Recovering Differential Operators on Spatial Networks

We give a short review of results on inverse spectral problems for ordinary differential operators on a spatial networks (geometrical graphs). We pay the main attention to the most important nonlinear inverse problems of recovering coefficients of differential equations from spectral characteristics provided that the structure of the graph is known a priori. In the first half of the review we provide results related to inverse Sturm-Liouville problems on arbitrary compact graphs. Further, results on inverse problems for arbitrary order differential operators on compact graphs are presented. At the end we provide the main results on inverse problems on noncompact graphs.Comment: 35 pages, 4 figure

Recovering Differential Operators with Nonseparated Boundary Conditions in the Central Symmetric Case

Inverse spectral problems for Sturm-Liouville operators on a finite interval with non-separated boundary conditions are studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We discuss statements of the problems, provide algorithms for their solutions along with necessary and sufficient conditions for the solvability of the inverse problems considered

On Quasi-periodic Differential Pencils with Jump Conditions Inside the Interval

Non-self-adjoint second-order differential pencils on a finite interval with non-separated quasi-periodic boundary conditions and jump conditions are studied. We establish properties of spectral characteristics and investigate the inverse spectral problem of recovering the operator from its spectral data. For this inverse problem we prove the corresponding uniqueness theorem and provide an algorithm for constructing its solution.Comment: 7 page

Recovering First Order Integro-Differential Operators from Spectral Data

First order integro-differential operators on a finite interval are studied. Properties of spectral characteristic are established, and the uniqueness theorem is proved for the inverse problem of recovering operators from their spectral data.Comment: 5 pages. arXiv admin note: text overlap with arXiv:1702.0078

Inverse Spectral Problems for Sturm-Liouville Operators on Hedgehog-type Graphs with General Matching Conditions

Boundary value problems on hedgehog-type graphs for Sturm-Liouville differential operators with general matching conditions are studied. We investigate inverse spectral problems of recovering the coefficients of the differential equation from the spectral data. For this inverse problem we prove a uniqueness theorem and provide a procedure for constructing its solution.Comment: 7 page

Recovering Variable Order Differential Operators with Regular Singularities on Graphs

We study inverse spectral problems for ordinary differential equations with regular singularities on compact star-type graphs when differential equations have different orders on diferent edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1410.2007; text overlap with arXiv:1309.5360 by other author

Inverse Problems for Systems of Variable Order Differential Equations with Singularities on Spatial Networks

Variable order differential equations with non-integrable singularities are considered on spatial networks. Properties of the spectrum are established, and the solution of the inverse spectral problem is obtained.Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1503.0175

File mapping Rule-based DBMS and Natural Language Processing

This paper describes the system of storage, extract and processing of information structured similarly to the natural language. For recursive inference the system uses the rules having the same representation, as the data. The environment of storage of information is provided with the File Mapping (SHM) mechanism of operating system. In the paper the main principles of construction of dynamic data structure and language for record of the inference rules are stated; the features of available implementation are considered and the description of the application realizing semantic information retrieval on the natural language is given.Comment: 17 pages, 3 figure

Recovering Dirac systems with singularities in interior points

We study the non-selfadjoint Dirac system on a finite interval having non-integrable regular singularities in interior points with additional matching conditions at these points. Properties of spectral characteristics are established, and the inverse spectral problem is investigated. We provide a constructive procedure for the solution of the inverse problem, and prove its uniqueness. Moreover, necessary and sufficient conditions for the global solvability of this nonlinear inverse problem are obtained.Comment: 22 page

On the dynamics of solitary wave solutions supported by the model of mutually penetrating continua

The model we deal with is the mathematical model for mutually penetrating continua one of which is the carrying medium obeying the wave equation whereas the other one is the oscillating inclusion described by the equation for oscillators. These equations of motion are closed by the cubic constitutive equation for the carrying medium. Studying the wave solutions we reduce this model to a plane dynamical system of Hamiltonian type. This allows us to derive the relation describing the homoclinic trajectory going through the origin and obtain the solitary wave with infinite support. Moreover, there exist a limiting solitary wave with finite support, i.e. compacton. To model the solitary waves dynamics, we construct the three level finite-difference numerical scheme and study its stability. We are interested in the interaction of the pair of solitary waves. It turns out that the collisions of solitary waves have non-elastic character but the shapes of waves after collisions are preserved.Comment: "Dynamical systems: Mechatronics and Life Sciences", Vol.2, Eds. J.Awrejcewicz et al., 2015 - PP. 453-460, 13-th International conference Dynamical Systems - Theory and Applications 7-10 December 2015, Lodz, Polan