The mathematical relations between the regular Coulomb function F η (ρ) and the irregular Coulomb functions H ± η (ρ) and G η (ρ) are obtained in the complex plane of the variables η and ρ for integer or half-integer values of . These relations, referred to as "connection formulas", form the basis of the theory of Coulomb wave functions, and play an important role in many fields of physics, especially in the quantum theory of charged particle scattering. As a first step, the symmetry properties of the regular function F η (ρ) are studied, in particular under the transformation → − − 1, by means of the modified Coulomb function Φ η (ρ), which is entire in the dimensionless energy η −2 and the angular momentum . Then, it is shown that, for integer or half-integer , the irregular functions H ± η (ρ) and G η (ρ) can be expressed in terms of the derivatives of Φ η, (ρ) and Φ η,− −1 (ρ) with respect to . As a consequence, the connection formulas directly lead to the description of the singular structures of H ± η (ρ) and G η (ρ) at complex energies in their whole Riemann surface. The analysis of the functions is supplemented by novel graphical representations in the complex plane of η −1 .

Different versions of the effective-range function method for charged particle collisions are studied and compared. In addition, a novel derivation of the standard effective-range function is presented from the analysis of Coulomb wave functions in the complex plane of the energy. The recently proposed effective-range function denoted as ∆ [Phys. Rev. C 96, 034601 (2017)] and an earlier variant [Hamilton et al., Nucl. Phys. B 60, 443 (1973)] are related to the standard function. The potential interest of ∆ for the study of low-energy cross sections and weakly bound states is discussed in the framework of the proton-proton 1 S0 collision. The resonant state of the protonproton collision is successfully computed from the extrapolation of ∆ instead of the standard function. It is shown that interpolating ∆ can lead to useful extrapolation to negative energies, provided scattering data are known below one nuclear Rydberg energy (12.5 keV for the protonproton system). This property is due to the connection between ∆ and the effective-range function by Hamilton et al. that is discussed in detail. Nevertheless, such extrapolations to negative energies should be used with caution because ∆ is not analytic at zero energy. The expected analytic properties of the main functions are verified in the complex energy plane by graphical color-based representations.

Entanglement in fishing gear is a significant threat to many cetaceans. For the 2 largest species, the blue whale Balaenoptera musculus and the fin whale B. physalus, reports of entangled individuals are rare, leading to the assumption that entanglements are not common. Studies of interaction with fisheries in other species often rely on the presence of scars from previous entanglements. Here, scar detection rates were first examined in humpback Megaptera novaeangliae, fin and blue whales using standard vessel-based photo-identification photographs collected between 2009 and 2016 in the Gulf of St. Lawrence, Canada. We then examined aerial images of fin whales collected with a drone in 2018 and 2019 and compared both methods. Entanglement rates were 6.5% for fin and 13.1% for blue whales using photo-identification images of individuals. Prominent scarring was observed around the tail and caudal peduncle, visible only when animals lifted those body sections above water when diving. For the small subset of pictures which captured the entire caudal peduncle, entanglement rates ranged between 60% for blue and 80% for fin whales. This result was similar to the 85% entanglement rate estimated in humpback whales. The assessment of aerial-based photography yielded an entanglement rate of 44.1 to 54.7% in fin whales. Scars were always around the peduncle, often the tail, rarely the dorsal fin and never around the pectoral fins, while the mouth cannot be examined from above. Thus, in species that do not regularly expose their tail or peduncle, aerial imagery is the preferred method to quantify entanglement rates by assessment of scars.

The hypothesis [1] that the particular linear tracks appearing in the measurement of a spherically-emitting radioactive source in a cloud chamber are determined by the (random) positions of atoms or molecules inside the chamber is further explored in the framework of a recently established one-dimensional model [2]. In this model, meshes of localized spins 1/2 play the role of the cloudchamber atoms and the spherical wave is replaced by a linear superposition of two wave packets moving from the origin to the left and to the right, evolving deterministically according to the Schrödinger equation. We first revisit these results using a time-dependent approach, where the wave packets impinge on a symmetric two-sided detector. We discuss the evolution of the wave function in the configuration space and stress the interest of a non-symmetric detector in a quantum-measurement perspective. Next we use a time-independent approach to study the scattering of a plane wave on a single-sided detector. Preliminary results are obtained, analytically for the single-spin case and numerically for up to 8 spins. They show that the spin-excitation probabilities are sometimes very sensitive to the parameters of the model, which corroborates the idea that the measurement result could be determined by the atom positions. The possible origin of decoherence and entropy increase in future models is finally discussed.

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