Convergence of the majorization method for multidimensional scaling

In this paper we study the convergence properties of an important class of multidimensional scaling algorithms. We unify and extend earlier qualitative results on convergence, which tell us when the algorithms are convergent. In order to prove global convergence results we use the majorization method. We also derive, for the first time, some quantitative convergence theorems, which give information about the speed of convergence. It turns out that in almost all cases convergence is linear, with a convergence rate close to unity. This has the practical consequence that convergence will usually be very slow, and this makes techniques to speed up convergence very important. It is pointed out that step-size techniques will generally not succeed in producing marked improvements in this respect.     

The draft genome of the transgenic tropical fruit tree papaya (Carica papaya Linnaeus)

Papaya, a fruit crop cultivated in tropical and subtropical regions, is known for its nutritional benefits and medicinal applications. Here we report a 3x draft genome sequence of 'SunUp' papaya, the first commercial virus-resistant transgenic fruit tree to be sequenced. The papaya genome is three times the size of the Arabidopsis genome, but contains fewer genes, including significantly fewer disease-resistance gene analogues. Comparison of the five sequenced genomes suggests a minimal angiosperm gene set of 13,311. A lack of recent genome duplication, atypical of other angiosperm genomes sequenced so far, may account for the smaller papaya gene number in most functional groups. Nonetheless, striking amplifications in gene number within particular functional groups suggest roles in the evolution of tree-like habit, deposition and remobilization of starch reserves, attraction of seed dispersal agents, and adaptation to tropical daylengths. Transgenesis at three locations is closely associated with chloroplast insertions into the nuclear genome, and with topoisomerase I recognition sites. Papaya offers numerous advantages as a system for fruit-tree functional genomics, and this draft genome sequence provides the foundation for revealing the basis of Carica's distinguishing morpho-physiological, medicinal and nutritional properties.