We introduce the notion of (metric) paracontact pair structure and establish certain properties of the characteristic foliations associated to it. We also consider normal paracontact pair structures and provide necessary and sufficient conditions for a paracontact pair structure to be normal. In particular, we formulate an analogue of Morimoto's theorem for product manifolds. Finally, we describe a way to obtain a metric paracontact pair structure on the total space of a principal $\mathbb{S}^1$-bundle via the Boothby-Wang construction.

The curved Josephson junction is described. In the framework of the Maxwell equations the equation that describes the influence of the curvature on the fluxion motion was obtained. The method of geometrical reduction of the sine-Gordon model from three to lower dimensional manifold was applied to the long Josephson junction. It was argued that the geometrical reduction describes the junctions with slowly varying curvatures.

We prove the Ptolemaean inequality and the Theorem of Ptolemaeus in the setting of $H$--type Iwasawa groups.

Here we discus the diffusion of particles in a curved tube. This kind of transport phenomenon is observed in biological cells and porous media. To solve such a problem, we discuss the three dimensional diffusion equation with a confining wall forming a thinner tube. We find that the curvature appears in a effective diffusion coefficient for such a quasi-one-dimensional system. As an application to higher dimensional case, we discuss the diffusion in a curved surface with thickness. In this case the diffusion coefficient changes to the tensor form depending on the mean and Gaussian curvatures. Then the diffusion flow can be interpreted as usual flow plus anomalous flow. The anomalous flow shows not only the diffusion but also the concentration depending on mean and Gaussian curvatures, and also it includes the flow proportional to the gradient of Gaussian curvature.

Discussed is the problem of the mutual relationship of differentially first-order and second-order field theories and quantum-mechanical concepts. We show that unlike the real history of physics, the theories with algebraically second-order Lagrangians are primary, and in any case more adequate. It is shown that in principle, the primary Schrödinger idea about Lagrangians which are quadratic in derivatives, and leading to second-order differential equations, is not only acceptable, but just it opens some new perspective in field theory. This has to do with using the Lorentz-conformal or rather its universal covering ${\rm SU}(2,2)$ as a gauge group. This has also some influence on the theory of defects in continua.

We study the 1-D motions of a charge under Coulomb force, within the electrostatic approximation. When the electrostatic force is attractive, no oscillating motion takes place. When repulsive, nonlinear oscillations will arise. In both cases dry friction has been taken into account and time equations have been solved providing time as elliptic integrals of first and second kind. A short phase plane analysis has been included. The oscillation period has been exactly computed and found to increase versus the initial speed of the mobile.

Asymmetric heavenly equation, presented in a two-component form, is known to be 3+1-dimensional bi-Hamiltonian system. We show that symmetry reduction of this equation yields a new two component 2+1-dimensional integrable bi-Hamiltonian system. We prove that this new 2+1-dimensional system admits bi-Hamiltonian structure, so that it is integrable according to Magri’s theorem.