The solvability of the Dirichlet problem for the soft Laplacian on a stratified set is proved using a modification of the well-known Perron method.
The solvability of the Dirichlet problem for the soft Laplacian on a stratified set is proved using a modification of the well-known Perron method.
A higher integrability of the gradient of a solution to the Zaremba problem in a bounded strictly Lipschitz domain is proved for an inhomogeneous p-Laplace equation with lower terms.
The paper presents a method for extracting provably random bit sequences from several independent Markov chain trajectories, each having an arbitrary finite order. In implementing quantum random number generators, the combined use of several trajectories makes it possible to significantly increase the speed of generating output bit sequences.
We obtain an asymptotic formula for the sum (Q(x) = sumlimits_{substack{ n leqslant x r(n + 1) ne 0 } } frac{{r(n)}}{{r(n + 1)}};;(x to + infty ),) where (r(n)) denotes the number of representations of n as a sum of two squares.
A new formula is given for the companion matrix of the composition of two polynomials over a commutative ring. The results obtained are used to provide a constructive proof of Plans’ theorem for 2-bridge knots, which states that the first homology group of an odd-fold cyclic covering of a three-dimensional sphere branched over a given knot is the direct sum of two copies of some Abelian group. A similar result is also true for the homology of even-fold coverings factored by the reduced homology group of two-fold coverings. The structure of the above-mentioned Abelian groups is described through Chebyshev polynomials of the second and fourth kinds.
We consider totally balanced cooperative games in which the characteristic function takes integer values. It is proved that any three- or four-person game has an integer imputation from the core. A similar result, with the exception of one degenerate case, is obtained for five-person games. An example of a five-person game is given in which the core consists of a single noninteger imputation.
We consider a pursuit differential game in a Hilbert space. The game dynamics is described by two semilinear evolutionary equations with a not necessarily bounded operator in the Hilbert space; each of these equations is controlled by its own player. The controls appear linearly on the right-hand sides of the equations, and their norms are bounded by given constants. Sufficient conditions for the solvability of the pursuit game are established in both linear and nonlinear cases. For this purpose, we use the Minty–Browder theorem and a chain technology of successive continuation of the solution to a controlled system to intermediate states. As examples of reduction to the abstract operator equation under study, we consider Oskolkov’s system of equations and a semilinear wave equation.
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