Том 11, № 1 (2017)

Under consideration is the uniform Euler–Bernoulli beam whose left end is fixed, and some load elastically fixed by two springs is concentrated at the right end. If the beam is hit then it begins vibrating. The aim of the article is to determine the parameters of fixedness (rigidity coefficients of springs) and loading (the mass andmoment of inertia of the load) of the right end of the beam by natural frequencies of its flexural vibrations. It is shown that the four unknown parameters of the boundary conditions at the right end of the beam are uniquely determined from the five natural frequencies of its flexural vibrations. Some counterexample is presented showing that four natural frequencies are insufficient for the unique identification of these four nonnegative parameters.

We consider the problem of processing some identical jobs with a complicated technological route on some production line in presence of parallel machines. Under some constrains on the number of jobs processed simultaneously, a cyclic schedule is desired with minimum cycle duration. Some algorithm for construction of an exact solution is proposed and substantiated. Also, we found the case of pseudopolynomially solvable problem.

We suggest the two new discrete differential sine and cosine Fourier transforms of a complex vector which are based on solving by a finite difference scheme the inhomogeneous harmonic differential equations of the first order with complex coefficients and of the second order with real coefficients, respectively. In the basic version, the differential Fourier transforms require by several times less arithmetic operations as compared to the basic classicalmethod of discrete Fourier transform. In the differential sine Fourier transform, the matrix of the transformation is complex,with the real and imaginary entries being alternated, whereas in the cosine transform, the matrix is purely real. As in the classical case, both matrices can be converted into the matrices of cyclic convolution; thus all fast convolution algorithms including the Winograd and Rader algorithms can be applied to them. The differential Fourier transform method is compatible with the Good–Thomas algorithm of the fast Fourier transform and can potentially outperform all available methods of acceleration of the fast Fourier transform when combined with the fast convolution algorithms.

A graph is called a 1-triangle if, for its every maximal independent set I, every edge of this graph with both endvertices not belonging to I is contained exactly in one triangle with a vertex of I. We obtain a characterization of 1-triangle graphs which implies a polynomial time recognition algorithm. Computational complexity is establishedwithin the class of 1-triangle graphs for a range of graph-theoretical parameters related to independence and domination. In particular, NP-completeness is established for the minimum perfect neighborhood set problem in the class of all graphs.

Under study is the process of heating a moving medium by high-frequency electromagnetic radiation in presence of heat exchange with the environment, in the approximation of a thermally thin layer. It is shown that, due to the competitive nature of the processes of heat emission in the heated media and heat exchange with the environment, the temperature profiles are realized in the form of autowaves. Inspecting analytical and numerical solutions, we determine the basic regularities of the dynamics of temperature waves.

The notions of boundary and minimal hard classes of graphs, united by the term “critical classes,” are useful tools for analysis of computational complexity of graph problems in the family of hereditary graph classes. In this family, boundary classes are known for several graph problems. In the paper, we consider critical graph classes in the families of strongly hereditary and minor closed graph classes. Prior to our study, there was the only one example of a graph problem for which boundary classes were completely described in the family of strongly hereditary classes. Moreover, no boundary classes were known for any graph problem in the family of minor closed classes. In this article, we present several complete descriptions of boundary classes for these two families and some classical graph problems. For the problem of 2-additive approximation of graph bandwidth, we find a boundary class in the family of minor closed classes. Critical classes are not known for this problem in the other two families of graph classes.

Under study is the Cauchy problemfor the nonstationary radiative transfer equation with generalized matching conditions that describes the diffuse reflection and refraction on the interface. The solvability of the initial-boundary value problem is proved. Some stabilization conditions for the nonstationary solution are obtained.

We consider k-threshold functions of n variables, i.e. the functions representable as the conjunction of k threshold functions. For n = 2, k = 2, we give upper bounds for the cardinality of the minimal teaching set depending on the various properties of the function.