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Vol 210, No 7 (2019)
The Plis metric and Lipschitz stability of minimization problems
Abstract
We study the metric introduced by Plis on the set of convex closed bounded subsets of a Banach space. For a real Hilbert space it is proved that metric projection and (under certain conditions) metric antiprojection from a point onto a set satisfy a Lipschitz condition with respect to the set in the Plis metric. It is proved that solutions of a broad class of minimization problems are also Lipschitz stable with respect to the set. Several examples are discussed.Bibliography: 18 titles.
Matematicheskii Sbornik. 2019;210(7):3-20
3-20
Quantum system structures of quantum spaces and entanglement breaking maps
Abstract
This paper is devoted to the classification of quantum systems among the quantum spaces. In the normed case we obtain a complete solution to the problem when an operator space turns out to be an operator system. The min and max quantizations of a local order are described in terms of the min and max envelopes of the related state spaces. Finally, we characterize min-max-completely positive maps between Archimedean order unit spaces and investigate entanglement breaking maps in the general setting of quantum systems.Bibliography: 34 titles.
Matematicheskii Sbornik. 2019;210(7):21-93
21-93
Smoothness of functions and Fourier coefficients
Abstract
We consider functions represented as trigonometric series with general monotone Fourier coefficients. The main result of the paper is the equivalence of the $L_p$ modulus of smoothness, $1< p< \infty$, of such functions to certain sums of their Fourier coefficients. As applications, for such functions we give a description of the norm in the Besov space and sharp direct and inverse theorems in approximation theory.
Bibliography: 34 titles.
Matematicheskii Sbornik. 2019;210(7):94-119
94-119
Two-sided estimates for domains of univalence for classes of holomorphic self-maps of a disc with two fixed points
Abstract
We investigate the class of holomorphic maps of a disc into itself that have an interior and a boundary fixed point, as well as the class of holomorphic maps of a half-plane into itself that have a fixed point in the interior of the domain and a fixed point at infinity. Two-sided estimates for domains of univalence are obtained for these function classes in terms of the values of the angular derivative at the boundary fixed point and the position of the interior fixed point.Bibliography: 21 titles.
Matematicheskii Sbornik. 2019;210(7):120-144
120-144
Properties of the spectrum of an elliptic boundary value problem with a parameter and a discontinuous nonlinearity
Abstract
An elliptic Dirichlet boundary value problem is studied which has a nonnegative parameter $\lambda$ multiplying a discontinuous nonlinearity on the right-hand side of the equation. The nonlinearity is zero for values of the phase variable not exceeding some positive number in absolute value and grows sublinearly at infinity. For homogeneous boundary conditions, it is established that the spectrum $\sigma$ of the nonlinear problem under consideration is closed ($\sigma$ consists of those parameter values for which the boundary value problem has a nonzero solution). A positive lower bound and an upper bound are obtained for the smallest value of the spectrum, $\lambda^*$. The case when the boundary function is positive, while the nonlinearity is zero for nonnegative values of the phase variable and nonpositive for negative values, is also considered. This problem is transformed into a problem with homogeneous boundary conditions. Under the additional assumption that the nonlinearity is equal to the difference of functions that are nondecreasing in the phase variable, it is proved that $\sigma=[\lambda^*,+\infty)$ and that for each $\lambda\in\sigma$ the problem has a nontrivial semiregular solution. If there exists a positive constant $M$ such that the sum of the nonlinearity and $Mu$ is a function which is nondecreasing in the phase variable $u$, then for any $\lambda\in\sigma$ the boundary value problem has a minimal nontrivial solution $u_\lambda(x)$. The required solution is semiregular, and $u_\lambda(x)$ is a decreasing mapping with respect to $\lambda$ on $[\lambda^*,+\infty)$. Applications of the results to the Gol'dshtik mathematical model for separated flows in an incompressible fluid are considered. Bibliography: 37 titles.
Matematicheskii Sbornik. 2019;210(7):145-170
145-170