PhylomeDB v3.0: an expanding repository of genome-wide collections of trees, alignments and phylogeny-based orthology and paralogy predictions

The growing availability of complete genomic sequences from diverse species has brought about the need to scale up phylogenomic analyses, including the reconstruction of large collections of phylogenetic trees. Here, we present the third version of PhylomeDB (http://phylomeDB.org), a public database for genome-wide collections of gene phylogenies (phylomes). Currently, PhylomeDB is the largest phylogenetic repository and hosts 17 phylomes, comprising 416 093 trees and 165 840 alignments. It is also a major source for phylogeny-based orthology and paralogy predictions, covering about 5 million proteins in 717 fully-sequenced genomes. For each protein-coding gene in a seed genome, the database provides original and processed alignments, phylogenetic trees derived from various methods and phylogeny-based predictions of orthology and paralogy relationships. The new version of phylomeDB has been extended with novel data access and visualization features, including the possibility of programmatic access. Available seed species include model organisms such as human, yeast, Escherichia coli or Arabidopsis thaliana, but also alternative model species such as the human pathogen Candida albicans, or the pea aphid Acyrtosiphon pisum. Finally, PhylomeDB is currently being used by several genome sequencing projects that couple the genome annotation process with the reconstruction of the corresponding phylome, a strategy that provides relevant evolutionary insights

Remarks on the Cauchy functional equation and variations of it

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.Comment: To appear in Aequationes Mathematicae (important remark: the acknowledgments section in the official paper exists, but it appears before the appendix and not before the references as in the arXiv version); correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of Theorem 2.1; a few small improvements in various sections; added thank