Let M be an open Riemann surface and A be the punctured cone in C n \ {0} on a smooth projective variety Y in P n−1 . Recently, Runge approximation theorems with interpolation for holomorphic immersions M → C n , directed by A, have been proved under the assumption that A is an Oka manifold. We prove analogous results in the algebraic setting, for regular immersions directed by A from a smooth affine curve M into C n . The Oka property is naturally replaced by the stronger assumption that A is algebraically elliptic, which it is if Y is uniformly rational. Under this assumption, a homotopy-theoretic necessary and sufficient condition for approximation and interpolation emerges. We show that this condition is satisfied in many cases of interest.

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A melon (Cucumis melo L.) genomic library of near-isogenic lines derived from the cross between the Spanish cultivar Piel de Sapo and the exotic accession PI 161375 has been evaluated for fruit quality traits in four different locations. Traits evaluated were fruit weight, soluble solids content, maximum fruit diameter, fruit length, fruit shape index, ovary shape index, external color, and flesh color. Among these traits, soluble solids content showed the highest genotype × environment interaction, whereas genotype × environment interactions for fruit shape and fruit weight were low. Heritability was high for all traits except soluble solids content, with the highest value for fruit shape and ovary shape. Ten to 15 quantitative trait loci were detected for soluble solids content, fruit diameter, fruit length, and fruit shape; and four to five for ovary shape, external color, and flesh color. Depending on the trait, between 13% and 40% of the detected quantitative trait alleles from PI 161375 increased the trait, and between 60% and 87% of them decreased it, resulting in some PI 161375 alleles of interest for breeding. Most of the quantitative trait loci detected in previous experiments could be verified with the near-isogenic line population. Future studies with the melon near-isogenic line genomic library will provide a better understanding of the genetic control of melon fruit quality in a wider context related to agronomy, genetics, genomics and metabolomics studies.

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In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in R n and null holomorphic curves in C n for any n ≥ 3. With this tool in hand we construct complete conformally immersed minimal surfaces in R n which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of R n which extend continuously up to the boundary; for n ≥ 5 we find embeddings with these properties.