in Fosterella christophii (Bromeliaceae) to investigate the genetic diversity and population structure within the F. micrantha group, comprising F. christophii, F. micrantha, and F. villosula. • Methods and Results: Full-length cDNAs were isolated from F. christophii and sequenced on a Pacific Biosciences RS platform. A total of 1590 high-quality consensus isoforms were assembled into 971 unigenes containing 421 perfect microsatellites. Thirty primer sets were designed, of which 13 revealed a high level of polymorphism in three populations of F. christophii, with four to nine alleles per locus. Each of these 13 loci cross-amplified in the closely related species F. micrantha and F. villosula, with one to six and one to 11 alleles per locus, respectively. • Conclusions: The new markers are promising tools to study the population genetics of F. christophii and to discover species boundaries within the F. micrantha group.