A note on strictly cyclic Shifts on ℓ

In this paper the author shows that a well known sufficient condition for strict cycliclty of a weighted shift on ℓp is not a necessary condition for any p with 1<p<∞

On perfect neighborhood sets in graphs

AbstractLet G = (V, E) be a graph and let S ⊆ V.. The set S is a dominating set of G is every vertex of V − S is adjacent to a vertex of S. A vertex v of G is called S-perfect if |N[ν]∩S| = 1 where N[v] denotes the closed neighborhood of v. The set S is defined to be a perfect neighborhood set of G if every vertex of G is S-perfect or adjacent with an S-perfect vertex. We prove that for all graphs G, Θ(G) = Γ(G) where Γ(G) is the maximum cardinality of a minimal dominating set of G and where Θ(G) is the maximum cardinality among all perfect neighborhood sets of G

Adaptations of Pseudoxylaria towards a comb-associated lifestyle in fungus-farming termite colonies

DATA AVAILABILITY: Supporting Information of this article is free of charge and contains list of accession numbers of sequences used for analysis, phylogenetic trees, cultivation studies including co-cultivation, analyses of genomic and metabolomic data, NMR and MS-data of isolated metabolites and data of insect feeding studies including statistical analyses.Characterizing ancient clades of fungal symbionts is necessary for understanding the evolutionary process underlying symbiosis development. In this study, we investigated a distinct subgeneric taxon of Xylaria (Xylariaceae), named Pseudoxylaria, whose members have solely been isolated from the fungus garden of farming termites. Pseudoxylaria are inconspicuously present in active fungus gardens of termite colonies and only emerge in the form of vegetative stromata, when the fungus comb is no longer attended (“sit and wait” strategy). Insights into the genomic and metabolic consequences of their association, however, have remained sparse. Capitalizing on viable Pseudoxylaria cultures from different termite colonies, we obtained genomes of seven and transcriptomes of two Pseudoxylaria isolates. Using a whole-genome-based comparison with free-living members of the genus Xylaria, we document that the association has been accompanied by significant reductions in genome size, protein-coding gene content, and reduced functional capacities related to oxidative lignin degradation, oxidative stress responses and secondary metabolite production. Functional studies based on growth assays and fungus-fungus co-cultivations, coupled with isotope fractionation analysis, showed that Pseudoxylaria only moderately antagonizes growth of the termite food fungus Termitomyces, and instead extracts nutrients from the food fungus biomass for its own growth. We also uncovered that Pseudoxylaria is still capable of producing structurally unique metabolites, which was exemplified by the isolation of two novel metabolites, and that the natural product repertoire correlated with antimicrobial and insect antifeedant activity.The German Research Foundation (DFG, Deutsche Forschungsgemeinschaft), the Germany´s Excellence Strategy, the European Research Council and The Danish Council for Independent Research. Open Access funding enabled and organized by Projekt DEAL.https://www.nature.com/ismejBiochemistryGeneticsMicrobiology and Plant Patholog

Critical Graphs With Respect to Vertex Identification

We explore various types of criticality with respect to differentiatingdominating sets, or identifying codes. Existence and characterization results are included. We conclude with open problems

REDUCING THE ADJACENCY MATRIX OF A TREE

Let T be a tree, A its adjacency matrix, and a scalar. We describe a linear-time algorithm for reducing the matrix In + A. Applications include computing the rank of A, finding a maximum matching in T, computing the rank and determinant of the associated neighborhood matrix, and computing the characteristic polynomial of A

Reducing the adjacency matrix of a tree

Let T be a tree, A its adjacency matrix, and a scalar. We describe a linear-time algorithm for reducing the matrix In + A. Applications include computing the rank of A, nding a maximum matching in T , computing the rank and determinant of the associated neighborhood matrix, and computing the characteristic polynomial of A