The Cosmic-Ray Extremely Distributed Observatory (CREDO) is a newly formed, global collaboration dedicated to observing and studying cosmic rays (CR) and cosmic-ray ensembles (CRE): groups of at least two CR with a common primary interaction vertex or the same parent particle. The CREDO program embraces testing known CR and CRE scenarios, and preparing to observe unexpected physics, it is also suitable for multi-messenger and multi-mission applications. Perfectly matched to CREDO capabilities, CRE could be formed both within classical models (e.g., as products of photon–photon interactions), and exotic scenarios (e.g., as results of decay of Super-Heavy Dark Matter particles). Their fronts might be significantly extended in space and time, and they might include cosmic rays of energies spanning the whole cosmic-ray energy spectrum, with a footprint composed of at least two extensive air showers with correlated arrival directions and arrival times. As the CRE are predominantly expected to be spread over large areas and, due to the expected wide energy range of the contributing particles, such a CRE detection might only be feasible when using all available cosmic-ray infrastructure collectively, i.e., as a globally extended network of detectors. Thus, with this review article, the CREDO Collaboration invites the astroparticle physics community to actively join or to contribute to the research dedicated to CRE and, in particular, to pool together cosmic-ray data to support specific CRE detection strategies.

An iterative method for reconstructing mass distribution in spiral galaxies using a thin disk approximation is developed. As an example, the method is applied to galaxy NGC 4736; its rotation curve does not allow one to employ a model with a massive spherical halo. We find a global mass distribution in this galaxy (without nonbaryonic dark matter) that agrees perfectly with the high-resolution rotation curve of the galaxy. This mass distribution is consistent with the I-band luminosity profile with the mean mass-to-light ratio M /L I ¼ 1:2, and it also agrees with the amount of hydrogen observed in the outermost regions of the galaxy. We predict the total mass of the galaxy to be only 3:43 ; 10 10 M . It is very close to the value predicted by the modified gravity models and much less than the currently accepted value of 5:0 ; 10 10 M (with %70% of the mass in a dark matter halo).

We study functions related to the experimentally observed Havriliak-Negami dielectric relaxation pattern in the frequency domain ∼ [1 + (i ωτ0) α ] −β with τ0 being some characteristic time. For α = l/k < 1 (l and k positive integers) and β > 0 we furnish exact and explicit expressions for response and relaxation functions in the time domain and suitable probability densities in their "dual" domain. All these functions are expressed as finite sums of generalized hypergeometric functions, convenient to handle analytically and numerically. Introducing a reparameterization β = (2 − q)/(q − 1) and τ0 = (q − 1) 1/α (1 < q < 2) we show that for 0 < α < 1 the response functions f α,β (t/τ0) go to the one-sided Lévy stable distributions when q tends to one. Moreover, applying the self-similarity property of the probability densities g α,β (u), we introduce two-variable densities and show that they satisfy the integral form of the evolution equation.