Abstract. We investigate the inverse problem of determining the time and space dependent kernel of the integral term in the n-dimensional integro-differential equation of heat conduction from the known solution of the Cauchy problem for this equation. First, the original problem is replaced by the equivalent problem where an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of this kernel. Next, assuming that there are two solutions k₁(x, t) and k₂(x, t) of the stated problem, it is formed an equation for the difference of this solution. Further research is being conducted for the difference k₁(x, t) - k₂(x, t) of solutions of the problem and using the techniques of integral equations estimates.

We present a new logic called SPL, embedded into Solovay’s provability logic S using a translation that embeds Visser’s formal logic FPL into G¨odel-L¨ob’s provability GL. SPL is formulated in the form of sequent and natural deduction calculi, a relational semantics is proposed.

The author previously put forward the hypothesis that in the n-dimensional Euclidean space Eⁿ, curves, any two oriented arcs of which are similar, are rectilinear. The same statement was proven for dimensions n = 2 and n = 3. In a space of arbitrary dimension, the hypothesis found its confirmation in the class of rectifiable curves. The work provides a complete solution to the problem, and in a stronger version:

a) a curve in Eⁿ, any two oriented arcs of which with a common origin (not fixed) are similar, is rectilinear;

b) if a curve in En has a half-tangent at its boundary point and any two of its oriented arcs with a beginning at this point are similar, then the curve is rectilinear;

c) if a curve in En has a tangent at an interior point and all its oriented arcs starting at this point are similar, then the curve is rectilinear. Examples of curves in E² and E³ are given, in which all arcs with a common origin are similar, but they are not rectilinear, and a complete description of such curves in E² is also given. Research methods are topological, set-theoretic, using the apparatus of functional equations.

This paper presents a calculation method and algorithm, as well as numerical results of studying chemically reacting turbulent jets based on three-dimensional parabolic systems of Navier-Stokes equations for multicomponent gas mixtures.

Continuity equations are used to calculate the mass imbalance when solving with constant pressure, and with variable pressures, with the equations of motion and continuity.

Diffusion combustion of a propane-butane mixture flowing from a square-shaped nozzle in a submerged flow of an air oxidizer has been numerically studied. Pressure variability significantly affects the velocity (temperature) profiles in the initial sections of the jet, and when moving away from the nozzle exit, the pressure effect can be considered imperceptible, but the flame length is longer than at constant pressure, but it does not significantly affect the shape of the flame.

The saddle-shaped behavior of the longitudinal velocity in the direction of the major axis is numerically obtained for large initial values of the turbulence kinetic energy of the main jet.

Given method allows the study of non-reacting and reactive turbulent jets flowing from a rectangular nozzle.

In the present paper lattice optimal cubature formulas are constructed by the variational method in the Sobolev space. In addition, the square of the norm of the error functional of the constructed lattice optimal cubature formulas in the conjugate Sobolev space is explicitly calculated.

The Lebesgue constant of the classical Fourier operator is uniformly approximated by a logarithmic-fractional-rational function depending on three parameters; they are defined using the specific properties of logarithmic and rational approximations. A rigorous study of the corresponding residual term having an indefinite (non-monotonic) behavior has been carried out. The obtained approximation results strengthen the known results by more than two orders of magnitude.

In this paper we consider a weakened version of the spectral synthesis for the differentiation operator in nonquasianalytic spaces of ultradifferentiable functions. We deal with the widest possible class of spaces of ultradifferentiable functions among all known ones. Namely, these are spaces of Ω-ultradifferentiable functions which have been recently introduced and explored by A.V. Abanin. For differentiation invariant subspaces in these spaces, we establlish conditions of weak spectral synthesis. As an application, we prove that a kernel of a local convolution operator admits weak spectral synthesis. We also show that a conjunction of kernels of convolution operators admits weak spectral synthesis if all generating ultradistributions have the same support equaled to {0} and there exists one generated by an ultradistribution which characteristic function is a multiplier in the corresponding space of entire functions.

For the problem on eigenvibrations of the plate with an attached oscillator, the new symmetric linear variational statement is proposed. It is established the existence of the sequence of positive eigenvalues of finite multiplicity with limit point at infinity and the corresponding complete orthonormal system of eigenvectors. The new symmetric scheme of the finite element method with Hermite finite elements is formulated. Error estimates consistent with the solution smoothness for the approximate eigenvalues and approximate eigenvectors are proved. The results of numerical experiments illustrating the influence of the solution smoothness on the computation accuracy are presented.

In this paper, we study an inhomogeneous Riemann boundary value problem with a finite index and a boundary condition on the real axis for a generalized equation Cauchy–Riemann with supersingular coefficients. To solve the problem, it was necessary to derive a structural formula for the general solution of this equation and to investigate the solvability of the Riemann boundary value problem of the theory of analytic functions with an infinite index due to the power-order vorticity point.