REALIZING PROFINITE REDUCED SPECIAL GROUPS

Special groups are an axiomatization of the algebraic theory of quadratic forms over fields. It is known that any finite reduced special group is the special group of some field. We show that any special group that is the projective limit of a projective system of finite reduced special groups is also the special group of some field.

Toward a resolution of Campanulid phylogeny, with special reference to the placement of Dipsacales

Broad-scale phylogenetic analyses of the angiosperms and of the Asteridae have failed to confidently resolve relationships among the major lineages of the campanulid Asteridae (i.e., the euasterid II of APG II, 2003). To address this problem we assembled presently available sequences for a core set of 50 taxa, representing the diversity of the four largest lineages (Apiales, Aquifoliales, Asterales, Dipsacales) as well as the smaller ""unplaced"" groups (e.g., Bruniaceae, Paracryphiaceae, Columelliaceae). We constructed four data matrices for phylogenetic analysis: a chloroplast coding matrix (atpB, matK, ndhF, rbcL), a chloroplast non-coding matrix (rps16 intron, trnT-F region, trnV-atpE IGS), a combined chloroplast dataset (all seven chloroplast regions), and a combined genome matrix (seven chloroplast regions plus 18S and 26S rDNA). Bayesian analyses of these datasets using mixed substitution models produced often well-resolved and supported trees. Consistent with more weakly supported results from previous studies, our analyses support the monophyly of the four major clades and the relationships among them. Most importantly, Asterales are inferred to be sister to a clade containing Apiales and Dipsacales. Paracryphiaceae is consistently placed sister to the Dipsacales. However, the exact relationships of Bruniaceae, Columelliaceae, and an Escallonia clade depended upon the dataset. Areas of poor resolution in combined analyses may be partly explained by conflict between the coding and non-coding data partitions. We discuss the implications of these results for our understanding of campanulid phylogeny and evolution, paying special attention to how our findings bear on character evolution and biogeography in Dipsacales.

On cohomologies and extensions of cyclic groups

Cohomology groups H(s)(Z(n), Z(m)) are studied to describe all groups up to isomorphism which are (central) extensions of the cyclic group Z(n) by the Z(n)-module Z(m). Further, for each such a group the number of non-equivalent extensions is determined. (C) 2011 Elsevier B.V. All rights reserved.

On E-functions of semisimple Lie groups

We develop and describe continuous and discrete transforms of class functions on a compact semisimple, but not simple, Lie group G as their expansions into series of special functions that are invariant under the action of the even subgroup of the Weyl group of G. We distinguish two cases of even Weyl groups-one is the direct product of even Weyl groups of simple components of G and the second is the full even Weyl group of G. The problem is rather simple in two dimensions. It is much richer in dimensions greater than two-we describe in detail E-transforms of semisimple Lie groups of rank 3.