This paper is devoted to the derivation of field equations in space with the geometric structure generated by metric and torsion tensors. We also study the geometry of the space generated jointly and agreed on by the metric tensor and the torsion tensor. We showed that in such space the structure of the curvature tensor has special features and for this tensor we obtained analog Ricci-Jacobi identity and evaluated the gap that occurs at the transition from the original to the image and vice versa, in the case of infinitely small contours. We have researched the geodesic lines equation. We introduce the tensor παβ which is similar to the second fundamental tensor of hypersurfaces Yn-1, but the structure of this tensor is substantially different from the case of Riemannian spaces with zero torsion. Then we obtained formulas which characterize the change of vectors in accompanying basis relative to this basis itself. Taking into considerations our results about the structure of such space we derived from the variation principle the general field equations (electromagnetic and gravitational).

In this article, we study the phase-space distribution of the quantum state as a framework to describe the different properties of quantum systems in continuous-variable systems. The natural approach to quantum systems is given the Gaussian Wigner representation, to unify the description of bosonic and fermionic quantum states, we study the structure of the Kahler space geometry as the geometry generated by three forms under the agreement conditions depended on the nature of the state bosonic or fermionic. Multimode light is studied, and we established that the Fock space vacuum corresponds to a certain homogeneous Gaussian state.

This article is dedicated to expanding our comprehension of the regularity of the solutions to the Cauchy problem for the quasilinear second-order parabolic partial differential equations under fair general conditions on the nonlinear perturbations. In this paper have been obtained that the sequence of the weak solutions u z  V1,0 2 , z = 1,2,….. to the Cauchy problems for the Equations ( 15) under the initial conditions u z (0,x) = 0 z converges to the weak solution to the Cauchy problem for the Equation (1) under the initial condition u(0, x) = u0 in V1,0 2 .

This article establishes the Holder continuity of the solutions to a quasi-linear system of elliptic partial differential equations with singular coefficients under the assumption of its form-boundary.

In this article, we establish new characterizations of convex functions, prove some connected convex type integral inequality; consider the pair of convex functions as the dual semi-norms in functional space. The properties of the integral operators are considered in the scales of the convex semi-norm under the standard conditions on singular kernels.