Doklady Mathematics最新文献

Using the recoupling theory, we define a representation of the pure braid group and show that it is not trivial.

We prove that for any (varepsilon > 0) and ({{n}^{{ - frac{{e - 2}}{{3e - 2}} + varepsilon }}} leqslant p = o(1)) the maximum size of an induced subtree of the binomial random graph (G(n,p)) is concentrated asymptotically almost surely at two consecutive points.

The article proves the following statement: in any hyperelliptic field L defined over the field of algebraic numbers K which having non-trivial units of the ring of integer elements of the field L, there is an element for which the period length of the continued fraction is greater any pre-given number.

New cases of integrable seventh-order dynamical systems that are homogeneous with respect to some of the variables are presented, in which a system on the tangent bundle of a three-dimensional manifold can be distinguished. In this case, the force field is divided into an internal (conservative) and an external component, which has dissipation of different signs. The external field is introduced using some unimodular transformation and generalizes previously considered fields. Complete sets of both first integrals and invariant differential forms are given.

Machine learning (ML) methods are applied to optimal resource control for Network Powered by Computing Infrastructure (NPC)—a new generation computing infrastructure. The relation between the proposed computing infrastructure and the GRID concept is considered. It is shown how ML methods applied to computing infrastructure control make it possible to solve the problems of computing infrastructure control that did not allow the GRID concept to be implemented in full force. As an example, the application of multi-agent optimization methods with reinforcement learning for network resource management is considered. It is shown that multi-agent ML methods increase the speed of distribution of transport flows and ensure optimal NPC network channel load based on uniform load balancing; moreover, such control of network resources is more effective than a centralized approach.

An extremal problem for positive definite functions on ({{mathbb{R}}^{n}}) with a fixed support and a fixed value at the origin (the class ({{mathfrak{F}}_{r}}({{mathbb{R}}^{n}}))) is considered. It is required to find the least upper bound for a special form functional over ({{mathfrak{F}}_{r}}({{mathbb{R}}^{n}})). This problem is a generalization of the Turán problem for functions with support in a ball. A general solution to this problem for (n ne 2) is obtained. As a consequence, new sharp inequalities are obtained for derivatives of entire functions of exponential spherical type.

A generalized solution of a mixed problem for the wave equation is constructed under minimal conditions on the right side of the equation. The solution is represented as a series from the Fourier method, and its sum is found. The form of a generalized solution of a mixed problem for an inhomogeneous telegraph equation is given.

According to Berezin and Faddeev, a Schrödinger operator with point interactions –Δ + (sumlimits_{j = 1}^m {{alpha }_{j}}delta (x - {{x}_{j}}),X = { {{x}_{j}}} _{1}^{m} subset {{mathbb{R}}^{3}},{ {{alpha }_{j}}} _{1}^{m} subset mathbb{R},) is any self-adjoint extension of the restriction ({{Delta }_{X}}) of the Laplace operator ( - Delta ) to the subset ({ f in {{H}^{2}}({{mathbb{R}}^{3}}):f({{x}_{j}}) = 0,;1 leqslant j leqslant m} ) of the Sobolev space ({{H}^{2}}({{mathbb{R}}^{3}})). The present paper studies the extensions (realizations) invariant under the symmetry group of the vertex set (X = { {{x}_{j}}} _{1}^{m}) of a regular m-gon. Such realizations H_B are parametrized by special circulant matrices (B in {{mathbb{C}}^{{m times m}}}). We describe all such realizations with non-trivial kernels. А Grinevich–Novikov conjecture on simplicity of the zero eigenvalue of the realization H_B with a scalar matrix (B = alpha I) and an even m is proved. It is shown that for an odd m non-trivial kernels of all realizations H_B with scalar (B = alpha I) are two-dimensional. Besides, for arbitrary realizations ((B ne alpha I)) the estimate (dim (ker {{{mathbf{H}}}_{B}}) leqslant m - 1) is proved, and all invariant realizations of the maximal dimension (dim (ker {{{mathbf{H}}}_{B}}) = m - 1) are described. One of them is the Krein realization, which is the minimal positive extension of the operator ({{Delta }_{X}}).

We prove that the size of the maximum induced forest (of bounded and unbounded degree) in the binomial random graph (G(n,p)) for ({{C}_{varepsilon }}{text{/}}n < p < 1 - varepsilon ) with an arbitrary fixed (varepsilon > 0) is concentrated in an interval of size (o(1{text{/}}p)). We also show 2-point concentration for the size of the maximum induced forest (and tree) of bounded degree in (G(n,p)) for p = const.

In a 1985 commentary to his collected works, Kolmogorov informed the reader that his 1932 paper On the interpretation of intuitionistic logic “was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types—propositions and problems.” We construct such a formal system as well as its predicate version, QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC. The axioms of QHC are obtained as a result of a simultaneous formalization of two well-known alternative explanations of intiuitionistic logic: (1) Kolmogorov’s problem interpretation (with familiar refinements by Heyting and Kreisel) and (2) the proof interpretation by Orlov and Heyting, as clarified and extended by Gödel.