Secondary homotopy groups

Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of homotopy types with homotopy groups concentrated in two consecutive dimensions.Comment: We added further commets and references to make the paper more easily readabl

Highly diverse nirK genes comprise two major clades that harbour ammonium-producing denitrifiers

Background: Copper dependent nitrite reductase, NirK, catalyses the key step in denitrification, i.e. nitrite reduction to nitric oxide. Distinct structural NirK classes and phylogenetic clades of NirK-type denitrifiers have previously been observed based on a limited set of NirK sequences, however, their environmental distribution or ecological strategies are currently unknown. In addition, environmental nirK-type denitrifiers are currently underestimated in PCR-dependent surveys due to primer coverage limitations that can be attributed to their broad taxonomic diversity and enormous nirK sequence divergence. Therefore, we revisited reported analyses on partial NirK sequences using a taxonomically diverse, full-length NirK sequence dataset. Results: Division of NirK sequences into two phylogenetically distinct clades was confirmed, with Clade I mainly comprising Alphaproteobacteria (plus some Gamma- and Betaproteobacteria) and Clade II harbouring more diverse taxonomic groups like Archaea, Bacteroidetes, Chloroflexi, Gemmatimonadetes, Nitrospirae, Firmicutes, Actinobacteria, Planctomycetes and Proteobacteria (mainly Beta and Gamma). Failure of currently available primer sets to target diverse NirK-type denitrifiers in environmental surveys could be attributed to mismatches over the whole length of the primer binding regions including the 3' site, with Clade II sequences containing higher sequence divergence than Clade I sequences. Simultaneous presence of both the denitrification and DNRA pathway could be observed in 67 % of all NirK-type denitrifiers. Conclusion: The previously reported division of NirK into two distinct phylogenetic clades was confirmed using a taxonomically diverse set of full-length NirK sequences. Enormous sequence divergence of nirK gene sequences, probably due to variable nirK evolutionary trajectories, will remain an issue for covering diverse NirK-type denitrifiers in amplicon-based environmental surveys. The potential of a single organism to partition nitrate to either denitrification or dissimilatory nitrate reduction to ammonium appeared to be more widespread than originally anticipated as more than half of all NirK-type denitrifiers were shown to contain both pathways in their genome

Exact sequences of tensor categories

We introduce the notions of normal tensor functor and exact sequence of tensor categories. We show that exact sequences of tensor categories generalize strictly exact sequences of Hopf algebras as defined by Schneider, and in particular, exact sequences of (finite) groups. We classify exact sequences of tensor categories C' -> C -> C'' (such that C' is finite) in terms of normal faithful Hopf monads on C'' and also, in terms of self-trivializing commutative algebras in the center of C. More generally, we show that, given any dominant tensor functor C -> D admitting an exact (right or left) adjoint there exists a canonical commutative algebra A in the center of C such that F is tensor equivalent to the free module functor C -> mod_C A, where mod_C A denotes the category of A-modules in C endowed with a monoidal structure defined using the half-braiding of A. We re-interpret equivariantization under a finite group action on a tensor category and, in particular, the modularization construction, in terms of exact sequences, Hopf monads and commutative central algebras. As an application, we prove that a braided fusion category whose dimension is odd and square-free is equivalent, as a fusion category, to the representation category of a group.Comment: 39 page

Molecular phylogeny and evolution of <i>Parabasalia</i> with improved taxon sampling and new protein markers of actin and elongation factor-1α

Background: Inferring the evolutionary history of phylogenetically isolated, deep-branching groups of taxa—in particular determining the root—is often extraordinarily difficult because their close relatives are unavailable as suitable outgroups. One of these taxonomic groups is the phylum Parabasalia, which comprises morphologically diverse species of flagellated protists of ecological, medical, and evolutionary significance. Indeed, previous molecular phylogenetic analyses of members of this phylum have yielded conflicting and possibly erroneous inferences. Furthermore, many species of Parabasalia are symbionts in the gut of termites and cockroaches or parasites and therefore formidably difficult to cultivate, rendering available data insufficient. Increasing the numbers of examined taxa and informative characters (e.g., genes) is likely to produce more reliable inferences. Principal Findings: Actin and elongation factor-1a genes were identified newly from 22 species of termite-gut symbionts through careful manipulations and seven cultured species, which covered major lineages of Parabasalia. Their protein sequences were concatenated and analyzed with sequences of previously and newly identified glyceraldehyde-3-phosphate dehydrogenase and the small-subunit rRNA gene. This concatenated dataset provided more robust phylogenetic relationships among major groups of Parabasalia and a more plausible new root position than those previously reported. Conclusions/Significance: We conclude that increasing the number of sampled taxa as well as the addition of new sequences greatly improves the accuracy and robustness of the phylogenetic inference. A morphologically simple cell is likely the ancient form in Parabasalia as opposed to a cell with elaborate flagellar and cytoskeletal structures, which was defined as most basal in previous inferences. Nevertheless, the evolution of Parabasalia is complex owing to several independent multiplication and simplification events in these structures. Therefore, systematics based solely on morphology does not reflect the evolutionary history of parabasalids

Growth of maps, distortion in groups and symplectic geometry

In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism: the uniform norm of the differential of its n-th iteration and the word length of its n-th iteration. In the latter case we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms. We find lower bounds on the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology. Applications to the Zimmer program are presented. We prove non-existence of certain non-linear symplectic representations for finitely generated groups including some lattices and Baumslag-Solitar groups.Comment: Latex, 45 pages, revised version: new results on the Zimmer program added; proofs of the distortion bounds rewritten in a more geometric language; section on maps with the slow growth removed and will appear separatel

Isolation and linkage mapping of NBS-LRR resistance gene analogs in red raspberry ( Rubus idaeus L.) and classification among 270 Rosaceae NBS-LRR genes

Plant R genes confer resistance to pathogens in a gene-for-gene mode. Seventy-five putative resistance gene analogs (RGAs) containing conserved domains were cloned from Rubus idaeus L. cv. ‘Latham' using degenerate primers based on RGAs identified in Rosaceae species. The sequences were compared to 195 RGA sequences identified from five Rosaceae family genera. Multiple sequence alignments showed high similarity at multiple nucleotide-binding site (NBS) motifs with homology to Drosophila Toll and mammalian interleukin-1 receptor (TIR) and non-TIR RNBSA-A motifs. The TIR sequences clustered separately from the non-TIR sequences with a bootstrap value of 76%. There were 11 clusters each of TIR and non-TIR type sequences of multiple genera with bootstrap values of more than 50%, including nine with values of more than 75% and seven of more than 90%. Polymorphic sequence characterized amplified region and cleaved amplified polymorphic sequence markers were developed for nine Rubus RGA sequences with eight placed on a red raspberry genetic linkage map. Phylogenetic analysis indicated four of the mapped sequences share sequence similarity to groupTIR I, while three others were spread in non-TIR groups. Of the 75 Rubus RGA sequences analyzed, members were placed in five TIR groups and six non-TIR groups. These group classifications closely matched those in 12 of 13 studies from which these sequences were derived. The analysis of related DNA sequences within plant families elucidates the evolutionary relationship and process involved in pest resistance development in plants. This information will aid in the understanding of R genes and their proliferation within plant genome