Vol 100, No 2 (2019)

Let \(\mathcal{G}\) be a metric, finite, noncompact, and connected graph with finitely many edges and vertices. Assume that the length of at least one of the edges is infinite. The main object of this paper is the Hamiltonian \({{{\mathbf{H}}}_{\alpha }}\) associated in \({{L}^{2}}(\mathcal{G};{{\mathbb{C}}^{m}})\) with a matrix Sturm–Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian \({{{\mathbf{H}}}_{\alpha }}\) and any other self-adjoint realization of the Sturm–Liouville expression is empty. We also indicate conditions on the graph ensuring that the positive part of \({\mathbf{H}_{\alpha }}\) is purely absolutely continuous. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of the operator \({{{\mathbf{H}}}_{\alpha }}\) is obtained. Additionally, a formula is found for the scattering matrix of the pair \(\{ {{{\mathbf{H}}}_{\alpha }},{{{\mathbf{H}}}_{D}}\} \), where \({{{\mathbf{H}}}_{D}}\) is the operator of the Dirichlet problem on the graph.

We consider three related problems of partitioning an \(N\)-element set of points in \(d\)-dimensional Euclidean space into two clusters balancing the value of (1) the intracluster quadratic variance normalized by the cluster size in the first problem; (2) the intracluster quadratic variance in the second problem; and (3) the size-weighted intracluster quadratic variance in the third problem. The NP-completeness of all these problems is proved.

In the paper we discuss a new bound of the total variation distance in terms of L² distance for random variables that are polynominals in log-concave random vectors.

Assuming the continuum hypothesis CH, it is proved that there exists a perfectly normal compact topological space Z and a countable set \(E \subset Z\) such that \(Z{\backslash }E\) is not condensed onto a compact space. The existence of such a space answers (in CH) negatively to V.I. Ponomarev’s question as to whether every perfectly normal compact space is an \(\alpha \)-space. It is proved that, in the class of ordered compact spaces, the property of being an \(\alpha \)-space is not multiplicative.

The collocation method is justified for the integral equation of the impedance boundary value problem for the Helmholtz equation. A sequence is constructed that converges to the exact solution of this boundary value problem, and an error estimate is deduced.

Properties of infinite-dimensional pseudodifferential operators (PDO) are discussed. In particular, the connection between two definitions of PDO is considered: one given in terms of the Hamiltonian Feynman measure and the other introduced in this work in terms of the Kolmogorov integral.

We propose a general method for parameter estimation of a distribution tail that does not depend on the fulfillment of the conditions of the Gnedenko theorem. We prove the consistency of the proposed estimator and its asymptotic normality under stronger conditions imposed on the parametric family of distribution tails. Additionally, the proposed method is adapted for estimating the Weibull and log-Weibull tail indices.

We introduce the following classes of integrable billiards: elementary billiards, topological billiards, billiard books, billiards with a potential, with a magnetic field, and geodesic billiards. These classes are used to test Fomenko’s conjecture about the realizability, up to Liouville equivalence, of integrable nondegenerate Hamiltonian systems with two degrees of freedom by billiards. In the class of book billiards, topological obstacles to realizability are found.

Non-convex Free Disposal Hull (FDH) model was proposed in the scientific literature at the end of the 20th century for performance measurement of complex multidimensional production units. FDH model was proposed almost simultaneously with the DEA (Data Envelopment Analysis) model. However, as distinct from the DEA models, production possibility set of FDH models are non-convex ones, which significantly refrained the development of these models. As far as we know, the necessity for such an approach has been noted in the world scientific literature for a long time. In this paper, an approach is proposed for three-dimensional visualization of FDH models. An approach was tested using real-life data sets from different areas. Computational experiments confirm reliability and effectiveness of the proposed approach.

We have proven that the maximum size k of an induced subgraph of the binomial random graph \(G(n,p)\) with a given number of edges \(e(k)\) (under certain conditions on this function), with asymptotic probability 1, has at most two values.

A nonlinear integral equation arising in the spatial model of biological communities developed by Austrian scientists Ulf Dieckmann and Richard Law is studied. Sufficient conditions for the existence of a solution of this equation (a fixed point of the integral operator) are found. The uniqueness of the solution is also analyzed.

The seismic response from the marine bottom is numerically simulated. Acoustic equations are used to describe the dynamic behavior of a water layer. Bottom sediments are described as a porous fluid-filled medium by using the Dorovsky model. A unified algorithm for full wave modeling based on the grid-characteristic method is proposed for solving all hyperbolic systems of equations in the entire computational domain. A distinctive feature of this approach is that the necessary contact conditions at the interface of media with different rheology are set in explicit form.

For the system of electrodynamics equations with anisotropic permittivity, the inverse problem of determining the permittivity is studied. It is supposed that the permittivity is characterized by a diagonal matrix \(\epsilon = {\text{diag}}({{\varepsilon }_{1}},{{\varepsilon }_{1}},{{\varepsilon }_{2}}),\) where \({{\varepsilon }_{1}}\) and \({{\varepsilon }_{2}}\) are positive constants everywhere outside of a bounded domain \({{{\Omega }}_{0}} \subset {{\mathbb{R}}^{3}}\). Time-periodic solutions of Maxwell’s equations related to two modes of plane waves coming from infinity and impinging on a local inhomogeneity located in \({{{\Omega }}_{0}}\) are considered. For determining functions \({{\varepsilon }_{1}}(x)\) and \({{\varepsilon }_{2}}(x),\) some information on the magnitudes of the electric strength vectors of two interfering waves is given. It is shown that this information reduces the original problem to two inverse kinematic problems with incomplete data regarding travel times of electromagnetic waves. The linearized statement for these problems is investigated. It is shown that, in the linear approximation, the problem of determining \({{\varepsilon }_{1}}(x)\) and \({{\varepsilon }_{2}}(x)\) is reduced to two X-ray tomography problems.

In this paper, novel spectral decompositions are obtained for the solutions of generalized Lyapunov equations, which are observed in the study of controllability and observability of the state vector in deterministic bilinear systems. The same equations are used in the stability analysis and stabilization of stochastic linear control systems. To calculate these spectral decompositions, an iterative algorithm is proposed that uses the residues of the resolvent of the dynamics matrix. This algorithm converges for any initial guess, for a non-singular and stable dynamical system. The practical significance of the obtained results is that they allow one to characterize the contribution of individual eigen-components or their pairwise combinations to the asymptotic dynamics of the perturbation energy in deterministic bilinear and stochastic linear systems. In particular, the norm of the obtained eigen-components increases when frequencies of the corresponding oscillating modes approximate each other. Thus, the proposed decompositions provide a new fundamental approach for quantifying resonant modal interactions in a large and important class of weakly nonlinear systems.