We consider the submanifolds in the unit tangent bundle of the pseudo-Riemannian manifold generated by the unit vector fields on the base. We have found the second fundamental form of this type of submanifolds with respect to the normal vector field of a special kind. We have derived the equations on totally geodesic non-isotropic unit vector field. We have found all the two-dimensional pseudo-Riemannian manifolds which admit non-isotropic totally geodesic unit vector fields as well as the fields.

Method of tangents and its iteration for the plane algebraic 3rd-order curves has been applied to the Diophantine equations family which depends on the parameter. The family is a generalization of well-known Fermat's example of tangents iteration. This family has been investigated. Formulas for the solutions for the first two steps of tangents iteration have been obtained. Analysis of iterations admissibility conditions depending on family parameter has been done. Examples and its graphic illustrations have been given also.

The interaction between the two eddy streams of a gas of rough spheres is investigated. A bimodal distribution with a Maxwellian modes of a special form is used. Different sufficient conditions for the minimization of the uniform-integral discrepancy between the sides of the equation Bryan-Piddaсk is obtained.

We prove the correctness of the Cauchy problem for pseudodifferential operators that are perturbations of the exponentially-correct differential operators of constant strength in the spaces of Schwartz and spaces of infinitely differentiable functions in an infinite layer.

The synthesis problem for the nonlinear system $\dot x_1=u(x), \dot x_2=x_1, \dot x_3=x_2^{3}$ is considered in this paper. The solution of this problem is given according to the Controllability Function method introduced by V.I. Korobov. This method gives us the possibility to find a set of positional controls that steers our system from arbitrary initial point $x_0$ to the origin in finite time $T(x_0)$.