Once a wheat sheaf has been sealed and tied up, its packed down straws display the same orientation and zero-divergence. This observation brings us to the mathematical notion of presheaf, i.e., a topological structure in which diverging functions are locally superimposed. We show how the concepts of presheaves and the correlated globular sets, borrowed from category theory and algebraic topology, allow a well-founded mathematical approach to otherwise elusive activities of the brain. The mathematical assessment of brain functions in terms of presheaves: a) explains why spontaneous random spikes synchronize; b) leads to the counterintuitive intuition of antidromic effects in neuronal spikes: when an entrained oscillation propagates from A to B, changes in B lead to changes in A. We provide testable previsions: a) we suggest the proper locations of transcranial magnetic stimulation’s coils to improve the clinical outcomes of drug-resistant epilepsy; b) we advocate that axonal stimulation by external sources backpropagates and alters the neuronal electric oscillatory frequency. Further, we describe how the hierarchical information transmission inside globular sets provides fresh insights concerning different issues at various coarse-grained scales, such as object persistence, memory reinforcement in spite of random noise, Bayesian inferential circuits.

Leather waste was carbonized at 800 °C in inert atmosphere. The resulting biochar was coated in situ with polypyrrole nanotubes produced by the oxidation of pyrrole in the presence of methyl orange. The composites of carbonized leather with deposited polypyrrole nanotubes of various composition were compared with similar composites based on globular polypyrrole. Their molecular structure was characterized by infrared and Raman spectra. Both conducting components formed a bicontinuous structure. The resistivity determined by four-point van der Pauw method was monitored as a function of pressure applied up to 10 MPa. The typical conductivity of composites was of the order of tenths to units S cm−1 and it was always higher for polypyrrole nanotubes than for globular polypyrrole. The conductivity decreased by 1–2 orders of magnitude after treatment with ammonia but still maintained a level acceptable for applications operating under non-acidic conditions. The composites were tested for dye adsorption, viz. cationic methylene blue and anionic methyl orange, using UV-spectroscopy. The composites are designed for the future use as functional adsorbents controlled by the electrical potential.

This article is a logical continuation of the work of Nuritdinov et al. (2021). We analyzed space observations of the apparent surface density of 81 globular clusters (GCs) observed by GAIA DR2 and published in de Boer et al (2019). First, we studied the features of the quality of the GC surface density data, including comparing them with the HST data. Using our 3-parameter model, previously tested on the example of HST space observations, we found the values of the stellar concentration parameter towards the center of the cluster. As a result, we found good correlations of this parameter with the physical characteristics of the GCs. We have also found the corresponding empirical formulas for this parameter.