Vol 220, No 5 (2017)

We study spectral properties of the one-dimensional Schrödinger operators \( {\mathrm{H}}_{\mathrm{X},\alpha, \mathrm{q}}:=-\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+\mathrm{q}(x)+{\varSigma_x}_{{}_n}\in X{\alpha}_n\delta \left(x-{x}_n\right) \) with local interactions, d_* = 0, and an unbounded potential q being a piecewise constant function, by using the technique of boundary triplets and the corresponding Weyl functions. Under various sufficient conditions for the self-adjointness and discreteness of Jacobi matrices, we obtain the condition of self-adjointness and discreteness for the operator H_X,α,q.

The conditions of regularization and the structure of generalized Green’s operator for a re-gularized linear matrix differential-algebraic boundary-value problem are found. To solve the problem of regularization of a generalized matrix differential-algebraic boundary-value problem, the original conditions of solvability and the structure of the general solution of a matrix equation of the Sylvester type are used.

We study statistical experiments with a random change of time, which transforms a discrete stochastic basis in a continuous one. The adapted stochastic experiments are studied in continuous stochas-tic basis in the series scheme. The transition to limit by the series parameter generates an approximation of adapted statistical experiments by a diffusion process with evolution.

The homeomorphisms of the Orlicz–Sobolev class W_loc^1,φ under a condition of the Calderón type on φ in ℝⁿ, n ≥ 3 are considered. For these classes of mappings, a number of theorems on the local behavior are established, and, in particular, an analog of the famous Gehring theorem on a local Lipschitz property, as well as various theorems on estimates of a distortion of the Euclidean distance are proved. In particular, the results hold for the homeomorphisms of the Sobolev classes W_loc^1,p with p > n − 1.