The paper considers a method of stabilized hard thresholding in the problem of inverting linear homogeneous operators using wavelet decomposition. In a data model with additive Gaussian noise, an analysis of the unbiased estimate of the mean square risk of this method is carried out. Under the assumption of a long-term dependence between noise coefficients, conditions are given under which the unbiased risk estimate is strongly consistent and asymptotically normal.

The problem of Hausdorff approximation by finite sets of the solution and the value of a multicriteria mixed strategy bimatrix game using a representation based on linear scalarization is considered. For the case of 2×2 matrices, explicit formulas are obtained for constructing nodes of a δ-net on the product of simplices of scalarization parameters. It is proved that the set of the equilibrium values obtained for the net converges in the Hausdorff metric to the solution of the initial game at δ →0. Possibile appearance of degenerate bimatrix games in scalarization is taken account. Examples are given for two-criteria 2×2×2 games.

In this paper, the definitions of generalized Student’s distributions are extended to a wider set of parameters of these distributions and multiplication theorems are given that allow the generalized Student’s and Lomax’s distributions to be represented as scale mixtures of the same distributions but with larger parameters. A similar result is obtained for beta distributions. Analogs of multiplication theorems are obtained for the classical Student’s and Lomax’s distributions as corollaries; in particular, it is shown that the Student’s distribution can be represented as a scale mixture of the Student’s distribution with a large number of degrees of freedom. A representation of strictly stable distributions concentrated on the positive semiaxis is also obtained as scale mixtures of a special distribution that is not stable. This alternative representation complements the multiplication theorem for such strictly stable laws.

We present an efficient algorithm for checking language equivalence of states in top-down deterministic finite tree automata (DFTAs). Unlike string automata, tree automata operate over hierarchical structures, posing unique challenges for algorithmic analysis. Our approach reduces the equivalence checking problem to that of checking the solvability of a system of language-theoretic equations which specify the behavior of a DFTA. By constructing such a system of equations and systematically manipulating with it through substitution and conflict detection rules, we develop a decision procedure that determines whether two states accept the same tree language. We formally prove the correctness and termination of the algorithm and establish its worst-case time complexity as O(n²) under the RAM (Random Access Machine) model of computation augmented with pointers.