The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex do...The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex domain Er,ρ such that F((z,r)) eiαEr,ρ = {eiαz : z ∈ Er,ρ} holds for every z ∈ D, w = ρeiα and harmonic mapping F with F(D)D and F(z) = w, where △(z,r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρeiα and w ∈ eiαEr,ρ, there exists a harmonic mapping F such that F(D) D, F(z) = w and F(z ) = w for some z ∈ △(z,r). (II) The author establishes a Finsler metric Hz(u) on the unit disk D such that HF(z)(eiθFz(z) + e-iθFz(z)) ≤1 /(1- |z|2)holds for any z ∈ D, 0 θ 2π and harmonic mapping F with F(D)D; furthermore, this result is precise and the equality may be attained for any values of z, θ, F(z) and arg(eiθFz(z) + e-iθFz(z)).展开更多
[ Objective] To provide effective primers for the rapid and accurate ISSR analysis of the germplasm materials of Mallotus oblongiolus （Miq.） Muello-Arg.. [Method] The modified CTAB method was used in the extraction ...[ Objective] To provide effective primers for the rapid and accurate ISSR analysis of the germplasm materials of Mallotus oblongiolus （Miq.） Muello-Arg.. [Method] The modified CTAB method was used in the extraction of the genomic DNA. 99 ISSR primers were used in the ISSR-PCR amplification for 20 germplasm materials from 10 populations in Hainan Island, so that some primers, which were suitable to all gerplasm materials of M. oblongiolu, could be selected. [ Result] 15 effective primers with characteristics of rich polymorphism, clear bands, and good repeatability were selected from 99 test primers. The 15 primers selected were used in the ISSR-PCR amplification for 66 germplasm materials of M. oblongiolus. From all of which the abundant and distinct DNA fingerprintings could be obtained. 286 DNA bands were obtained, and of which 231 bands were polymorphic, which amounted to 80.77% of the total bands amplified. And 19.1 bands could be obtained with each primer, averagely. [ Conclusion] The 15 primers selected could be effectively applied to ISSR analysis of the germplasm resources of M. oblongiolus.展开更多
基金supported by National Natural Science Foundation of China (Grant No.10671093)
文摘The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex domain Er,ρ such that F((z,r)) eiαEr,ρ = {eiαz : z ∈ Er,ρ} holds for every z ∈ D, w = ρeiα and harmonic mapping F with F(D)D and F(z) = w, where △(z,r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρeiα and w ∈ eiαEr,ρ, there exists a harmonic mapping F such that F(D) D, F(z) = w and F(z ) = w for some z ∈ △(z,r). (II) The author establishes a Finsler metric Hz(u) on the unit disk D such that HF(z)(eiθFz(z) + e-iθFz(z)) ≤1 /(1- |z|2)holds for any z ∈ D, 0 θ 2π and harmonic mapping F with F(D)D; furthermore, this result is precise and the equality may be attained for any values of z, θ, F(z) and arg(eiθFz(z) + e-iθFz(z)).
基金Supported by Special Fund for Key Disciplines Program of HainanUniversity~~
文摘[ Objective] To provide effective primers for the rapid and accurate ISSR analysis of the germplasm materials of Mallotus oblongiolus （Miq.） Muello-Arg.. [Method] The modified CTAB method was used in the extraction of the genomic DNA. 99 ISSR primers were used in the ISSR-PCR amplification for 20 germplasm materials from 10 populations in Hainan Island, so that some primers, which were suitable to all gerplasm materials of M. oblongiolu, could be selected. [ Result] 15 effective primers with characteristics of rich polymorphism, clear bands, and good repeatability were selected from 99 test primers. The 15 primers selected were used in the ISSR-PCR amplification for 66 germplasm materials of M. oblongiolus. From all of which the abundant and distinct DNA fingerprintings could be obtained. 286 DNA bands were obtained, and of which 231 bands were polymorphic, which amounted to 80.77% of the total bands amplified. And 19.1 bands could be obtained with each primer, averagely. [ Conclusion] The 15 primers selected could be effectively applied to ISSR analysis of the germplasm resources of M. oblongiolus.
