The relations between the Newton–Cartan theory, Newton’s classical gravitational theory and General Relativity are discussed. It is shown that the limit \(c \to \infty \) of General Relativity becomes identical with the Newton–Cartan theory, provided the so-called time function — defined as proportional to the singular part of the covariant Einsteinian metric in the transition \(c \to \infty \) — satisfies a certain boundary condition at spatial infinity. Using this limiting process, one obtains an asymptotic representation of Einsteinian fields “near” the Newtonian limit. This should allow us to identify post-Newtonian corrections for the Newton–Cartan theory.

All possible \(N = 1\) and \(N = 2\) superconformal transformations are presented and classified into two (for \(N = 1\)) and three (for \(N = 2\)) types. Only the first one corresponds to the superconformal group, all others are elements of a semigroup. They are noninvertible and do not admit an infinitesimal form. The set of them is the ideal of the full superconformal semigroup. The permanent is used when classifying \(N = 2\) superconformal transformations and finding the \(N = 2\) Berezinian. Also the transformations from \(N = 1\) to \(N = 2\) and from \(N = 2\) to \(N = 1\) are found. The structure of the superfields which are conformal outside super Riemann surface is obtained.

Two types of extension of the Minkowski space-time are compared. It is shown that the composite fermions model can be considered in (\(N = 2\))-superspace without torsion, with additional coordinates transforming independently from main coordinates. A set of supermanifolds corresponds to a set of solutions of the model. Their number and character of constraints determine an internal symmetry group, while in supersymmetrical models this group is determined by the extension degree \(N\). Use of anticommuting coordinates leads to appearance of scalar SU(2)-doublets in the model.

The problem of distribution of the spacing of compound nucleus resonances is studied by using an identity which expresses a determinant in terms of the trace of log of the matrix. An explicit connection between the two-point correlation function and the fluctuation property of Gaussian Orthogonal Ensemble is shown.

We show that the proton impurity in a neutron matter can create an inhomogeneity in density which acts as a potential well localizing the proton’s wave function. At low densities this inhomogeneity is a neutron bulge, whereas at high densities a neutron deficiency (bubble) occurs. We calculate variationally the proton’s energy using a Gaussian wave function. The neutron background is treated in a Thomas–Fermi approximation. The Skyrme interactions are used. We find that the localized proton has lower energy than the plane wave proton for densities below the lower critical density \(n_1 \cong 0.3n_0\), and above the upper critical density \(n_{\rm u} \cong 2.2n_0\), where \(n_0 = 0.17\) fm\(^{-3}\). We discuss some implications of the proton localization for magnetic properties of neutron matter containing a small admixture of protons.