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Vol 211, No 5 (2020)
- Year: 2020
- Articles: 5
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7465
3-30
The Cauchy problem for an abstract second order ordinary differential equation
Abstract
We prove the existence and uniqueness of a solution for the Cauchy problem for a linear abstract second order differential equation, obtain its representation, and prove that it is continuously dependent on the time at which the initial conditions are specified. Based on these results, we prove the existence and uniqueness of a solution of the Cauchy problem for a nonlinear abstract second order differential equation. This result is applied to show that the initial-boundary value problem for a nonlinear hyperbolic divergence structure equation has a unique solution. Bibliography: 49 titles.
Matematicheskii Sbornik. 2020;211(5):31-77
31-77
The statistical properties of 3D Klein polyhedra
Abstract
Let $\Gamma$ be a rank-$s$ lattice in $\mathbb R^s$. The convex hulls of the nonzero lattice points lying in orthants are called the Klein polyhedra of $\Gamma$. This construction was introduced by Klein in 1895, in connection with generalizing the classical continued-fraction algorithm to the multidimensional case. Arnold stated a number of problems on the statistical and geometric properties of Klein polyhedra. In two dimensions the corresponding results follow from the theory of continued fractions. An asymptotic formula for the mean value of the $f$-vectors (the numbers of facets, edges and vertices) of 3D Klein polyhedra is derived. This mean value is taken over the Klein polyhedra of integer 3D lattices with determinants in $[1,R]$, where $R$ is an increasing parameter. Bibliography: 27 titles.
Matematicheskii Sbornik. 2020;211(5):78-97
78-97
Kripke semantics for the logic of problems and propositions
Abstract
In this paper we study the propositional fragment $\mathrm{HC}$ of the joint logic of problems and propositions introduced by Melikhov. We provide Kripke semantics for this logic and show that $\mathrm{HC}$ is complete with respect to those models and has the finite model property. We consider examples of the use of $\mathrm{HC}$-models usage. In particular, we prove that $\mathrm{HC}$ is a conservative extension of the logic $\mathrm{H4}$. We also show that the logic $\mathrm{HC}$ is complete with respect to Kripke frames with sets of audit worlds introduced by Artemov and Protopopescu (who called them audit set models). Bibliography: 31 titles.
Matematicheskii Sbornik. 2020;211(5):98-125
98-125
Waring's problem in natural numbers of special form
Abstract
Let $\mathbb N_0$ be the set of positive integers whose binary decompositions contain an even number of ones. We give a bound for the trigonometric sum of special form over numbers in $\mathbb N_0$; using this bound, we derive an asymptotic formula for the number of solutions to Waring's equation in positive integers in $\mathbb N_0$, and also a bound for the number of terms in the last equation, which is sufficient for the equation to be solvable in integers in $\mathbb N_0$. Bibliography: 9 titles.
Matematicheskii Sbornik. 2020;211(5):126-142
126-142