Vol 63, No 7 (2019)

We consider the problem of localizing (determination of position) the first kind discontinuities of a function of one variable and the problem of localizing q-jumps of a noisy function. In the first case, we assume that the exact function is smooth except for a finite number of discontinuities of the first kind. In the second case, the exact function is smooth except for a finite number of small segments of length 2q. It is required to determine the number of discontinuities (q-jumps) and approximate their positions using the function approximately specified in L₂(ℝ) and the level of disturbance. We construct a class of regular averaging methods and obtain estimates of the accuracy of localization, separability, and observability on classes of correctness.

We find sufficient conditions for the unique solvability of nonlocal problem for the abstract Bessel—Struve differential equation. Both nonlocal conditions are written with the help of Krdelyi—Kober operators.

We propose a penalty method for general convex constrained optimization problems, where each auxiliary penalized problem is solved approximately with a special composite descent method. Direction finding choice in this method is found with the help of an equivalent equilibrium type problem. This allows one to keep the complete structure of the initial problem, although without nonlinear constraints and to simply calculate the descent direction in separable problems. Convergence of the method in primal and dual variables is established under rather weak assumptions.

We establish some new effective oscillation conditions for solutions to linear delay differential equations of the first order. We develop a new approach to obtaining oscillation conditions in the form of the upper limit of a function of equation parameters. We apply the proposed approach to equations with one and several concentrated delays and to those with a distributed delay. We demonstrate the advantages of the obtained results over the well-known ones.

We investigate four-dimensional locally homogeneous pseudo-Riemannian manifolds with an isotropic Weyl tensor. An algorithm for obtaining a complete classification of such manifolds is described.