Geometry of balanced and absorbing subsets of topological modules

We define the concepts of balanced set and absorbing set in modules over topological rings, which coincide with the usual concepts when restricting to topological vector spaces. We show that in a topological module over an absolute semi-valued ring whose invertibles approach [Formula: see text], every neighborhood of [Formula: see text] is absorbing. We also introduce the concept of total closed unit neighborhood of zero (total closed unit) and prove that the only total closed unit of the quaternions [Formula: see text] is its closed unit ball [Formula: see text]. On the other hand, we also prove that if [Formula: see text] is an absolute semi-valued unital real algebra, then its closed unit ball [Formula: see text] is a total closed unit. Finally, we study the geometry of modules via the extreme points and the internal points, showing that no internal point can be an extreme point and that absorbance is equivalent to having [Formula: see text] as an internal point.

Hub connectivity, neuronal diversity, and gene expression in the C. elegans connectome

Studies of nervous system connectivity, in a wide variety of species and at different scales of resolution, have identified several highly conserved motifs of network organization. One such motif is a heterogeneous distribution of connectivity across neural elements, such that some elements act as highly connected and functionally important network hubs. These brain network hubs are also densely interconnected, forming a so-called rich-club. Recent work in mouse has identified a distinctive transcriptional signature of neural hubs, characterized by tightly coupled expression of oxidative metabolism genes, with similar genes characterizing macroscale inter-modular hub regions of the human cortex. Here, we sought to determine whether hubs of the neuronal C. elegans connectome also show tightly coupled gene expression. Using open data on the chemical and electrical connectivity of 279 C. elegans neurons, and binary gene expression data for each neuron across 948 genes, we computed a correlated gene expression score for each pair of neurons, providing a measure of their gene expression similarity. We demonstrate that connections between hub neurons are the most similar in their gene expression while connections between nonhubs are the least similar. Genes with the greatest contribution to this effect are involved in glutamatergic and cholinergic signalling, and other communication processes. We further show that coupled expression between hub neurons cannot be explained by their neuronal subtype (i.e., sensory, motor, or interneuron), separation distance, chemically secreted neurotransmitter, birth time, pairwise lineage distance, or their topological module affiliation. Instead, this coupling is intrinsically linked to the identity of most hubs as command interneurons, a specific class of interneurons that regulates locomotion. Our results suggest that neural hubs may possess a distinctive transcriptional signature, preserved across scales and species, that is related to the involvement of hubs in regulating the higher-order behaviors of a given organism.Author summarySome elements of neural systems possess many more connections than others, marking them as network hubs. These hubs are often densely interconnected with each other, forming a so-called rich-club that is thought to support integrated function. Recent work in the mouse suggests that connected pairs of hubs show higher levels of transcriptional coupling than other pairs of brain regions. Here, we show that hub neurons of the nematode C. elegans also show tightly coupled gene expression and that this effect cannot be explained by the spatial proximity or anatomical location of hub neurons, their chemical composition, birth time, neuronal lineage or topological module affiliation. Instead, we find that elevated coexpression is driven by the identity of most hubs of the C. elegans connectome as command interneurons, a specific functional class of neurons that regulate locomotion. These findings suggest that coupled gene expression is a highly conserved genomic signature of neural hubs that may be related to the specific functional role that hubs play in broader network function.

φ-Multipliers on Banach Algebras and Topological Modules

We prove some results concerning Arens regularity and amenability of the Banach algebraMφAof allφ-multipliers on a given Banach algebraA. We also considerφ-multipliers in the general topological module setting and investigate some of their properties. We discuss theφ-strict andφ-uniform topologies onMφA. A characterization ofφ-multipliers onL1G-moduleLpG, whereGis a compact group, is given.

Maximal independent, analytic sets in Abelian polish groups

The paper deals with the questions: (a)whether a topological module admits maximal linearly independent subsets that are analytic(b)whether an Abelian topological group admits maximal independent subsets that are analytic(c)whether a topological field extension admits transcendence bases that are analytic.