Canonical lossless state-space systems: Staircase forms and the Schur algorithm

A new finite atlas of overlapping balanced canonical forms for multivariate discrete-time lossless systems is presented. The canonical forms have the property that the controllability matrix is positive upper triangular up to a suitable permutation of its columns. This is a generalization of a similar balanced canonical form for continuous-time lossless systems. It is shown that this atlas is in fact a finite sub-atlas of the infinite atlas of overlapping balanced canonical forms for lossless systems that is associated with the tangential Schur algorithm; such canonical forms satisfy certain interpolation conditions on a corresponding sequence of lossless transfer matrices. The connection between these balanced canonical forms for lossless systems and the tangential Schur algorithm for lossless systems is a generalization of the same connection in the SISO case that was noted before. The results are directly applicable to obtain a finite sub-atlas of multivariate input-normal canonical forms for stable linear systems of given fixed order, which is minimal in the sense that no chart can be left out of the atlas without losing the property that the atlas covers the manifold

Recombination confounds the early evolutionary history of human immunodeficiency virus type 1: Subtype G is a circulating recombinant form

Human immunodeficiency virus type I (HIV-1) is classified in nine subtypes (A to D, F, G, H, J, and K), a number of subsubtypes, and several circulating recombinant forms (CRFs). Due to the high level of genetic diversity within HIV-1 and to its worldwide distribution, this classification system is widely used in fields as diverse as vaccine development, evolution, epidemiology, viral fitness, and drug resistance. Here, we demonstrate how the high recombination rates of HIV-1 may confound the study of its evolutionary history and classification. Our data show that subtype G, currently classified as a pure subtype, has in fact a recombinant history, having evolved following recombination between subtypes A and J and a putative subtype G parent. In addition, we find no evidence for recombination within one of the lineages currently classified as a CRF, CRF02 -AG. Our analysis indicates that CRF02_AG was the parent of the recombinant subtype G, rather than the two having the opposite evolutionary relationship, as is currently proposed. Our results imply that the current classification of HIV-1 subtypes and CRFs is an artifact of sampling history, rather than reflecting the evolutionary history of the virus. We suggest a reanalysis of all pure subtypes and CRFs in order to better understand how high rates of recombination have influenced HIV-1 evolutionary history.</p